Quantum Mechanics in Hilbert Space, 2nd ed

Quantum Mechanics in Hilbert Space SECOND EDITION

This is a volume in PURE AND APPLIED MATHEMATICS A Series of Monographs and Textbooks Editors: SAMUEL EILENBERG AND HYMAN BASS A list of recent titles in this series appears at the end of this volume.

QUANTUM MECHANICS IN HILBERT SPACE SECOND EDITION

Eduard Prugovetki DEPARTMENT OF MATHEMATICS UNIVERSITY OF TORONTO TORONTO, CANADA

ACADEMIC PRESS A Subsidiary of Harcourt Brace Jovanouich, Publishers

New York London Toronto Sydney San Francisco

COPYRIGHT @ 1981, BY ACADEMIC PRESS,INC. ALL RIGHTS RESERVED. NO PART OF THIS PUBLICATION MAY BE REPRODUCED OR TRANSMITTED IN ANY FORM OR BY ANY MEANS, ELECTRONIC OR MECHANICAL, INCLUDING PHOTOCOPY, RECORDING, OR ANY INFORMATION STORAGE AND RETRIEVAL SYSTEM, WITHOUT PERMISSION IN WRITING FROM THE PUBLISHER.

ACADEMIC PRESS, INC.

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United Kingdom Edition published by ACADEMIC PRESS, INC. (LONDON) LTD.

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Library of Congress Cataloging i n P u b l i c a t i o n Data

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PrugoveEki , Eduard Quantum mechanics i n H i l b e r t space, second e d i t i o n . (Pure and applied mathematics, a series of momgraphs and textbooks;

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v. 9 2 )

Bibliography : p Includes index. 1. H i l b e r t space. 2. Quantum theory. I. Title. XI. S e r i e s . QA3.P8 v O l . 92 [QC174.17.H55] 510s 80-534 ISBN 0-12-566060-X [530.1'2'01515733]

PRINTED IN THE UNITED STATES OF AMERICA

81 82 83 84

9 8 7 6 5 4 3 2 1

To My Parents

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Contents

XV

Preface to the Second Edition Preface to the First Edition List of Symbols

xvii XiX

1

Introduction I. Basic Ideas of Hilbert Space Theory 1.

Vector Spaces 1.1 Vector spaces over fields of scalars 1.2 Linear independence of vectors 1.3 Dimension of a vector space 1.4 Isomorphism of vector spaces Exercises 2. Euclidean (Pre-Hilbert) Spaces 2.1 Inner products on vector spaces 2.2 The concept of norm 2.3 Orthogonal vectors and orthonormal bases 2.4 Isomorphism of Euclidean spaces Exercises 3.

4.

11 13 14 16 17 18 20 21 23 24

Metric Spaces 3.1 Convergence in metric spaces 3.2 Complete metric spaces 3.3 Completion of a metric space Exercises

25 26 27

Hilbert Space 4.1 Completion of a Euclidean space 4.2 Separable Hilbert spaces

30

29

32

* One asterisk indicates sections in which all proofs can be skipped at a first reading.

** T w o asterisks indicate sections that are introductory to original papers and research material. vii

Contents

viii

4.3 la spaces as examples of separable Hilbert spaces 4.4 Orthonormal bases in Hilbert space 4.5 Isomorphism of separable Hilbert spaces Exercises Wave Mechanics of a Single Particle Moving in One Dimension 5. 5.1 The formalism and its partial physical interpretation 5.2 The wave mechanical initial-value problem 5.3 Bound states of the system 5.4 A particle moving in a square-well potential Exercises References for Further Study

33 36 41 43

44 47 49 52 54 56

11. Measure Theory and Hilbert Spaces of Functions

* 1.

*2.

*3.

Measurable Spaces 1.1 Boolean algebras and u algebras of sets 1.2 Boolean algebras of intervals 1.3 Bore1 sets in R” 1.4 Monotone classes of sets Exercises

58 61 62 63 65

Measures and Measure Spaces 2.1 The concept of measure 2.2 Basic properties of measures 2.3 Extensions of measures and outer measures 2.4 Cartesian products of measure spaces Exercises

66 68 70 76 79

Measurable and Integrable Functions 3.1 The concept of a measurable function 3.2 Properties of measurable functions 3.3 Positive-definite integrable functions 3.4 Real and complex integrable functions 3.5 Infinite sequences and sums of integrals 3.6 Integration on Cartesian products of measure spaces Exercises

80 81 85 89 92 94 99

4.

Spaces of Square-Integrable Functions 4.1 Square-integrable functions 4.2 Hilbert spaces of square-integrable functions *4.3 The separability of L2 spaces 4.4 Change of variables of integration Exercises

5.

The Hilbert Space of Systems of n Different Particles in Wave Mechanics 5.1 The Schroedinger equation of n-particle systems 5.2 The center-of-mass frame of reference

101 103 109 115 118

119 122

Contents

5.3 The bound states of n-particle systems 5.4 Properties of the n-particle Schroedinger operator 5.5 The initial-value problem Exercises 6. Direct Sums and Tensor Products of Hilbert Spaces 6.1 Direct sums of Euclidean spaces 6.2 Separability and completeness of direct sums of Hilbert spaces 6.3 Bilinear forms on vector spaces *6.4 Algebraic tensor products of vector spaces *6.5 Hilbert tensor products of Hilbert spaces Exercises 7.

The Two-Body Bound-State Problem with a Spherically Symmetric Potential 7.1 Two particles interacting via a spherically symmetric potential 7.2 The equation of motion in spherical coordinates 7.3 Spherical harmonics on the unit sphere 7.4 The completeness of trigonometric functions 7.5 The completeness of Legendre polynomials 7.6 Completeness of the spherical harmonics 7.7 The two-body problem with a Coulomb potential Exercises References for Further Study

ix 126 128 131 132 132 134 137 140 144 147

147 149 151 153 157 162 164 169 171

III. Theory of Linear Operators in Hilbert Spaces 1.

2.

3.

Linear and Antilinear Operators on Euclidean Spaces 1.1 Linear and antilinear transformations 1.2 Algebraic operations with linear transformations 1.3 Continuous and bounded transformations 1.4 Examples of bounded and unbounded operators Exercises

172 174 178 179 180

Linear Operators in Hilbert Spaces 2.1 Linear functionals on normed spaces 2.2 The dual of a Hilbert space 2.3 Adjoints of linear operators in Hilbert spaces 2.4 Bounded linear operators in Hilbert spaces 2.5 Dirac notation for linear operators 2.6 Closed operators and the graph of an operator 2.7 Nonexistence of unbounded everywhere-defined self-adjoint operators Exercises

193 195

Orthogonal Projection Operators 3.1 Projectors onto closed subspaces of a Hilbert space 3.2 Algebraic properties of projectors 3.3 Partial ordering of projectors

197 200 202

182 183 186 188 190 191

Contents

X

3.4 Projectors onto intersections and orthogonal sums of subspaces

* 3.5 4.

Appendix: Extensions and adjoints of closed linear operators Exercises

Isometric and Unitary Transformations Isometric transformations in between Hilbert spaces Unitary operators and the change of orthonormal basis The Fourier-Plancherel transform Cayley transforms of symmetric operators Self-adjointness of position and momentum operators in wave mechanics Exercises

4.1 4.2 4.3 *4.4 4.5

5.

212 214 216 219 224 226

Spectral Measures

5.1 The point spectrum of a self-adjoint operator 5.2 Spectral resolution of self-adjoint operators 5.3 5.4 5.5 *5.6 *6.

203 209 21 1

with pure point spectrum Weak, strong, and uniform operator limits Spectral measures and complex measures Spectral functions Appendix: Signed measures Exercises

The Spectral Theorem for Unitary and Self-Adjoint Operators Spectral decomposition of a unitary operator Monotonic sequences of linear operators Construction of spectral families for unitary operators Uniqueness of the spectral family of a unitary operator Spectral decomposition of a self-adjoint operator The spectral theorem for bounded self-adjoint operators Exercises References for Further Study

6.1 6.2 6.3 6.4 6.5 6.6

226 227 229 23 1 235 236 240 24 1 242 243 247 249 253 25 5 256

IV. The Axiomatic Structure of Quantum Mechanics 1.

2.

Basic Concepts in the Quantum Theory of Measurement 1.1 Observables and states in quantum mechanics 1.2 The concept of compatible observables 1.3 Born’s correspondence rule for determinative measurements 1.4 Born’s correspondence rule for preparatory measurements 1.5 The stochastic nature of the quantum theory of measurement Exercises

257 260 26 1 264 267 268

Functions of Compatible Observables 2.1 Fundamental and nonfundamental observables 2.2 Bounded functions of commuting self-adjoint operators 2.3 Algebras of compatible observables

269 270 274

Contents

*2.4 Unbounded functions of commuting self-adjoint operators Exercises

3.

4.

277 284

The Schroedinger, Heisenberg, and Interaction Pictures 3.1 The general form of the Schroedinger equation 3.2 The evolution operator 3.3 The Schroedinger picture 3.4 The Heisenberg picture and physical equivalence of formalisms 3.5 The formalism of matrix mechanics 3.6 The interaction picture Exercises

285 286 29 1 293 297 298 300

State Vectors and Observables of Compound Systems Superselection rules and state vectors The Hilbert space of compound systems Tensor products of linear operators The observables of a system of distinct particles Symmetric and antisymmetric tensor products of Hilbert spaces The connection between spin and statistics Spin and statistics for the n-body problem Exercises

301 302 303 304 305 306 308 310

Complete Sets of Observables The concept of a complete set of operators Cyclic vectors and complete sets of operators The construction of spectral representation spaces Cyclicity and maximality Exercises

31 1 315 321 324 328

4.1 4.2 4.3 4.4 4.5 4.6 4.7 *5.

xi

5.1 5.2 5.3 5.4 6.

Canonical Commutation Relations 6.1 The empirical significance of commutation relations 6.2 Representations of canonical commutation relations 6.3 One-parameter Abelian groups of unitary operators 6.4 Representations of Weyl relations *6.5 Appendix: Proof of von Neumann’s theorem Exercises

7.

The General Formalism of Wave Mechanics A derivation of one-particle wave mechanics Wave mechanics of n-particle systems The Schroedinger operator Closures of linear operators The Schroedinger kinetic energy operator The Schroedinger potential energy operator The self-adjointness of the Schroedinger operator The angular momentum operators Time-dependent Hamiltonians Exercises

7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 * *7.9

329 33 1 334 339 342 347 348 351 354 355 357 360 366 369 372 373

Contents

xii

* 8.

Completely Continuous Operators and Statistical Operators 8.1 Completely continuous operators 8.2 The trace of a linear operator 8.3 Hilbert-Schinidt operators 8.4 The trace norm and the trace class 8.5 Statistical ensembles and the process of measurement 8.6 The quantum mechanical state of an ensemble 8.7 The von Neumann equation in Liouville space 8.8 Density matrices on spectral representation spaces * *8.9 Appendix: Classical and quantum statistical mechanics in master Liouville space Exercises References for Further Study

375 380 383 385 390 392 396

400

404 41 1 412

V. Quantum Mechanical Scattering Theory Basic Concepts in Scattering Theory of Two Particles 1.1 Scattering theory and the initial-value problem 1.2 Asymptotic states in classical mechanics 1.3 Asymptotic states and scattering states in the Schroedinger picture 1.4 Meller wave operators 1.5 The scattering operator 1.6 The differential scattering cross section 1.7 The transition operator *1.8 The T-matrix formula for the differential cross section Exercises

1.

2.

3.

General Time-Dependent Two-Body Scattering Theory 2.1 The intertwining property of wave operators 2.2 The partial isometry of wave operators 2.3 Properties of the S operator 2.4 Initial and final domains of wave operators 2.5 Dyson’s perturbation expansion 2.6 Criteria for existence of strong asymptotic states 2.7 The physical asymptotic condition Exercises General Time-Independent Two-Body Scattering Theory 3.1 The relation of the time-independent to the time-dependent approach 3.2 Lippmann-Schwinger equations in Hilbert space 3.3 Spectral integral representations of wave operators 3.4 The transition amplitude 3.5 The resolvent of an operator 3.6 The resolvent method in scattering theory 3.7 Appendix: Integration of vector- and operator-valued functions

414 416 418 42 1 423 425 426 430 436 438

440 442 445 448 45 1 454 457

458 463 47 I 472 474 417 479

Contents

* *3.8

Appendix: Scattering theory in Liouville space Exercises Eigenfunction Expansions in Two-Body Potential Scattering Theory 4. 4.1 Free plane waves in three dimensions 4.2 Distorted plane waves 4.3 Free and distorted spherical waves 4.4 Eigenfunction expansions for complete sets of operators 4.5 Green's operators and Green functions 4.6 Lippmann-Schwinger equations for eigenfunction expansions 4.7 The on-shell 2"-matrix operators 4.8 The off-shell T-matrix and **4.9 Appendix: Scattering theory for long-range potentials **4.10 Appendix: Eigenfunctions and transition density matrices in statistical mechanics Exercises

xiii 486 489 49 1 494 496 498 501 503 506 510 513 516 519

5.

Green Functions in Potential Scattering 5.1 The free Green function 5.2 Partial wave free Green functions 5.3 Fredholm integral equations with Hilbert-Schmidt kernels 5.4 The full Green function 5.5 Fredholm expansion of the full Green function 5.6 Symmetry properties of the full Green function *5.7 Appendix: The spectrum of the Schroedinger operator *5.8 Appendix: Relations between resolvents and spectral functions Exercises Distorted Plane Waves in Potential Scattering 6. 6.1 Potentials of Rollnik class 6.2 Fredholm series expressions for distorted plane waves 6.3 Asymptotic completeness and the generalized Parseval's equality 6.4 The scattering amplitude 6.5 The Born series 6.6 Distorted plane waves as solutions of the Schroedinger equation *6.7 Appendix: Analytic operator-valued functions Exercises

520 524 525 528 529 53 1 533 539 542 543 545 550 553 557 560

564 567

7.

Wave and Scattering Operators in Potential Scattering The existence of strong asymptotic states The completeness of the Msller wave operators Proof of asymptotic completeness Phase shifts for scattering in central potentials The general phase-shift formula for the scattering operator Partial-wave analysis for spherically symmetric potentials Exercises 8. Fundamental Concepts in Multichannel Scattering Theory 8.1 The concept of channel 7.1 7.2 *7.3 7.4 7.5 7.6

569 576 580 585 588 592 594 596

xiv

Contents

8.2 Channel Hamiltonians and wave operators 8.3 The uniqueness of channel strong asymptotic states 8.4 Interchannel scattering operators 8.5 The existence of strong asymptotic states in n-particle potential scattering **8.6 Two-Hilbert space formulation of multichannel scattering theory 8.7 Multichannel eigenfunction expansions and T-matrices **8.8 Multichannel Born approximations and Faddeev equations *8.9 Appendix: von Neumann’s mean ergodic theorem Exercises References for Further Study

598 603 607

609 612 615 62 1 626 628 630

Hints and Solutions to Exercises

63 1

References

669

Index

679

Preface to the Second Edition

Since the appearance in 1971 of the first edition, Chapters 111, IV, and V of this book have served, with certain modifications, as the basis of a two-term course on functional analysis in quantum mechanics offered at the University of Toronto at the advanced undergraduate and first-year graduate level. The aforementioned modifications have now been incorporated in the present edition, together with additional material which brings the book up to date, and in some instances provides an introduction to topics of contemporary research interest. Practically all alterations occur in the second half of the book, and especially in the part dealing with quantum scattering theory. In addition to reorganized exposition, they consist of substantial simplifications in the proofs of various theorems. Hence it is hoped that the text has now become in its entirety easy to follow for any student in mathematics and/or physics interested in the subject. Furthermore, a number of completely new sections have been included summarizing research results which have been published during the past decade and that directly pertain to the subject matter of this book. Since a detailed coverage of all the material contained in Chapters 111-V would be unfeasible in a typical one-year course, asterisks have been liberally employed to indicate proofs of theorems, and even full sections, that can be skipped without making the subsequent material incomprehensible. As was the case with the first edition, the goal of the present edition is not to supplant but rather to supplement any of the good textbooks on nonrelativistic quantum mechanics, which as a rule have a purely physics orientation. More precisely, the chief aim is to provide in a xv

xvi

Preface to the Second Edition

readable and self-contained form, which would be accessible even to students without an extensive mathematical background, rigorous proofs of all the main statements of a functional-analytic nature that one encounters in standard textbooks on quantum mechanics. I t should be emphasized that no attempt has been made to include any material dealing with complex analysis or with group theory. Indeed, in the case of complex analysis a reasonable level of mathematical rigor is maintained in many of the better physics textbooks on quantum scattering theory, whereas the subject of group theory in quantum mechanics has already received ample attention in the form of monographs and compilations of review articles on the subject at all conceivable levels of mathematical sophistication. Albeit a few monographs dealing in part with some of the subject matter of this book have also been published since the first edition made its initial appearance, as a rule they tend to lean heavily toward the purely mathematical aspects of the subject. I t is therefore hoped that the present edition may still perform a useful function by filling the gap between purely physics and purely mathematics oriented textbooks and monographs pertaining to Hilbert space theory as applied to nonrelativistic quantum mechanics.

E. PRUGOVEEKI Toronto, 1981

Preface to the First Edition

This book was developed from a fourth-year undergraduate course given at the University of Toronto to advanced undergraduate and first-year graduate students in physics and mathematics. It is intended to provide the inquisitive student with a critical presentation of the basic mathematics of nonrelativistic quantum mechanics at a level which meets the present standards of mathematical rigor. It should also be of interest to the mathematician working in functional analysis and related areas, who would like to see some of the applications of the basic theorems of functional analysis to quantum mechanics. With these aims in mind, I have tried to make the book self-contained. A knowledge of advanced calculus, linear algebra, the basics of ordinary linear differential equations, and the very basic concepts of classical mechanics (such as the notions of force, momentum, angular momentum energy, etc. of an n-particle system) provide an adequate background for the reader of this work. The material is organized in the form of definitions, theorems, and proofs of theorems. Such a format has the advantage of enabling the reader to grasp immediately the basic concepts and results, separating them from the sometimes tedious techniques. In order to .facilitate the perusing of the book preceding a more thorough reading, marginal asterisks indicate those theorems whose proofs can be skipped at a first reading. I consider the material contained in the present book to stand in the same relation to quantum physics, as the material of conventional mathematical physics stands in relation to classical physics. Therefore, the prevailing style and approach is similiar to texts on mathematical xvii

xviii

Preface to the First Edition

physics: the mathematical methods are in the foreground ; the physical examples appear only as illustrations of these methods. Emphasis is placed on general concepts and basic techniques of proving theorems, rather than on computational methods. Consequently, the reader who is not already familiar with the material that can be found in a conventional book on quantum mechanics will find it extremely useful to take at least an occasional glance at any of the standard textbooks on quantum mechanics (some of which are listed in the References) which cover the physical aspects of quantum theory in more detail. Since the text is oriented toward physical application, I have given the minimal amount of mathematics necessary for a good understanding of the main mathematical aspects of nonrelativistic quantum mechanics. Consequently, basic material on measure theory and functional analysis has been introduced in a selective manner, always keeping in mind its application to quantum mechanics. However, I believe that the interested reader can easily expand his knowledge of the areas of mathematics introduced in this book by using the reference books and the present text as a starting point. The list of references given at the end of each chapter is meant to help him in this task. Furthermore, since the last two chapters contain mostly material which until now was published only in original papers but not in textbooks or monographs, I have supplemented these references with a few historical notes. However, many of the mathematical tools used in the present book have already become standard. Hence, no effort has been made to provide an exhaustive bibliography or to give credit to the original source, with the exception of names which are now customarily associated with certain theorems, formulas, equations, etc., and of references related to topics of current research interest. Most of the exercises form an integral part of the text. Consequently, though many of these exercises are relatively easy to solve, it has been deemed necessary to provide the solutions or detailed hints appearing at the end of the text. However, the reader is urged to try and solve these exercises by himself before looking up the solutions. The book should appeal to the increasing number of students in theoretical physics who desire to understand the justification of the multitude of heuristic procedures and shortcuts ordinarily employed in the typical books and courses on nonrelativistic quantum mechanics. Such procedures, which until a few decades ago had to be taken at face value, can be now rigorously justified, due to advances in functional analysis in general, and in the Hilbert space theory in particular. E. PRUGOVEEKI Toronto, 1971

List of Symbols

This list contains only the symbols which consistently refer to the same concept, and which are frequently employed, and the page numbers where they first appear.

%m(

W")

complex conjugate of the complex number a, 19 linear or antilinear operator, 173 adjoint of the operator A, 187 Boolean algebra generated by the family X of sets, 60 Boolean u algebra generated by X, 60 the algebra of bounded operators on S, 175, 181 Borel set, 62 the family of Borel sets on R", 62 the family of Borel sets on Wn which are finite unions of intervals, 61 the family of Borel subsets of Q C UP,111 the family of characteristic functions of subsets of Q, 101 the family of all complex-valued continuous functions defined on Wn, 17 the family of m-times continuously differentiable complex functions on Rn, 45 a family of square-integrable and vanishing at infinity functions in Vm(Wn),45 the subset of %'*(Wn) of functions of compact support, 129 the family of n-tuples of complex numbers, 12 domain of definition of the operator A, 186 metric function, 25 Laplacian, 121 phase shift of lth partial wave, 497, 589, 593 xix

xx

List of Symbols

Kronecker delta, 22 E(S) spectral measure, 231 E,, spectral function, 235 EM projector onto space M, 197 d Euclidean space, 18 ei vector in orthonormal system, 22 9 field of scalars, 12 fk(e,+) scattering amplitude, 494, 553, 556 @p)(a), @@(a) eigenfunctions, 496, 499, 500 Qk(r) plane wave, 491 Qp)(r) distorted plane waves, 494, 503 GA(a,a'; f ) , GY'(a,a'; f ) Green functions, 501, 502 graph of an operator A, 191 Hamiltonian, 294 free Hamiltonian, 298 Schroedinger operator, 122, 368 Hilbert space, 2, 31 Hilbert ray, 269 Planck's constant, 8, 46 limit in the Hilbert-Schmidt norm on 9 ( X ), 399 characteristic function of S, 31 infimum of F, 84 family of intervals in Rn, 61 kernel of an integral operator, 428 class of sets, 58 Banach space of p-integrable functions on 52, 181 Hilbert space of p-square-integrable functions on 52, 103 family of p-square-integrable functions on 52, 103 limit in the mean, 105, 463 Hilbert space of one-column complex matrices, n = 1,..., co 23 Liouville space over X , 397 metric space, 25 monotone class on 9, 63 closed linear subspace of a Hilbert space X , 197 intial domains of 52, , 438, 441 measure of S, 67 Lebesgue measure of B, 67, 68 extension of measure p, 70 normed space, 30 the family of n-tuples of real numbers, 2, 7, 12 resolvent of the operator A at complex value t;, 452

List of Symbols

range of operator A, 186 set of rational numbers, 35 final domains Q*, 441 density operator, 391, 404 scattering operator, 423, 438 the spectrum of the operator A, 252,475 the continuous spectrum of A, 253, 475 the point spectrum of A, 48, 253, 475 the exceptional set of energy values, 546 the set exceptional set of momentum values, 546 strong limit, 230 supremum of F, 84 support of F, 128 spin variable, 308 transition operator, off-shell .T operators, 511 trace of operator A, 380 step function, 130 unitary operator, 214 time evolution operator, 286 one-parameter group of unitary operators, 288 uniform limit, 230 potential, 121 vector space, 11 a set over which a measurable space is defined, 58 weak limit, 230 incoming and outgoing Meller wave operators, angles in spherical coordinates, algebraic tensor product, 141 antisymmetric tensor product, 306 symmetric tensor product, 306 projector onto f, 199, 376 inner product, 18 Hilbert-Schmidt inner product on S(.W),397 norm, 20 Hilbert-Schmidt norm, 384 Rollnik norm of a potential V, 545

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Introduction

The beginning of the era of modern quantum mechanics is marked by the year 1925 and the two almost simultaneous papers of Heisenberg [1925] and Schroedinger [1926]. The first of these papers proposes essentially the formalism of matrix mechanics, while the second one proposes the formalism of wave mechanics. It was first indicated by Schroedinger that these two formulations are physically equivalent. Both can be embraced in a more general formulation of quantum mechanics, which was first proposed in a somewhat heuristic form by Dirac [1930]. A mathematically rigorous development of this general formalism of quantum mechanics can be achieved by taking advantage of Hilbert space theory, which is done in this book.* Though classical and quantum mechanics are very dissimilar in many respects, they do share a lot of common features when viewed from a general structural point of view. To find these common features, one must have some insight into the mechanism of a physical theory from an abstract point of view, at which one arrives by ignoring the technical details of the theory. One can dissect any physical theory into the following main constituents: (1) formalism, (2) dynamical law, (3) correspondence rules. Viewed from an abstract point of view, the formalism consists of a set of symbols and rules of deduction. With these symbols one can build statements or propositions. The rules of deduction enable us to deduce new statements from those already given. Ideally, every scientific theory starts with a set of basic statements

* For a detailed historical survey of the origins and development of quantum theory consult any of the many standard textbooks on quantum mechanics (e.g., Messiah [19621). 1

2

Introduction

called the axioms of the theory. However, in practice, while a theory is in the growing process, it very often happens that no set of axioms is clearly stated and the axiomatic formulation of the theory is left to the future. Nonrelativistic quantum mechanics has an accepted axiomatic formulation (given in Chapter IV) which covers all the present practical needs, though proposals for new sets of axioms are still making their appearance in scientific journals. These proposals have mainly the virtue of- greater generality over the conventional Hilbert space formulation adopted here (so that they contain this formulation as a special case). However, from the practical point of view they have not yet produced any new results not derivable from the conventional approach. The two classes of basic objects in the formalism of both classical and quantum mechanics are the states and observables. We shall illustrate these concepts by a few examples. In the Newtonian approach to classical physics, the state of a system is given by the family of trajectories of all particles constituting the system. For instance, the state of one particle moving in three dimensions is given by a vector-valued function r ( t ) , r ( t ) E R3 (Rn denotes an n-dimensional real Euclidean space), in the time-parameter t. I n the Hamiltonian (canonical) approach to classical mechanics, the state of a single particle is also a vector-valued function

defined now in a six-dimensional real Euclidean space called the phase space of the system. I n the Schroedinger formulation of quantum mechanics, the state is again a vector-valued function Y(t),only this time the function assumes values from an infinite-dimensional complex Euclidean space X , which bears the name of Hilbert space. T h e observables of the formalism are symbols related to specific experimental procedures for measuring them. For instance, in Newtonian classical physics of a one-particle system, r ( t ) is the position of the particle at time t; p(t) = m r ( t ) its momentum ( r ( t ) = d r ( t ) / d t ) ; mk2(t)/2 its kinetic energy (mdenotes the mass of the particle), etc. I n the Hamiltonian pz(t), and p 3 ( t ) are, in general, related approach Ql(t), QB(t), q3(t), by some functional relation to the Cartesian position and momentum components of the particle. I n quantum mechanics, observables are represented by so-called self-adjoint operators in the Hilbert space X . Every physical theory contains a particularly important ingredient called the dynamical law (frequently referred to as the “dynamics”). This dynamical law is, in general, a relation which some of the basic objects of the formalism must satisfy. I t is the key component of the theory

Introduction

3

because it gives the theory its predictive power. From a formal logical point of view, the dynamical law could be considered a part of the formalism. However, due to its central role in predicting the future behavior of the system from a knowledge of its present or past behavior, we prefer to single it out. From the mathematical point of view, the dynamical law is very frequently a differential equation which has to be satisfied by the state of the system, and in that case it is called the equation of motion. For instance, in Newtonian classical mechanics of a single particle of mass m moving under the influence of a force F,the equation of motion is given by Newton's second postulate d2r(t) r(t) = mr(t) = F(t), dt2 ' In the canonical formalism of a single particle we have

where H(qk ,P k , t ) is the Hamiltonian of the system. In the Schroedinger formulation of quantum mechanics, the equation of motion governs the time evolution of the state of the system, and is called the Schroedinger equation. However, in the Heisenberg formulation of quantum mechanics, the equation of motion, called the Heisenberg equation, does not involve the state (which in this case is time independent) but rather the observables of the system (see Chapter IV, 93). It must be realized that the dynamical law need not always be expressed by a differential equation. Very frequently, it is expressed by an integral equation, sometimes derivable from a differential equation. In modern quantum scattering theory, the dynamical law is given by analyticity conditions that certain key functions (the so-called S-matrix elements) appearing in the theory have to satisfy. The correspondence rules are the rules which assign empirical meaning to some of the symbols appearing in the formalism. As such, they provide the link between theory and experiment. The body of all correspondence rules of a theory is in physics better known under the name of the physical interpretation of the theory. In classical mechanics, which has a very direct intuitive appeal, the correspondence rules are very straightforward, and they leave no room for plausible alternatives. For instance, in Newtonian one-particle mechanics, it is well known that for a particle in the state r ( t ) , t E R', the components of the vectors r ( t ) , mi-(t), etc. represent the length of the

4

Introduction

projections of the position, momentum, etc. vectors on the three axes of an inertial frame” with Cartesian coordinates. In the case of quantum mechanics, which does not yield itself to visualization as readily as classical mechanics, the matter of physical interpretation is much more complex and therefore open to discussion. In this case the desirability and empirical consistency of the entire body of correspondence rules is still open to scrutiny and critical appraisal, being treated in its own right in the theory of measurement of quantum mechanics. However, the main features of the now widely accepted interpretation, sometimes called the Copenhagen school interpretation (see also Chapter IV, §1), are quite clear. The most striking of these features is that this interpretation is essentially statistical; it does not provide us with definite statements about the future behavior of a quantum mechanical system, but rather with probabilistic statements about the likelihood of different patterns of behavior. It should be mentioned that this feature has given rise to much controversy as to whether quantum mechanics provides the ultimate tool in describing the behavior of atomic and subatomic systems. The ranks of physicists who have adopted a critical attitude when considering this subject include notably the names of some of the pioneers in the field, such as Einstein, Schroedinger,t and de Broglie. Part of the physical interpretation of the theory consists in giving correspondence rules which assign to observables experimental procedures for measuring them. From the mathematical point of view, these observables are represented (in the conventional formalism adopted here) by self-adjoint operators in a Hilbert space. These operators will be studied from a general point of view in Chapter 111. At present we shall describe some experimental procedures for measuring, in principle,: the most common of observables: those of position, momentum, and spin of an object of atomic or subatomic size (we shall call such an object We remind the reader that an inertial frame is a physical body (usually the laboratory of the experimenter) in which Newton’s equation of motion is valid for macroscopic bodies moving at speeds which are low by comparison with the speed of light. It is an empirical fact that a .frame of reference moving during the duration of the experiment at uniform speed with respect to the sun is a very good approximation to an inertial frame. t It is interesting to mention that one of the earlier physical interpretations of the formalism of wave mechanics (which constitutes a special case of the Hilbert space formalism of quantum mechanics) was proposed by Schroedinger, was in total disagreement with the Copenhagen school interpretation, and lacked any statistical features. The experimental procedures for measuring these observables “in principle” may be conceptually very simple, but from the purely technical point of view they may not be the most efficient or the easiest to carry out with a high degree of accuracy and at a given cost.

*

5

Introduction

a microparticle, so as not to confuse it with the more special term, elementary particle). Measurements for determining the position of a particle can be carried out by a number of detectors (Wilson chambers, bubble chambers, Geiger-Muller counters, etc.). These detectors signal the presence of a particle within the volume enclosed by the detector by a directly noticeable (macroscopic) change of their state. Conceptually the simplest of such detectors is a photographic plate on which a landing microparticle leaves a characteristic mark. The essential parts of an apparatus for determining the momentum of a charged particle are depicted in Fig. 1. If particles of known charge e

FIG. 1. Experimental arrangement for determinative measurement of the momentum of a charged microparticle.

interact to the left of the screen+ S1, and if a particle is detected by the detector depicted in Fig. 1 (which can be any of the types mentioned, or some other type best suited to the needs of a particular experiment), then the determined momentum p of particle is of magnitude (in suitable units) (0.2)

IP I

= eHr0,

and in the direction indicated in Fig. 1; in the above formula, H is the strength of a uniform magnetic field H,which is present to the right of the screen S, and orthogonal to the plane of drawing. Both screens S1and S, should be impenetrable to all the particles used in the experiment. This impenetrability can be established by closing the apertures in the screens and checking that a detector placed to the right of each screen does not detect anything except the “background noise,” which has a definite pattern and is due to the everpresent cosmic rays.

6

Introduction

It must be stressed that the above-mentioned experimental arrangements are the ones which give an empirical meaning to the concepts of position and momentum of a microscopic particle. In other words, it is not meaningful to ask the question: “How do we know that the macroscopic change of state taking place in a detector (the black mark on a photoplate, or the track of condensed vapor in a Wilson chamber) is “in reality” due to a particle, and that the momentum of that particle is indeed given by Eq. (0.2) ?” Such a question is empirically meaningless because, in the ultimate analysis .(see Heisenberg [1925, p. 174]), the only knowledge that we possess about the microscopic world is the one we can extract from macroscopic phenomena. The above type of measurement determines the value of observables at the instant the measurement is carried out. More precisely, from such a measurement one can compute, within bounds of accuracy inherent in the particular apparatus, what would be the value of a particular observable if there were no disturbances due to the interaction of the system (which in the above cases is the microparticle) and the apparatus. This type of measurement will be called a determinatiwe measurement. A determinative measurement does not tell us, in general, what the value of the measured observable, or even what the fate of the system will be after the measurement. For example, the system might be completely destroyed or cease to be an independent entity (for instance, when the detector is a photoplate and the particle is absorbed by it) after the measurement. In general, the measurement process will disturb the system or even change its nature (particle “creation and annihilation”), and such a disturbance must be taken into account if the measurement is used as a preparation of a system with a certain value of the measured observable. This phenomenon is characteristic of microphysics and cannot be ignored, as is done in macrophysics; in principle it exists also in macrophysics, but from the practical point of view it is completely negligible. T o realize this, one should recall that measurements in classical mechanics require “seeing” the system, i.e., the reflection or emission of light from the objects constituting the system. Now, light (i.e., photons) has a certain momentum, which is imparted to the objects on which it impinges or by which it is emitted, but that momentum is completely negligible when dealing with objects of macroscopic size. Thus, in quantum mechanics it is important to distinguish between determinative measurementsof the above type and preparatory measurements.

* It is more common to call a determinative measurement simply a “measurement,” and a preparatory one a “preparation of state.” We prefer to avoid the term “preparation of state,” since it suggests that such a measurement will prepare a quantum mechanical state, which is by no means always true.

7

Introduction

SOURCE OF PARTICLES

FIG.2. Experimental arrangement for a preparatory measurement of the momentum of a charged microparticle.

For instance, in the modification in Fig. 2 of the apparatus in Fig. 1, we have a device which is meant to prepare charged particles of agiven momentum. If we keep the shutter open for a very short‘5nstant”” from to to to At, then any particle that might have emerged during this period tothe right of the screen S (i.e., in the interaction region, where it would interact with other particles adequately prepared) would have a momentum of magnitude I p I given by the formula (0.2), and in the direction indicated on Fig. 2. A striking feature of the above experimental procedure is that it does not tell us, by itself, whether there was indeed a particle which had passed through the aperture while the shutter was open. This feature is common to many preparatory measurements. It is in the determinative (later) stage of the experiment that the presence or absence of a particle is established.+ We note that the above preparatory procedure cannot prepare a sharp value p of the momentum, but rather, due to the finite size of the apertures 0, , 0, , and 0, , it prepares a whole range A C 88, of momenta. When the size of the last aperture 0, is such that so-called “diffraction effects” can be neglected, this prepared range d consists of all the momenta that a particle of charge e would possess when traveling along all the imaginable paths that such a particle would have to follow according to classical mechanics in order to pass through the apertures 0, , 0, , and 0,. The above-mentioned diffraction effects are again characteristic of

+

* By a very short “instant” is meant a time interval of duration A t which is negligible in comparisonwith theother errors of measurement occurring in the particular experiment in which the described preparatory measurement is included. + We could, of course, add a detector to the above apparatus, but such a detector would significantly disturb the prepared value of the momentum.

8

Introduction

microphysics, and constitute the so-called “wave nature” of microparticles. They correspond to the experimental observation that if a detector (e.g., a photoplate) is placed behind two parallel impenetrable screens each having one aperture, a “particle” might be detected not only along any straight line passing through the two apertures, but also at other points. If we imagine that a microparticle travels along a trajectory, then it would seem that the trajectory is in such cases bent after the particle has passed through the aperture. If we have a beam of particles (i.e., many independent particles), then the net effect which will be observed in a photoplate placed behind the screens is a diffraction pattern, qualitatively very similar to the diffraction pattern of a beam of light. We note that the experimental arrangement in Fig. 2 prepares not only a range d of momenta, but also a range A’ of the position observables, where d’ is the region enclosed by the aperture 0,. However, while classical mechanics assumes that the linear dimensions of d and d’ could be made arbitrarily small by building sufficiently precise apparatus, it is a direct consequence of the diffraction effects that this is not the case with microparticles. This experimental finding is embodied in Heisenberg’s uncertainty principle which states that no apparatus can be built which would prepare a particle to have the x coordinate (in some Cartesian frame of reference) within the interval d, and the p,-momentum component within the interval d P z ,so that the inequality

is not satisfied, where I d, I and I Ape I denote the lengths of the respective intervals, and h is a universal constant called the Planck constant; similar relations

hold for the y and z Cartesian components. It should be mentioned that the experimental arrangements in Figs. 1 and 2 determine and prepare, respectively, also the energy of the microparticle on which the measurement is carried out. By definition, if the momentum of a free particle of mass” m is p, then its energy is E

= p2/2m.

The mass of a particle can be measured by the experimental arrangement in Fig. 1 if we introduce an electric field E orthogonal to the existing magnetic field H. Quantities like mass, charge, and intrinsic spin are characteristicof each kind of microparticle, and, in fact, provide the only means of differentiating types of microparticles, like electrons, positrons, neutrons, protons, hydrogen atoms, water molecules, etc.

9

Introduction

We turn now to measurements of spin. Assume that we have in classical mechanics a body which rotates, with respect to an inertial frame of reference, around a certain axis passing through it, which is stationary or moves at a uniform speed with respect to that frame of reference. Such a body has then an angular momentum s with respect to that axis. If the dimensions of the body are negligible with respect to other dimensions in the experiment, so that the body can be called a particle, then s is called the spin of the particle. If the particle carries a charge, then due to its spin it will have a magnetic (dipole) moment P proportional to s. When such a particle travels through an area in which there is an inhomogeneous magnetic field H, this field will act on it with a force F = grad@ * H). In the light of the above remarks, each one of the particles of equal velocity (i.e., equal momentum) and equal spin in a beam passing through the experimental arrangement depicted in Fig. 3 would be deflected

SOURCE OF PARTICLES

FIG. 3. The Stern-Gerlach experiment.

from a straight path by an amount proportional to the projection pe of the magnetic moment P of that particle in the direction of the gradient H of the field H. If the spins of the particles were randomly oriented in all directions (as we could expect them to be if the source were a gas of such particles), then one would predict on the basis of the above considerations a continuous distribution of marks on the photoplate. However, the Stern-Gerlach experiment carried out with molecular and atomic beams reveals instead, in general, n discrete lines (in Fig. 3, n = 2)-a phenomenon which is usually described by saying that the spin is quantized.

10

Introduction

In quantum mechanics, by definition, the spin of the above particles is taken to be s=- n - 1 2

(though, strictly speaking, for reasons to become clear when a systematic theoretical study of the spin is undertaken, the total spin is [s(s 1)]1/2). Thus, s can assume only integer and half-integer values 4, 1, #, 2, ...; in Fig. 3 we have depicted the case of spin 3. The experimental arrangement of Stern and Gerlach can be used as an apparatus for a determinative measurement of the spin component in the direction H[ In that case the source of particles would originate in the interaction region, where particles of known spin are interacting. If a particle of integer spin* s leaves a mark at 0, then, by definition, it has spin zero in the Hdirection; the first, second, ..., ( n - 1)/2 mark above 0 correspond to spin components in the H direction equal to 1, 2,,,., ( n - 1)/2, respectively, while the first, second, ..., (n - 1)/2 marks below 0 correspond to spin components -1, -2,..., - ( n - 1)/2, respectively. In case of a particle of half-integer spin, there will be no middle mark; the first, second, ..., (n - 1)/2 marks above or below 0 correspond to spin components 4, 4,..., (n - 1)/2, or -4, -8 ,..., -(n - 1)/2, respectively. Hence we see that according to the very definition of the spin projection onto a certain axis, that projection can assume only integer values in case of integer-spin particles, and only half-integer values in case of halfinteger-spin particles. The above experimental arrangement can be easily transformed into an apparatus for preparatory measurements of spin by replacing the photoplate with a screen which has apertures at the spots where a beam of particles from the given source had left tracks. I t has to be mentioned that no simultaneous measurements of spin in two different directions can be carried out on microparticles-a feature which is in complete agreement with certain properties (noncommutativity of spin-component operators) of the formalism of quantum mechanics. Here we end this short survey of some of the experimental procedures for measuring some of the basic observables which occur in quantum mechanics, and which will frequently appear in the pages of this book. I n Chapter I we start our systematic study of the Hilbert space formalism of quantum mechanics, and related mathematics.

+

+

2s

This means that a beam of such particle with random-oriented spins would have

+ 1 tracks on the photoplate.

CHAPTER

I

Basic Ideas of Hilbert Space Theory

T h e central object of study in this chapter is the infinite-dimensional Hilbert space. T h e main goal is to give a rigorous analysis of the problem of expanding a vector in a Hilbert space in terms of an orthogonal basis containing a countable infinity of vectors. We first review in $1 a few key theorems on vector spaces in general, and in 52 we investigate the basic properties of vector spaces on which an inner product is defined. In order to define convergence in an inner-product space, we introduce in $3 the concept of metric. I n $4 we give the basic concepts and theorems on separable Hilbert spaces, concentrating especially on properties of orthonormal bases. We conclude the chapter by illustrating some of the physical applications of these mathematical results with the initial-value problem in wave mechanics.

1, Vector Spaces

1.1.

VECTOR SPACES OVER FIELDS OF SCALARS

A mathematical space is in general a set endowed with some given structure. Such a structure can be given, for instance, by means of certain operations which are defined on the elements of that set. These operations are then required to obey certain general rules, which are called the postulates or the axioms of the mathematical space.

Definition 1.1. Any set V on which the operations of vector addition and multiplication by a scalar are defined is said to be a vector 11

12

I. Basic Ideas of Hilbert Space Theory

space (or linear space, or linear manifold). T h e operation of vector addition is a mapping,* (f,g)+f+g,

(f,dEY

x Y,

f+geY,

of Y x Y into Y , while the operation of multiplication by a scalar a from a field+ F is a mapping

4, (%f)E

(a,f)*

F x

v,

afe v,

of F x Y into V . These two vector operations are required to satisfy the following axioms for anyf, g, h E V‘ and any scalars a, b E F :

+

(1) f g = g +f(commutativity of vector addition). (2) (f + g) h = f + (g + h) (associativity of vector addition). (3) There is a vector 0, called the zero vector, such that g satisfies the relationf + g = f if and only if g = 0. (4) 4 f g) = af at?. ( 5 ) ( a b ) f = af (6) ( a h ) f = a(bf). (7) If = f, where 1 denotes the unit element in the field.

+

+

+-

+

+ w.

By following a tacit convention, we denote a mathematical space constructed from a set S by the same letter S, except where ambiguities might arise. Thus, we shall denote by 9”the vector space consisting of a set V together with the vector operations on Y also by Y . When in a vector space the multiplication by a scalar is defined for scalars which are elements of the field F , we say that we are dealing with a vector space over thefield F . If the field F is the field of real or complex numbers the vector space is called, respectively, a real or a complex vector space.

* We remind the reader that a mapping M of a set S into a set T is any unambiguous rule assigning to each element 6 of S a single element M ( 6 ) of T,M ( 6 ) is called the image of 6 under the mapping M. The set S is the domain of definition of M, while the subset T , C T of all image points M ( f ) ,T , = {v = M(5): 6 E S } , is the range of M. If T, = T, then we say that M is a mapping of the set S onto the set T. If S, ,..., S, are sets, then S, x ... x S, denotes the family (6, ,..., 6,) of all n-tuples of elements 6, E S, ,..., 6, E S , , and is called the Cartesian product of the sets S, ,...,S , . + A field is a set on which field operations of summation and multiplication are defined, i.e., operations satisfying certain axioms. We do not give these axioms because in the sequel we are interested only in two special well-known fields: the field of real numbers (wl and the field of complex numbers consisting, respectively, of the set of real numbers (wl and the set of complex numbers on which the field operations are ordinary summation and multiplication of numbers (see Birkhoff and MacLane [1953]).

c1

13

1. Vector Spaces

As an example (see also Exercises 1.1, 1.2, and 1.3) of a real vector space consider the family (a,) of one-column real matrices and define for a =

("), /?=(s,) bn

an

vector summation by the mapping

and for any scalar. a E R1 define multiplication of a by a as the mapping (a, a) ++ aa

=

(""). aarl

It is easy to check that Axioms 1-7 in Definition 1.1 are satisfied. Analogously we can define the complex vector space (C") by introducing in the set @" of one-column matrices vector operations defined by the mapping (1.1) and (1.2), where now a, /3 E @", and therefore a, ,..., a, , b, ,..., b, , as well as the scalar a, are complex numbers. 1.2.

LINEAR INDEPENDENCE OF VECTORS

Theorem 1.1. Each vector space V has only one zero vector 0, and each element f of a vector space has one and only one inverse (-f ). For any f E V , Of

(-1)f

= 0,

=

(-f).

Proof. If there are two zero vectors 0, and 0, , they both have to satisfy Axiom 3 in Definition 1.1,

f=f+O,

=f+O,

+ +

for all f. Hence, by taking f = 0, we get 0, = 0, 0, , and then by 0, = 0, 0, = 0,. Now taking f = 0, we deduce that 0, = 0,

+

f and therefore Of

=

=

lf=(l + o ) j

=

lf+Of=f+Of

0. We have

(-1)f + f

=

(-l)f

+ If = (-1 + 1)f = Of = 0 ,

I. Basic Ideas of Hilbert Space Theory

14

which proves the existence of an inverse (-f) = (-l)f for f. This inverse (-f) is unique, because if there is another f, E V such that f f, = 0, we have

+

(-f) = (-f) + 0 = (4)+(f +fl) = K-f) +fl +fi = 0 +fl =fl . Q.E.D. Definition 1.2. T h e vectors f, ,...,f, are said to be linearly independent if the relation clfi

+ + * * a

cnfn

= 0,

~1

,a*.,

F,

cn E

hast, = ... = c, = 0 as the only solution. A subset S (finite or infinite) of a vector space V is called a set of linearly independent vectors if any finite number of diferent vectors from S are linearly independent. T h e dimension of a vector space V is the least upper bound (which can be finite or positive infinite) of the set of all integers v for which there are v linearly independent vectors in V . 1.3.

DIMENSION OF

A

VECTORSPACE

When the maximal number of linearly independent vectors in the vector space V is finite and equal to n, then by the above definition V is n dimensional; otherwise the dimension of V is 00, and V is said to be infinite dimensional.

+

+

Theorem 1.2. If the vector space V is n dimensional ( n < co), then there is at least one set f, ,...,f, of linearly independent vectors, and each vector f E V can be expanded in the form (1 -3)

f = % f l + + %fn, a , ,..., a , (which are scalars) **.

where the coefficients mined by f.

are uniquely deter-

Proof. Iff = 0, (1.3) is established by taking a , = For f # 0, the equation ( 1*4)

Cf

+

C,fl

+ -.. +

cnfn

=

a, = 0.

=0

should have a solution with c # 0 due to the assumption that fl ,...,f, are linearly independent, while f,fi ,...,f, have to be linearly dependent because V is n dimensional. From (1.4) we get

f = (-C,/C>f,

+ ... +

(-C*/Clfn

9

15

1. Vector Spaces

which establishes (1.3). If we also had

f = hf, + ... + bJn

(1.5)

9

then by subtracting (1.5) from (1.3) we get

+ ... +

(a,--l)f,

(an -

=0

U f n

*

As f, ,...,fn are linearly independent we deduce that

a, - b,

=

0, ...,

0, thus proving that a, ,..., a, are uniquely determined when f is given. Q.E.D. a, - b, =

Definition 1.3. We say that the (finite or infinite) set S spans the vector space Y if every vector in Y can be written as a linear combination f = a&,

+ + anhn, .*'

h,

,..., h, E s

of a finite number of vectors belonging to S; if S is in addition a set of linearly independent vectors, then S is called a vector basis of Y .

, g is a vector basis of the n-dimenTheorem 1.3. If the set {g, ,...,} sional (n < +a)vector space Y , then necessarily m = n. Proof. As 9'-is n-dimensional, there must be n linearly independent vectors f,,...,fn . If the set {g, ,...,gm}is a vector basis in Y ,we can write

Thus, if we try to satisfy the equation (1.7)

Xlf,

+ + *.-

Xnfn

= 0,

we get by substitutingf, ,...,f, in (1.7) with the expressions in (1.6) (1.8)

(allxi

+ + ainxnlgi + +

+ +

(ami~i

**.

amnXn)gm

= 0-

Since g, ,..., g, are assumed to be linearly independent, the above equation has a solution in x1 ,..., x, if and only if a,,Xl

(1.9) a,,x,

+ + a,,xn + ... + ***

UmnXn

=0

= 0.

However, as f,,...,f, are also linearly independent, (1.7) or equivalently = (1.8) or (1.9) should have as the only solution the trivial one x1 =

I. Basic Ideas of Hilbert Space Theory

16

<

xn = 0. Now, m n because V is n dimensional and g, ,..., g, are linearly independent (see Definition 1.2); therefore, (1.9) has only a trivial solution if and only if m = n. Q.E.D.

Definition 1.4, A subset V, of a vector space V is a vector subspace (linear subspace) of V if it is closed under the vector operations, i.e., if f + g E Vl and af E V, whenever f,g E V, and for any scalar a. A vector subspace Vl of V is said to be nontrivial if it is different from V and from the set (0). From the very definition of the dimension of a vector space V we can conclude that the dimension of a vector subspace V, of V cannot exceed the dimension of V . 1.4. ISOMORPHISM OF VECTOR SPACES

Definition 1.5. Two vector spaces V, and V, over the same field are isomorphic if there is a one-to-one mapping V, onto Vzwhich has the properties that if f, and g, , f, ,g, E V, , are the images of fl and g, , f, , gl E V, , respectively, then for any scalar a , afz is the image of afl

-

and f,

.f2

J

+ g, is the image of fl + g, fl + g, -fz + gz.

The importance of the isomorphism of two vector spaces Vl and V, lies in the obvious fact that two such spaces have an identical vector structure. It is easy to see that the relation of isomorphism is transitive (see Exercise 1.6), i.e., if Vl and V, as well as V, and Vaare isomorphic, then Vl and V3are also isomorphic.

+

Theorem 1.4. All complex (real) n-dimensional ( n < co) vector spaces are isomorphic to the vector space (P)[(Rn) in case of real vector spaces].

Proof. Consider the case of an n-dimensional vector space V . According to Theorem 1.2 there is a vector basis consisting of n vectors f, ,...,f, , and each vector f E V can be expanded in the form (1.3), where a, ,..., a, E C1are uniquely determined by f. Consequently f H OLf

=

(?) an

E(P)

17

1. Vector Spaces

is a mapping of V into (C"). Furthermore, this is a one-to-one mapping of 7cr onto (C") because to any

corresponds a unique f = b, f, easy to see that

+ *.. + b, f, such that /3 = cuJ . It is also

f + g I--t a f + g U f H

= af

+

a,f = Ucuf

a g 9

.

Since isomorphism of vector spaces is a transitive relation (see Exercise 1.6) we can conclude that all n-dimensional complex vector spaces are mutually isomorphic, because each of them is isomorphic to (a=,). Q.E.D.

EXERCISES 1. I . Check that the set of all m x n complex matrices constitutes an m * n dimensional complex vector space if vector addition is defined as being addition of matrices, and multiplication by a scalar is multiplication of a matrix by a complex number. 1.2. Show that the set C1 of all complex numbers becomes a twodimensional real vector space if vector addition is identical to addition of complex numbers, and multiplication by a scalar is multiplication of a complex number (the vector) by a real number (the scalar). 1.3. Show that the family %'O(W) of all complex-valued continuous functions defined on the real line is an infinite-dimensional vector space if the vector sum f g off(.), g(x) E %?O(W) is the function (f g)(x) = f ( x ) g(x), and the product uf of f ( x ) E %?O(R1) with u E C' is the function ( u f ) ( x ) = u f ( x ) . T h e zero vector is taken to be the function

+

+

f

(x)

+

= 0.

1.4. Prove that if X is a family of linear subspaces L of a vector space V,then their set intersection OLE%L is also a vector subspace of V .

1.5. Show that if S is any subset of a vector space V , then there is a unique smallest vector subspace Vscontaining S (called the vector subspace spanned by S). 1.6.

Verify that the relation of isomorphism of vector spaces is:

(a) reflexive, i.e., every vector space V is isomorphic to itself; (b) symmetric, i.e., if V, is isomorphic to V2, then V2is isomorphic to V,;

I. Basic Ideas of Hilbert Space Theory

18

(c) transitive, i.e., if Vl is isomorphic to Vzand Vzis isomorphic to V 3 then , Vl is isomorphic to V3.

1.7. Prove that the following subsets of the set Uo(R1) (see Exercise 1.3) are vector subspaces of the vector space Uo(R1): (a) the set 8, of all polynomials with complex coefficients; (b) the set 8, of all polynomials of at most degree n. Show that 8,3 gJn.

2. Euclidean (Pre-Hilbert) Spaces 2.1.

INNER PRODUCTS ON VECTORSPACES

A Euclidean (or pre-Hilbert or inner product or unitary) space d is a vector space on which an inner product is defined. T h e Euclidean space is called real or complex if the vector space on which the inner product is defined is, respectively, real or complex.

Definition 2.1. An inner (or scalar) product I -)on the complex vector space 7cr is a mapping of the set V x V into the set C1of complex numbers (a

(f,g)Cf(fIg),

(f,g)EvXv,

(fIg>E@l,

which satisfies the following requirements: for all (1) (f I f ) > 0, (2) = (g If >*, (3) (f I ag) = a < f I g>, (4) ( f l g h ) =
+

f # 0, Q

E @',

+ (fl h ) .

Note that by inserting f = g = h = 0 in Point 4 we get (0 10) = 0. Following a notation first introduced by Dirac [1930] and widely adopted by physicists, we denote the inner product off and g by (f I g). Mathematicians often prefer the notation (f, g) and replace Point 3 in Definition 2.1 by

(4g) = a(f9 g). The above definition can be easily specialized to real vector spaces, in which case the inner product (f I g) is a real number, and Point 2 of Definition 2.1 becomes (f I g) = (g If). As in quantum physics we deal almost exclusively with complex Euclidean spaces, we limit ourselves from now on to the complex case. Consequently, if not otherwise stated,

19

2. Euclidean Spaces

whenever we talk about a Euclidean space, we shall mean a complex Euclidean space.

Theorem 2.1. I n a Euclidean space 8,the inner product (f I g) satisfies the relations (a) (af I g> = a*, (b) ( f + g I h ) = ( f l h ) (g I h ) . The proof is obtained by a straightforward application of Points 1-4 in Definition 2.1 :

+

(af I g> =

As an example c l a finite-dimensional Euclidean space, we can take the vector space (Cn)defined in the preceding section, in which we introduce as the inner product of the vectors a and fl with the kth components a k and bk , ( a 1 /3> = a,*b,

+ a,*b, + + an*b, . .**

It is easy to check that the above mapping of (P)x (Cn)into @' satisfies the four requirements of Definition 2.1. We shall denote the above Euclidean space with the symbol 12(n). An example of an infinite-dimensional Euclidean space is provided by the vector space [Up2,(W)] of all continuous complex-valued functions f ( x ) on the real line which satisfy

in which the inner product (see Exercise 2.1) is

Theorem 2.2. Any two elementsf, g of a Euclidean space d satisfy the Schwarz-Cauchy inequality

I
< .

Proof. For any given f,g E 8 and any complex number a we have, from property 1 in Definition 2.1 and the comment following it, (f -t- ag I f

+ ag> 2 0.

20

I. Basic Ideas of Hilbert Space Theory

I n particular, if we take in the above inequality

we easily show that the inequality = A2(g I g>

+ 2h I < f l g)l + z 0

is true for all real values of A. A necessary and sufficient condition that g(A) 0 is that the discriminant of the quadratic polynomial g(h) is not positive Kf I g>I2- < 0, from which the Schwarz--Cauchy inequality follows immediately. Q.E.D.

2.2. THECONCEPTOF NORM T h e family of all Euclidean spaces is obviously contained in the family of vector spaces. There is another family of vector spaces with special properties which is of great importance in mathematics: the family of normed spaces.

Definition 2.2.

A mapping

f

-+

Ilfll,

f v,

llfllE R',

of a complex vector space Y into the set of real numbers is called a norm if it satisfies the following conditions:

IlJll >. 0 for f f 0, II OII = 0, (3) 11 afll = I a I I/f \I for all a~ Cl, (the triangle inequality). (4) Ilf -tg II < llfll II g II We denote the above norm by 11 * 11. (1) (2)

+

For a real vector space, we require in Point 3 that a E R'. The last requirement in Definition 2.2 is known as the triangle inequality because it represents in a two- or three-dimensional real vector space a relation satisfied by the sides of a triangle formed by three vectors f,g andf g. A real (complex) vector space on which a particular norm is given is called a real (complex) normed vector space. A Euclidean space is a special case of a normed space; this can be seen from the following theorem.

+

21

2. Euclidean Spaces

Theorem 2.3. I n a Euclidean space d with the inner product

(f I g) the real-valued function

llfll

=

mm-

is a norm.

Proof. T h e only one of the four properties of a norm which is not satisfied by (2.3) in an evident way is the triangle inequality. We easily get Ilf+g1I2 (2.4)

+ + (g If>+ (g lg)

(f+ g I f + & = (f + f > = (fI f i 2 W f I g> (g I g>. =

+

+

From the Schwarz-Cauchy inequality we have

I Re(f I g>l < I(f

Ig)I

< llfll

Ilg I I Y

which when inserted in (2.4) yields llf+g112
=(Ilfll+Ilg11)2*

The above relation leads immediately to the triangle inequality.

Q.E.D.

VECTORS AND ORTHONORMAL BASES 2.3. ORTHOGONAL Some elementary geometrical concepts valid for real two- or threedimensional Euclidean spaces can be generalized in a straightforward manner to any Euclidean space.

Definition 2.3. I n a Euclidean space d two vectors f and g are called orthogonal, symbolically f 1g , if (f I g ) = 0 . Two subsets R and S of E are said to be orthogonal (symbolically, R 1S) if each vector in R is orthogonal to each vector in S. A set of vectors in which any two vectors are orthogonal is called an orthogonal system of vectors. A vectorf is said to be normalized if l l f l l = 1. An orthogonal system of vectors is called an orthonormal system if each vector in the system is normalized. Theorem 2.4. If S is a finite or countably infinite set of vectors in a Euclidean space d and (S) is the vector subspace of d spanned by S, then there is an orthonormal system T of vectors which spans (S),i.e., for which (T) == (S);T is a finite set when S is a finite set. Proof. As the set S is at most countable we can write it in the form

s=

{fl

, . f z ,*-I

by assigning each vector in S to a natural number. I n general some of the vectors in S might be linearly dependent. We can build from S

I. Basic Ideas of Hilbert Space Theory

22

another set So of linearly independent vectors spanning the same subspace (S), i.e., such that (So)= (S), by the following procedure (which should be applied consecutively on n = 1, 2,...) : iff, is the zero vector or is linearly dependent on f, ,...,fn-l, then discard it; otherwise include it in S o . Thus we get a set So of linearly independent vectors

so

= {g, ,g2

,...I,

(So) = (S).

We can obtain from So an orthonormal set T such that (T) = ( S o ) by the so-called Schmidt (or Gram-Schmidt) orthonormalization procedure. Since g, # 0, we can introduce the vector

which is normalized. Proceeding by induction, assume that we have obtained the orthonormal system of vectors el ,..., e,,-, . Then en is given by en =

gn - (en-1 I gn) II gn - < e n 4 I gn)

- . * * - <el - * * . - (el

en-1 en-1

I gn) e l I En) e l II

*

The above vector is certainly well defined, since the denominator of the above expression is different from zero; namely, if it were zero, then we would have gn - (en-1 I gn) en-1 - * * * - (el I gn) e l = 0, i.e., g, would depend on el ,..., en-l . However, by solving the equations for el ,..., efi-.l , it is easy to see that we have El gn-1

= C1.A

= cn-1.1e1

-1- cn-1.2e2

+ + *-*

Cn-1.n-1%-1

,

and therefore if g, depended on el ,..., e n - l , then it would also depend on g, ,..., gnPl, contrary to the fact that So consists only of linearly independent vectors. The vectors of T are obviously normalized. I n order to prove that T is an orthonormal system, assume that we have proved that (ei I e j ) = Sii for i, j = 1,..., n - 1. Then we have for m < n

which proves that (ei I ej) is orthonormal.

=

6, for i,j

=

1, ..., n. Thus, by induction T

23

2. Euclidean Spaces

g,

As we have for any n that el ,..., en can be expressed in terms of ,..., g, , and vice versa, we can conclude that (T) = (So). Q.E.D.

2.4. ISOMORPHISM OF EUCLIDEAN SPACES We introduce now a concept of isomorphism of Euclidean spaces, which makes two isomorphic Euclidean spaces identical from the point of view of their vector structure as well as from the point of view of the structure induced by the inner product.

Definition 2.4. T w o Euclidean spaces 8,and 8,with inner products

and I .),, respectively, are isomorphic (or unitarily equivalent) if there is a mapping of 8,onto 8: ( -I *),

(a

fl H f 2 , f l E 4 such that if for anyf, ,g, E 8,the vectorf, vector g, E 8.is the image of g, , then

fie 8 2

9

-

fl f g 1 - f z 4-1

(fl I g1)1

E

8,is the image off, and the

+gz, 4 2

= (fz

I

a E @',

I gzh *

A mapping having the above properties is called a unitary transformation of 8,onto 8..

Theorem 2.5. All complex Euclidean n-dimensional spaces are isomorphic to Z2(n), and consequently (see Exercise 2.8) mutually isomorphic. Proof. If d is an n-dimensional Euclidean space, there is according to Theorem 1.2 a set of n vectors fl ,...,fn spanning B. According to Theorem 2.4, we can find an orthonormal system of n vectors el ,..., en which also spans 8.It is easy to check (see Exercise 2.7) that the mapping

provides an isomorphism between B and Z2(n). Q.E.D. Obviously, a similar theorem can be proved for real Euclidean spaces.

Theorem 2.6. A unitary transformation (2.6)

fl-fz,

fie47

f2E82,

24

I. Basic Ideas of Hilbert Space Theory

of the Euclidean space 8,onto the Euclidean space E2 has a unique inverse mapping which is a unitary transformation of g2onto 8,. Proof.

We note that since

llfl - g1 111 = Ilf*- g 2 112

Y

the images f2 and g, of fl and g , , respectively, are distinct whenever f, # g, . Since the unitary map of dlis onto d2,we conclude that the inverse of the mapping (2.6) exists. We leave to the reader the details of the remainder of the proof. EXERCISES 2.1.

Show that for a finite interval I

1

= I f*b)

dx

is an inner product on the vector space q 0 ( I ) . 2.2. Show that the vector space q2)(R1) introduced in Section 2.1 is a subspace of the vector space Wo(R1). 2.3.

Prove that (2.2) is an inner product in q2)(R1).

2.4.

Show that

II2= , Ilf+gll = llfll + llgll, if and only if either f is a multiple of g , i.e., iff = ag, a E @', or g = 0, and if in addition a 2 0 in case of the second relation. 2.5. Show that if T is an orthonormal system of vectors, then all the vectors in T are necessarily linearly independent. 2.6. space.

Prove that a subspace of a Euclidean space is also a Euclidean

2.7. Show that the mapping (2.5) is a mapping of d onto Z2(n),and that it satisfies the requirements of isomorphism given in Definition 2.4. 2.8. Show that the relation of isomorphism of inner-product spaces is an equivalence relation, i.e., it is (see Exercise 1.6) reflexive, symmetric, and transitive.

25

3. Metric Spaces

3. Metric Spaces 3.1.

CONVERGENCE IN METRIC SPACES

In an n-dimensional Euclidean space d we can always find, due to Theorems 1.2 and 2.4, a basis of n vectors e, ,..., en which constitute an orthonormal system. We can then expand any vector f of d in that basis n

f = k1 akek =l We easily see that ak = (ek If). In an infinite-dimensional Euclidean space not every vector can be expanded in general in terms of a finite number of vectors. We can hope, however, to replace (3.1) with the formula (D

=

1

akek

3

k=l

but then we meet with the problem of giving a precise meaning to the convergence of the above series. This problem is solved in its most general form in topology, but for our purposes it will be sufficient to solve it within the context of metric spaces.

Definition 3.1. If S is a given set, a function d ( 4 , q ) on S x S is a metric (or distance function) if it fulfills the following requirements for any t,7,t; E S : (1) d(4, rl) > 0 if 4 f 77, (2) 4) = 0, (3) 4'571)= 4%41, (triangle inequality). (4) d(E, t;) d ( t , 7)t- 471, t;) A set S on which a metric is defined is called a metric space. A metric space does not have to be a linear space. For instance, a bounded open domain in the plane becomes a metric space if the metric is taken to be the distance between each pair of points belonging to that domain; such a domain obviously is not closed under the operations of adding vectors in the plane, but it provides a metric space. Generalizing from the case of one-, two-, or three-dimensional real Euclidean spaces, we introduce the following notions.

a,

<

Definition 3.2. An infinite sequence t1 , t2,... in a metric space A is said to converge to the point $, E A?if for any E > 0 there is a positive

26

I. Basic Ideas of Hilbert Space Theory

number N(c) such that d(5, En) < c for all n > N(c). An infinite sequence 5, , 5, ,... is called a Cauchy sequence (or a fundamental sequence) if for any c > 0 there is a positive number M ( c ) such that d ( [ , , 5 ,) < E for all m,n > M(c).

el, 5, ,... in a metric space A! con-

Theorem 3.1. If a sequence verges to some { E &, then its limit a Cauchy sequence.

5 is

unique, and the sequence is

Prooj. If 5, , 5, ,... converges to 4 E A! and to 9 E A, then by definition, for any c > 0 there are N,(c) and N,(c) such that d(5, 5,) < c for n > N,(c) and d(7, 6,) < c for n > N2(c). Consequently, for n > max(N,(c), N,(r)) we get by applying the triangle inequality of Definition 3.1, Point 4, 45,T)

<

4 5 9 5 , )

+W n

3

77)

< 2c.

As c > 0 can be chosen arbitrarily small, we get d ( 5 , ~= ) 0, which, according to Definition 3.1, can be true only if 5 = r]. Similarly we get d(5m

if m, n

,I n )

<

d(5m9

I)

+ d(E,

En)


> N1(c/2);i.e., the sequence 6, , 5, ,... is also a Cauchy sequence. Q.E.D.

3.2. COMPLETEMETRICSPACES In case of sequences of real numbers, every Cauchy sequence is convergent, i.e., the set R' of all real numbers is complete. We state this generally in Definition 3.3.

Definition 3.3. A metric space .& is complete if every Cauchy sequence converges t o an element of A!. Not every metric space is complete, as exemplified by the set R of all rational numbers with the metric d(rn,/n, ,mz/nz)= I ml/nl - m2/nz 1, which is incomplete. However, we know that the set R is everywhere dense in the set R1; we state this generally as follows: Definition 3.4. A subset S of a metric space A! is (eserywhere) dense in A! if for any given E > 0 and any 5 E A! there is an element r] belonging to S for which d ( 5 , q ) < c. We can reexpress the above definition after introducing a few topological concepts, generalized from the case of sets in one, two, or three real dimensions.

27

3. Metric Spaces

Definition 3.5. If .$ is an element of a metric space .X, then the set of all points 7 satisfying the inequality d(5, 7) < E for some E < 0 is called the E neighborhood of 4. If S is a subset of A, a point 5 E A is called an accumulation (or cluster or limit) point of S if every E neighborhood of contains a point of s.T h e set consisting of all the cluster points of S is called the closure of S. Obviously always S C S ; if S = S then S is called a closed set. We say that the subset S of a metric space d?is (everywhere) dense in A? if and only if A? is its closure, i.e., if and only if .X. T h e procedure of completing the set % of rational numbers by embedding it in the set of all real numbers can be generalized.

<

s

s

2

Definition 3.6. A metric space .X is said to he densely embedded in the metric space M- if there is an isometric mapping of A into JZ, and if the image set M'of .A!in A? is everywhere dense in A. A one-to-one mapping .$ f--1 g of a metric space ,X into another metric space A- is called isometric if it preserves distances, i.e., if d,(.$, 7) = d,([, ij) for 5, q E ,A!and <,ij E .4?whenever f H .$ and 9 c+ 3. 3.3.

COMPLETION OF A h'IETRIC SPACE

*Theorem 3.2. Every incomplete nietric space A?can bc embedded in a complete metric space J&, called the compl&m of .A'. T h e proof of this theorem can be given by generalizing Cantor's construction, by which one builds the set of real numbers from the rational numbers. Denote by Asthe family of all Cauchy sequences in .&. If [' = .$*",...} are two such sequences, we say that .$2'l...} and &" = they are equivalent if and only i f lirn d(g,',

n-ta

5,")

=

0.

It is easy to see that we have thus introduced an equivalence relation in Ma(see Exercise 3.1) if we recall (see Exercises 1.6 and 2.8) the general definition of an equivalence relation.

-

Definition 3.7. A relation holding between any two ordered elements of a set S is called an equivulence relation if it is (1)

--

reflexive: (2) symmetric: .$ (3) transitive: [

- -

5 for all f E S;

implies that 7 4; 5 implies that

q and q

4

-

5.

I. Basic Ideas of Hilbert Space Theory

28

-

A subset X of S having the property that all the elements of X are equivalent and that if q 5 and 5 E X then q E X is called an equivalence class (with respect to the equivalence relation -). We denote the family of all equivalence classes in A,[with respect to the equivalence relation given by (3.2)] by the symbol A?, and agree to denote the equivalence class containing the Cauchy sequence [ also by [. Consequently if p, p E A,then p = f" if and only if the Cauchy sequences p, f" E A?, are equivalent, i.e., satisfy (3.2). by defining We introduce the real function d,([, ij) on A,x for if = {tl , t2,...} and ij = {ql , q2 ,...1

(3-3)

&,

ij)

=

E ! 45,

3

71,).

In order to see that the above limit exists for any [, ij E A8we employ the relation (see Exercise 3.2)

to show that d ( t l , ql), d ( t 2 , q2),... is a Cauchy sequence of numbers, and therefore has a limit; namely as fl, t2,... and ql , q2 ,... are Cauchy sequences, we can make d(&, , 5,) < E if m,n > N1(e),and d(qm,7), < E if m, n > N2(e),which, used in conjunction with (3.4), proves the statement. We can show that dJ[, i j ) also defines a real function on A? x 2 by establishing thzt d,(g', 7') = d,(f", ij") if = p and ij' = ij" for p, f", ij', ij" E A?.We first obtain that d,(g', 3') = d , ( p , ij') from the inequality (see Exercise 3.3)

because d(t,', fntt) + 0 as n + co due to the fact that the Cauchy sequences p and f" belong to the same equivalence class. Similarly we can show that d,(f", i j ' ) = d,(g", ij"), and thus prove that d , ( p , ij') = d,(g", 4"). It is easy to check that the function d,([, ij) defines a metric on A? (see Exercise 3.4). We show now that the ensuing metric space, which we denote also by A?, is complete. Assume that [(l), [(2),... is a Cauchy sequence in A?, where [ ( k ) is the equivalence class containing the Cauchy sequence {tik), tik),...} of E elements of A, Choose for each integer k an element q k = such that d(tLt), q k ) = d ( f g ) , 511"))< I/k for all m greater than some Nk; this is certainly possible because tik), tkk), ... is a Cauchy sequence in A.

29

3. Metric Spaces

Consider now the elements i j k of A, . We obviously have dS(!Q),

ijk)

=

ds([(k)',

7 j k ) since Iim ds([i:)3 [ ( k ) )

d(S::',

=

If we let in the above relation m

{qk , r]k ,...} and

[g) = {(g),fy),...}

a)< I l k .

then we find that d s ( [ z ) , i j k ) = O as m -+ 00 (see Exercises 3.5 and 3.6) -+ CO,

and consequently dS("",

We can now deduce that i j by writing

(3.5)

d(7m

Y

17n)

ijk)

< l/k.

= { q l , q2 ,...}

is a Cauchy sequence in

= ds(ijm Y i j n )

< ds(7jm,

[(m))

+ ds(F"), P ) )+ ds(l("),

ijm)

Since [(l), [(2),... is a Cauchy sequence in A?,we can make d,([(m),[(,)), and consequently the entire right-hand side of (3.5) arbitrarily small for all sufficiently large m and n. Thus, i j E dS . We can establish that the equivalence class i j E A? containing the Cauchy sequence i j = {ql , q 2 ,...} is the limit of [ ( l ) , [(2),... if we write (3.6)

d*(ij,

P )< ds(ij,

ijk)

+

dS(7jk

3

4 9 .

The right-hand side of (3.6) can he made arbitrarily small for sufficiently large k because d y ( i j k ,[ ( k ) ) 1/K and limk+md6(7j,i j k ) = 0 (see Exercise

<

3.6). In order to finish the proof of the theorem, we have to embed &? into the complete metric space d#. T o that purpose we map 5 E A! into the equivalence class [ containing the sequence (5, 5,...}. This mapping is obviously one-to-one and isometric, as d(5, q) = ds([, ij). Furthermore, the image A%" of M in A? is everywhere dense in A?; namely if f j EJ# contains {ql , v2 ,...} E A s , then for arbitrary E > 0 we can choose an qk in A'' containing { q k ,qk ,...} and such that d,(7j, ijk) < c.

EXERCISES

-

Show that the relation [ i j between any two Cauchy sequences [ = (6, , t2,...} and i j = {vI, q 2 ,...} of a metric space M , defined to mean that lim,,,,fl d ( t n , q,) = 0, satisfies the three requirements given in Definition 3.7 for an equivalence relation. 3.1.

I. Basic Ideas of Hilbert Space Theory

30

3.2. Prove that any four elements satisfy the relation

I 461

9

-

52)

tl, 6, , ql , 7,of a metric space A

4% 7741 < 4 5 1 Y

Y

?1)

+

4 5 2

Y

772).

3.3. Prove that if f , 7,5 are elements of a metric space A, then

I

4

5

9

17)

-

45YOI

< 4%5).

3.4. Show that the function ds(.f,i j ) defined on M-x A? by (3.3) satisfies the four requirements for a metric (those requirements are formulated in Definition 3.1).

3.5. If in a metric space A the sequence 5, , f , ,... converges to f , prove that for any 7 E A!, limn+md(tn , 7) = d ( f , 7).

3.6. If is the equivalence class of A? (introduced in Theorem 3.2) containing the Cauchy sequence { f , , f 2 ,...} EM, , then for any E > 0 there is an N ( E )such that d,([, .fk) < E for all k > N ( E ) ,where [k = { f k , f k ,...}. Prove this statement! 3.7. Show that if S, is an everywhere dense subset of a metric space A, and S, is an everywhere dense subset of S , , then S, is everywhere dense in A. 4.

4.1.

Hilbert Space OF COMPLETIQN

A

EUCLIDEAN SPACE

It is easy to establish (see Exercise 4.1) that in normed space JV

is a metric. Therefore, we can define in J1’ convergence, completeness, etc. in the metric (4.l), which is then called convergence, completeness, etc. in the norm. A complete normed space bears the name of Banach space. The above concepts can also be applied to Euclidean spaces, because according to Theorem 2.3 we can introduce in such spaces a norm, and therefore also a metric. A Euclidean space which is complete in the normXis called a Hilbert space. Not every Euclidean space is a Hilbert space. For instance, the

* T h e concept of completeness can be defined and considered for other topologies besides the norm topology.

31

4. Hilbert Space

Euclidean space U$,(R1) introduced in 52 is not complete. To see this, note that the sequence f,,f 2 ,... of continuous functions

is a Cauchy sequence in %7!21(R1)(see Exercise 4.2) but it does not converge to an element of UYz@l). I n fact, it is easy to establish that with increasing n, the functions in the above sequence approximate arbitrarily closely in norm the discontinuous step function

which, however, does not belong to %7f2)(R1).

Definition 4.1. We say that the Euclidean space 8 can be densely embedded in the Hilbert space X if there is a one-to-one mapping of d into #', such that the image &' of d is everywhere dense in X , and the mapping represents an isomorphism between the Euclidean spaces d and 8'. Theorem 4.1. Any incomplete Euclidean space B can be densely embedded in a Hilbert space.

Proof. Denote by d the complete metric space built from the set 4 of Cauchy sequences in & according to the procedure used in proving Theorem 3.2. Define in c& the operations

P+ d = {fl +

(4.3)

a ?= (49

g1 ,f2

4 2

9.-

>

+

g2

9.-1,

for any two sequences 3 = {fl , f 2 ,...}, g = {g, ,g, ,...} from . It is easy to check that the above operations are operations of vector addition and multiplication by scalar. Furthermore, if f' = J", where f' = {fl', f2' ,...} and J" = {fl",f2",...}, i.e., iff and J" belong to the same equivalence class in d and therefore

6

lim llfn'

- fnn

n-m

then

11

=

lim d(fn',fn")= 0, n+a

(1 ufn' - ufn" 11 = I a I llfn' - f," 11 -+0; thus, we also have that + 2 and uJ' = uJ". Consequently, (4.3) defines vector

f' + 2 = J"

operations on d,' which thus becomes a vector space. We now introduce the complex function on c& x Rs defined by

(4.4)


=

A+ ic

(fn

I En>.

32

I. Basic Ideas of Hilbert Space Theory

T h e limit in (4.4) exists because (fiI gl), (fiI g2),... is a Cauchy sequence of numbers, as can be seen from the following inequality: I(fm

namely,

I gm)

II~, II

+

- (fn

I gn)l

I(fm - f n I gwJ + (fn I Ern - gn)l < llfm - f n II II gm II 4-llfn II II gm - gn II; =

W,0) = IIJII, and

IIg,

II

-

112 IIR for

fi

-+

=, where

while Ilf, & -fn 11 and 1) g , - g, )I can be made arbitrarily small for sufficiently large m and n. Furthermore, if p' = f" E 4 , then (J' I g"), = (p" I g"), , as can be concluded from the inequality I
I gn)

- (fn"

I gn)l

< llfn'

-fn"

II 1I gn It*

Hence, (31g"), is a uniquely defined function on d x 8'.This function determines an inner product on d (see Exercise 4.4). It is obvious that the m a p p i n g f t t f = {f,f, ...} of € into d has an image 8'which is a linear subspace of 8;according to the construction, &' is everywhere dense in d and the above mapping provides an isomorphism between d and 8'. Q.E.D. A similar theorem can be proved for normed spaces (see Exercise 4.5).

4.2. SEPARABLE HILBERTSPACES I n quantum mechanics we deal at present almost exclusively with a special class of Hilbert spaces which are called separable.

Definition 4.2. T h e Euclidean space d is called separable if there is a countable everywhere dense subset of vectors of €. In the early days of research on Hilbert spaces, separability was taken to be an integral part of the concept of a Hilbert space. In quantum mechanics we are concerned primarily with separable complex Hilbert spaces. We shall agree that in the future whenever we refer to a space as a Hilbert space we mean a complex Hilbert space, except if otherwise stated. Theorem 4.2. Every subspace of a separable Euclidean space is a separable Euclidean space. of a Euclidean space d is also a Proof. T h e fact that a subspace Euclidean space is easy to check (see Exercise 2.6). In order to establish

33

4. Hilbert Space

the separability of 8,,construct a countable subset S = {g,, ,g,, , g,, ,g,, ,...} of 8, in the following way. Let R = {fi ,f 2 ,...I be a countable everywhere dense subset of 8; there has to be such a set because of the separability of &. Let g,,, denote a vector of 8,satisfying (1 g, - f, 11 < l/m,in case there is at least one such vector, or the zero vector in case there is no vector of 8,in the l/m neighborhood of f n . The set S is everywhere dense in 8' because for any given h E and m > 0 we can find an f n E R such that 11 h - fn 11 < l / m . Thus, by the above rule of constructing S we certainly have g, # 0 and therefore

II h - gmn II d II h

-fn

II -t- Ilfn

-grim

II < 2im.

This proves that S is everywhere dense in 8,. Q.E.D.

4.3. 1'

SPACES AS

EXAMPLES OF SEPARABLE HILBERTSPACES

As an important example of an infinite-dimensional separable Hilbert space we give the space Z2(co), which is basic in matrix mechanics.

Theorem 4.3. T h e set Z2(m)of all one-column complex matrices with a countable number of elements

01

for which (4.5) becomes a separable Hilbert space, denoted by Z2( a), if the vector operations are defined by

(4.7)

and the inner product by

c a,*b, W

(4-8)

(a

I P>

=

k=l

*

I. Basic Ideas of Hilbert Space Theory

34

The operation (4.7) maps C1 x Z2(m) into Z2(co) because a k l2 < +00 if (4.5) is satisfied. In order to see that (4.6) maps Z2(w)x Z2(m) into Z2(a), apply the triangle inequality on the v-dimensional space Z2(v), v < f a , in order to obtain Pyoof.

xr=lI auk l2 = I a l2 xg1I

The above inequality shows that when v -+ 00, the left-hand side converges if a,p E Z2( 00). Similarly, we prove that (4.8) converges absolutely by applying the Schwarz-Cauchy inequality on the v-dimensional space Z2(v), v < + 00, in order to obtain

We leave as an exercise for the reader (see Exercise 4.6)to check that we deal indeed with a Euclidean space. , T o prove that this Euclidean space is complete, assume that &), d 2 )... is a Cauchy sequence, where

As we obviously have that for any k = 1, 2,...

1

k

- a‘n’

I

< I(

a(m)

- Cdn) I19

we deduce that for fixed k the sequence a:’), aka),... is a Cauchy sequence of complex numbers; hence, this sequence has a limit bk . We shall prove that the one-column infinite matrix

is an element of Z2(m),and that dl),d 2 )... , converges in the norm to #I. By applying again the triangle inequality on the v-dimensional space Z2(v),v <: +a, we obtain (4.9)

35

4. Hilbert Space

the above inequality is true for any m = 1, 2,... . As ,... is a Cauchy sequence, there is for any given E > 0 an No(€)such that for any m,n > No(€)and any positive integer v

On the other hand, as b, = li%+m u(km),we can find for tixed v an IVY(€) such that I b, - uim)I < ~ / 2 ( , + l )for / ~ any m > N V ( c ) and all k = 1, 2, ..., v. Thus, we get from (4.9) that for all n > No(€)and all positive integers v (4.10)

As the right-hand side of the above inequality is independent of v and the inequality itself is true for any n > No(€),we can let v + co in (4.10) to derive

[x I m

(4.11)

for all n

bk - u p ) Iz]l”4

> No(€).

k=l

By returning again to Z2(v),v

< + co,to obtain

we establish that fl E Z2( co) by letting v + co. The relation (4.1 1) tells us now that a(,),... converges to fl. Finally, in order to prove the separability of Z2( a),consider the set D of all the one-column matrices 01 from Z2(+co) with kth components ( a ) , = a, , where a, has rational numbers as its real and imaginary part, i.e., Re a, , I m a, E ‘93, k = 1, 2,..., and in addition (4.12)

a,,,

= a,,,

=

.*

=0

for some integer n. The set D is countable (see Exercise 4.7). I n order to prove that D is everywhere dense in Z2(co), take any one-column

I. Basic Ideas of Hilbert Space Theory

36

+

matrix y E 12( 00) with kth component ( Y ) = ~ ck . As y E 12( a), there is for any given E > 0 an integer n such that m

1 ck 1' < c2/2.

1

k=n+l

Furthermore, as the set % of rational numbers is dense in the set [w1 of real numbers, we can choose an a E D which satisfies (4.12) and is such that 1 ck - ak I < E / ~ / % z for all k = 1,..., n. Thus, we have

which proves that 12( 00) is separable.

Q.E.D.

4.4. ORTHONORMAL BASESIN HILBERTSPACE In an infinite-dimensional Euclidean space it is important to distinguish between the vector space (S) spanned by a set S, and the closed vector space [S]spanned by S.

Definition 4.3. T h e vector space (or linear manifold) ( S )spanned by the subset S of a Euclidean space &' is the smallest" subspace of € containing s. The closed vector subspace [ S ] spanned by S is the smallest closed vector subspace of & containing S. In the finite-dimensional case (8)= [S]because all finite-dimensional Euclidean spaces are closed (see Exercise 4.8). That this is not so in the infinite-dimensional case can be deduced from the following theorem, whose simple proof we leave to the reader (see Exercise 4.9). Theorem 4.4. T h e subspace (S)of the Euclidean space & spanned by the set S is identical with the set of all finite linear combinations adl anfn of vectors from S, i.e., in customary set-theoretical notation,

+ +

( S ) = {alfl

+ -..+ anfn:

,...,f, E S , a, ,..., a, E C1, n

fl

=

1, 2,...}.

The closed linear subspace [S]spanned by S is identical to the closure (s) of ( S ) .

Definition 4.4. An orthonormal system S of vectors in a Euclidean space & is called an orthonormal basis (or a complete orthonormal system) in the Euclidean space d if the closed linear space [ S ]spanned by S is identical to the entire Euclidean space, i.e., [S] = 6'. That is, if Y is a subspace of 8 and S C Y ,then necessarily ( S ) C Y .

37

4. Hilbert Space

It must be realized that an orthonormal basis T in an infinite-dimensional Euclidean space & is not a vector basis for the vector space 8,i.e., (T) is in general different from &. For instance, this is so with the basis el , e2 ,... of Z2( a), where em is the vector whose nth matrix component is (%In

= %?,a

*

Theorem 4.5. A Euclidean space & is separable if and only if there is a countable orthonormal basis in B. Proof. (a) T o prove that in a separable Euclidean space B there is a countable orthonormal basis, note that due to the separability of 6 there is a countable set S = {fl ,f2,...} which is everywhere dense in B. According to Theorem 2.4, there is a countable orthonormal system T = {el, e2 ,...} such that (S) = (T). Due to Theorem 4.4 we have then [TI = (T)=

T

(s> = [ S ] = 8.

(b) Conversely, to show that if there is a countable orthonormal basis = {el , e2 ,...}, then & is separable, consider the set

+

R = {riel ... rnen: Re yl ,..., Re r , , Im y1 ,..., Im r, E%, n = 1, 2,...} which is countable, as can be established by using the technique for solving Exercise 4.7. T h e set R is also everywhere dense in B; namely, if j~ 8 and E > 0 is given, then as [el , e2 ,...I = 8,there is a vector anen such that g = alel

+ +

[If

- alel

-

..- - anen II < ~ / 2 .

Furthermore, we can choose complex numbers rl real and imaginary parts so that

1 Y k - a, I < E / 2 &, Thus, we have that for h

=

rlel

k

=

,..., r,

with rational

l,..., %!.

+ + rnen E R *-.

There are a few very important criteria by means of which we can establish whether an orthonormal system S is a basis in a Euclidean space.

Theorem 4.6. Each of the following statements is a suficient and necessary condition that the countable orthonormal system T = {el, e2,...} is a basis in the separable Hilbert space* 2:

* The theorem applies to the finite-dimensional as well as the infinite-dimensional case, though it is stated and proved here for infinite-dimensional 2.In the finite-dimensional case, co should be replaced by the dimension of 2.

I. Basic Ideas of Hilbert Space Theory

38

(a) The only vector f satisfying the relations (4.13)

If)

(ek

= 0,

is the zero vector, i.e., (4.13) implies f (b) For any vector f E A?‘,

1, 2 9 . e .

k

=

=

0.

(4.14)

or symbolically written m

f

1

=

(ekIf>ek,

k-1

where (ek 1 f ) is sometimes called the Fourier coeflcient off. (c) Any two vectors f , g E A?‘ satisfy Parseval’s relation: m

(4.15)

(f I g>

*

= k-1

(d) For anyfE A?‘ (4.16)

We start by proving that the criteria (a) and (b) are equivalent to the requirement that T = { e l , e2 ,...} is an orthonormal basis, as that requirement was formulated in Definition 4.4. T o do that we shall prove that (a) implies (b), (b) implies “ T is a basis” (as formulated in Definition 4.4) and “T is a basis” implies (a). I n order to show that (a) implies (b) we need the following lemma.

Lemma 4.1. For any given vector f of a Euclidean space B (not necessarily separable) and any countable orthonormal system { e l , e2 ,...} in 8,the sequence f l ,f 2 ,... of vectors n

(4.17)

1 If>

fn =

(ek

ek

k=l

is a Cauchy sequence, and the Fourier coefficients (ek If) inequality n

(4.18)

llfn

11’

=

I(ek k=l

If>[’< I l f 1.’

satisfy Bessel’s

4. Hilbert Space

39

Proof. Write hn

=

f

-

fn

3

where f , is given by (4.17). We have

because (ei 1 h,)

=

0 for i

and consequently, since (h, (4.19)

By using (4.17) and (ei I ej)

=

1, 2,..., n,

1 h,) 2 0, (fn

Ifn)

=

< .

Sij we derive

which shows in conjunction with (4.19) that Bessel’s inequality (4.18) is true. From (4.18) we can deduce that

i.e., the above series with nonnegative terms is bounded and therefore it converges. Since we know that for m > n

I. Basic Ideas of Hilbert Space Theory

40

we easily see from (4.20) and (4.21) that fl ,f2 ,... is a Cauchy sequence. We return now to proving Theorem 4.6. If T = {el, e2 ,...} is a countable orthonormal system in the Hilbert space X , then according to Lemma 4.1 for any given f E 3 the sequence fl ,f2 ,..., where n fn

=

1

If) ek

k=l

is a Cauchy sequence. Since X is complete, this sequence has a limit gEX. We can now show that if the statement (a) about T in Theorem 4.6 is true, then (b) is also true, due to the fact that (a) implies f = g; namely, for any K = 1, 2, ... we have



=

E !

=

if I ek>

n

-

JjFm1 <ei 8

I ek>

= 0.

4

Thus, if (a) is true, we must have f - g = 0. I t is obvious that statement (b) implies that T = {el, e2 ,...} is a basis, because according to Theorem 4.4 any f E 2 is the limit of elements fl ,f2 ,... from the linear space (T) spanned by T, where f, E (T) is of the form (4.17). We show now that the fact that T = {el, e2 ,...} is an orthonormal basis implies that (a) is true. Assume that some f E 3 is orthogonal on the system { e l , e2 ,...}. Since f E [el , e2 ,...] = 3, there is a sequence g, ,g2,... E (el , e2 ,...), i.e., for some integer sn 8"

gn

1

akek

3

k=l

which converges to f. Consequently, as (f 1 e k )

=

0,

and therefore f = 0. We shall demonstrate that statement (c) is equivalent to (a) or (b) by showing that (b) implies (c), and (c) implies (a), and thus finish the proof of Theorem 4.6. If (b) is true, then we have (see Exercise 4.10) (4.22)

4. Hilbert Space

41

where

From the relation

we immediately obtain Parseval’s relation (4.15). If we assume (c) to be true, then (a) is also true, because if some vectorf is orthogonal on {e, , e, ,...}, i.e., (f I e k ) = 0, k = 1, 2,..., then by insertingf = g in (4.15) we get m

(f If> =

C cf I

ek>
if>= 0,

k=l

which implies thatf = 0. Finally, (d) follows from (c) by taking again in (4.15) that f = g. Vice versa, if (d) is true then (a) has to be true, because if (f I e k ) = 0 for K = 1, 2, ..., then we get from (4.16) that l i f i 1 2 = 0, which implies t h a t f = 0. Q.E.D. It is easy to see that, due to the fact that every Euclidean space can be embedded in a Hilbert space (Theorem 4.1), the criteria (b) (c), and (d) are also necessary and sufficient criteria for T to be an orthonormal basis in a Euclidean space in general, while (a) is necessary but not sufficient (see Exercise 4.13).

4.5. ISOMORPHISM OF SEPARABLE HILBERT SPACES We can now demonstrate for infinite-dimensional Hilbert spaces a theorem analogous to the Theorem 2.5 for finite-dimensional Hilbert spaces.

Theorem 4.7. All complex infinite-dimensional separable Hilbert spaces are isomorphic to P( a),and consequently mutually isomorphic. Proof. If X is separable, there is, according to Theorem 4.5, an orthonormal countable basis {el , e, ,...} in #, which is infinite when &‘ is infinite dimensional. According to Theorem 4.6 we can write for any f E 2 m

f=

Ckek k=l

9

ck

=

(‘k

If>,

I. Basic Ideas of Hilbert Space Theory

42

where by (4.16)

Therefore

Vice versa, if

fl

=

(:)

b, EP(CO),

n

fn =

1

bkek

k=l

is a Cauchy sequence, because for any m

>n

m

IIfm -fnI12

1

=

I b, I2

k=n+l

and

I b, l2 converges. Thus, due to the completeness of

X

f l ,f2 ,... converges to a vector f E # and we have

Therefore, the inverse mapping of the mapping j.- a, of i@ into P(c0) exists, and has P(c0) as its domain of definition. Hence the mapping f H a, is a one-to-one mapping of 2 onto P(CQ). It can be easily checked (see Exercise 4.11) that this mapping supplies an isomorphism between i@ and P( 00). Q.E.D.

As we shall see later, the above theorem provides the basis of the equivalence of Heisenberg's matrix formulation and Schroedinger's wave formulation of quantum mechanics.

Theorem 4.8.

If the mapping f-f',

f€&,

f'E&

is a unitary transformation of the separable Euclidean space into the Euclidean space F', and if (el , e2 ,...} is an orthonormal basis in F,

43

4. Hilbert Space

then {el’, e2‘, ...} is an orthonormal basis in g’,where en‘ denotes the image of e n .

Proof. Let d be infinite dimensional, and denote by ( -I .)l and the inner products in 8 and 8‘respectively. Then

(. I -)2

(ei’

I ei’>2 = (ei I ej>l = S i j ,

i.e., {el’, e2‘, ...} is an orthonormal system in 8’.Since each f ’ a unique inverse image f E 6 , we have

E

b’ has

which by Theorem 4.6(b) proves that {el’, e2’, ...} is a basis. The case when d is finite dimensional can be treated in a similar manner. Q.E.D.

EXERCISES 4.1. Show that in a normed space N the real function d ( f , g ) = (1 f - g 11 on JV x N is a metric, i.e., it satisfies all the requirements of

Definition 3.1. 4.2.

for m,n

Prove that for any E

> 0 there is an N ( E )such that

> N ( E ) ,where f n is given

by (4.2).

4.3. Check that the operations (4.3) satisfy the axioms in Definition 1.1. 4.4.

Check that (4.4) satisfies the requirements of Definition 2.1.

4.5. Show that if N is a normed space and j is the completion of N in the norm, then: is a linear space with respect to the operations (a)

i+i={f1 +g1,fz+gz,-*l, ai = kfl , af2

,...l;

(b) the limit I I J I I , = limn+mI(f n J exists for every Cauchy sequence ,f.L }., and defines a norm in N ; (c) N is a Banach space and the image N’of N in 2 defined by the mappingfw {f,f , ...} is a linear subspace of M which is everywhere dense in N.

{fl

44

I. Basic Ideas of Hilbert Space Theory

4.6. Show that (4.6), (4.7), and (4.8) satisfy the axioms for vector addition, multiplication by a scalar, and inner product respectively. 4.7. Show that the subset D of Z2( co)is countable, where D consists of all vectors a which have the properties: (1) a finite number of components a, ,..., a, (for some integer n = 1, 2,...) of ix are complex numbers with real and imaginary parts which are rational numbers; (2) the rest of the components vanish. 4.8. Show that every finite-dimensional Euclidean space is a separable Hilbert space. 4.9. Prove Theorem 4.4. 4.10. Show that if in a Euclidean spacef, ,f2,...converges in norm tof and g, ,g, ,... to g, then (f I g> = limn+a, (f,I g,). 4.11. Show that the mapping ft,a, of X' onto Z2(m) satisfies the requirements for an isomorphism, given in Definition 2.4. 4.12. Prove that if one orthonormal system {el , e, ,...} in a Euclidean space 8 satisfies either (4.14), or (4.15), or (4.16), for every vectorf (or, in case of (4.15), for any two vectors f and g) from 8,then {e, , e, ,...) is a basis in B. 4.13. Verify that the criterion of Theorem 4.6(a) is not sufficient to insure that an orthonormal system {el , e, ,...} in a Euclidean space 8 satisfying that criterion is a basis by showing the following: Let {h, , h, ,...} be an orthonormal basis in a Hilbert space 2,and let B be the vector subspace spanned by (Ckm,,(l/k)hk), h, , h, ,..., i.e., B = (Ern k = l (Ilk) h, , h, ,...); then d is a Euclidean space. Prove that: (a) {el = h , , e2 = h, ,..., e, = h,+, ,...} is not an orthonormal basis in 8. (b) I f f € d is orthogonal to {el , e, ,...} C 8,thenf = 0.

5. Wave Mechanics of a Single Particle Moving in One Dimension 5.1.

THEFORMALISM AND ITS (PARTIAL) PHYSICAL INTERPRETATION

As an illustration of a physical application of the preceding results, we shall consider the case of a particle restricted to move in only one space dimension within a potential well. We denote the space-coordinate variable by x and the time variable by t. Assume that on our system there acts a force field F(x) which can be derived from a potential V(x), i.e.,

45

5. A Single Particle Moving in One Dimension

F(x) = -(d/d x) V(x).I n classical mechanics, if we denote the momentum of the particle by p, we have the following expression for the total energy E of a particle of mass m: E

(5.1)

= p2/2m

+ V(x).

Classically the state of the particle is described by its trajectory x(t), where at any moment t, x ( t ) E W. As we mentioned in the Introduction, one of the postulates of quantum mechanics is that the state of a system is described by a function Y(t), where Y(t)is a vector in a Hilbert space. I n the wave mechanics version of quantum mechanics, the state of a one-particle system is postulated to be described at time t by a “wave function” +(x, t ) which is required to satisfy the condition

/

+m

I +(x,

t)12 dx =

1.

-m

As a function oft, +(x, t ) is assumed to be once continuously differentiable in t ; in addition we require for the present that +(x, t ) have a piecewise continuous second derivative in x. Thus, we can consider #(x, t ) to be at any fixed time t an element of the Euclidean space W:z)(R1) (see Exercise 5.2) of all complex functions f ( x ) which vanish at infinity and are square integrable, i.e., If(x)12 dx --m

< +a,

as well as once continuously differentiable with limz..,foof’(x) = 0. In Wt2)(R1) the inner product is taken to be

and consequently we recognize (5.2) to be the normalization condition

II W)l12=

-m

I +(x, t)l”X

=

1,

where Y(t)E ‘%tZ)(R1) denotes the vector represented by the function

fr(4 = +(x, 9.

As a dynamical law we have in classical mechanics an equation of motion derivable from Newton’s second law, which in the present case is

I. Basic Ideas of filbert Space Theory

46

In wave mechanics it is postulated that the wave function #(x, t) satisfies the Schroedinger equation* [we assume throughout that V ( x ) is piecewise continuous], (5.4)

A heuristic recipe which leads to the above differential equation is the following: replacep in (5.1) by the differential operator -ifi(a/ax) and E by the differential operator ifi(a/at). If we deal with these operators ) ~@/ax2,we get the operator relation formally and write ( a / a ~=

. a = -f i 2 a2 - + V(X)

Ifi

at

2m

ax2

which yields (5.4) when applied to #(x, t). We note the important fact that Schroedinger's equation (5.4) preserves the normalization condition (5.2):

This is important in view of the correspondence rule proposed by Born [ 1926a, b], which constitutes the generally accepted physical interpretation of quantum mechanics. A very special case of this interpretation, which will be stated in its most general form in Chapter IV, $1, is in the present case given by the rule that for any interval I in 08'

represents the probability of detecting the system (ie., the one particle) within the interval I, if a measurement of position is performed at time t. Thus Pt(R1) =

--m

I $(x,

t)I2 dx

represents the probability of finding the system anywhere and, by ti

=

h / 2 r , h is the Planck constant; numerically ti

=

1.054 x lo-*' erg sec.

47

5. A Single Particle Moving in O n e Dimension

definition, this has to be equal to one.* I n view of this interpretation the necessity of (5.2) being fulfilled at all times is obvious. As we shall see in Chapter IV, another consequence of Born’s interpretation is that whenever Y(t) represents a physical state and c is a constant for which I c I = I , the vector-valued function (c!P(t)) represents the same physical state.

5.2. THEWAVEMECHANICAL INITIAL-VALUE PROBLEM In the case of classical mechanics the basic problem which is encountered is the initial-condition problem: given the position xo and velocity wo of the system at some initial time t o , find the state of the system at all times; i.e., find the unique trajectory x ( t ) which satisfies the initial conditions *(to) = W,

x(to) = x 0 ,

(S = dx/dt).

In practice this problem reduces to solving the initial-condition problem of the differential equation (5.3), which is of the second order in the variable t. The analogous initial-value problem in the quantum mechanical case consists in giving the state t,bo(x) at some time t o , and requiring to find a state $(x, t) at all times t, which is such that (5.5)

#(% to) = #o(x)*

Since the dynamics is now given by the Schroedinger equation (5.4), which is of the first order in t, the above initial-condition problem has a unique solution if a,ho(x) E % ‘ ~ 2 1 ( ~ 1 ) . The general procedure of solving the above problem consists in reducing it to an “eigenvalue” problem. One tries first to find a solution #(x, t) of the form

4

#(4exP[-(i/4

Etl, where E is a yet undetermined constant. When (5.6) is inserted in (5.4), we get a so-called eigenoalue equation for $(x) and E

(5.6)

’

#(x,

=

A somewhat naive interpretation of the statement that an event E (in this case the event is the detection of the. system within the interval Z at time t) has the probability P, is the following “frequency interpretation”: if the same experiment (in our case, the measurement of the position of the particle) is repeated under identical conditions a large number N of times, then the number of times the event E occurs should be v % PN, i.e., P w v/N, where the approximation w should be increasingly better with increasing N (on different concepts of probability see, e.g., Pap [1962], Fine [1973]).

I. Basic Ideas of Hilbert Space Theory

48

which in physics is called the time-independent Schroedinger equation. This equation has in general a family &,., of functions +(x) E Wt2)(R1) as solutions for a set S, of values of the parameter E. The numbers belonging to the set S, are called, from the general mathematical point of view, the ezgenvalues of the differential equation (5.7).

Theorem 5.1. If the functions IcI1(x), +,(x) and their first derivatives &,(x)/dx, d+,(x)/dx, as well as V ( x ) + , ( x ) and V(x)+,(x), are from V:,,( W),then

If each solution +(x) of (5.7) has the property that +(x), &(x)/dx, V ( x )+(x) E %tz)(R1), then each eigenvalue E of the time-independent Schroedinger equation (5.7) is a real number, and if +l(x) and +,(x) are two eigenfunctions of (5.7) corresponding to two different eigenvalues El # E, , then 1,5~(x)and +,(x) are orthogonal. Integrating by parts twice we get PYOO~. (5.9)

kz dZ#, -a

+ w,)

When we let above a -+ +m, we obtain (5.8) because both integrals converge due to the fact that (CI1(x), +,(x), d+,(x)/dx, d+,(x)/dx, V(x)+l(x), V(x)+,(x) E %:,)(Rl), while the first two terms on the right-hand side of (5.9) vanish due to the fact that, by definition, any functionf(x) in Wt2)(R1) has to vanish at infinity together with its first derivative. If +l(x) and +,(x) satisfy the conditions of the theorem so that (5.8) is true, and if in addition they are solutions of (5.7) corresponding to eigenvalues El and E, , respectively, then we can write (5.8) in the form (5.10)

(4d-4 I &lw> = (%lh(4 I#&)>.

By taking in (5.10) +l(x) = +,(x) = +(x) and consequently El= E, =

E, we obtain EC#(x) I #(4>= E*(Jl(x) IJl(4).

5. A Single Particle Moving in One Dimension

49

If #(x) is a nontrivial solution of (5.10) [i.e., if + ( x ) # 01, we have (I) I #) > 0 and therefore we must have E = E * ; i.e., we have proved that the eigenvalues of (5.7) are real under the specified conditions. In order to prove the last point of the theorem, namely that +1 I#, when El # E, , it is sufficient to note that due to the fact that El and E, must be real numbers, we can write (5.10) in the form

Since El # E 2 , the above implies that (+l(x)

I t,h2(x))

=

0. Q.E.D.

We shall establish in Chapter IV that the above theorem is true under more general circumstances than the ones mentioned so far.

5.3. BOUNDSTATES OF

THE

SYSTEM

From the physical point of view the wave functions in gbare considered to be bound states" and the numbers in S, are taken to be (as a special case of Born's correspondence rule given in Chapter IV) the only possible energy values that a system in a bound state can assume; they are therefore said to be the energy eigenvalues of the bound states, and the set S, is called the point energy spectrum. We note (see Exercise 5.3) that for a given eigenvalue E E S, the corresponding set of eigenfunctions is a linear subspace ME of %f2)(R1). If the subspace ME is one dimensional then the energy eigenvalue E is called nondegenerate; otherwise E is said to be degenerate. It is easy to see that the Euclidean space %'f?)(R1)is not complete (see Exercise 4.2). According to Theorem 4.1 there is a completion of %f2)(R1) which we denote by &(I). We shall establish in Chapter I1 that &(l) is a separable Hilbert space. I t is assumed that each vector of &(l) can represent a physical state at a certain time. We shall limit ourselves in the rest of this section to the closed subspace spanned byt gb.By definition, each one of the vectors in gb represents a bound state at a given time. As XA1)is a subspace of 2 ( l ) which is separable, it has to be also separable according to Theorem 4.2. Each one of the closed subspaces ME of corresponding to the eigenvalue E must also be separable. Consequently, according to

)1A'&

&A1)

Classically, a particle moving in a central force field is said to be in a bound state if its trajectory is a closed curve. t To simplify the notation we denote by the same symbols the incomplete Euclidean spaces and their isomorphic images in their completion; we do the same with the elements of these spaces and the image points of these elements in the completion.

50

I. Basic Ideas of Hilbert Space Theory

Theorem 4.5 there is in each ME an at most countably infinite orthonorma1 system TEspanning ME . The closed subspace spanned by

is obviously identical to 2';').The set T is an orthonormal system because each TE , E E S, , is an orthonormal system, while for El # E2 the sets MEl and ME* I and therefore also TE,and TE,, are orthogonal. Since all the vectors in T are linearly independent (see Exercise 4.12) while X;') is separable, it follows (see Exercise 5.5) that T is a countable set. In practice the elements of each TEcan be chosen to belong to %?t2)(Wl) and therefore satisfy (5.7). Consequently, if we write T in the form

the vector Yk~2'i') will be related by the construction outlined in proving Theorem 4.1 to a wave function I,~~(x)E %'tz)(R1) satisfying the time-independent Schroedinger equation lrr2 2m

d2hc(X)

dx2

+

V(X)#,(X)

= Ek$hk(X).

St),

As T is an orthonormal basis in we can expand according to Theorem 4.6 every vector in 2';')in the form

c m

Y'

=

(Y'k

I y >v/,

k=l

It must be remembered that the precise meaning of the above expansion is

We claim that the general solution of the initial-value problem for bound states, when the intial state at some time to is represented by a vector YoE Xi'),is given by m

(5.1 1)

y ( t )=

1exp [ k= 1

1 -

-

(Y'k

I

yk

*

The above series is again required to be a limit in the norm of its partial

5. A Single Particle Moving in One Dimension

51

sums; Ek is the eigenvalue corresponding to the eigenfunction &(x) E %'f2)(R1)which represents the vector Yk. First we have to establish that the series in (5.1 1) really converges for every t to an element Y(t)of HA1).

Theorem 5.2,

For any fixed t the sequence Gl(t), G2(t),... n

=

1

~ k ( t lU ) ,3

k=l

is convergent in the norm to some vector Y(t)E a?;').For t limit "(to) satisfies the initial condition Y(to)= Yo.

= to its

Proof. As Yl, Yz,... is an orthonormal basis in Xkl), we have according to Theorem 4.6 m

1I k=l

m

Ck(t)I2 =

c

I W k

I 'U,>lZ= II yo It2 < +a.

k=l

Thus, the sequence Gl(t),G2(t),... is a Cauchy sequence, and consequently it has a limit Y(t), since ZA1)is complete. As ck(tO)= (YkI Yo), R = 1, 2, ..., are the Fourier coefficients of Yo, we get from Theorem 4.6 Y(to)-- Yo. Q.E.D. In order to justify" the statement that (5.11) represents the solution of the initial value problem, assume that the series

converges in the norm as well as point by point for every value of x and t to a limit function +(x, t), and that az+(x, t ) / a x z and a+(x, t ) / a t can be obtained by differentiating the series (5.12) term by term twice in x and once in t, respectively. If we insert this +(x, t) into the Schroedinger equation(5.4) and interchange the order of summation and differentiation, then we easily find that +(x, t) satisfies (5.4) by taking into consideration that each a,hk(x, t ) satisfies (5.7) for E = Ek .

* A complete and general proof will be given in Chapter IV.

I. Basic Ideas of Hilbert Space Theory

52

5.4.

A PARTICLE MOVING IN A SQUARE-WELL POTENTIAL

In order to provide a simple example of the procedure outlined above, let us consider the case of a particle moving in a so-called square-well potential V(x),given by for 0 < x < L V ( x )= >O for x < 0 and x > L.

,"I

This potential represents the idealized case of a force which is everywhere zero except at the "walls" x = 0 and x = +L of the "potential well," where it is infinite. T h e time-independent Schroedinger equation (5.7) becomes in this case

+ 2m E*(x) = 0,

d2*(X)

-@-

dx2

(5.13) d2+(X)

dx2

+

g

( E - V,) *(x)

= 0,

0 <x x


< 0,

x

> L.

The general solution of the system (5.13) is

I

a2eik"s

+ bze--ik"r,

k"

=

-[ 2m(E - V,) ]

lP2

ti2

,

x>L.

The above expressions should represent the same wave function $(x) in the three different regions. Due to the normalization condition (5.2) a necessary condition for this is that $(x) -+0 as I x I -+ 00. This is possible only if k' = k" is a pure imaginary number:

+

k'

= k" = j K ,

K

=

(l/fi)[2m(Vo - E)]''2

>0

and b, = b, = 0. Since # ( x ) ~ % ~ ~ ~ ( l$(x) W ) , also has to be continuous and have a continuous first derivative at x = 0 and x = L; i.e., we have to impose the following conditions:

$44lz+-o

=

a1

=

$w

I+-r+O

=c

+ d9

5. A Single Particle Moving in One Dimension

53

By eliminating a, from the first pair and u2 from the second pair of equations (5.14) we are led to the following linear system of equations for c and d: ( K - ik)c ( K + ik)d = 0, (5.15) (K ik) ePikLc ( K - ik)d = 0.

+ +

+

In order to have a nontrivial solution (5.15), the determinant

I

(5.16)

I

+IdI

(I

c

K

- ik

> 0) of

the system

- ik

K (K

+ ik) ePikL

must vanish. Since (5.17)

k

=

(l/fi) d%E,

K

= (l/ti)

d2m(V0 - E),

and h, m, and Vo are fixed constants of the problem, the condition that (5.16) should be zero is a transcendental equation

1-( K-ik + tk

e2ikL =

(5.18)

K

to be satisfied by

E. As we have

we can write where K~ -

E

k2

cosc$=---=1-2-, (5.19)

K~

+ k2

sin4 = --= K~

2~k

+ k2

VO -2

The relation (5.18) is equivalent to

where (5.19) is determined by (5.18) up to a multiple of 277, i.e.,

I. Basic Ideas of Hilbert Space Theory

54

Thus, we have an entire family of eigenvalues En ,constituting the point spectrum of energy levels of a single particle moving inside the squarewell potential. The energy values En can be obtained by solving numerically the equation

for each n. Each eigenvalue E,, is nondegenerate, and the corresponding eigenfunction $,(x) is 2ik, en"", K, ik,

+

(5.20)

$,(x) =

(eiknz-

Kn

Kn

x

- ikn

+ ikn

<0

e-iknz),

0 <x


where the expressions for k, and K, are obtained by taking in (5.17) E = En .The absolute value 1 c, 1 of the constant c, has to be determined from the normalization condition

I+m I

$,(x)l2 d x = 1.

-m

The remarkable feature of the eigenfunction (5.20) is that +%(x) does not vanish outside the region 0 x L. This means, in view of the physical interpretation of I $,(x)12, that there is a nonvanishing probability of finding the particle outside the potential well-a typical quantum mechanical phenomenon which is not met in the classical treatment of the same problem, where the particle in a bound state is trapped inside the potential well.

< <

EXERCISES

5.1. Check that the set Vn([-u, + a ] ) of n-times continuously differentiable functions f ( x ) defined on the finite interval [ - a , +a] becomes a Euclidean space if the vector operations are defined by (af

+ W(4 = af(4 + k(4

and the inner product by -a

55

5. A Single Particle Moving in One Dimension

5.2. Show that the set %~zl(R1) of once continuously differentiable functions f ( x ) that vanish at infinity together with their first derivatives and are square-integrable is a complex Euclidean space if the vector operations are defined for f,g E ViZ@l) by

(f +g)(x)

=

f ( 4+ g(x),

( a f ) ( x )= Q f ( X ) ,

a E Q=',

and the inner product by

5.3. Show that if 1,4~(x),&(x) E Wt21(R1) are eigenfunctions of (5.7) corresponding to the eigenvalue E, then for any given complex numbers u and b the function q b l ( x ) b&(x) is also an eigenfunction with the eigenvalue E.

+

5.4. A linear harmonic oscillator in one dimension is a particle of mass m whose potential energy is

where o denotes in classical mechanics the characteristic frequency of oscillations which the particle performs. Show that in the quantum mechanical case the energy spectrum S, of bound states is nondegenerate and equal to

s,

= {pitiw,

#L ,..., (n + a) ffw ,...}.

Verify that the energy eigenvalue En = (n normalized eigenfunction

+ 8) fiw corresponds to the

where Hn(u), n = 0,1, 2, ..., are the Hermite polynomials

= (24"

- n(n

- 1)

1

(2u)n-Z

+ n(n - I)(#1 -. 22)(n - 3) (2uy-4 - ...

5.5. If T is an orthonormal system in a separable Hilbert space, prove that T is a countable set.

56

I. Basic Ideas of Hilbert Space Theory

References for Further Study On finite-dimensional vector spaces: Birkhoff and MacLane [1953], and Nering [1963].

For 552-4 consult Dennery and Krzywicki [1967], von Neumann [1955, Chapter 111, and for a more advanced approach, Naimark [1959, Chapter I]. For $5 see Landau and Lifshitz [1958], Messiah [1962], or any other of the many standard textbooks on quantum mechanics.

CHAPTER

11

Measure Theory and Hilbert Spaces of Functions

In the last chapter we met examples of Euclidean spaces whose elements were square-integrable functions. These spaces were not complete, and in order to find their completion we had to employ the constructions outlined in 53 and Theorem 4.1 of Chapter I. We can embed, however, Euclidean spaces of functions of a very general type in Hilbert spaces whose elements are also functions. I n order to study these spaces it is necessary to familiarize ourselves with the basics of the modern theory of integration in general, and the theory of the Lebesgue integral in particular. In $51, 2, and 3 we define and study the basic constituents of measure spaces in order to be able to define in 53 integration on such spaces. The knowledge of the methods used in proving the theorems formulated in these three sections is not absolutely essential for an understanding of the rest of the material in this book. Consequently, at the first reading, the reader may skip the proofs and limit himself to reading the various definitions and theorems in order to familiarize himself with the basic concepts and results. In $4 we introduce Hilbert spaces of square-integrable functions and in 55 we study the application of the theory of these spaces to the wave mechanics of n particles of different kinds. T h e introduction of the concepts of direct sums and tensor products of Hilbert spaces in 96 proves desirable before a more detailed study of the two-body boundstate problem is carried out in 97.

11. Measure Theory and Hilbert Spaces of Functions

58

*1. Measurable Spaces

ALGEBRAS AND a ALGEBRAS OF SETS 1 .l. BOOLEAN While discussing the physical interpretation of wave mechanics, we have seen that for each interval I (1.1)

Pt(0

=

I

I +(% t)l”x

is the probability of finding the system [which is in the state $(x, t ) ] within I at time t . T h e formula (1.1) provides an example of a real function whose domain of definition is a family of sets. Such a function is ‘called a set function. In 92 we shall be concerned with a special class of set functions, which are called measures, and which are basic in the modern theory of integration. In order to define the concept of measure we need first the concept of measurable space. In case of measurable spaces we deal again with a set S in which a certain type of structure is specified. The structure is now determined by giving a family of subsets of X which satisfies certain axioms, and which is then called a Boolean a algebra.

Definition 1.1. A nonempty class X of subsets of a set S is called a Boolean algebra (or j e l d or additive class) if we have (1.2)

R uS E X ,

( 1.3)

R‘EX,

whenever

R, SEX-,

whenever R E X ,

where R‘ = S - R denotes the complement of R with respect to I (i.e., the set of all elements of I which do not belong to R).”T h e class X of subsets of S is called a Boolean a algebra (or afield, or completely additiwe class) if in addition to being a Boolean algebra it has the property that the union (JgElS , of any countable number of sets S, , S , ,... from X also belongs to X .

Theorem 1.1.

If the class X of subsets of a set S is a Boolean

algebra, then (a) the entire set S and the empty set 0 belong to X , (b) the intersection R n S belongs to X whenever R, S E X , (c) the difference R - S and symmetric difference, R A S = ( R - S ) u (S - R) belongs to X whenever R, S E X . + We prefer the name Boolean algebra to that of field because thus no confusion with the concept of field of scalars can arise.

59

1. Measurable Spaces

The fact that S E X if X is a Boolean algebra follows from the requirement that there is a subset R of S which belongs to .X. Consequently, R' E &- and therefore S = R u R' E X , as well as 0 = S'E &In order to prove the other two statements of Theorem 1.1 we need to establish some of the general rules of dealing with sets.

Lemma 1.1.

If 9 is a family of sets, and R is any given set, then

usEF

Proof. If 5 E R n ( S) then by definition 5 E R and 5 E UsEF S, i.e., 4 belongs to at least one set S, from 9. Thus, 5 E R n S , , S, E 9, and consequently 5 E U,,,(R n S ) . Conversely, if 7 E USESF(R n S),then 7 belongs to at least one of the sets (R n S), i.e., 7 E R and 7 E USEFS. Thus, 7 E R n S).

Q.E.D.

It is interesting to note that if we look upon the operation u as set addition and n as set multiplication, then we obviously have that these operations are commutative R u S

=

SuR,

R n S

=

SnR,

and associative R u ( S u T ) = (R u S ) u T ,

R n ( S n T) = ( R n S ) n T,

like the ordinary addition and multiplication of numbers. Furthermore, as follows from the above lemma, they are also distributive: R n (S u T ) = (R n S) u (R n T).

Lemma 1.2. If 9 is any family of subsets of a set X,and if for any given set S we denote by S' the complement of S with respect to S, then (1.4)

1'

s

S',

= S E F

Proof. T o establish the first relation, assume that 5 is an element of (USESF S)'. Then 5 4 UsGF S, which means that 5 does not belong to i.e., 4 E S' for all S E 9; thus 5 E S'. any of the sets S E 9, Conversely, if 7 E S', then 7 E S' for each S E 9, i.e., 7 4 S for all S E 9. Therefore, 7 4 S, which implies that 7 is an element of S)', thus proving completely the first of the relations (1.4).

nSEF usEF

nsEF

11. Measure Theory and Hilbert Spaces of Functions

60

We obtain the second of the relations (1.4) by taking in the first of the relations (1.4) the complement of the left- and right-hand side, and noting that for any subset R of %, (R')' = R" = R. Q.E.D.

It is now easy to finish the proof of Theorem 1.1. I n order to prove Theorem l.l(b), write R nS

= (R'

v S')'.

Since we have R, S E X , we also have by (1.3) R ,S' quently from (1.2) we get R' U S' E X . Similarly we get R -S

=R

n S'

= (R'

v S")'

= (R'

v S)'

E

EX

, and conse-

X,

by noting that R - S = R n S'. T h e result that R - S , S - R E X whenever R, S E X yields RAS

= (R - S)U

( S - R)E.%.

Theorem 1.2. For any given nonempty family S of subsets of a set 3 there is a unique smallest" Boolean algebra d(S) and a unique smallest Boolean u algebra du(S) containing S ; d(9) and du(S) are called, respectively, the Boolean algebra and the Boolean u algebra generated by the family 9.

Proof. We give the proof for d ( 9 )the ; proof for the case of d U ( 9 ) is completely analogous. Denote by 8 the family of all Boolean algebras d containing 9, i.e., if d E 8 then S C d ;8 is not empty because it certainly contains the family Gx of all subsets of X,which is obviously a Boolean algebra. The family

&(F)=

nG

(365

is a Boolean algebra; namely, d(9) is not empty since it contains 9, and if R, S E d(9), then for each 6 E 5 we have R u S , R' E G, and consequently R u S, R' E d(9). In addition d(S) is the smallest Boolean algebra containing 9, since according to its construction, it lies within any other Boolean algebra containing 9. There is only one smallest Boolean algebra d(9) containing

* T h e set S belonging to a family 9 of sets, is said to be the smallest set in that family .F if it is contained in any other set belonging to 9.

61

1. Measurable Spaces

9; namely, if dl(F)and d2(9) were two such algebras, then all(%) n d2(S) would also be a Boolean algebra containing 9. That is possible only if d1(F) = d%(S). Q.E.D.

1.2. BOOLEANALGEBRAS OF INTERVALS For our purposes the most important Boolean algebra is the one generated by the family P of all intervals- including the degenerate intervals, which contain only one point, and the empty set a - i n the n-dimensional vector space (Rn).

Theorem 1.3.

T h e family 9;of all unions 11 ,..., Z k E y n ,

z1u".uzk,

k = 1 , 2,...

of intervals in P is identical to the Boolean algebra d(Yn) generated by P.

Proof. We obviously have C d ( R n ) . Therefore, in order to is a Boolean show that 4?: 3 d ( R n ) it is sufficient to prove that algebra; namely, if we had that result established, then would be a Boolean algebra containing P, and since d ( R n ) is the smallest Boolean algebra containing 9, we would also have 3 d ( R n ) , i.e., L q = d(W). If R, S E S;, we can write S=

R=I1U*.*UIk,

J1U***U Jl,

where Il ,..., J l E Yn. Hence RUS

= Z1 U

T o prove that R'

k = 1 we have R

***

V

Ik

V Jl V

U Jl E

we proceed by induction in k. I n the case 11'can

= Il is an interval, and it is easy to check that

be written in the form 11' = Z(') U

U

I("), I"), ...,I(")E .fn,

i.e., R' E at. Assume now that the statement R' of the form Il u u Ik , i.e., that we can write

u V

(1.5)

(11 v

'**

u Ik)'

=

m=l

Then in the case k

+ 1, by writing

11

,...,

Y

< 2n3"-',

E9 9; is

true for all R

E 3".

11. Measure Theory and Hilbert Spaces of Functions

62

we get from (1.5) and (1.6), with the help of the Lemmas 1.1 and 1.2, (Il u

Since ](I) n Jnl

u Ik+J

,...,

](v)

= (Il

n Jnl

u

u I&)'n Ii+l

E P, we conclude that (Ilu.--u Ik+l)' E AP.

Q.E.D. it would have been sufficient to Notice that in constructing consider only disjoint finite unions of intervals (see Exercise 1.5). 1.3.

BORELSETSIN Rm

A set belonging to the Boolean u algebra dq(.fn) generated by P bears the name of a Borel set in n dimensions. We cannot generalize Theorem 1.3 to Borel sets by saying that every Borel set is a countable union of intervals, because there are Borel sets which are not that. An example of such a Borel set in dv(R1) is the set 3 of all irrational numbers; 3 is a Borel set because it is the complement of the set % of all rational numbers, and % is the countable union of all degenerate intervals containing only one rational number, and therefore is a Borel set. The fact that the family of A@ of all Borel sets in Rn is very large, though by no means contains every subset of Rh, is confirmed by the following statement.

Theorem 1.4. Every open and every closed set in the Euclidean space Rn is a Borel set. Proof. Assume that 0 is an open set* in Rn. For every natural number m we consider the following open intervals in Rm

I

= x = (xl

,..., x,):

k1 - 1

k1+1 k,-1 <x,<m "*" m kl ,..., k, = 1, 2,...,

<XI<m

which obviously cover Rn. The union

0 is an open set in a metric space if for every point x E 0 there is an c neighborhood of x which lies within 0.

63

1. Measurable Spaces

of all intervals Ii:!..,kn lying within 0 is evidently countable. Furthermore

u R(m); m

0=

m=l

namely, if x E 0 is any given point of 0, then there is an E neighborhood of x contained in 0,and consequently for a sufficiently large m, m > 2n/e, there will be an interval I;:! containing x and lying within that E neighborhood. Since we have written 0 as a countable union of Borel sets Rm), it follows that 0 is a Borel set. Any closed set C in Iwn is the complement (with respect to Rn) of the open set C' = 0. Since 0 E an, it follows that C = 0' E an.Q.E.D. Given the Boolean algebra d(9) generated by some family 9, we can construct the Boolean u algebra &(9generated ) by 9, by means of monotone sequences of sets from d(9).

1.4.

MONOTONE CLASSES OF SETS

Definition 1.2. An infinite sequence S,, S, ,... of sets is called monotonically increasing (or expanding) if S, C S, C -..,and monotonically decreasing (or contracting) if S , 3 S, 3 ; in the first case we write m

while in the second case we write m

A nonempty class .X of subsets of a set 3 is called a monotone class if

every monotone sequence S, , S, ,... E X has its limit in X , i.e., 1imk+,, S, E X .

Theorem 1.5. If 9 is a nonempty family of subsets of a set A?-, there is a unique smallest monotone class ) 1 0 2 ( containing 9 ) ip, which is called the monotone class generated by 9. The proof of the above theorem can be carried out in the same fashion as the similar proof for d ( 9 )in Theorem 1.2;by noting that the family Gx of all subsets of 3 is a monotone class. *Theorem 1.6. Every Boolean u algebra is a monotone class, and every Boolean algebra which is a monotone class is a Boolean u algebra.

XI. Measure Theory and Hilbert Spaces of Functions

64

Proof. (a) If duis a Boolean (I algebra and R, , R, ,... E duis monotonically increasing, then by Definition 1.1 of a Boolean u algebra,

u m

limRk =

(1.7)

k-m

RkEdu.

k=l

I n the case that S, , S, ,... is a monotonically decreasing sequence in du, then S, - S, , S, - S , ,... is a monotonically increasing sequence in du, and we have according to (1.7) (see also Exercise 1.3) S, - lim Sk k-m

= lim(S, - s k ) E du k-m

Therefore lim Sk = Sl - ( S , - lim Sk)E du.

k-m

k+ 00

(b) If d is a Boolean algebra and a monotone class, and S, , S, is an infinite sequence of sets from d , then R, ,R, ,...,

,...

is a monotonically increasing sequence in d . Since d is a monotone class, we have m

and therefore d is a Boolean u algebra.

Q.E.D.

*Theorem 1.7. If &’ is a Boolean algebra and W ( d )is the monotone class generated by &’, then W ( d ) is identical with the Boolean u algebra du(d) generated by the family d of sets. Proof. Since, according to Theorem 1.6, du(d) is a monotone class, while by definition W ( d )is the smallest monotone class containing d , we must have ( 1-8)

W(d) c 4W)-

Since du(d) is the smallest Boolean u algebra containing d , by proving that 9u1(&’)is a Boolean u algebra, we could also deduce that dX.4 c Wd),

which would imply in conjunction with (1.8) that du(d) =W ( d ) . Now, in order to prove that W(&’)is a Boolean u algebra, it is sufficient,

65

1. Measurable Spaces

due to the second part of Theorem 1.6, to establish that W ( d ) is a Boolean algebra. For a given R E W ( d )denote by %(R)the family of all sets S in W ( d ) which are such that S', R u S E W(d). T h e family %(R)is a monotone class. As a matter of fact, if S, , S, ,...E %(R)is a monotone sequence, then since 9X(d)is a monotone class (see Exercise 1.6), (lim S,)' n+m

=

lim Sn' E 9X(d), n+m

while the sequence of the sets R u S , , R u S, ,... E !UZ(dis) also a monotone sequence and we have (see Exercise 1.6) (lim S,) u R = lim(S, n+m

n+m

u R)E W ( ~ ) .

In addition, if R E & , then %(R) is certainly not empty because it contains d.Since W ( d ) is the smallest monotone class containing d, we must have %(R)G W ( d ) . Thus, %(9) = 9X(d),and therefore S' ~%?l(d) for every S E %(9) = W ( d ) . Furthermore, if R ~ dthen, for an arbitrary S E Y . R ( d ) we have S E %(R)= 'Xn(d), i.e., R U S = S U R E !UZ(d); consequently, we also have R E %(S) for any R E d , i.e., %(S)3 d.Since %(S) is a monotone class, while W ( d ) is the smallest monotone class containing d,it follows that %(S) = %I(& for arbitrary S E Y . R ( d ) . This means that R u S, R' E !UZ(d whenever ) R, S E %(S) = !UZ(d), i.e., !UZ(dis) indeed a Boolean algebra. Q.E.D.

Definition 1.3. T h e ordered pair (9,d)consisting of a set 9 and a Boolean u algebra d of subsets of 9 is called a measurable space. Every set S belonging to d is said to be a measurable set in the measurable space (9,d). By following the tacit convention adopted from the beginning with regard to mathematical spaces, we shall denote the measurable space (%, d), consisting of the set 9 and a structure on 9 given by the Boolean u algebra d , also by the letter %-except where ambiguities might arise. Thus, due to the properties of Boolean u algebras, we can say that if R is a measurable subset of 9, then R' is also measurable, and if S, , S, ,... is a countable family of measurable subsets of 9, then their union Skis measurable in S.

ukm,l

EXERCISES 1.1.

Show that if R and S are any two sets, then R nS

=

( R u S) - (R

S).

11. Measure Theory and Hilbert Spaces of Functions

66

1.2. A nonempty class X of subsets of a set X is called a Boolean ring (or additive class of sets) if R u S, R - S E X whenever R, S E X ; a Boolean ring 9 is a Boolean u ring if S, u S, u -.-E 9 whenever S, , S, ,... E 9. Explain why every Boolean algebra (u algebra) is a Boolean ring (u ring). 1.3. Show that if 9 is a Boolean ring (see Exercise 1.2) of subsets oftheset%, t h e n R - S , R A S , RnSEWwheneverR,SEW. 1.4. Prove that if F is a family of subsets of a set X,there is a smallest Boolean ring 9(F)and a smallest Boolean u ring 9-(F) containing 9;these rings are said to be generated by 9. 1.5. Show that the family gt of all unions of intervals is identical with the family of all disjoint unions of intervals. 1.6. Show that if S, , S, ,... is a monotone sequence of subsets of a set X, then for any given set R, the sequences R u S, , R u S, ,..., R n S , , R n ,... and s,', ... are monotonic and R U limk++ms k = limk++m(Ru s k ) , R n limk++ms k = limk++mR n S k , (limk++ms k ) ' = limk++@s k ' .

s,

s,',

1.7. Show that Theorem 1.5 is true not only for Boolean algebras, but also, more generally, for Boolean rings. 1.8. Prove that if d is a Boolean algebra of subsets of X,and R is any given subset of 9, then ' d = {R n S: S E d}is a Boolean algebra of subsets of R. Formulate and demonstrate the same result for Boolean u algebras. Show that if d generates the Boolean u algebra du of subsets = { R n S : S E do} of X,then d' generates the Boolean u algebra dU,' of subsets of R.

1.9. Let d be a Boolean u algebra and S, , S, ,... a countable family of sets from d .Show that fig1s k belongs to d .

*2. Measures and Measure Spaces 2.1.

THECONCEPTOF MEASURE

In the last section we mentioned the concept of a set function which is, in general, a mapping of a family 2- of sets into W or C'. A set function F ( S ) , S E 2-, is said to be additive if for any disjoint S , , S , E X

w,u S,)

=

Wl) + W,);

67

2. Measures and Measure Spaces

it is said to be u additive or countably additive if for any countable family S, , S , ,... of disjoint sets in X we have F(S1u

s,u

- * ) = F(S,)

+ F(S,) + . **-

Definition 2.1. A measure p(S) is an extended real-valued set function whose domain of definition is a Boolean algebra d , and which satisfies the following requirements: (1) P ( 0 ) = 0, (2) p(S) 0 for all

S~ d , (3) P((J;=~S,) = CkZlp ( S k ) if the sets S, , S, ifS,nSj= 0 whenifj. m

,... are

disjoint, ie.,

We see that a measure is an extended real-valued nonnegative a-additive set function. T h e adjective “extended” indicates that for some sets S Ewe ~ might have p(S) = +co. T h e symbol +co is thus formally adjoined to the set R’, and in the extended set R1 u {+m} we define m = co,a co = m if a > 0, and 0 00 = 0, the formal operations a a / m = 0, for all a E R1 u {+a}.

+

-

-

Definition 2.2. A measure space (3,d ,p ) is a measurable space (9,d )for which a measure p(S) is defined on the Boolean u algebra d. For any S E d,the number p(S) is said to be the measure of the set S. In the case that p(S) < + co, S is said to have a finite measure. If the set S is a countable union (J:=’ S , of sets S , of finite measure, then the measure p(S) of S is said to be ufinite. A measure p(S) on d for which the set 3 has a finite (a-finite) measure is called a finite (aJinite) measure; the measure space (T,d, p ) is then also called finite (u finite). As a simple example of a measure whose domain of definition is (see §l), consider the set function p\”(B) which for an interval I (open or closed) from a to b on the real line is equal to the length b - a of the interval. Since any element B E &?ican be written as a finite union of disjoint intervals (see Exercise 1.5) (2.1)

B=Ilu...uI,,

IinIj=

0

for i#j,

we can define p\l)(B) by

The number p\l)(B) is called the Lebesgue measure of B E&?:. It is obvious that the value pl1)(B)is uniquely determined regardless of the

11. Measure Theory and Hilbert Spaces of Functions

68

way we split B into disjoint intervals. I t is easy to see that the set function p ( t ) ( B )is u additive. Hence, if we define pi1)(0 ) = 0, we have a measure on 93: ; p\’)(B) is a a-finite measure because we can write Rf = u::-,[n, n + 11 and the Lebesgue measure of any bounded interval is finite. The type of measure with which we shall frequently have to deal is the probability measure. A probability measure pJS) is a measure on a measurable space (9,d), which assigns to each S ~d a number which represents the probability that an event, whose possible outcome could be any point in 9 and oni) a point of X,will happen to be in S. Due to this interpretation it is required that a probability measure satisfy the “normalization” condition p J 9 ) = 1; thus, every probability measure is necessarily finite. An example of a probability measure-of the type we shall encounter very frequently in the future-is the measure P,(B), B E 93:, defined on at, whose value for B = I E J1is given by (1.1); for a general B E A?;, which can always be written in the form (2.1), we define (see Exercise 2.3) Pt(B) = pt(I1) f

2.2.

‘**

+

Pt(lk)*

BASICPROPERTIES OF MEASURES

Definition 2.3. An extended real-valued set function F ( S ) is said to be continuousfrom aboae (below) if for every monotonically decreasing (increasing) sequence S, , S, ,... of sets from the domain of definition of F( S) we have F(ljm S,)

=

+m

limF(Sk)

k-m

whenever F(limk+ms k ) is defined and, in the case that S,, S, ,... is decreasing, whenever I F(S,)I < co for at least one value of n (see Exercise 2.4).

+

Theorem 2.1 + Every measure is continuous from above and below. Proof. If R, , R, ,... is a monotonically increasing sequence from the measure space (Z, d,p ) and limk+mR, E d ,then we can write m

lim Rk k+ w

=

u

(&

- Rk-l),

R,

= 0,

k=l

where the sets R, - R, , R, - R,

,... E d (Rk - Rk-, E d

by Theorem

69

2. Measures and Measure Spaces

1.1) are disjoint. Hence, if limk+wRk E d,

proving that p is continuous from below. If S , , S , ,... E at is monotonically decreasing and p(Sno)< 00 for some n o , we have that Sno- S , , Sno- S, ,... is monotonically increasing. According to (2.3), if 1irnk+, s k E d and therefore

+

lim(Sno- S k )

k+m

= Sno- lim

Sk

k-3m

eat’,

we have p(jm(sno -3W - Sk)) =

pz

dSn0

- Sk)-

Since p(Sno- sk) = p(Sno)- p(Sk) for k >, no (see Exercise 2.1), we get from the above relations dSn0

-

fiES k )

=

dS,)

-

Fz

dSk)*

Therefore, p ( S ) is continuous from above, as can be seen from the following:

dim h+w S k )

= d S n 0 - (Sno

-

!z

Sk))

= p(Sno)- p(Sno- lim &) k+m

= limp(&). h+w

Q.E.D.

Theorem 2.2. Every finite, nonnegative and additive set function F(S) defined on a Boolean algebra d and satisfying F( 0)= 0, which is either continuous from below at every R E & , or continuous from above at 0 E d , is necessarily also u additive, i.e., it is a measure. Proof. Let S, , S, ,... be any infinite sequence of disjoint sets from d , which is such that R = Ukm,l s k E d.Then the sequence R, , R, ,... with n

Rn

=

u

Sk

k=l

is monotonically increasing. Since F ( S ) is additive, we have n

F(Rn) =

F(Sk). k=l

11. Measure Theory and Hilbert Spaces of Functions

70

If F ( S ) is continuous from below at R F

u

= F(1im

S k

=

R,)

Ug=l S , , then

= limF(R,) n-rm

n+m

which proves that F ( S ) is u additive. IfF(S) is continuous from below at 0 , then since R - R, ,R - R, ,... is a monotonically decreasing sequence with lim,+m(R - R,) = 0 , we have 0 = limF(R - R,) = F(R) - limF(Rn), n+ m

n+m

i.e., m

F(R) = limF(R,) n-rm

=

1

F(sk).

Q.E.D.

k=l

2.3. EXTENSIONS OF MEASURES AND OUTERMEASURES We saw in $1, Theorem 1.6, that a Boolean algebra is a monotone class if and only if it is a Boolean u algebra. It would obviously be convenient if we could extend a measure p defined on a Boolean algebra d of subsets of 9? to the Boolean u algebra du(d) generated by d,and thus obtain a measure space (S,du(d), p ) in the sense of Definition 2.2.

*Theorem 2.3. Let p(S) be a measure defined on the Boolean algebra d of subsets of a given set S.T h e set function (2.4) p(R) = inf

1

m

1

u m

p(sk):

k=l

s k

r> R,

S,

1

, S, ,... E cd ,

k=l

R E cdu,

defined on the Boolean u algebra du= du(d) generated by d,i.e., for is a measure on duwhich coincides with p(S) on d: all R E dU,

s Ed,

p(s) = p(S),

If p(S) is a a-finite measure, then p(S) is also u finite, and p(S) is the only measure on duwhich coincides with p(S) on d.T h e measure p(S) is called the extension of p(S). Consider the extended real-valued set function (2.5)

p+(R) = inf

1

m

1 k-1

m

p(sk):

RC

u

k=l

s k

, S, , S,

,... ~

I

c d

71

2. Measures and Measure Spaces

for any given subset R of %. Since for any such R there are countable families S, , S, ,... E sd which cover R, i.e., whose unions contain R, we can always take the infimum (greatest lower bound) of Cz=,p(Sk) for all such families, i.e., p + ( R ) is defined on the family Gx of all subsets of %, it is obviously nonnegative, and it satisfies the relation

for any R, ,R,

,... E Gz .

In order to see that (2.6) is true, note that for each given E > 0 and each R, there is, according to the definition in (2.5), a sequence s,, ,S,, ,... E d such that R, C S,k , for which

u,”=,

m

(2.7)

p+(Rn)

< 1p ( S n k ) < p+(Rn) + & k=l

Since { S n k , n, k = 1, 2, ...} is a countable family of sets from sd and U;=I Rn c U:,k=l s,,, We have by (2.7)

The relation (2.8) is true for an arbitrarily small quently (2.6) follows.

E

> 0 and conse-

Definition 2.4. A nonnegative set function M ( R ) , defined on each subset R E Gz of a set %, for which M ( 0 ) = 0 and which is countably subadditive, i.e., for which

for any sequence S , , S , ,... E G x ,is called an outer measure in 3. A subset S of % is said to be measurable with respect to the outer measure M(R), or simply M measurable, if it satisfies the relation (2.10)

M(R) = M ( R nS )

for any subset R of %.

+ M ( R n S’)

72

11. Measure Theory and Hilbert Spaces of Functions

We have proved thus far that p+(R) is an outer measure in I I. n order to demonstrate that p+(R) is a measure on do= do(&) we have to prove the next two lemmas.

, then *Lemma 2.1. If M ( R ) , R E Gr , is an outer measure in I the class dMof all M-measurable sets S E Gr is a Boolean u algebra, and the outer measure M ( S ) , S E dM, restricted to dMis a measure.

Proof. We start by proving first that dMis a Boolean algebra. Since for any R E Gr ,

+

M ( R ) = M ( R n S)= M ( R n 0 ) M ( R n 9),

we have 0 ~d’ . From the relation (2.10) it is evident that if S is M measurable, then S‘ is also M measurable. Therefore, we only have to prove that S, u S, is M measurable whenever S , and S, are M measurable. If S , , S , E dM, then for any R E 6% M ( R ) = M ( R n S,)

(2.1 1)

since also R n S , , R n S,’ (2.12)

M ( R n S,)

(2.13)

M ( R n S,’)

E Gr

we have

+ M((R n S,) n S;), M ( ( R n S,’) n S,) + M ( ( R n S,’) n S,’).

= M ( ( R n S,) =

+ M ( R n S,‘);

n S,)

By substituting (2.12) and (2.13) into (2.1 l), and noting that S,’ n S,‘

= (S,u

S,)’,

we arrive at

we , can replace Since the above relation is true for any subset R of I R in (2.14) by R n (S, u S,). Then the last term on the right-hand side of (2.14) becomes M ( 0 ) = 0, while the terms in the brackets remain unchanged, because we easily get by means of Lemmas 1.1 and 1.2 [ R n ( S , u S,)] n S, n S , = R n S, n S, , [ R n ( S , u S,)] n S, n S,‘

=R

n S, n S,‘,

[ R n ( S , u S,)] n S,‘ n S,

=R

n S,’ n S,

.

73

2. Measures and Measure Spaces

Therefore, we have (2.15) M(R n (S, u S,)) = M(R n Sl n S,)

+ M ( R n S, n S2))+ M ( R n Sl' n S2),

i.e., (2.14) reads (2.16)

M(R) = M ( R n ( S , u S,))

+ M ( R n ( S , u S,)'),

proving that S, u S , is M measurable. In order to prove that dMis also a Boolean u algebra it is sufficient to consider sequences of disjoint sets S , , S , ,... E dM; namely if R, , R, ,... E dMis any sequence, then it is easy to show that m

u

m

Rn=

n=l

u

n=l

.

(Rn-

n-1

u

Rk),

k=l

u;~:

where the sets S , = R , R k , n = 1, 2,..., also belong to d M , which was shown to be a Boolean algebra, and are obviously disjoint. If S, and S , are disjoint, we have S, n S , = 0 , S , n S,' = S, , S,' n S , = S , , and (2.15) becomes M ( R n ( S , u S,))

= M ( R n S,)

+ M ( R n S,).

By induction we get

From the subadditivity of M ( R ) it easily follows that (2.18)

(u;==, (u,"=,

u;=,

because S,)' 3 s k ) ' . Since S, is M measurable due to the established fact that dMis a Boolean algebra, we get by using (2.17) and (2.18)

74

11. Measure Theory and Hilbert Spaces of Functions

Due to the countable subadditivity of M ( R ) , we have (2.20)

( u ))

M R

Sk

=

M

(kI1

u ( Rn

W

sk)

< C M(Rn

(ky1

sk).

k=l

Since (2.19) is true for any n, we obtain by letting n using (2.20),

-+

00

in (2.19) and

Due to the subadditivity of M ( R ) , we also have

u;='=,

The relation (2.21) in conjunction with (2.22) shows that s k is M measurable, i.e., s k E dM, and hence dMis a Boolean u algebra! In order to prove that M ( R ) is u additive on d ,, we insert in (2.21) R= s k to derive

u,"=,

u,"=,

which in conjunction with (2.9) yields the desired conclusion.

Q.E.D.

generated *Lemma 2.2. Every set R in the Boolean u algebra .dc by d' is pf measurable, i.e., dois contained in d@+ .

Proof. Take any set R E GS . By the definition (2.5) of p+(R), for any E > 0 there is a sequence s, , s, ,... E & such that R c s k , for which

u,"=,

For any set S E & we have [since s k n S and s k n S' are disjoint] m

(2.24)

1d S k ) k=l

s k =( s k

m

=

c

n S ) u (8, n S'), where

m

d S k

k=l

3 P+(R n S )

s, f kc= l p(sk

+ P+(R n S'),

Ly')

75

2. Measures and Measure Spaces

where the last inequality

u,”=,

is a consequence of the fact that ( S , n S’). By combining

R n SC (8,n S) and R n S’ C (2.23) and (2.24) we obtain (2.25)

p+(R)

+

E

u,”=,

2 p+(R n S )

+ p+(R n S‘),

which holds for any E > 0. Since p+( R) is an outer measure and therefore is subadditive, we get p+(R)

< p+(R n S ) + p+(R n S’),

which in conjunction with (2.25) shows that p+(R) = p+(R n S )

+ p+(R n S’),

i.e., that S E d is measurable. Thus we have arrived at the result that the Boolean algebra d is of all p+-measurable subsets of X. contained in the Boolean a algebra d@+ Since duis the smallest Boolean o algebra containing d,we must have that .Iu C du+ . Q.E.D. Lemma 2.2 tells us, de fucto, that the set function p(R) defined by (2.4) is indeed an extension of p(S), S E & , to du. From the very definition (2.4) of F(R) it is obvious that F(R)is a finite in case that p(S) is a finite. In order to see that F(S) is the only extension of p(S) when p(S) is 0 finite, assume that P1(R),R E do , is also an extension of p(S), i.e., iil(S) = p(S) = ii(S),

S E d .

Let R E dube a set for which at least one of the two extensions is finite, say p(R) < f m . Denote by m(R)the family of all sets in duwhich are subsets of R, i.e., they are of the form R n S, S E du, and for which iiland ii are equal, j&(R n S ) = p ( R n S ) ,

R n S E %I(R).

9X(R) is a monotone class, because if S , , S, sequence then, since ii(Sn)is finite, we have ii(lim S,) n+m

=

,... E %l(R)is a monotone

lim ii(S,) = lim jil(S,) = Pl(lim S,). n+m

n-m

n+m

m ( R ) obviously contains the Boolean algebra (see Exercise 1.8) dR= { R n S : S E d}. Since 1132(R)is a monotone class, it must coincide = {R n S : S E du} generated (see with the Boolean o algebra dU,,

11. Measure Theory and Hilbert Spaces of Functions

76

Exercise 1.8) by d . Obviously R E du,R and therefore R E W(R), which means that P(R) = p1(R). If R O € d uis a set of infinite F; measure, it can be covered by an infinite sequence R, , R, ,...E du, m

ROC

u

Rk,

n-1

of sets R, of finite F; measure because F;(R,) is assumed u finite. I t is easy to check that

6

Ro=

n=l [ R o n ( R n - ~ l k=l R k ) ] ~

and since Sn) = Ro n (R, - (JiiiRk) is obviously a set from duand subset of R, , its F; measure is finite. According to the above we have then j i l ( S ( R ) ) = F;(S(n)),

and therefore m

i w o ) =

m

1 F;l(Scn))= 1 iVn)) = F;(Ro), n=l n=l

which proves the uniqueness of F;(R). Theorem 2.3 is of great importance to us because it tells us that all the u-finite measures defined on L4?: have a unique extension to the family 9P of all Bore1 sets. This conclusion applies in particular to the Lebesgue measure introduced at the beginning of this section for the sets in at,as well as to any probability measure (which is always finite) that has been defined on at. Due to Theorem 2.3, when studying measures we can restrict ourselves from now on, without loss of generality, to measures defined on Boolean u algebras, i.e., to the study of measure spaces (see Definition 2.2). 2.4.

CARTESIAN PRODUCTS OF MEASURE SPACES

An important way of building a new measure space from two or more measure spaces Xl,..., X , is by introducing the product measure x X , of the sets Xl,..., Xn. on the direct product 3, x

Theorem 2.4. If (Xl ,dl, p l ),..., ( X , ,dn,pn) are n measure spaces, there is a measure p ( S ) defined on the Boolean u algebra d of subsets of the set 9 = Xlx x Xn, such that d is generated by

77

2. Measures and Measure Spaces

the family dlx x dnof subsets of S,and has the property that for any S, E d,,..., S, ~ d .

~ (x 4

(2.26)

x Sn)

*.*

~(sn);

= P(&)

thismeasure is unique if the measure spaces (3,,d,,p1),..., (S,, d,,p,) are a finite. The above measure p is called the product of the measures p, ,...,p, , and the measure space (S,d, p) is called the Cartesian product of the measure spaces (319 4 9 ~ 1 ) , * - *(, S n ,d n 9 pn). In order to prove the above theorem, we establish first the following lemma.

*Lemma 2.3. If dl ,..., d,are Boolean algebras of subsets of the ,...,d,)of finite sets %, ,..., S,, respectively, then the family @(dl unions of sets belonging to dlx ..- x d, is a Boolean algebra of x S,, and 9(dl,...,d,)is identical with the subsets of S = Slx family of disjoint unions of sets from d, x .--x d, . Proof. It is obvious that 9(dl,...,d,)is closed under the operations of taking unions of sets from @(dl ,..., dm). Furthermore, if S E @(dl ,..., dn), then S can be written in the form m

S=

u

(S: x

5'; E .pl:

x Si),

..., St E dn,

k

,..., m.

=1

k=l

Since for any R (2.27)

R'

=

= R,

x R, E d, x

x

u

u

j=1 Z l ~ ( R 1 . R 1 ' )

E p ( d 1 7.e.9

-*.

(Z, x

x d,, we have

... x Rj'

x *.. x Z,)

Z,€(R,.R,')

dn),

we get by application of Lemmas 1.1 and 1.2

n (S: m

S' =

k-1

x

..- x st)'=

u T', 7

i-1

where r is some positive integer, and each Tiis an intersection of a finite number of sets from d, x .*- x dn. But, since for an intersection T of sets R{ x ... x R', , j = 1,..., s, from d, x ... x d, we have

78

11. Measure Theory and Hilbert Spaces of Functions

[i.e., T is also an element of 9(dl,...,dn)], we conclude that S' E 9(dl,..., dn). Thus, 9(dl,..., dn) is a Boolean algebra. Any element

s=

u m

Sk,

s1, ..., S m € dl x

***

x dn

k=l

of 9(d.,..., dn)can be written as the disjoint union of sets Ri = Si - Ui-l k = l Sky

s = S' u (Sl - S2) u ... u ( s m yjl S k ) -

k=l

By applying (2.27) and (2.28) on each of the sets R1, ..., Rm in the above union, we arrive by induction at the conclusion that R1, ..., Rm are disjoint unions of sets from dl x -.* x dn, i.e., that S can be written x dn. Q.E.D. as a disjoint union of sets from dl x T h e Boolean algebra 9(dl,..., dn) is obviously the Boolean algebra generated by dl x x dn. I t is easy to check that the set function defined on a set R E 9(dl,..., dn)

---

m

R

=

(JRk,

Rk = Rt x

x RE€&' x

***

x dn,

k-1

R i n Rk = 0

for i # R,

by the expression (2.29)

is a measure. Due to Lemma 2.3, p ( R ) is defined by (2.29) on the entire Boolean algebra 9(d1 ,...,dn). Evidently, p ( R ) is u additive if pl(R1),..., pn(Rn)are u additive. x dnis obviously The Boolean u algebra d generated by dl x identical with the Boolean u algebra generated by 9(dl,..., dn). Consequently, by Theorem 2.3, there is an extension of p(R),R E 9(dl,..., dn), to d , which is unique when p ( R ) is u finite. Thus, the proof of Theorem 2.4 is completed. Note that we can apply Theorem 2.4 to p l ( B ) = = p n ( B ) = p:')(B), B E ~ Fin, order to arrive at a measure p\")(B), B €an, called the Lebesgue measure in 9 (see also Exercise 2.3). For an interval

79

2. Measures and Measure Spaces

x (a, , b,) = ( a , , b,) x the definition of p\’)(B)

I

#(Z)

=

E

an,we

(b, - U l )

have according to (2.29) and

... (b, - u,).

EXERCISES 2.1. Show that every measure p(S) on a Boolean algebra at‘ is a monotone set function, i.e., if S C R, R, S E d , then p ( R ) p(S), and a subtractiwe set function, i.e., p ( R - S) = p ( R ) - p(S) whenever S C R, R, S E ~ . 2.2. Define a set function pLn)(B)on the Boolean u algebra anof all Borel sets in Bn by writing pim)(B)= 1 if B E 9P contains the origin of UP, and pkn)(B) = 0 if B does not contain the origin. Show that this set function is a finite measure; we shall call this measure the 6 measure.

2.3. Let p(xl ,..., x,) be any nonnegative Riemann integrable function on UP. Show that the set function p,,(B) on the Boolean algebra Bt, which is defined for intervals I E 9 n by the Riemann integral

and for a general element B = Ilu if i # j) of a;by PAW = PLP(Z1)

u Ik ,(I, ,...,I k E P, Is n I j = 0

+ + *-*

PD(Ik)

is a measure (called a Lebesgue-Stieltjes measure induced by p) if we take PI( 0 ) = 0. Why is p l ( B ) necessarily u finite ? Note. If p(x, ,...,xn) = 1, the corresponding measure p I ( B )= p y ) ( B ) is called the Lebesgue measure on an. 2.4. Is it true for p\l)(B) defined by (2.2) that p(l)(limk+mBk) = limk+mpi1)(Bk) for every monotonically decreasing sequence of Borel sets in B 1 ? 2.5.

If (X, A?’,p ) is a measure space, the set function p+(R) = sup{p(S): S C R, S ~ d }

is obviously an extended real-valued function defined for every subset

R E Gis of X;this set function is called the inner measure induced by p. Show that if R, , R, ,..., E 6% is any disjoint sequence of sets, then

80

II. Measure Theory and Hilbert Spaces of Functiona

*3. Measurable and Integrable Functions

3.1. THECONCEPT OF

A

MEASURABLE FUNCTION

In the present section our main purpose is to define and study the integral

of a function f (g), 6 E S,defined on a measure space* S = (S,d ,p). T o do this we need the concept of a measurable function.

Definition 3.1. A real-valued function f (g), E S,defined on a measurable subset R of the measurable space S = (S,d ) is said to be measurable on R if for every open interval I C R1 the subset

f-W

(3.1)

= {I:I E W ( I )€1)

of S is measurable, i.e., it belongs to d . If f ( g ) is an extended realthenf(8) is valued function, i.e., if it can also assume the values fa, said to be measurable on R if in addition to the sets (3.1)the sets f-'(+m)

= {I:I E R , f ( I ) = +a),

f-'(-00)

= {I:I € R , f ( f )= -00)

are also measurable. An extended complex-valued function h( 6) is measurable if its real part Re h(g) and imaginary part Im h(6) are measurable. i.e., In case that the functionf(g), 6 E Rn, is measurable in (Wn,S), with respect to the family of all Borel sets, thenf(g) is said to be Borel measurable; if f ( 6 ) is measurable with respect to the larger class of Lebesgue measurable sets-i.e., with respect to the sets S C Rn which are measurable (see Definition 2.4) with respect to the Lebesgue outer measure P ; ( ~ ) ( RR ) ,E GRm,defined by (2.5)from the measure p\*)(B), B E A?;, pp'(B) =

-thenf(g)

I dx, B

dr,

(Riemann integration on B)

is said to be Lebesgue measurable. Note that the class of

all Lebesgue measurable sets is, according to Theorem 2.3, a Boolean According to the agreement we have followed until now,we denote the measure space (I d, ,p)-consisting of the set Ion which a Boolean u algebra d of subsets of Iand a measure AR),R E d, are defined-also by the letter 9.

81

3. Measurable and Integrable Functions

u algebra which contains" the family 97, of all Borel sets in Rn;therefore, every Borel measurable function is also Lebesgue measurable. The characteristicfunction x s ( f ) of any subset S of a set I is defined by

As a simple example of a measurable function we can take the characteristic function x R ( f )of a measurable set R. A slightly more complicated example is provided by a simple function on a measurable space I. Definition 3.2. A function f ( f ) on a measurable space is called simple if it is of the form n

f ( 4 ) = C ukXSk(6),

Si n Sj = 0

for i + j ,

k=l

where a, ,..., a, are, in general, complex numbers and xs,(5) is the characteristic function (3.2) of the set S , . As an example of Borel measurable functions on R" we have the continuous functions. Namely, if f ( x ) is a real continuous function defined on an open subset of R m and I is an open interval in R', then f - l ( I ) is an open set in R"; therefore f - l ( I ) is a Borel set in R" by Theorem 1.4. Since the real and imaginary parts of a continuous complex function defined on an open subset of Rn are also continuous, we can state the following.

Theorem 3.1. Every continuous complex function defined on an open subset R of Rn is Borel measurable on R.

3.2. PROPERTIES OF MEASURABLE FUNCTIONS The measurable functions are important to us only to the extent that an integrable function has to be first of all measurable. I n the theory of integration we shall need certain general properties of measurable functions, which we state in Theorems 3.2-3.5.

Theorem 3.2. If the extended real-valued function f ( f ) defined on the measurable subset R of the measurable space I = (S, d ) is measurable, and if B is a Borel set on the real line, thenf-'(B) is a measurable set in I. It can be proved [Halmos, 1950, $131 that every Lebesgue measurable set is of the form B A S,where B E 9"and S is a subset of a Borel set B, , with B, of zero Lebesgue measure, i.e., py)(B1)= 0.

82

11. Measure Theory and Hilbert Spaces of Functions

Proof. Denote by S the family of all subsets S of R1 whose inverse images f - l ( S ) in S are measurable sets. We will show that S is a Boolean 0 algebra. Since we can write for any S, , S , ,... E S

f-'(U

k)

-

k=l

we conclude that have f-'(S')

=

u,"=,S ,

[f-'(S)]',

E

Uf-l(sk), k-1

9. Furthermore, for any subset S of

S' = R'

-

s,

[wl we

[f -1(S)]' = 9- - f - l ( S ) .

Consequently, if S E 9, then we see that f - l ( S ' ) E d , i.e., f-l(S') is measurable, and therefore S' E 9. Thus, 9is a Boolean 0 algebra which contains the family of all open intervals in R1. Every closed interval I, in R' has as a complement either an open interval (if I, is infinite) or the union of two open intervals (if I, is finite) and therefore it has to belong to the Boolean 0 algebra 9. Since any other element of P can be written as the difference of two closed intervals (e.g., [a, b) = [a,b] - {b}), we conclude that 4l C S.Consequently, S must contain the Boolean 0 algebra 9 generated by P . Q.E.D.

Theorem 3.3. Let f ( 0 , 4 E R, be a real measurable function on the measurable set R C 3, and let g(x), x E R', be a Borel measurable real function defined on a subset of the real line, which contains the range off( 4). Then the composite function h( 5) = g[f( 5)] is a measurable function. Proof. Since g(x) is Borel measurable, for every open interval I E Y1 the set g-l(I) is a Borel set. C~nsequentlyf-'[g-~(I)] is measurable in S by Theorem 3.2, becausef(6) is measurable, i.e., V ( 1 ) = f - ' [ g-1(1)]

is measurable.

Q.E.D.

Note that it follows immediately from the above theorem that for any real number a the functions af(t), a + f ( f ) and If(4)l" are measurable whenever f ( 5 ) is measurable; we get this result by taking g(x), x E W,to be equal to, respectively, the continuous (except at x = 0 in the case of I x la, a < 0) and therefore Borel measurable functions ax, a x, and 1 x 1". In order to prove Theorem 3.4 we need Lemmas 3.1 and 3.2 (see also Exercise 3.1).

+

3. Measurable and Integrable Functions

83

Lemma 3.1. If f(() and g ( 0 , 6 E R, are real-valued measurable functions on the measurable set R, then the set

{I: f ( 0 > g ( 0 ) is a measurable set. Proof. Since the set R! of all rational numbers is countable, we can write it in the form of a sequence !R

= {Yl

,Y 2 ,...}.

For each rk E !R the sets f-l((Yk

+m))

9

g-'((--Co,

= {I:f(O> Yk),

4)= { I :g ( 0 < Tk)

are measurable on X, because they are inverse images of open intervals. It is easy to see that

u ( { I : f ( I>) m

(3.3)

{ I : f ( I> ) g(IN

=

Yk)

n{E: g ( 0

< YkN

k=l

by noting that whenever for some 5 E R we have f(5) > g ( [ ) , there must be a rational number r for which f(5) > r > g ( f ) . Since {[:f(5) > rk} n {t:g(5) < rk} as the intersection of two measurable sets is also measurable, we conclude from (3.3) that { ( : f ( f ) > g ( [ ) } is measurable. Q.E.D.

Lemma 3.2. A real-valued function h(e), 4 E X,on the measurable set R is measurable if for any real number c h-'((--co,

4) = { I :h($) < c>

is a measurable subset of X .

P ~ o o f . For any finite open interval (a, b) we have (3.4)

< h(5) < b} { I :h ( I ) < b) - { I :h(5)

h-'((a, b ) ) = {I:a =

< 4-

<

Since we can write {E: h(5) a} as an intersection of a countable number of sets which are measurable according to the assumption

w)<

{I:

m

=

n (5: h ( ~<) + i~

n=l

84

11. Measure Theory and Hilbert Spaces of Functions

<

we conclude that {g: h(g) u} is measurable (see Exercise 1.9), i.e., both sets appearing on the right-hand side of (3.4) are measurable, and therefore h-l((a, 6)) is measurable. Q.E.D.

Theorem 3.4. If gl(x),gz(x), ... is a sequence of extended complexvalued functions which are measurable on the measurable set R C %, and if g(4 = !zgn(4

exists, then g(x) is an extended complex-valued function which is measurable on R.

Proof. It is easy to check with the help of Lemma 3.1 (see Exercise 3.3) that the functions hn(x) =

SUP k=n,n+l

,...

Reg4x)

are measurable on R. [Im gk(x) are treated similarly]. Since we have Reg(x) = lim Reg,(x) = lim h,(x) = n-m

n-tm

inf

n=1,2,

... h,(x)

because hl(x) >, hZ(x)2 * - - , and as inf,,-l,z,... h,(x) is measurable if h,(x), hz(x),... are measurable functions (see Exercise 3.3), we conclude that g(x) is measurable on R. Q.E.D.

Theorem 3.5. If f(() and g ( ( ) , ( E R, are extended real-valued measurable functions defined on the measurable set R,and if (3.5)

{f(I)=

then the functions f(()

n {&?(I) = Ffool = 0 ,

+ g( 4) and f(6) g( () are also measurable.

Proof. Due to the requirement (3.5), the functionsf(() f(() g ( f ) are defined everywhere. The sets

+ g(6) and

{ f ( O + g ( I ) = *a>= {f(I)= *.o> u { g ( I ) = *4, { f ( I ) g ( l ) = *a) = [{f(I) = -+a) nM I )> 011 u W(I)> 01 n { g ( O = k 4 l obviously are measurable. Therefore, we can concentrate on the set of points at which f(() and g( 5) are finite. For any real number c we can write (3.6)

{I:-a -4

85

3. Measurable and Integrable Functions

and since c - g(5) is measurable, we conclude from Lemma 3.1 that the set on the left-hand side of (3.6) is measurable. Therefore,f(() g(5) is measurable by Lemma 3.2. The measurability o f f ( () g( 6) follows immediately from the relation

+

f(t)g(l)= W

( 5 ) + S(l>lZ- If(l)- g(t)I”,

which is true wheneverf(4) # fa, g ( ( ) # fa. Q.E.D.

3.3. POSITIVE-DEFINITE INTEGRABLE FUNCTIONS In discussing the concept of measurable functions it was sufficient to consider only measurable spaces 9 = (9,-01). For the study of the general theory of integration we need functionsf((), 5 E 3, defined on measure spaces 9 = (9,d ,p). I n order to introduce the general concept of integral of such functions we have to start gradually by defining first integration for the simple functions introduced in Definition 3.2.

Definition 3.3.

We say that the simple function (see Definition 3.2) n

40 =

c

akxs,(o,

k=l

defined on a measurable set R of the measure space 9 = (9,d ,p) is integrable on R if p ( R n {l:45) # 0)) <

Go.

Its integral with respect to p. is then defined* to be

(3.7) Next we define the concept of an integral on 9 for nonnegative functions:

Definition 3.4. An extended real-valued nonnegatiwe function f(() defined on a measurable subset R of a measure space 9 = (9,d ,p) is said to be integrable on R iff (8) is measurable on R, and if there is a nondecreasing sequence (3.8)

SAl)

< 4 6 ) < ... +

In the case that p(R n 15 : s(6) = 0))= 00, the sum appearing in (3.7) is still well defined, because we have agreed in 92 to write 0 * ( 0 0 ) = 0.

+

11. Measure Theory and Hilbert Spaces of Functions

86

of simple functions, each integrable on R, which converges at each point f E R tof(f), (3.9)

f(f)

=

;z

sn(E),

and is such that limn+@Jsn( f) dp( 5) exists and is finite. The integral of

f ( f ) on R is defined as

J f ( 6 ) 445')

(3.10)

R

= n-tw lim

J

sn(0

446).

Note that the above definition by whichf(f) is the limit of simple integrable functions implies that the measure of R n { S : f ( f ) > 0) is u finite, because m

{(:f('8 > O> =

u [R

k=l

(6: s k ( 6 ) > o}],

and since s k ( [ ) is integrable, p ( { f : s k ( f ) > 0)) < +a. if So= R n { f : f ( f )> 0} is u finite, we can write Conversely, 00 So = Un=lR, , where R, , R, ,... is a monotonically increasing sequence of sets of finite measure. T h e sequence s,(f), n = 1, 2,..,, of simple functions defined by

sl(:),zn

=

Rn

{5:f(l) 24

is a sequence of integrable simple functions which satisfies the requirements (3.8) and (3.9). Thus, if p ( R ) is u finite, we can always define the limit appearing in (3.10). Now we have to establish that if the limit (3.10) exists, then it is independent of the choice of the sequence of simple functions. For that we need the following lemma.

<

<

<

, f E R, is a nondecreasing Lemma 3.3. If 0 s l ( f ) s 2 ( f ) sequence of simple functions integrable on the measurable set R in the measure space 9 = (9,d ,p), and if s( f), f E R, is also a simple function integrable on R, and such that (3.12)

0

< s ( f ) < limsn(f), n+m

87

3. Measurable and Integrable Functions

then

lR

<

s ( t ) +(t) n+m lim

(3.13)

J

s n ( t ) MI).

Since s ( 4 ) is integrable on R, the set

Proof.

Ro

=

{t*s ( t ) > 01 *

is a measurable subset of R, and we obviously have by (3.7)

Assume that p(Ro) > 0-because true. For each n = 1, 2, ... the set

if p(Ro) = 0, (3.13) would already be

<

is p measurable (see Lemma 3.1 and Exercise 3.1). Since sl(() sz() ... , we have S, C S, C ; due to (3.12) we can deduce that S , = R, . By the continuity from below of p(S) (Theorem 2.1), we get (3.14) lim P(Sn) = P(RO)*

<

n+m

<

The simple function s(() is necessarily bounded: s(5) M. Thus, by taking into consideration the rules for dealing with integrals of simple functions (see Exercises 3.5 and 3.6), and since JR-sn s,(4) dp(() 2 0 because s(, () >, 0, we get (3.15)

I

R

sn(t)

~

t2 / ) s n ( t ) ~

t

)

Sn

When n+ co, we find from (3.14) that the second term on the righthand side of (3.15) vanishes because ~Ro-S,,

4 4 ) 4-40< MP(R0 - &a),

11. Measure Theory and Hilbert Spaces of Functions

88

while the third term converges to E . Thus we have arrived at the conclusion that lim

n-rm

J

S,(l)

4-432 JR 44)4-43- E

> 0; therefore, (3.13) holds. Q.E.D. Theorem 3.6. If sl((), s2(() ,... and si((), sZf(() ,... are

is satisfied for every

e

two nondecreasing sequences of nonnegative simple functions converging at each 4 E R tof(() 2 0,then the limits limn+mon and limn+wunf of the integrals

are equal in case that at least one of the sequences u1 ,u2 ,... and u1I ,u2f ,... converges. Proof. $,I(()

Assume that

u l , u2

,...

converges. For any n we have


< k+m lim u, .

all, u2',... is a monotonic nondecreasing sequence, which consequently converges because it is bounded. We have lirn un' < lim u, . n-rm k-r m

It is easy to establish from (3.7) that

By noting that again according to Lemma 3.3 u,

< limo,' n-m

for any k = 1, 2, ..., we conclude that lim u,

k+m

Hence 1imk+- (lk

= lim,+,w 0%'.

< lim u,'. n-rm

Q.E.D.

Lemma 3.4. If f((), ( E R, is an extended real-valued measurable function, then the functions (3.16)

are also measurable.

89

3. Measurable and Integrable Functions

The above lemma follows immediately from Theorem 3.3 if we note that the function g(x) = max{x, 0} is a continuous function on R'. 3.4. REALAND COMPLEX INTEGRABLE FUNCTIONS The extended real-valued measurable functionf( 5) defined on a measurable subset R of a measure space % = (3,d ,p) is p integrable on R if its positive part f + ( f ) and its negative par't f - ( f ) , defined by (3.16), are p integrable on R. Then its integral is defined to be

Definition 3.5.

An extended complex-valued measurable function h ( f ) , f E R, is integrable on R if Re h(5) and Im h(5) are integrable on R, and its integral is taken to be (3.18)

/,W) 445)

=:

/

R

Re 45')4 4 4 )

+i/

R

Im W

)444').

We have thus arrived at a very general concept of an integral. When we consider integration on the real line, i.e., S = R1, and we take R = [a, b], we obtain a generalization of the Riemann integral

s"

f(x)dx

of the functionf(x) if we take the measure space ( W , B1, pi')), with the Lebesgue measure p'jl)(B), B E B1. For instance, assume that f ( x ) 2 0 is continuous in [a, b ] . Then by Theorem 3.1 f ( x ) is measurable [note that p\l)({a}) = p{')({b}) = 01. If we build for a partition a = x,, < x1 < .-.< x, = b of [a, b] the Riemann sum n U,

=

C uk(xk - x k J ,

uk = i n f { f ( x ) :xk-1

k=l

we note that

corresponds to the integral (3.7),

( I ,

of the simple function

< < xk}, E

90

11. Measure Theory and Hilbert Spaces of Functions

If we take a sequence of finer and finer partitions of [a, b] in which the (n + 1)st partition is a subpartition of the nth one, so that maxk=l,...,n I xk - xk+l 1 -+ 0 when n -+ co, then obviously the nondecreasing sequence sl(x) s2(x) converges to f ( x ) , and by Definition 3.4

<

(3.19)

<

f (x) d#(x)

e..

=

lim u,

n-m

.

On the other hand, from the definition of the Riemann integral we also have /:f(x) dx

=

lim a,,

n-rw

proving the equality of these two integrals. T h e integral

of a function on [wl is called the Lebesgue integral. It is ordinarily denoted by

like the Riemann integral, to which it is equal in the case that the Riemann integral exists. However, as we can see in simple examples (see Exercise 3.4), the Lebesgue integral exists also in cases in which the Riemann integral does not exist. I n fact, the following theorem (first proved by Lebesgue, whose proof we do not reproduce”) is true.

Theorem 3.7. A bounded function defined on [ a , b ] CUP is Riemann integrable on [a, b] if and only if it is continuous almost everywhere on [a, b]. In that case the Lebesgue integral off(x) on [a, b] exists and is equal to the Riemann integral off(.) on [a, b]. In Theorem 3.7 the expression “almost everywhere” appears. This expression is very frequently used in the theory of measure. Definition 3.6. A statement concerning every element 8 E S of a measurable set S in measure space T is said to be true almost everywhere in S , if it is true on a set R C S whose complement S - R with respect to S is of measure zero, i.e., p ( R ) = p(S). T h e fact that sets of measure zero are not essential in computing integrals can be observed in the following theorem.

* See

Munroe [1953, Theorems 24.4, 24.51.

91

3. Measurable and Integrable Functions

Theorem 3.8. If h ( [ ) , [ E R, is any measurable" function on the set R of measure zero, then h ( [ ) is integrable on R and

Consequently, i f f ( 5) = g(8) almost everywhere on a measurable set S, and f (6) is integrable on S, then g([) is integrable on S and

Proof. I n case that h ( [ ) 2 0, (3.20) can be very easily proved by resorting to sequences (3.8) of simple functions, and noting that in our case (3.7) is always zero. By means of Definition 3.5 this result can be immediately generalized to any real, and consequently any complex measurable function. T h e second part of the theorem and (3.21) follow Q.E.D. by taking h ( [ ) =f([) - g([).

Theorem 3.9, If f([) and g([) are measurable functions defined g([) almost everywhere on R, and on the measurable set R, if If([)l if g(5) is integrable on R, thenf([) is also integrable on R, and

<

R o o f . From Definition 3.5 of the integral for real and complex functions it is obvious that the theorem is true in general if it is true for nonnegative functions f ( 0 ,since (Ref([))* lf(4)1, (Imf(E))* lf(5)I. Consider therefore the case whenf(8) >, 0. Since l f ( [ ) I g([) on a set R, C R, where p ( R - Ro) = 0, we have

<

[Ro n { 5 : f ( 5 ) > OH

<

<

c [R, n {E: g ( 0 > 011.

Therefore, as R n {[: g([) > O} has a o-finite measure because g([) is integrable on R, the set R n { e : f ( [ )> 0} has to be also of o-finite measure. Consequently, the sequence s,() s2([) of simple func=f ( 0 , tions (3.1 1) which are integrable on R and for which limn+m)[(,s can be built.

<

*

<

In measure theory the measure spaces are usually extended to so-called complete measure spaces, having the property that any subset of a set of measure zero is also measurable and of measure zero. This device makes any function h(g), 6 E R, automatically measurable on the set R when p ( R ) = 0.

92

11. Measure Theory and Hilbert Spaces of Functions

By Lemma 3.3, given any nondecreasing sequence sl’((), s2’([), simple functions converging to g ( ( ) for every ( E R, we have on

=

1

R

SJO

2 4 J~ ~~ i (dp(t) t) =J

~

5G )

R

R

... of

AO

for every n = 1,2,... . Consequently, the sequence ul \< u2 \< is bounded from above and therefore it has a finite limit, which by definition is limo, = f ( f ) d p ( t ) . Q.E.D.

1

n-,m

R

3.5. INFINITE SEQUENCES AND SUMS OF INTEGRALS *Theorem 3.10 (Lebesgue monotone convergence theorem). Let

fl(t),fi(t),...

be a monotone sequence of nonnegative functions, each integrable on the measurable set R. T h e extended real-valued function (3.22)

f(5) =! i

5 E R,

fn(0

is integrable on R and

provided that in case of a monotonically increasing sequence the limit in (3.23) is finite.

<

Proof. Consider first the case when fl(() . For each n = 1, 2, ..., let s,,((), sn2((), ... be a nodecreasing sequence of simple functions for which

Fz

Snd5)

= In(4),

and consequently

For each positive integer k the function

is evidently a simple integrable function. From the above definition and the fact that snk(() s ~ ~ + ~ we ( ( )get

<

Skk(5)

<

Skk+l(5)

G

Sk+lK+I(6),

3. Measurable and Integrable Functions

93

By using (3.22) we deduce from the above inequality that

Since &( 4) ,
lims;,(S)

k-bm

< k+m limfk(E) < +m.

The function f(() is measurable over R because f(5) = SUP^+^^^(^) (see Exercise 3.3). Furthermore, (3.24) and (3.25) show that f(5) is integrable over R and that (3.26)

J f ( l )C ( 5 ) = lim J R

k+m

s;k(5)

4.45) G ;z JR f n ( 0

dP(0

On the other hand, sincefn(() <j((),we have according to Theorem 3.9

J’R

(3.27)

< J’R j-(t)~ t ) ,n = 1, 2,e.m

jn(O

*

The inequalities (3.26) and (3.27) yield (3.23). In case that fi(O 2 f2(f) 2 --- , the sequence g,(Q),gdt),..., g,(l)

=fi(5) -fn(O,

n

= 1,

2,***,

is nondecreasing, and according to the above, its limit g(l) = $Egn(l)

=fi(O

- iZfn(5)

is integrable over R:

J

R

g(t)

+(o = !z JR gn(0 MI) < JR fi(f)MO

Consequently, the function

!,!Zfn(t) = fi(l)

-

is integrable over R,and since JR

$Zfn(l)4 4 5 ) = J’R fi(04-43- JR g(5) dP(49,

we see that (3.23) is again true.

Q.E.D.

94

11. Measure Theory and Hilbert Spaces of Functions

*Theorem 3.11. Let R, , R, measurable sets, and let

,... be an infinite sequence of

disjoint

ag

R=URn n=l

be their union. If the extended real-valued function f(5) is integrable on each set R , , thenf(5) is integrable on R if and only if

and in that case

Proof.

I n case thatf(5) 3 0, we can introduce the functions

if([)

for

[ER~V--.VR,

for

[ E R-

u n

R,,

k=l

<

< ---

which are obviously integrable on R. T h e sequence fl(8) fi(() hasf(() as a limit, and satisfies the conditions of Theorem 3.10. Thus we have shown thatf([) is integrable on R if and only if

which proves the theorem for the case whenf(5) 2 0. T h e more general case whenf(5) is not nonnegative everywhere in R can be reduced to the already considered case by treating f+([) andf-(() separately. Q.E. D.

3.6.

CARTESIAN PRODUCTS MEASURE SPACES

INTEGRATION ON

OF

We shall study now the problem of integration on Cartesian products of measure spaces in relation to integration on each of the spaces entering in such a Cartesian product. I n order to simplify the notation, we shall limit ourselves to the statement and proof of results for products of two measure spaces. These results can very easily be generalized to the case of products of any finite number of measure spaces.

95

3. Measurable and Integrable Functions

*Theorem 3.12.

If the extended real-valued function f ( f , q),

5 E 3, q ~ gis , measurable on the Cartesian product (Xx Y, d ) of then the functions the measurable spaces X = (3,dl)and g = (g,d2), fn(5) =

fd7)

f(5Y 4,

=f(t,v),

5 E s, 7 E g>

are measurable functions on X and g,respectively.

Proof. We shall give only the proof forf,,(f) since the proof forft(q) proceeds along identical lines. Let I E J1be a one-dimensional interval. We have for fixed q E g (3.28)

fi;'(l)= {5:f,(O € 4= { I : f ( t ,7)€ 4 = (5: (5,d E f - l ( o l =

rf-l(n?;

we have introduced above the notation in which, for any subset S of

S x W ,we denote by S,, the subset of X containing the points f for which ( f , 7) belongs to S, i.e.,

s, = {t:(t,?)E S}, s c s x 9. We shall prove now that if S C X x @Y is measurable in 9"x g, is measurable in I . then for any given q E ?Y the set S,, C I If S C I x 9Y is of the form S1x S2 and if S is measurable, i.e., if S E d ,then by definition S1E dl and S2E d2. Since in this case

; ;1

s,,=

if 7 E S2 if 7 $ S2,

and 0 E dl , it follows that S , is measurable in X. Denote by F the family of all subsets S of S x @ which are such that S,, E d1for all 7 E g.I t is easy to verify that 9is a Boolean u algebra. Since, according to the above, 9contains dl x d 2 while , by definition d is the smallest Boolean u algebra containing dl x d 2 ,it follows that d C 9. Thus, we have proved that S,, E dl for all q E aY whenever S ~ d . In particular we can deduce from (3.28) that

f 3 4 = [f-'(Z)In

E

4

for all I E Y1, since f(f , q) is measurable, and consequently f - l ( I ) E d . Q.E.D.

11. Measure Theory and Hilbert Spaces of Functions

96

*Theorem 3.13 (Fubini's theorem). Let (S, dl,pl) and (Y, d2,p2) be o-finite measure spaces and (3 x g , d , p ) the Cartesian product of these spaces. If the functionf(e, 7) is integrable on the measurable set S E d ,then the functions

are almost everywhere defined and integrable on S I C X and S 2 C Y respectively, and

Proof. We shall first prove the theorem for the case whenf(f, 7) is the characteristic function xs( g, 7) of a measurable set S E d. Consider the case when S = S' x S2, S' E dl , S2E dz. We have R(5)

=

x d 5 ) P2(S2),

h(7) = xse(7) P l ( L W ,

and consequently g(5) dPl(5) = P l ( m Pa(S2) =

h(7)dPA7). SZ

Since

I,

x&7)

df d7 = P(S) = P(S' x S Z )= PdS') P2(S2),

the relation (3.30) is verified for the case where S E dl x d2. Denote by X the class of sets S E which ~ are such that for their characteristic functions xs(e, 7)the theorem is true. It can be inferred directly from Theorem 3.10 that X is a monotone class. We have seen that each S E dl x .at2 belongs to . X . Furthermore, we can extend this result to any finite (or even countably infinite) disjoint union of sets from dl x d2by using Theorem 3.10. Since the family of all finite disjoint unions of sets from dl x d2is identical to the Boolean algebra generated by dl x dz(see Lemma 2.3), which is therefore contained

3. Measurable and Integrable Functions

97

in .X, we arrive at the desired result that .d C X by employing Theorem 1.7. Thus, we have proved that the theorem is true whenever f ( f , q ) is the characteristic function of a measurable set; this result is correct even where the integration is carried out over a set S which is different from the set R whose characteristic function is being integrated, as can be easily derived from the relation

From this statement we can infer immediately that the theorem is true for any simple function which is integrable on S E d. Now let!([, 7) be a nonnegative function integrable on S E d.Choose a sequencef,(f, q) < f i ( f , q) -.-of simple functions integrable on R, for which almost everywhere

<

n+m limfn(l, 7 ) =f ( 5 , 7 ) ,

and consequently (3.31)

The sequence of functions g,(f), g2(f),...,

is nondecreasing, and by Theorem 3.10 we have, almost everywhere,

A second application of Theorem 3.10 yields the facts that g ( f ) is integrable on S1and that (3.32)

Since fn(f, v), n = 1, 2, ..., are simple functions for which the theorem has been already established, we have (3.33)

98

11. Measure Theory and Hilbert Spaces of Functions

hence, by combining (3.31)-(3.33) we get

The equivalent result for h(5) can be derived by the same procedure. With the theorem established for nonnegative functions, it is trivial to extend it to real and then complex functions by using Definition 3.5. Q.E.D. I t is essential to realize that when we are presented with an iterated integral (3.34)

we cannot apply immediately Fubini's theorem to invert the order of integration in 5 and 7. We could justify such a procedure only if we knew not only that (3.34) exists, but also that f((, q ) is integrable on S , x S, in the measure p, x p,. The following theorem frequently provides help in such situations.

*Theorem 3.14 (Tonelli's theorem). Let (9,d,,p,) and (CY,d 2,p,) be o-finite measure spaces and (9x Y, d ,p ) their Cartesian product. Suppose the function f((,77) on S , x S, C 9 x CY is measurable on S, x S, , that ,[(fI q)1 is integrable on S , for all ( E S, , with the possible exception of a set R, C S, with p,(R,,) = 0, and that the iterated integral

exists. Then the integral

also exists. Proof. Let g(5, q ) be a function which is positive and integrable on S, x S, ; for example, if S(l),S(,),... are sets of finite p measure which = S, x S, , then the function cover+ S , x S, , i.e., S(l)v S2)v

At, 7)=

(l/n2)[P(~'~')I-'XS d

6 ,>I.

provides an adequate choice if p(S(n))> 0 for all n.

* Recall that I x Q, and therefore also S, x S, ,is of o-finite p measure, and therefore such a covering exists.

99

3. Measurable and Integrable Functions

Let us introduce the functions fn(5 ,

77) = inf(lf(5,

7711, %(l,7))’

11

=

1,2,... *

The functionf,([, r ] ) is measurable, since it is equal to the measurable function ng((, r ] ) on the measurable set (see Lemma 3.1) ((5977):

ng(5977)


and equal to f((,r ] ) on the complement of that set. Moreover, by Theorem 3.9f,(f, r ] ) is integrable in S, x S, , since I fn((, q)1 ng((,r]) and g ( f , v ) is integrable on S , x S , . Hence, by Fubini’s theorem, the functionf,(f, r ] ) is integrable in r] over S, , with the possible exception of a set R, C S, with pl(R,) = 0. of p, measure zero. Let us introduce the set R, = R, U R, U All functions f,([, 7) are integrable on (S, - R,,) x S, . Since jn(&r ] ) If(, r ] ) l , we obtain by applying Fubini’s theorem and Theorem 3.9

<

<

which shows that the integrals of f,([, r ] ) on S , x S, are bounded. Hence, Theorem 3.10 can be applied to infer that since If([, r])I = limf,([, r ] ) , the function If([, r])1 is integrable on S , x S, . Q.E.D.

If the conditions of the above theorem are fulfilled, then the order of integration in (3.34) can be inverted. This conclusion can be reached by using Tonelli’s theorem to infer thatf((, r ] ) is integrable on S, x S, , and then applying Fubini’s theorem to justify the reversal of the order of integration.

EXERCISES 3.1. By employing Lemma 3.1, show that if the functionsf(() and g(() are measurable, then the sets

are measurable.

3.2. By employing the same procedure as in proving Lemma 3.2, show that an extended real-valued function is measurable if and only if

11. Measure Theory and Hilbert Spaces of Functions

100

for every real nu nber c the set (5: - 00
<

<

<

3.3. Prove that iff,((),f,((),. . is a sequence of extended real-valued measurable functions on the measurable space X,then the functions ,... fa(() and inf,,,,, ,... fa(() are also measurable. 3.4.

Show that for the simple function defined on R = [0, 11 by '(x)

=

4 12

for x irrational for x rational

the Lebesgue integral over R exists and

J,m 4 p ( x )

= 4.

3.5. Derive directly from the Definition 3.3 that if sl(() are two simple functions integrable on R, then

and s,(()

3.6. Show, by using only the Definition 3.3, that if R, and R, are two disjoint measurable sets and the simple function ~ ( 5is) p integrable on R, and R, , then s(() is p integrable on R, u R, and

3.7. Explain why a real functionf(x) is p integrable on a set R if and only if it is absolutely p integrable on R, i.e., if J R If(x)l dp(x) exists.

3.8. Show that iff(() and g(() are p integrable functions on R, then for any a, b E C1,the function af(() bg(() is p integrable in R and

+

101

4. Spaces of Square-Integrable Functions

3.9. Show that i f f ( f ) > , g ( [ ) on the measurable set R, and iff([) and g([) are integrable on R, then

4. Spaces of Square-Integrable Functions

4.1. SQUARE-INTEGRABLE FUNCTIONS Let 52 be a Borel subset of W. Denote by a: the family of all Borel is a Boolean u algebra sets in Rn which are subsets of 52. Clearly, (see Exercise 1.8). Let p ( B ) , B E 99; , be a measure on , and denote by 52 the measure space (52, 99:, p).

Definition 4.1. We say that an extended complex-valued function f ( x ) , x E 52, defined almost everywhere on 52 is square integrable whenever f ( x ) is measurable [i.e., If(x)lz is also measurable by Theorem 3.31 and If(x)lz is integrable on L?,i.e.,

5

R

If(x)12d P ( 4

exists and is finite.

If a square-integrable function f ( x ) is not defined everywhere on 52, we can extend it to 52 by writingf(x) = 0 at the points x where it was not previously defined. Hence, we shall assume in the sequel that each square integrable function on 52 is thus suitably defined everywhere in 52. A function that is square integrable is not necessarily integrable ! Witness the case of the function f ( x ) = r 3 I 4 , x E 52 = [l, co), for integrals in the Lebesgue measure on a: :

+

I,,

.m)

f ( x ) dpi"(x)

=

Jmx-814 %.!a

=

+m.

1

However, in the case p(52) < +a,a square-integrable function will also be integrable.

Theorem 4.1. If the measure p(R) of a measurable set R is finite, then every function!([) on R.

which is square integrable on R is also integrable

102

11. Measure Theory and Hilbert Spaces of Functions

Theorem 4.1 can be deduced immediately from Theorem 3.9 and the inequality lf(5)I < Hl lf(5)I”-

+

Theorem 4.2. The set L(,,(Q,p ) of all complex-valued functions which are square integrable on Q is a complex vector space with respect to the vector operations (4.1 )

(4.2)

(af)(x) = af(4,

f ( x >EL(Z)(Q, P),

a E @’,

(f + d ( x ) =f ( 4 + g(4,

fb),g(x) EL(Z,(Q,

P).

Proof. Iff(x) is square integrable on 52 then u f ( x ) is obviously also square integrable. In order to establish that f ( x ) g(x) E L(z)(Q,p) whenever f ( x ) , g(x) E L(,)(Q,p ) it is sufficient to use the easily verifiable inequality

+

If@)+g(4l2

< 2(1f(x)lP+ I g(x)l2)

in conjunction with Theorems 3.5 and 3.9. Once it is established that the operations (4.1) and (4.2) leave L(,)(Q,p) invariant, it is a trivial task to check that all the axioms for a vector space, given in Chapter I, Definition 1.1, are satisfied, if in addition we take as the zero vector the functionf(x) = 0, x E Q. Q.E.D. The family of all spaces &)(Q, p ) includes as a special case the vector space P(m) introduced in Chapter I, $4; P(m) is obtained when we take SZ = (1, 2,...} and when the measure p is defined (uniquely) as the o-additive set function for which p({n}) = 1,

n

=

1,2,...

We get by applying Theorem 3.1 1

m

.

c1

= n=l

(n)

m

m

If(x)l244x) =

c

n=1

lf(n)I2CL({n}) =

c lf(n)l2*

n=1

Lemma 4.1. Two extended complex-valued measurable functions f ( f ) and g ( 0 , defined on the measurable set R in a measure space, are equal almost everywhere in R if and only if (4.3)

I03

4. Spaces of Square-IntegrableFunctions

Proof. If the above two functions are equal almost everywhere, then If (5) - g(01 = 0 almost everywhere and (4.3) follows from Theorem 3.8. Conversely, if (4.3) is true, then for any integer n > 0 the set S,, = (5: I f ( 5 ) - g(5)I > l/n} n R has to be of measure zero, because since If (5) - g(5)1 > (l/n) xs,(f), we have as a special case of Theorem 3.9, 0 G P(&)

= J R Xs,(O dP(gr)

< n2 JR If([)- g(5)I2d P ( 0 = 0.

Consequently m

PL(@

If (5) -g(I)I > 0 ) ) = p

p(&) = 0.

Q.E.D.

4.2. HILBERT SPACES OF SQUARE-INTEGRABLE FUNCTIONS It is easy to check (see Exercise 4.1) that the relation of “being almost everywhere equal” that can hold between pairs of functions from L(,)(Q,p) is an equivalence relation. It is customary to denote the family of all equivalence classes of functions which are almost everywhere equal and belong to L(z,(Q,p) also by the symbol L(,,(Q,p), but in order to avoid confusion, we shall denote that family by L2(Q,p). We denote the equivalence class which contains the square integrable functionf(x) by f. Thus, iff(x) and g(x) are almost everywhere equal, then f and g are identical equivalence classes, and we can write f = g; conversely, f = g implies thatf(x) = g(x) almost everywhere.

Q , belong to the equivalence Theorem 4.3. If f ( x ) , g(x) E L ( ~ ) (p) classes f,g €L2(Q,p), and a is a complex number, denote by uf and f g the equivalence classes containing the function a f ( x ) and f ( x ) g(x), respectively. The operations

+

(4.4)

+

(%f)

-- +

(4f) E @’ x L2(Q,PI,

af,

(f,g) f

g,

(f,g) E LYQ, PI

.fEL2(Q’ p)

x L2(Q,P),

f + g E L2(fA PI

are vector operations, and the set L2(Q,p) with these vector operations becomes a Hilbert space when the inner product of f,g czL2(Q,p) is defined by (4.5)

(f I g )

=

I f*(4g(4 I2

dP(X)*

The proof of this theorem, which is the central theorem of this section, will be given in a few stages.

104

11. Measure Theory and Hilbert Spaces of Functions

First we have to note that (4.4) are indeed mappings, i.e., that uf and f g are uniquely determined by f andg. That is quite obvious, because if fl = f, and g, = g, , then afl(x) = uf,(x) almost everywhere, i.e., ufi = uf2 ; and since

+

(4.6)

{x:f1(4

+ gl(4 z

f2(4

+ g2(41

c ( { x : f d x ) # f 2 ( 4 1 u {x: gl(4 zg Z ( m the measure of the set on the left-hand side of (4.6) is zero because the measures of both sets on the right-hand side of (4.6) are zero; therefore, fl(x) gl(x) = f2(x) g2(x) almost everywhere, i.e.,fi gl = f2 g2. From Theorem 4.2 we know that a f ( x ) , f ( x ) g(x) E L(2)(Q, p) whenever f (x), g(x) E L ( ~ ) ( p), Q , so that both mappings (4.4) are defined for all ( a , f )E C1 x L2(SZ,p) and (f, g) €L2(Q,p) x L2(Q,p). Theorem 4.2 implies in an obvious way that we deal indeed with vector operations, if we adopt as the zero vector in L2(Q,p ) the equivalence class of square integrable functions that vanish almost everywhere on Q. From the inequality

+

+

+

+

+

(If(4l- lg(41)2 3 0 we can immediately derive (4.7)

If*(4g(x)l

=

If(4lI g(4l G 4(If(41Z + I g(4I2).

Therefore, the function If *(x)g(x)l is integrable on SZ iff(x) and g(x) are square integrable on Q, as a consequence of Theorems 3.3 and 3.9. Thus, f*(x)g(x) is integrable on Q (see Exercise 3.7), and (4.5) is defined for any f (x), g(x) €L(,)(Q,p). Furthermore, ( f I g) is uniquely defined for any f , g E L2(Q,p) because iffl(.) = f,(x) almost everywhere, then

R

= {x:f1(4

#ffi(x)1,

by Theorem 3.8. It is easy to check (see Exercise 4.2) that (4.5) satisfies the axioms for an inner product. Thus far we have proved that L2(Q,p ) is a Euclidean space. We turn now to proving the completeness of L2(Q,p).

Definition 4.2. A sequence of functions fl(x), f,(x), ... E L ( ~ ) ( p Q), is said to be fundamental in the mean if for any E > 0 there is a positive integer N ( E )such that (4.8)

J

R

If m ( 4

-f

n ( W 444 <

105

1. Spaces of Square-Integrable Functions

and in that case we write f ( x ) = I.i.m.fn(x), n4+m

where "1.i.m." stands for "limit in the mean." According to (4.5) for anyf(x) E L(z)(sL,p )

Ilfl12

=

(f If>=

J

R

If(x)l2dCLW

Hence, iffl(x),fz(x), ... is fundamental in the mean, thenf, ,fz ,... €L2(Q,p) is a Cauchy (or fundamental) sequence in the norm in L2(Q,p), and convergence in the mean in Le,(Q,p) corresponds to convergence in the norm in L2(Q,p).

*Theorem 4.4 (Riesz-Fischer theorem). If a sequence fl(x), f2(x),... E L(,)(Q,p ) is fundamental in the mean, then it converges in the mean to a functionf(x) EL(~)(SZ, p).

... is fundamental in the mean, by taking in (4.8) k = 1,2,..., nK = 1 + N(1/8k), we get a nondecreasing sequence n, < n2 < for which Proof. Iffl(x),fz(x),

(4.9)

J

Ifnk+l(")

< 118~.

- f n k < x > ~ ~2 C L ( X )

R

Since fi(x), f z ( x ) ,... are p-measurable functions, the sets (4.10)

Sn = {x:

Ifn,+,(x)

-fnk(x)I

are p measurable. Furthermore, by (4.9)

2 1/W

106

11. Measure Theory and Hilbert Spaces of Functions

we have (4.11)

Therefore, the sets Rk

(4.12)

=

Sk U Sk+, U ..*,

k

=

1, 2,...

***

=

1/2k-'.

have measures (4.13)

p(Rk)

< 1/2k+ 1/2k+'+

A point x1 E 52 belongs to R, if and only if for some m

I fn,+,(xJ Therefore, a point x Ifn,+l(x)

E .Q

- fn,(x1)I

3

>k

wm-

does not belong to R, if and only if

-fn,(x)I

< 1/2",

= k, k

+ I,...

.

converges absolutely and uniformly on R,'. Since this is true for every k = 1, 2, ..., we conclude that (4.15)

f ( 4 =g y n , ( x )

everywhere on

It is obvious from (4.12) that R, 3 R, 3 . Consequently, due to the continuity from above of every measure (Theorem 2.1), we get by using (4.13) p ( R ) =m ! p ( R k ) < lim 1/2k = 0, +cv k-m i.e., R is of measure zero. The function f ( x ) is defined everywhere on R' by (4.15), and being the limit of functions measurable on R' it is also measurable on R' by Theorem 3.4. We can extend the domain of definition of f ( x ) to 52 by writingf(x) = 0 for x E R. The function so derived is still measurable because R is measurable.

4. Spaces of Square-Integrable Functions

107

Since fn,(x),fn,(x),... converges uniformly to f ( x ) on RL, the convergence

(4.16)

gnljx)

=

I f n ( x ) -fnl(x)I2, +

gn(x> =

I

gnz(x)

= fn(x)

-fn,(x)12,***

I f n ( x ) -f(x)lz

for any fixed n = 1, 2, ... . By assumption are integrable on 52, and therefore also on

is also uniform on R,' = Ifn(x) -fn,(x)I2 Rk), so that the set

g,,(x)

m

> 0} - R,

T F ) = (J {x:g,,(x) m=l

is of o-finite measure, and can be therefore written as the disjoint union of sets Ti;), T g ) ... , of finite measure. By Theorem 3.11 we can write

Due to the uniform convergence on R,' of the sequence (4.16), for each > 0 there is an N,(E)such that

E

I gnm(4

-

gn(4I

< t,

m

> Nn(4

for all x E R,'. Thus, for every i = 1, 2,...,

J

,n)

I gnm(x)

-

gn(x)l ~

x

<)

4 ~ k ) ) 7

m

>

~ n ( ~ ) 7

'ki

and consequently

(4.18) Therefore, by using (4.17), (4.18), and Theorem 3.1 1 we get

108

11. Measure Theory and Hilbert Spaces of Functions

On the other hand, since the sequence f l ( x ) , f 2 ( x )... , is fundamental in the mean, for any c > 0 there is an N(c) such that when n, n, > N ( E ) , we have

Consequently, by using (4.19) we conclude that for n

for every k

=

> N(c)

1, 2, ... . As we have m

52 - R

(J R,‘

=

= R,‘

u (Rz‘ - R1’)U (R3’ - RZ‘) U .*.,

k=l

we obtain by employing Theorem 3.8 (note that p ( R ) = 0) and Theorem 3.1 1

i.e., for n

> N(E),

Since we can write Ifn(X)l2

< If(x)12 + If&) -f(X)l2,

(4.20) implies that f ( x ) is square integrable and that fl(x), f 2 ( x ) ... , converges tof(x) in the mean. Q.E.D. From Theorem 4.4 we can immediately conclude that L2(Q,p ) is complete, because if fi ,f 2 ,... eL2(Q,p) is a Cauchy sequence, then

109

4. Spaces of Square-Integrable Functions

fl(x), fz(x),... ~ L ( ~ ) (p5)2 is , a sequence which is fundamental in the , converges mean; therefore, by the Riesz-Fisher theorem, fl(x), f 2 ( x ) ... in the mean to a function f ( x ) E L ( ~ ) ( p), Q , and consequently fl,fz ,... converges in the norm to f €L2(52,p). We note that in the course of proving Theorem 4.4 we have also obtained the following results.

Theorem 4.5. If f l ( x ) , f 2 ( x ) ... , EL(~)(SZ, p ) is a sequence which converges in the mean to f ( x ) EL(~)(Q, p ) then there is a subsequence f, (x), fn,(x), ... of the original sequence which converges almost everywhere point by point tof(x), i.e., there is a set R of measure zero such that x E R' = 52 - R. lim fn,(x) = f ( x ) , m-m

Furthermore, there is a monotonically decreasing sequence of sets

which is such that fn,(x), fn,(x),

... converges uniformly tof(x)

on each

Rk) = 8 - R,. *4.3.

THESEPARABILITY OF L2 SPACES

Thus far we have proved that LZ(52, p ) is a Hilbert space, i.e., we have proved the key Theorem 4.3. There are, however, spaces L2(Q,p) which are not separable (see Exercise 4.6). We would like to investigate under what conditions L2(52, p ) is separable. In order to do that, we need the following theorem.

Theorem 4.6. T h e vector subspace (C,) of Le)(52,p), spanned* by the family C, = (xB(x),B E ah? ,p ( B ) < +a} of all characteristic functions xB(x), x E 52, of Bore1 sets B C 52 of finite measure, is everywhere dense in the mean in Le)(Sz,p), i.e., for a given f ( x ) E Le)(52,p ) and any E > 0 there is a function s(x) E C, such that (4.21)

Proof.

Jn Ifb) - M24.44 <

Q.

Due to Definition 3.5, it is obviously sufficient if we prove

Note that Cn is identical to the family of all complex simple functions which are integrable on Q.

11. Measure Theory and Hilbert Spaces of Functions

110

the theorem for nonnegative functions f ( x ) 2 0 from Le,(Q, p). Given such a functionf(x), the set = { x : f ( x ) > 0}

R,

has to be of a-finite measure because If(x)lz is integrable on SZ, and therefore on R, . Thus, we can write R, as a disjoint union

of sets of finite measure. By Theorem 3.1 1

Consequently, given an

E

> 0, there is an integer N ( E )>, 0 such that

m

-

(4.22)

Since f ( x ) is square integrable on R, and p(Rk) < +a,by Theorem 4.1 f ( x ) is also integrable on R, . Therefore, according to Definition 3.4, there is a nondecreasing sequence sik)(x),sik)(x),... of integrable simple functions vanishing outside Rk such that

- sik)(x)12, ).(fI Consequently,) . ( f I sequence on R, for which

Therefore, by Theorem 3.10

so that there is an n

= no for

which

... is

- sik)(x)lz,

a nonincreasing

111

4. Spaces of Square-Integrable Functions

The function

c

Nk)

s(x) =

sE'(x)

k=l

is obviously an element of C,, for which we have, because of (4.22) and (4.23),

+ f

k=N(fl+l

1

I f ( x ) 1 2 d p ( x )< z.

Q.E.D.

Rk

Consider now the family 3: of all closed intervals I2;:::;ba,,

...,

b1, bn IU1....,Un

(4.24)

-

- [a1 >

bll x *.. x

[an

9

bnl

with rational end points, i.e., a, ,..., a, , b, ,..., b, E %. Denote by C i the subset of C, consisting of all characteristic functions ,yR(x), x E 52, corresponding to all sets R = ( I , u u In()n 52, where I, ,..., IV1E Yp, i.e., (4.25) CL= {xR(x): R = (I,u

**.

u Im) n SZ, Zl ,..., I,,, EYF,m

=

1, 2 ,...}.

We claim the following.

*Lemma 4.2. If the measure p ( B ) , B E g;,is u finite in 52, the family CL defined in (4.25) is everywhere dense in the set C, C L(,,(52,p), i.e., for every B E 37; and any E > 0, there can be found a x ~ ( x E ) CA such that (4.26)

I xIdx)

R

- XR(x)12

dp(x) <

'.

Proof. Note that I x B ( x ) - xR(x)I can assume only the values 0 and 1 and consequently (4*27)

1I n

Xdx)

- XR(x)12

ddx)= =p

n

IXdx) ( ~ R)

- XR(X)I

+p

dp(x)

( ~ B) =p(B

A R).

Consider now the Boolean u algebra do(aF) generated by the family of all finite unions of intervals with rational ends. Since any interval I E P can be written as a countable union of disjoint intervals with 3 gn.Consequently rational end points, we have du(a',n)

A'?.:

dU(Lq!) = d(P) =a

n

112

11. Measure Theory and Hilbert Spaces of Functions

. Thus,

because d ( Y n )3

we also have

da(C;) = { B n Q: B E da(9:)} = { B n Q: B E an} = a;,

i.e., 93; is identical to the Boolean a algebra generated by CL . We can easily establish (see Exercise 4.7) that C; is a Boolean algebra. According to Theorem 2.3, the set function

is a measure on the Boolean a algebra 93; generated by the Boolean algebra CL . Furthermore, if p is a a-finite measure, then p ( B ) , B E CL , has a unique extension to so that p and G, are identical: B E 9;.

p(B) = p(B),

From the definition of F(B) it follows that for any

R,,R, ,... E CL so that

z

> 0 we can choose

m

dB)< 1 d R k ) < d B )f k=l

If we write n

sn

=

u

Rk,

k=l

k-1

we have S, C S, C

u W

=

Rk,

.--, and since B C S,

+ bzp(Sn - B ) = p ( S - B )

lim p ( B A S,) = lim p ( B - S), n+m

n-m

m

= p ( s ) - tL(B)

<1

p(Rk)

- d B )< c/2.

k=l

Consequently, there is an no for which p(B A SnJ

and therefore, for R

=

< 42,

Sn, we have p(B

R)

< c.

The above result in conjunction with (4.27) proves that xR(x) satisfies (4.26). Q.E.D.

113

4. Spaces of Square-Integrable Functions

*Lemma 4.3. If p is a a-finite measure, the family Osof all simple p-integrable functions

with a, )...,a, E @1 having rational numbers for their real and imaginary parts (i.e. Re a, I m a, )...,Re am, I m a,n E S),is a countable subset of L(,)(sZ,p ) which is everywhere dense in the mean in Lq2)(Q,p). )

Proof. The result that Os is countable can be deduced easily (see Exercise 4.4) from the statement that the set S of all rational numbers is countable. Iff(x) E L(,,(Q,p), according to Theorem 4.6 we can find an integrable simple function s(x) E (C,) m

s(x) =

1

Bi n Bj

Ckx&(x),

=

m

for i # j ,

k=l

x B I ( x ) , - * , XBm(s)

R '

such that (4.28)

Ilf

- s 11' =

J

R

If(.)

444 < 49.

- s(.)'l

We can now choose complex numbers a, parts which are rational numbers so that

)...)a,

1 a, - c, 1' < €/9mp(Bk),

with real and imaginary

I a k I > 0.

The simple function

is integrable because s(x) is integrable, and

Finally, according to Lemma 4.2, for each xB,(x) we can find a xR,(x) such that

k

= 1, ..., m.

114

11. Measure Theory and Hilbert Spaces of Functions

We see that ni

sr(x) =

1

akXRk(x)

k=l

belongs to Os , and since 11 h 11, h(x) EL(,)(SZ,p), is a norm in L2(Q,p), by applying the triangle inequality and using (4.30), we get

Ic

- sr

11

=

I
nr

11 S1

ak(XBk - X R k )

k=l

I 11 XBk

- XRk

11 < dT/3.

k=l

Thus, by combining this relation with (4.28) and (4.29), we arrive at

Ilf

- sr

< Ilf - s II + II

II

i.e., we have an s,(x)

\If

- S,

1,

E

S

- $1

II + II s1 - sr II < d:,

Dsfor which

=

J) . ( f I

- s,(Y))~

dp(x)

< E. Q.E.D.

R

Lemma 4.3 establishes the separability of L2(SZ,p ) when p is a finite because the subset d, of all equivalence classes s EL,(Q,p ) containing a function s(x) E Os is obviously a countable everywhere dense set in L2(SZ,p). Thus, we have proved the following theorem.

Theorem 4.7. T h e Hilbert space L2(SZ,pL) is separable when the measure p(B), B E B'z , is a finite. T h e following statement can be immediately deduced from Lemma 4.3 by noting that every finite union R, of finite intervals can be written as a finite union of disjoint finite intervals.

Theorem 4.8. If the set 52 is of a-finite p measure, the vector subspace spanned by all the characteristic functions xR(x), x E SZ, corresponding to the sets R = I n 0, where I E 9"is a finite interval, is everywhere dense in L2(SZ,p). A special class of Hilbert spaces L2(SZ,p) which are of great importance in quantum mechanics corresponds to the choice p ( B ) = p\")(B), B E B't , where p\") is the Lebesgue measure on 37: . T h e Hilbert space L2(SZ,pi")) is separable because p:") is a a-finite measure. Such a Hilbert space L2(SZ,p r ) ) is simply denoted by L2(SZ), and correspondingly pi")) is denoted by L(,)(SZ). In the case SZ = W, the set W)is the space of all Bore1 measurable functionsf(x), for which the Lebesgue integral

115

4. Spaces of Square-Integrable Functions

exists. T h e subset Li(Rn) of all g EL:(R%) corresponding to functions g(x) whose Riemann integral -.* I g(x)I2 dnx exists and is finite is obviously a linear subspace of Li(Rn). Due to Theorem 3.7 the Euclidean space L;(Rn) is isomorphic to the space Up2,(Rn)of all continuous functions which are square integrable on Rn. We know that W:2)(R1) is incomplete (see Chapter I, §4),and the same can be proved about %?$)(Rm),Thus, the space Li(R") is incomplete, which shows that the introduction of the Lebesgue integral is essential for the validity of the Riesz-Fischer theorem, and thus for the construction of complete Euclidean spaces of square integrable functions.

JT:

4.4.

JT:

CHANGE OF VARIABLES OF INTEGRATION

In order to prove the next theorem we need the following lemma.

*Lemma 4.4. If the functions +'(x) ,..., +,(x), x = (x, ,..., x,) E B, , mapping the Borel set B , E 9,into Rn,are Borel measurable, then the mapping x

+

x1

+(x)

(dl(x),*.*, +n(X))

is a Borel measurable mapping, i.e., the inverse image +-l(B) of any Borel set B E 9,is again a Borel set.

x I , ,Il , . . . , I , E J',

Proof. For an n-dimensional interval I = Il x we can write

Since y3k(x) is a Borel measurable function, the set +;'(Ik) is a Borel set; hence, +-'(I) is a Borel set for any interval I E P. Denote by Fdthe family of all sets S for which +-l(S)E 9,. F4is a Boolean u algebra: if S , , S, ,... E Fd , then +-'(S,

i.e., S, u S, u

u s,u ...) E

u

= +-l(S1) +-'(S,)

u

*.*

Ean,

Fb, and for any S E Fd

4-'(S')

=

(4-'(S))' E a n ,

i.e., S' E Sd . Since we have already established that FdI)P,and since 9P is the smallest Boolean u algebra containing P, it follows that InC Sd, i.e., +-l(B) E anfor any B E 9%.Q.E.D.

116

11. Measure Theory and Hilbert Spaces of Functions

Theorem 4.9. (a) Let l2 be a Bore1 subset of Rn whose boundary" is of Lebesgue measure zero and on which a Lebesgue-Stieltjes measure p(B) =

I P(4 B

dnx,

P(X)

B E a: I

2 0,

is defined, and let x ---t x1 = +(x) be a mapping of l2 onto the set l21 C R", given by

4=

41@1

x = (xl

>*.*>

xn),

...)x,) E sz,

x1 = (xi

)...)x;)

E

a,,

where $l(x),..., &(x) are almost everywhere in Q continuously differentiable. If the Jacobian of the transformation

is almost everywhere positive, then

determines a Lebesgue-Stieltjes measure on 23;' . (b) If the mapping x + x1 = +(x) has almost everywhere in l2 an inverse x1 -+ x = +-l(xl), then for each functionf(x), x E l2, the function

f'(4 = f ( 4 - W )

(4.33)

is defined almost everywhere on l2', and is p1 integrable on B1 iff(.) B = $-l(B1); we have

is

p integrable in

(4.34)

1 B'

f'(x1)

dp'(x1) =

1

f ( x ) dp(x),

B'

E

a;' .

B=d-'(B')

Recall that a boundary point of a subset of a metric space is any point which is not an interior point but it is an accumulation point of that subset; an interior point of a set sd is a point which has an c neighborhood which lies in sd.

117

4. Spaces of Square-Integrable Functions

The mapping (4.35)

f

+

f ',

f E L2(Q,PI,

f'

E L2(Q1,p'),

is a unitary transformation (see Definition 2.4 in Chapter I) of the Hilbert space L2(SZ,p) onto the Hilbert space L2(SZ1,p'). PYOOJ.(a) Denote by Gothe set of all interior points x E SZ at which + J x ) in (4.31) are continuously differentiable and at which J(x) > 0. If x E Do , then +'(x), ...,+Jx) are continuously differentiable and J ( x ) > 0 also in some neighborhood of x; thus, the set SZ, is open.

$l(x),...,

According to the assumptions, p(SZ - SZ,) = 0. The set SZ, is mapped by (4.31) onto a set SZ; C SZ1. Since J'(x') is continuous in SZ, , it is also bounded and therefore integrable on every compact set B C $2, of finite measure. According to Lemma 4.4 the inverse image B, under the mapping (4.31), of every Borel set B' C SZ; is also a Borel set, due to the fact that +'(x),..., + J x ) are continuous and therefore measurable by Theorem 3.1. Thus, (4.32)really determines a measure on SZ;, which can be extended to Q1 if we write pya1 -

a;)= 0.

(b) It is easy to see that the functionfl(xl) = f(+-'(x')) is p1 measurable on* SZ1 if f ( x ) is p measurable on SZ. In order to establish (4.34), note that for B' E a,"l , f'(x') = XBl(x'), we get from (4.32) that (4.34) is equivalent to the relation (4.36)

P1(B1)= P ( + - * ( w

If we choose B' to be an open subset of SZl, then the set B is also open if B denotes the inverse image +-l(B1) of B1 after a possible subtraction from +-l(Bf) of the set +-'(B1)n (SZ - Go)of measure zero. When B1 is an interval in SZ, we can deduce from well-known theorems of the calculus that (4.36) is true by noting (first for continuous p, then generalizing) that (4.36) can be written in the form (4.37) *fl(xl) is defined almost everywhere by the relationfl(xl) wt of p' measure zero we can define .fl(xl)

arbitrarily.

= f(d-l(d)).

On a remaining

11. Measure Theory and Hilbert Spaces of Functions

118

Thus, due to the additivity of pl(B1)and p ( B ) , (4.36) is also established for any set B1 which is a finite union of disjoint intervals lying in Q1, i.e., on the Boolean algebra gt,agenerated by the family 9;of all intervals lying in s2 (see Exercise 4.8). Since pl(B1)is a o-finite measure and the Boolean u algebra generated by is ~39:~ (see Exercise 4.8), we conclude from Theorem 2.3 that (4.36) is true for any B’ E . Thus, as the next obvious result, we have that (4.34) is true whenfl(xl)-and consequently also f(x)-are simple functions. From Definitions 3.4 and 3.5 we can infer in a straightforward manner that (4.34) is true wheneverf(x) is p integrable on B. If fl(x) = f 2 ( x ) almost everywhere in the p measure, then f:(xl) =ft(xl) almost everywhere in the p1 measure, where f:(xl), i = 1, 2, are defined almost everywhere by (4.33). We conclude that (4.35) defines a mapping of L2(Q,p ) onto L2(Q1,p’). It is quite evident that this mapping is linear, has an inverse and preserves the inner products of the spaces L2(s2,p ) and L2(Q1,pl), i.e., that it is a unitary transformation. Q.E.D.

EXERCISES 4.1. Show that the relation, “f(x) = g(x) almost everywhere,” is an equivalence relation in Le,(Q, p). 4.2.

Check that

(f I g> = J j*(XI

g(4

4-44,

f,g E WJ-4P),

satisfies the axioms for an inner product. 4.3. How can you deduce directly from Theorem 4.3 that Z2( is a separable Hilbert space ?

+ co)

4.4. Show that the set CL appearing in Lemma 4.2, and the set Ds introduced in Lemma 4.3 are both countable sets.

4.5. Show that the set a ; ,p(s2) < +a,in which we identify any two sets whose symmetric difference if of measure zero, is a metric space if the distance of R, S E g: is taken to be d ( R , S) = p ( R A S ) . 4.6. Check that the set function p(B), B E . @ , defined as being equal to the number of elements in B , in case B is finite, and +co in case that B is an infinite set, is a measure on 9?’. Show that L2(R1, p ) is not separable.

4.7. Prove that the set CL introduced in Lemma 4.2 is a Boolean algebra of subsets of s2.

119

5. Hilbert Space of Systems of n Different Particles

4.8.

Show that the family &?&

of all unions of disjoint intervals

I E P lying within an open set SZ, 52 C Rn, is a Boolean algebra. Prove that the Boolean a algebra generated by &?& is identical to

5. The Hilbert Space of Systems of in Wave Mechanics

n

a:.

Different Particles

5.1. THESCHROEDINGER EQUATIONOF PARTICLE SYSTEMS We shall study now the general framework used in wave mechanics to describe a system of n particles in which each one of the particles is of a different kind from the rest. T h e requirement that the particles are of different kinds is essential because systems containing two or more particles of the same kind of atomic or subatomic size have peculiarities of behavior reflected in certain statistical features, which require that they conform to either Bose-Einstein or Fermi-Dirac statistics-subjects that we will discuss in Chapter IV. For instance, a two-particle system of the kind we are considering now would be an electron-proton system which, when in a bound state, constitutes a hydrogen atom. An example of a three-body system of different particles is an electron-protonneutron system, which is called a deuterium atom when it is in a bound state. However, tritium, consisting of one electron, one proton, and two neutrons, already is not a system of the considered type, except if we treat its nucleus as one particle and the electron as another particle. We must mention that at present we ignore the possible internal degrees of freedom that the particles might have. These internal degrees of freedom are reflected in the existence of spin, a concept we shall introduce in Chapter IV. We assume that our n particles move in three dimensions, and denote by rkthe position vector of the kth particle. If we choose in the threedimensional real Euclidean space R3 a fixed inertial reference system of coordinates" characterized by three orthonormal vectors e, , ey , and e, , we can expand rk in terms of e, , ey , and e, , rk = xkez

+ ykey+ z k e z ,

k

=

1,2 ,..., n.

In wave mechanics it is postulated that the state of a system of n particles is described at any given time t by a function +(TI ,..., r, ; t ) In practice the reference frame is usually tied to the laboratory in which the experiments are performed. For the duration of the typical microscopic experiment such a frame moves uniformly with respect to the sun, i.e., it is approximatelyinertial. Such a system is called the laboratory frame of reference.

11. Measure Theory and Hilbert Spaces of Functions

120

defined on the configuration space [W3, of the coordinate vectors rl ,..., r, , It is required in addition that #(rl ,..., r, ; t) is once continuously differentiable in t and that for fixed t it is square integrable," i.e., #(rl ,..., r, ; t) EL(,)(IW~~), and normalized (5.1)

JRan

I +(rl,..., r,;

t)I2dr,

... dr,

=

1.

I n general, if a function #(rl ,..., r, ; t) can represent a state of the given system, then #(rl ,..., r, ; t) is called a wave function of that system. As in the one-particle, one-dimensional case treated in Chapter I, $5, the wave function #(rl ,..., r, ; t) does not have any direct physical meaning. Instead, we have again (as a special case of the Born correspondence rule, which will be given in Chapter IV) that the LebesgueStieltjes measure (5.2)

P,(B) = B

I +(r1,..., r,; t)ladr,

dr, ,

B E g3",

is interpreted as a probability measure: P,(B) gives, for every Bore1 set B, the probability of having within B the outcome of a measurement at time t of the positions r1 ,..., r, of the n particles of a system prepared in a state represented by #(rl ,..., r, ; t). In view of Born's interpretation, it turns out, as we shall prove in general in Chapter IV, that if a state is described at time t by the wave function #(rl ,..., r, ; t), then the function qhl(r1,..., r, ; t), where I c I = 1 and #(rl ,..., r, ;t ) = #l(rl ,..., r, ; t) almost everywhere (with respect to the Lebesgue measure on OX3,), describes the same state. This implies that each function in the equivalence class #(t) E L2([W3,), containing #(rl ,..., r, ; t) describes the same state as #(rl ,..., r, ; t). Thus, we can state as the first assumptiont of wave mechanics:

Postulate W1. The state of a system of n different particles is described at any time t by a normalized vector #(t) from the Hilbert space L2(IWSn).T h e time-dependent vector function c#(t), I c I = 1, represents the same state as #(t). Throughout this section integrability and measurability are only with respect to the p!")). measure space (IWsn, BSn, t The postulates we shall state in this section represent a convenient way of systematically formulating some of the basic assumptions of wave mechanics. These postulates are by no means exhaustive. A complete study of the postulates of quantum mechanics will be given in Chapter IV. The reader should find it easy to adapt those general axioms to the special wave mechanical case.

5. Hilbert Space of Systems of n Different Particles

121

As an equation of motion we have now a Schroedinger equation of a more general form than (5.4) of Chapter I. If the considered system is such that in the classical mechanics case the potential energy of the system" is V(rl ,..., r,), then we arrive at the Schroedinger equation by a generalization of the heuristic procedure given in Chapter I, $5. Take the energy conservation equation E = C - - -P k+2 V ,

(5.3)

k-1

2mk

where pk and mk are, respectively, the momentum and the mass of the kth particle, and make in (5.3) the following formal substitution:

a

+ ev- a + e,-. a aYk

azk

We arrive at the formal operator relation

(5.4)

Now we postulate the following.

Postulate W2. Each wave function +(rl ,..., r, ; t ) , +(t) EL~(R~,), in t. If +(rl ,..., r, ; t ) is twice differentiable in the configuration space R3" of the variables r1 ,..., r, , then it

11 $(t)ll = 1, is once differentiable

satisfies the time-dependent Schroedinger equation, (5.5)

a

ifi at $(rl ,..., rn;t )

whenever the function on the right-hand side of (5.5) belongs ~ o L ( , ) ( R ~ ~ ) . In view of the interpretation of P , ( B ) in (5.2), it is essential that a We are assuming no external force, so that the potential energy V(r, ,..., r,) is time independent. Classically, V(r, ,..., r,) completely describes the interaction. The force Fk with which the rest of the particles are acting on the kth particle is Fk = +

-MVW

+ e,(Wauk) + e,(aVPzdl.

122

11. Measure Theory a n d Hilbert Spaces of Functions

wave function satisfy (5.1) at all times. We leave to the reader (see Exercise 5.2) to prove the following theorem.

Theorem 5.1. If the function +(rl ,..., r, ; t ) , + ( t ) eL2([W3,),is once continuously differentiable in t and twice continuously differentiable in the configuration space variables, and such that ( a / a t )+(rl ,..., r, ; t ) and Ak$(rl ,..., r, ; t ) are square integrable, and if $(rl ,..., r, ; t ) satisfies the Schroedinger equation (5.5), then

xi=l

+

provided II #(O)ll = 1 and I v k + I =0(1 rk l--ak), "k > 2, as I rk I --t ao. Using the same procedure as in 55 of Chapter I, we shall try to find solutions of (5.5) of the form (5.6)

+

#(r1,*.*, rn) exp[-(i/4 Etl

where €L2([W3,).By inserting (5.6) in (5.5) we arrive at the timeindependent Schroedinger equation for our problem, which can be written in symbolic form: (5.7)

zfs#(rl).-,rn) = W(r1,..., r n ) )

where H , is a differential operator form"

called a Schroedinger operator. Thus we have arrived again at an eigenvalue problem (5.7). T h e same terminology applies now as in the one-dimensional, single-particle case treated in Chapter I.

FRAME OF REFERENCE 5.2. THECENTER-OF-MASS There is, however, an essential difference (besides dimension and number of particles) between the present case and the one already

* It must be realized that a differential operator form such as (5.8) is essentially only

.f

9 of functions into another class of functions by applying to each function from 9 the procedure symbolically embodied in the operator form. Once the class of functions 9 is specified, we deal with a well-defined mapping, which is customarily called an operator. However, the same differential operator form can be applied to different classes of functions, thus defining different operators. The importance, from the mathematical as well as physical point of view, of specifying in a given problem the class of functions on which the differential operator form acts will be explained in Chapter 111, 92, and (5.8) will be studied in Chapter IV, 97. a recipe for mapping a certain class

5. Hilbert Space of System of n Different Particles

123

treated in Chapter I: the one particle was moving inside an external field of force described by the potential V(x), while at present the interaction originates inside the system by having the particles interact with one another with forces described by the potential V(rl ,..., r,), and we assume no external forces. Thus, classically speaking, the total force on the center of mass

of the system will be zero (see Exercise 5.1), and the center of mass will be in uniform motion in any inertial reference frame. Quantum mechanically, this means that (5.7) does not have any solutions for which I,~EL~(IW~~~). T o see this, let us replace the variables rl ,..., r, with new variables rl', ..., r6-1, r,' = R,which include the center-of-mass position vector R.T o be specific, let us choose the following variables: rl'

=

r2 - rl ,

r2' = r3 - R, ,

(5.10) rh-l R

=

r, - RnP2, 1 M

= -(mlrl

+ ... + m,r,),

In order to establish how the Schroedinger operator form looks in these new variables, we shall proceed in a few stages. First note that if we make in $(rl ,..., r,) the substitution rl'

=

r2- r, ,

(5.1 1)

thus obtaining the function $(l)(rl',R, , r3 ,..., r,), we have

124

11. Measure Theory and Hilbert Spaces of Functions

where Ml is the reduced mass of the first two particles:

We shall proceed by induction in . By performing in the funcl Lon , Rkd1, rk+l,..., r,) the substitution $(k-l)(rl’,..., (5.14)

rk‘

=

R k

=

rk+l

- Rk-1

mk+lrk+l

3

+ + + (ml

m1

*’*

+ *.. +

mk) Rk-l 7

mk+l

we get the function $(k)(rl’ ,..., rk’,Rk , rk+2,..., rn),and we have

Thus, by induction in k we easily get from (5.12) and (5.15)

5. filbert Space of Systems of n Different Particles

125

since from (5.11) and (5.14) it follows that

Now, in physics the only potentials used are those which, from the classical point of view, correspond to forces which satisfy the actionreaction principle. This implies (see Exercise 5.3) that when V(rl ,..., r,) is expressed in terms of the variables (5.10), it does not depend on R. Thus, in the new variables the Schroedinger operator form is (5.18)

n-1

1

ti2

H s -- - -A,, -Aj’ 2M j=1 2Mj

+ V(r,’ ,..., rk-,).

If we seek now a solution of (5.7) of the form (5.19)

$(TI ,*--,

rn)

= +c(R) Y(r1‘,.*., 4-1)

we arrive at the following system of differential equations:

..., rhW1),

= EbY(rl’,

E

=

Ec

+Eb.

Following the classical tradition, E, is called the energy of the center of mass of the system, Eb is the internal energy of the system, and E is the total energy. The transformations (5.10) obviously map W n into itself. Because of the linearity in r1 ,..., r, of those transformations, the Jacobian

of the transformation is obviously a constant in our case. Thus, we can write (Theorem 4.9)

126

11. Measure Theory and Hilbert Spaces of Functions

where &(r1‘,..., R) is the function which we obtain when we express in +(rl ,..., r,) all rl ,..., r, in terms of rl’,..., R. If Gl is of the form (5.19), then we have by Fubini’s theorem (Theorem 3.13)

+

if ~and ~ ) only if + c eL2([W3)and Y EL~(IW~(,-~)). However, i.e., E L ~ ( [ W the equation (5.20) for the energy of the center of mass of the system E L2([W3). Its eigenfunctions does not have any eigenfunctions

2111 A R exp (fpcR)

- fi2

2 1 exp ~ (h z PcR) ,

= Pc2

Pc E [W3,

are not square integrable, corresponding to the fact that, as we shall see in Chapter IV, 97, the operator d, does not have a point spectrum, but only a continuous one. Physically, this is due to the fact that, because of the lack of external forces, the center of mass of the system is in free motion. 5.3.

THEBOUND STATES OF PARTICLE SYSTEMS

Postulate W3. T h e set S, of all eigenvalues of the equation (5.21) which correspond to square-integrable eigenfunctions are the only internal energy values which the n-particle system in a bound state can assume. T h e closed linear subspace X‘(bn)of L2([W3,),spanned by all €L2(IW3,)for which

+

YR(rl’ ,..., rkP1)= +(rl ,..., r,),

YR

1 9

is an eigenfunction of (5.21) for every R E R3, contains all the Hilbert vectors which can represent, at a given time t, a bound state of the n-particle system” interacting via the potential V(r, ,..., r,)

=

V(rl’,..., rh-,).

T h e interpretation of an eigenvalue Eb of (5.21), with eigenfunction E L2([W3(,-l)), as internal bound energy is possible only if Ebis a real number. This will be so if we can generalize Theorem 5.1 in Chapter I.

* We talk about bound states of the entire n-particle system, which leaves out, e.g., when n = 3, the case when two particles are bound, while the third one is free! We shall study this more general problem in scattering theory.

5. Hilbert Space of Systems of n Different Particles

127

Theorem 5.2. T h e Schroedinger operator form H , , the center-ofmass energy operator form (5.22)

Hc =

-

and the internal energy operator form n-1

(5.23)

Hi=

-

jj2

-Aj' j=1

2Mj

+ V(rl',..., rh-,)

define Hermitian operators when applied to twice continuously differentiable functions, which together with their first derivatives vanish faster than I rk I-l, k = 1,..., n, as I rk I --t a. A differential operator A defined on L2(Rm)is said to be Hermitian if, whenever Af and Ag are defined for f, g €L2(Rrn)and belong to L2(Rm),then (5.24)

(4I g>

=

(f4 4).

Proof. Denote by S, the sphere in R3 of radius r and with center at the origin, and by B, its surface. By applying Green's theorem twice on STn= S, x x S, E Wn, we get (5.25)

+ Jsr"f * ( H s g )

dr1.e.

drn

9

where a/ank is the derivative in the coordinates rk in the direction of the outer normal to B, . When we let I -+ a in (5.25), we get

+

because the surface integrals entering in the sum on the right-hand side of (5.25) have to vanish, since the functions I f 1, I g I, I af/ank I, and I ag/ank I are assumed to vanish faster than I rk 1-l when I rk I -P +a. The proof for H , and Hi is completely analogous. Q.E.D.

If Yl , Y2E L ~ ( R ~ ( are ~-~ eigenfunctions )) of (5.23) corresponding to the eigenvalues EL1) and EL2),respectively, then we also have, by (5.23),

11. Measure Theory and Hilbert Spaces of Functions

128

that HiYl , HiY2E L ~ ( R ~ ( ~ - Consequently, ~)). by Theorem 5.2 (in Chapter IV we shall see that the condition at infinity can be dropped): (5.26)

Elj"*(Yl I Y2)= ( H ! Y l I Y z )= (Y1 1 H i Y Z ) = EF)(Yl I Yz).

In case that Yl= Y 2 , and therefore EL1) = EL2) = E b , the above relation implies that Eb = Eb*. If EL1) # EL2) we get from (5.26) that (Yl I Y 2 )= 0. Thus, we have arrived at the following conclusion.

Theorem 5.3.

Each eigenvalue E of the internal energy operator

Hiin (5.23), belonging to an eigenfunction Y E L ~ ( R ~ ( ~is- ~a) )real , number. Eigenfunctions corresponding to different eigenvalues are mutually orthogonal. Thus, to each eigenvalue belongs at least one nonzero eigenvector, and the eigenvectors belonging to different eigenvalues are orthogonal and therefore linearly independent. Since L2(R3("-l)) is separable, any orthogonal system of vectors contains at most a countable number of elements. Hence, we can state Theorem 5.4.

Theorem 5.4. The number of bound-state energy eigenvalues is at most countably infinite, i.e., the point energy spectrum S, contains a countable number of elements. 5.4.

OPERATOR PROPERTIES OF THE 12-PARTICLE SCHROEDINGER

An immediately noticeable feature of the Schroedinger operator form in (5.8) is that it is not defined on all the elements of L 2 ( P n ) .The class of functions from L2(R3,) on which H , is defined depends on the particular form of the potential V(rl ,..., r,). Broadly speaking, the more singularities V(rl ,..., r,) has the narrower the class on which H, can be defined. In the next chapter we shall find Theorem 5.5 very useful.

Theorem 5.5. If V(rl ,..., r,) is square integrable on every bounded measurable set in R3,, then the differential form Hs when applied on (; R3), of infinitely many-times differentiable the functions of the family % functions with compact support" in maps linearly the set %;(Wn) (where obviously %;(Rsn) C L(2)(R3n))into L(2)(R3n).

Proof. If f ~ % c ( R ~ ~then ) , f can be represented by a function f(rl ,..., r,) which is certainly continuous and thus measurable, and vanishes outside a bounded open set S, i.e., it is integrable and square The support suppf of a function f is the closure of the set of all points at which the function is not zero, In R" a set is compact if and only if it is closed and bounded.

129

5. Hilbert Space of Systems of n Different Particles

integrable on R3,. By the hypothesis of the theorem, V(r, ,..., r,) is square integrable on S. Thus, V(r, ,..., r,)f(rl ,..., r,) is integrable and therefore also square integrable, as we can convince ourselves by using an inequality like (4.7) and applying Theorem 3.9. Furthermore, we evidently have that for every K = I, ..., n

wg(R3n) L(Z)(R3n),

Akf

and therefore we can conclude that H , ~ E L ~ ( R ~Q.E.D. ~). The importance of the preceding theorem is related to Theorem 5.6.

Theorem 5.6. T h e set '+?;(Rm)of all elements of L2(Rm)which can be represented by infinitely many times differentiable functions of compact support is a vector subspace ofL2(Rm),which is dense in L2(Iw"). Proof. The set W;(Rm) is obviously linear. That %;(Rim) is dense in LZ(Rm) can be demonstrated in the following way: According to Theorem 4.8, for any given f € L 2 ( R m ) we can find a simple function

c Y

s(x) =

ckxIk(x)

k=l

(where each

is a finite interval) such that

I l f - s II -=z 42. If we could prove that the characteristic function of any finite interval can be approximated in the mean arbitrarily well by an infinitely manytimes differentiable function of compact support, then for each I k we could find a g, E '+?;(Rm) such that (5.27)

Therefore, the fact that W;(Rm) we have g

=

= L2(Rm)would

1c k g k gc(Rm),

k=l

and for the arbitrarily chosen f E L2(Rm) llf-gll


tlls-gll

be established because

11. Measure Theory and Hilbert Spaces of Functions

130

Any finite interval I in Rm is a direct product of m finite intervals in R1. Consider a semiclosed finite interval

I = [a1 , b,) x ... x

(5.28)

[a,,

b,)

whose characteristic function can be written as

where O(x) is the step function

Assume that we have at our disposal a sequence hl(x), hg(x),..., x of functions which are monotonically nondecreasing h,(x’)

< h,(x”),

x’

< x”,

n

=

1, 2,...,

infinitely many times differentiable in x, and such that hn(x)

where

B,

for x

< -6,

1

> 0 and lim

n-,m

6,

= 0.

An example of such a sequence is provided by

0 exp

for x

(-

1n2x2 n2x2

1

for

< -l/n

-1in

for x

<x


> 0.

We have (5.30) J

T h e function

--m

I h,(x)

- O(x)12 dx

< E,

--f

0,

n

--f

+a.

E

R1,

131

5. Hilbert Space of Systems of n Different Particles

obviously is infinitely many times differentiable and of compact support. It is easy to derive from (5.29) and (5.30) that

Thus, we have proved that a g, E W;(Rm) satisfying (5.27) can always be found when I, is a semiclosed interval like (5.28). Since the difference between I in (5.28) and any other interval with the same end points as I is of Lebesgue measure zero, our result applies to any finite interval in R”. Q.E.D.

5.5. THEINITIAL-VALUE PROBLEM The time-dependent Schroedinger equation (5.5) is of the first order in t . Hence given the initial condition at some time t = to in the form of a twice differentiable function +(0)(rl,..., rn), there will be at most one solution +(rl ,..., r, ;t ) of (5.5) which satisfies the initial condition $(rl ,..., r,; to) = $(0)(rl ,..., rn).

(5.31)

In the case where YAP)~ L ~ ( [ w )9~ ( ~ - l ) Y$)(r,’,..., ri-l)

is a bound state, i.e.,

+(O) E

=

+(“)(rl ,..., rn),

Xp),we have the result

x

2M where (see Chapter 111, 94.3 on Fourier transforms) #(rl ,..., r,; t )

=

(YJ q p ) ~ ,

YR(rl’,..., rLl; t ) ,

qt)>= j,3,n-ll yv*(r1’ ,..., rk.-l)Y~ - (rl’,..., rhd1)dr,’


(0)

drA-l

,

is the solution of the initial value problem (5.31). I n (5.32), {Yl, Y2,...} is an orthonormal basis (in general, countably infinite) of the space 2c-I)(spanned by all the eigenvectors Y E L ~ ( W ( ~ -of~ )Hi) ) which consists of eigenvectors of Hi : H,Yv = E t ) Y v .

132

11. Measure Theory and Hilbert Spaces of Functions

One can arrive at the conclusion that such a basis exists by a straightforward generalization of the reasoning employed in $5 of Chapter I. By duplicating the proof of Theorem 5.2 in Chapter I, it can be established that if the series (5.32) is infinite, it has to converge in the norm, and that its limit # ( t ) has the property that $(to)= (cl0. T h e fact that (5.32) really satisfies the equation (5.21) can be checked by inserting (5.32) in (5.5) and differentiating term by term. Naturally, in this procedure we have to assume that the order of summation and differentiation can be interchanged-a very arbitrary, and sometimes incorrect assumption, when the series is infinite. However, it follows from the general theory of linear operators expounded in Chapter I11 that (5.32) really represents the general solution of the initial-value problem for bound states.

EXERCISES 5.1. Show that if there are no external forces acting on an n-particle system, then classically, the total force F = M# (R and M are given in, (5.9)) acting on the center of mass R of the system is a zero vector.

5.2. Prove Theorem 5.1, and justify each step in the proof by applying well-known theorems of the calculus. 5.3. Show that as a consequence of the action-reaction principle V,V = 0 in the absence of external forces.

5.4. Prove that the series in (5.32) converges in the norm, and that its limit YR(to)for t = to is equal to YLO).

6. Direct Sums and Tensor Products of Hilbert Spaces 6.1.

DIRECT SUMSOF EUCLIDEAN SPACES

I n this section we shall study two important procedures of constructing new Hilbert spaces from given Hilbert spaces. These constructionsthe direct sum and the tensor product of Hilbert spaces-have a great variety of applications to problems in quantum mechanics. With these applications in mind, we shall not treat the most general concept of such spaces that appears in the mathematical literature, but rather restrict ourselves to the level of generality which is sufficient for treating the mathematical problems to which nonrelativistic quantum mechanics gives rise. Such applications will occur frequently in 97 and in Chapter IV.

133

6. Direct Sums and Tensor Products of Hilbert Spaces

Theorem 6.1. Let g1,&, ,... be a finite or countably infinite sequence of Euclidean spaces, with inner products (. I .), , (. I .),,...,respectively. The family of all sequences f

=(f1, fi

,...) E 8;x 8*x

.*.

for which

1

(6.1)

Ifk)k <

(fk

k

is a Euclidean space-customarily @ 8 k k

denoted by

=81

@ 8,@

”’

and called the direct sum of 8,, 8,,...-in which the vector operations €” ,f = (fl ,f, ,...), g = (g, ,g, ,...), are and inner product off, g E defined in the following way:

Bk

f (6.2)

+g

=

(fl + g 1

af = ( a f 1

9

.fi

Y f i

+

g2

,-),

Y...),

(f I g> = 1 ( f k Igk)k k

*

Sequences (fl ,f, ,...) satisfying (6.1) are denoted by f, @ f2 @

.

Proof. If we have a finite number of Euclidean spaces c?’,..., &’, then each element f of &, x x &n obviously satisfies (6.1) and therefore f is also an element of g1 @ *.. @ &’ . I t is a straightforward task to check that f g, uf and (f 1 g) are defined for all f, g E c?, @ --.@ 8’ and scalars a, and satisfy all the axioms for vector operations and inner product, respectively. If we have a countably infinite number of Euclidean spaces g1,8,,..., we have to check first whether the expressions in (6.2: are really defined gk. Since we have for all f, g E

+

a;=,

it follows that uf E gkwhenever f E Gk. In order to establish that f g E @;=l gk when f, g E @ =;, apply the triangle inequality to the space @L=l cfk to obtain

+

&k

, we

134

11. Measure Theory and Hilbert Spaces of Functions

From the above inequality we can immediately conclude that 8 k by letting v 4 co. In order to show that

f + g E @=;,

+

is absolutely convergent (and therefore also convergent) whenever (g, ,g, ,...) belong to 8 k , we use the SchwarzCauchy inequality in each of the spaces cYk to obtain

(f,,f2 ,...) and

From (6.4) and (6.5) we can obviously deduce the absolute convergence of (6.3). Thus, we have established that the expressions (6.2) are really defined for every f,g E 8‘ . We leave to the reader the straightforward task of checking that all the eaxioms of a Euclidean space are fulfilled when in addition we take the sequence (0, , 0, ,...) of the zero vectors 0, , 0, ,... of E; , g2,... respectively, to be the zero vector of gn . Q.E.D.

ofl

As examples of direct sums of Hilbert spaces, note that Z2(m isomorphic to Z2(m) @ Z2(n), and that P+Co)

= P(1) @ P(1) @

+ n) is

**..

6.2. SEPARABILITY AND COMPLETENESS OF DIRECTSUMS OF HILBERT SPACES I n proving the next two theorems, we shall use the same method as in proving the separability and completeness of Z2( co) in Theorem 4.3 of Chapter I.

+

*Theorem 6.2. If &“,g2,... is a finite or infinite sequence of Euclidean spaces, their direct sum @ k &k is a separable Euclidean space if and only if each of the spaces 8,,g2,... is separable.

6. Direct Sums and Tensor Products of Hilbert Spaces

135

Proof. We shall consider only the case where we deal with an infinite sequence 8,, g2,... of Euclidean spaces. Let S,, S, ,... be countable everywhere dense subsets of g1,8, ,..., respectively. Now, all sets of the form R,

=

S, x

x Sv x {Ov+l} x {Ov+2} x

***

***

are countable subsets of @;=l 8 k (0, is the zero vector of W

=

u

8k).

Hence

W

@&k k=l

Rv

v=l

is a countable set, which is also dense in @= :, 8 k . Namely, for any given f = (f, ,fi ,...) E @= :, 8 k and E > 0, there is a positive integer n for which

c W

IlfkIIk2

< E/2*

*

k=n+l

Furthermore, there are g,

E

llfk -g k

S,

,...,g,

S, such that

E

1; <

=

I,***,n,

because S, is dense in gk.Therefore for g = (81

,...)gn

9

On+l

9

On+, ,*..) E Rn

we have n

IIf-gII' = ( f - g I f - g )

=

1 k=l

Ilfk - g k I 1 2

+

c 00

IlfkIIz

< 6.

Q.E.D.

k=n+l

*Theorem 6.3. If Z,, Z2,... is a finite or countably infinite sequence of Hilbert spaces, their direct sum Q k Z kis also a Hilbert space. Proof. We shall consider explicitly only the more difficult case of an infinite sequence of Hilbert spaces Z 1, X 2,... . IJetf'1),f(2),... E OF=, Z k ,

I36

11. Measure Theory and Hilbert Spaces of Functions

the sequence f i’),f i2), ... is a Cauchy sequence in @ k , and has a limit , because @ k is complete. By using the triangle inequality in P ( u ) we can derive that

f k E i@k

orTl is a Cauchy sequence,

for each m = 1, 2 ,... . Sincef ( l ) f, ( 2 ) ,... E for any E > 0 there is an integer N ( E )such that

fkm) for each fixed for all m,n > N ( E ) .Furthermore, since f k = limm+co positive integer, there is an integer N”(E)such that lifk

for nz

-f$’

//k

< 2--(k+23/26, k

=

1, 2 ,..., v,

> N ( E ) Thus, . from (6.6) we find that

for all n > IVY(€). By letting any n > N ( E )

u-+

co in the above relation we derive for

T h e triangle inequality in Z l @

[i k=l

llfk

llk2]1/2

< [i

llfk

- - a

@ X Vyields

-.fY llk’]1/2

k=l

+ [ i Ilf?’ k=l

From the above we can immediately infer that m

l l f k Ilk’

< +co

k=l

by choosing a fixed n = no > N ( E ) and employing (6.7). Thus, f ( n ) . Q.E.D. = (fl ,f 2 ,...) E z kand , by (6.7),f =

f

orTl

It is easy to see that the operation of constructing direct sums of Euclidean spaces is commutative and associative (see Exercises 6.2-6.4).

137

6. Direct Sums and Tensor Products of Hilbert Spaces

6.3. BILINEARFORMSON VECTOR SPACES In order to be able to study tensor products of Euclidean spaces, we need to familiarize ourselves with some of the properties of bilinear forms.

Definition 6.1. A bilinear form ( - I .) on a vector space" is a complex-valued function defined on V x V , having the property that for any f , g , h E V and any complex number a

(fI "g) = 4fI g) = (a*f I g),

+

(flg 4 (f+g I

=

=

+( f l 4 (fl4 + ( g I h).

(fig)

A bilinear form (f 1 g) is called Hermitian if for all f,g

E

Y

A Hermitian bilinear form is said to be positiwe definite if

(fIf) 2 0 for all f E V . If (. I .) is a bilinear form, the function ( f l f ) on Y is called the quadratic form associated with (. I .) (see also Exercise 6.6). We note that an inner product on V is a positive-definite bilinear form for which ( f If) = 0 only when f = 0. As an example of a bilinear form on 12(n)we can take (6.8)

(a I /?)A =

a,*A,,b,,

a =

i.k=l

where A is an n x n matrix A = ( A11 : An1

*.*

A1n !). Ann

Anticipating the results we shall obtain in the next chapter (see Chapter 111, Exercise 2.5), we can state that, vice versa, to each bilinear form ( * I .) on P(n) corresponds a unique matrix A in terms of which Throughout this section we consider only complex vector spaces and on them (* I .) is sometimes called a sesquilinear form. All the definitions and theorems can be easily specialized to real vector spaces by limiting the field of scalars to R'.

138

11. Measure Theory and Hilbert Spaces of Functions

( a I p) can be written in the form (6.8). Furthermore, (. I .) is Hermitian if and only if the matrix A is Hermitian. By replacing in the proof of Theorem 2.2 in Chapter I the inner product (f I g) by a positive-definite bilinear form (fI g), we arrive at the following conclusion.

Theorem 6.4. If (. I .) is a positive-definite bilinear form on the vector space V , then (fI g) satisfies the Schwarz-Cauchy inequality

I(f I g)I2G (fI f ) ( g I g) for any two vectors f,g E Y .

Theorem 6.5. If (f I g) is a positive-definite bilinear form on the vector space V , the set

6 = {f: (flf)= 01 is 3 vector subspace of V .

Proof. If f E Yoand a E C1, then (uf 1 u f ) = I u 1 2 ( f

f) = 0, i.e.,

Uf€V0.

Take any two vectors f,g E Yo. From Theorem 6.4we ;et

I(f I g)I2 < (fIfUI g) = 0 and therefore (f I g) = 0. Consequently (f+gIf+g) =(fIf)+(glf)+(fIg)+(gIg)=O,

i.e., f + g E V o .

Q.E.D.

Lemma 6.1, Let Y be a vector space and Vl a vector subspace of V ;then the family Y/Ylof all sets* 3 of the form f € Y ,

J={f+h:hEc},

is also a vector space-called the quotient space of V and -Y,-if vector operations are defined in the following way: uf = { a ( f

+ h): h E K},

f+g"={f+g+h:hE*trlJ

and the zero vector 0 of Y/Vlis chosen to be N

(6.10)

0 = (0

+ h: h

€

^y;}

=

Yl .

* Here, pis called the equivalence class of,f modulo Vl .

the

139

6. Direct Sums and Tensor Products of Hilbert Spaces

PYOO~. It is easy to see that the above operations (6.9) are not dependent on the chosen representative from V of an equivalence class belonging to V/V,. For instance, if f , =f z modulo V, i.e., )

fl = {f,+ h: h E <}= {fi thenf,

+ h: h E K } = fz ,

~ f so, that there is an h, E V, for whichf, + h, = fz. Therefore fl t

g

=fz

+ g (mod<)

because for any h E V, (f1+g)+h=(fz+g)+h',

h'=h--h

1,

where h' E Vl , since h, h, E V, and V, is a linear space. It is a straightforward task to check that the operations (6.9) satisfy all the axioms for vector operations if (6.10) is taken to be the zero vector i n y , . Q.E.D.

Theorem 6.6. If ( - I is a positive-definite bilinear form on the vector space V , and Vois the vector subspace on which the quadratic form ( f If) vanishes (see Theorem 6.5), then a)

(6.11)

(fig') = (fig),

L 2 E w L

f,geV,

is an inner product on the quotient space V / V o . Proof. First, we have to establish that (3I g") is uniquely defined for , i.e., that whenf, = fz(mod Vo) andg, = g, (mod Yo), everyf) g" E V/Vo then (f,I g,) = (fzI g,). To that purpose write (6.12)

(fl 181) = (fl -fz I g1) + (fz I g J = (fl - fz I g1) + (fz I'g1 - gz) + (fz I gz).

Since ( f i g ) is positive definite, we can use the Schwarz-Cauchy inequality (Theorem 6.4) to write

I(f1-f z I gJI2 G (fl -f z I fl - f d g 1 I gl), I ( f 2 I g1 - gz)I2G (fz I f d g 1 - gz I g1 - gz), and then note that the right-hand sides of the above inequalities vanish because (f,- f2 I fl - fz)= (8, - g, 1 g, - gz)= 0; thus, we get from (6.12) that (fl I g1) = ( J Z I gz). It is easy to check that (3I g") is a positive-definite bilinear form on V/Vo. Furthermore, if (f 13) = (fIf) = 0 it follows that f E Y o , i.e., f = Yo, where Yois the zero vector of V/Vo . This establishes the

140

11. Measure Theory and Hilbert Spaces of Functions

theorem, since a positive-definite bilinear form whose quadratic form vanishes only on the zero vector is an inner product. Q.E.D. It is worth noting that in 94 we implicitly used Theorem 6.6 when we constructed the Hilbert spaceL2(SZ,p) from the vector space L(,,(SZ,p).

TENSOR PRODUCTS OF VECTOR SPACES *6.4. ALGEBRAIC Consider now the Cartesian product V, x ..- x Vn of n sets Vl ,..., V n ,which are vector spaces. Introduce in V, x x Vn equivalence classes (see Exercise 6.7) by the identification

<

for all a E C1 and all h'l) x ... x h(")E x x Vn, and denote by (V, x .-. x Vn)the family of all such equivalence classes. We shall use the symbol h ( l ) @ @ h(") to denote the equivalence class containing h(l) x *.. x h("), i.e., h'"

@

. . I

@

h(n)

= {(c,h(l))x

Define in ("y; x scalar a E C1, (6.14)

x (c,h(")):c1 *.. c,

*..

... x

u(h(1)@

*-*

=

1, c1 )...,c,

E

Cl}.

Vn)-an operation of multiplication by a @ h("))= (ah'")

0 0h(fi).

for all h(l)E V, ,..., h(") E Vn. We shall call (6.15) the zero element of x V , ) - ,and denote it by 0. (Vl x Consider now the family of all v-tuples (fl ,...,f v ) of elementsf, ,...,f v from (Vl x .-. x Vn)-, and introduce in this family equivalence classes (see Exercise 6.8) by the following identifications: (6.16)

(fl

)***,f")

= ( f k l ,...,fk,)

141

6. Direct Sums and Tensor Products of Hilbert Spaces

for any permutation fkl

,...,fku

(6.17)

(fl

,"',f"

for any fl ,...,fu E (Vl x (6.18) (hc1) @

@ t(k)

= (h"'

of fl ,...,fv E (Vl x 9

0 ) = (fl

9.-

x Vn)-,

tfu),

x Vn)-, and

0 **.

@ A("), h(1) @

0 *.. @ ( h y + h y ) @ *..

... @ hLk) @

-*-

0 h("),fl ,..,fY)

@ h("),fl ,...,f")

for any fl ,...,fv, h(l) 0 -..@ E (Vl x ... x Vn)-.Denote the family of all such equivalence classes by Vl 0, @, Vn. Introduce in (Vl x x Vn)"operations defined in the following way:

---

(6.19)

+ (g1 ,-,g")

(fl ,-,fu)

4fl

(6.20)

= (fl

,"',fU

9

g1 ,...,gJ,

= E Q='.

,...,A ) = ( 4 1 ,..., afu),

It is a straightforward though tedious procedure to check that (6.19) and (6.20) leave the equivalence classes of Vl @, -..@, Vninvariant, and that they satisfy all the axioms imposed on vector operations. Theorem 6.7. The set Vl 0, 0, Vn of equivalence classes constructed from elements of the Cartesian product Vl x ... x Vnof vector spaces Vl ,..., Vnaccording to the equivalence relations (6.13)(6.14) and (6.16)-(6.18) is a vector space (called the algebraic tensor product of Vl ,...,Vn)under the operations (6.19) and (6.20)-having as a zero vector the equivalence class containing (0), where 0 is the zero element of (Vl x --.x Vn)". Since we can obtain for any f = (fl ,...,fu) E Vl @, ..-0, Vn, by repeated application of (6.19), that

+ *..+ (f"h

(fl ,...,f")= (fl)

we can state another lemma.

Lemma 6.2. The subset (Yl x *.. x Vn)-of "v; @a*.. spans the vector space Vl @, 0,Vn; every f G Vl 0,.*. can be expanded in the form

.-.

f=

(6.21)

@

0 . .

@h

y

+

+ hJ') @

.**

@aVn

0,Va

@At"' ,

hy),..., h!1) E "v; ,..., h y ,...)h y E "r,

.

In the above relation we have simplified the notation by agreeing to write h = (h) for any h E Vl x ..- x Vm.

11. Measure Theory and Hilbert Spaces of Functions

142

Assume now that the bilinear forms ( * I .)l ,..., ( * I are defined on the vector spaces Vl ,..., Vn, respectively. We can define a complex function (f I g) for all f,g E Vl x -.. x Vnin the following way: n

(6.22)

(f") @

... @ f W I g(1)@ ... @ g ( n ) ) = n (p) I g(k)), . k-1

Note

(Vl x

that (6.22) really determines Vn)"x (Vl x * * . x Vn)":

..- x

a

unique

function

on

I g'l' @ g'2' @ *-) I g ( 1 ' ) 1 ( f ( 2 ' I g'2')2 ... = (f"' I g ( l ' ) l ( u f ( z )I g'2'), ... - (f"' 0(Uf'") @ ..* I g'l' @g'2' @ --).

((uf"') Of'2' @

- 1 .

- (uf"'

@, Vnin the form

By Lemma 6.2 we can write any f,g E Vl @, v1

f

=

cf;"@

.**

Of!"',

i=l

(6.23)

c gY' YZ

g

=

@ ..*

Og:"'.

j=1

If we define

c c (fy) @ v1

(6.24)

(f I g )

=

vz

*-*

i=1 5=1

Of?) Igy) @

@ gjn)),

it is easy to check that the above expression does not change when any of the substitutions suggested by (6.16)-(6.18) are performed, i.e., that (6.24) extends the domain of definition of ( * 1 - ) t o (Vl 0, 0, Vn)x (Vl @, 0,Vn).Furthermore, it is very easy to verify that (. I is a bilinear form on Vl @, @, Vn. If the bilinear forms ( * I *)1 ,..., ( - I - ) n are Hermitian, it is evident from (6.22)-(6.24) that ( - I .) will also be Hermitian. I is a positive-definite bilinear form We shall prove now that whenever (. I *)1 ,..., I -)n are positive definite. We need Lemma 6.3. a)

( a

a)

( a

If (fl ,f2,...) is a finite or countably infinite system of vectors in a vector space V , then there is a countable system {el , e2 ,...}, such that (e, I ej) = aij and each vector fi can be expanded in the following way:

Lemma 6.3.

c aifej ni

(6.25)

fi

=

j=1

+fi',

( f i ' Ifi')

zz

0.

143

6. Direct Sums and Tensor Products of Hilbert Spaces

Proof. In the quotient space V/Vo,

6 = {f:(flf) = 0,f E Y C ) the bilinear form (f I g) induces, according to Theorem 6.6, the inner product (312) defined in (6.11). We can apply on {fl,f2 ,...} Theorem 2.4 in Chapter I to deduce the existence of a system {gl ,g2 ,...} for which (Ei I Zj> = 6, and (6.26)

If e, is a chosen vector of 9'- belonging to the equivalence class Ei, (6.26) implies, according to (6.9), fi

+ gi

ni

=

C aijej + hi i=1

hi gi E 6 *

We get (6.25) from the above by noting thatf,'

.

= hi - gi E Yo

Q.E.D.

In order to show that (f If) 2 0 for any given f~ Vl 0,*.. 0, Vn when (f I g), ,..., (f I g)nare positive definite, expand f in the form (6.21). According to Lemma 6.3 we can find for each K = 1,...,n vectors eik),..., ek: in terms of which (6.27)

where (gjk)I g:k))k = 0, and therefore (f'")I g;") Thus, we have

=

0 for all f ( k )E V k.

In particular, if ( - I -)1 ,..., (. I .)n are inner products in Vl ,..., Yn respectively, then we have in (6.27) gjk)= Ok . Consequently, from (6.28) we can deduce that (f If) = 0 if and only iff = 0. In summary, we can state the following.

144

11. Measure Theory and Hilbert Spaces of Functions

are bilinear forms on Vl ,..., Vn, Theorem 6.8. If (. I ,..., (. I respectively, then the complex function (f I g) defined by (6.22)-(6.24) is a bilinear form on Vl 0,- * - 0, V, ; 1 .) is Hermitian, positive definite, or an inner product if all (. I .)l ,..., (. I .), are, respectively, Hermitian, positive definite, or inner products. a),

( a

*6.5. HILBERTTENSOR PRODUCTS OF HILBERT SPACES Definition 6.2. Let X1,..., X , be Hilbert spaces with inner products ( * I .)l ,..., (. I .), respectively. I n the algebraic tensor product Xl 0, @,X% of X1,..., Sndenote by (. I -) the inner product which has the value n

( f I g>

(6.29)

=

(f'k'

Idk')k

k=l

for f and g of the form f

=fa'

@

- * a

Of'"', g

= g"'

0 x ... Og(n);

(6.29) defines a unique inner product by Theorem 6.8. T h e Hilbert

tensor product X I @ @ X , of the Hilbert spaces S1 ,..., Snis the Hilbert space which is the completion of the Euclidean space d with the inner product (6.29). Consider now two Hilbert spaces L2(Ql ,pl)and L2(Q,,p,). Denote by pl x p, the product of the measures p1 and p2 (as defined in Theorem 2.4) on the Bore1 subsets of SZ, x SZ, Iff €L2(SZ1,pl) and g €L2(SZ2,p2), then f(xl) g(x,) represents an element of L2(SZl x SZ, , p1 x p,), which we denote by f g:

.

-

(f

(6.30)

*

g)(x1

Y

x2) =

f(4 g(x2).

Theorem 6.9. The linear mapping (6.31)

n

n

=

akfk k=l

gk

3

=

akfk

'gk

9

k=l

of the algebraic tensor product of Sl = L2(SZl , p l ) and X , = L2(SZ2, p2) into X , = L2(SZl x SZ, ,p1 x p,) can be extended uniquely to a unitary transformation of Xl @ X , onto X 3.

Proof. It is easy to see that (6.31) is a linear mapping of 2P10, X 2 into &',.

145

6. Direct Sums and Tensor Products of Hilbert Spaces

According to Theorem 4.8, the linear set

(where 52, = 52, x 52, , and I, are intervals in the Euclidean space containing the set 52, , i = 1,2) is dense in 8,. Therefore (see Exercise 6.9), Vl 0, V, is dense in X10, if,, and consequently also in 2

1 0x2.

V, into V 3 .Furthermore, for any The mapping (6.31) maps Vl 0, xRi ,xs, E Vi we can write, if we denote by (. I .) the inner product in Xl@*Z, (xR1

8 X R , I x S l 0X S , ) =

=

J

XR1(Xl) XS,(Xl) Ql

J,

44%)J

XR,(XZ) XS,(X2) 4 2 ( X z )

%

X R , ( X l ) X R 2 ( X 2 ) X S , ( X l ) XS,(XZ)

From the above we can deduce that if f

(f I g> = J4fw &) 4%x Pz)(X),

4% x PdX1

~ andf g I-+

9

Xz).

6, then

f,g E "y;

@a

-u;

3

i.e., the mapping (6.30) is an isomorphism of Vl 0V2onto V 3 .Since Vl @ Y2is dense in X102,and Y, is dense in &, , this isomorphism can be extended uniquely (cf. Exercise 6.10) to an isomorphism between X l 0Zz and 2 , . Q.E.D.

Theorem 6.10. The Hilbert tensor product Xl 0 - - * 03?, of separable Hilbert spaces S1 ,..., X , is separable; if {elk):i E Uk}is an orthonormal basis in the X k, then {eit) @ 0e(,nn):il E U, ,..., in E U,} is an orthonormal basis in Z1 @ @ &, . Proof. If X kis separable, then there is a countable orthonormal basis {e(lk):i E Uk}in X k The set

.

T

= {e2)

0

8 e,':': il E u1,..., inE u,)

is also countable. In addition, T is an orthonormal system:

11. Measure Theory and Hilbert Spaces of Functions

146

In order to see that T is a basis in Zl 0 --.0Z, note that the linear manifold &k = (elk),ep), ...) spanned by {elk),ep), ...} is dense in Z, , and therefore cY1 0,*.- 0, 8, , which is contained in the linear space (T) spanned by T, is dense (see Exercise 6.9) in Zl @ -.-@ 2, . Q.E.D. In the next section we shall find useful the following generalization of the preceding theorem.

Theorem 6.1 1. If {e4:i E U} is an orthonormal basis in the separable Hilbert space and {eij :j E B} is for each value of the index i E U an orthonormal basis in the separable Hilbert space Z 2 ,then T = {e, 0e:j :j E B, i E U} is an orthonormal basis in Zl x H 2. Proof. T is an orthonormal system because <eil

0eiljl I elz 0eizjz> = <eil I eiz)l<eiljl I ei,jz)2 = 6.

= aili2<eiljl I eiljZ>2

.a. .

WZ 3122 ’

Letf be any given element of Z1 0H 2.Since {ei @ eik : i E U, K is by Theorem 6.10 a basis in Zl , for any given B > 0 there is a

E

B}

As each T$ = {eij :j E B} is a basis in Z2 we can find finite linear combinations bfkjeijapproximating e i k :

xy~~

Consequently

Thus, (T) is also dense in 2, 0.Z2 , i.e.,

[TI = Zl 0Z2.

Q.E.D.

147

7. Two-Body Problem: Spherically Symmetric Potential

EXERCISES 6.1.

Prove that L2((- co,01) @ Ly[O,

+

‘30))

is isomorphic to L2(Iw1).

6.2. Show that if 8,,..., 8,are Euclidean spaces and k, ,..., k, is any permutation of the indices I ,..., n, then g13 --.@ gn is isomorphic to &kl @ .*- @ 8k, . Prove the same result for the countably infinite case, when instead of k, ,..., k, one takes an infinite sequence k, ,k, ,... in which each positive integer appears once and only once.

6.3. and

Show that the Euclidean spaces 8,@ (g2@ G), g1@ @ 8’) @ g3are isomorphic to one another.

6.4. Show that 8k’) @ (@= :, isomorphic Euclidean spaces.

8;) and

5 @4

(8;08:) are

+-

Show that if the complex function (f I g) satisfies ( f g h) I(fld (f I h) and (fI ag) = 4fI g), (f I g) = (g If)*.then (f i g> is a Hermitian bilinear form. 6.5.

+

~

6.6. Prove that a bilinear form ( . 1 .) is Hermitian if and only if its quadratic form Q(f ) is real for all f E Y . 6.7. Show that if we state that “h(l)x x -gC1) x * * . x gcn) if and only if h(i) = cg(i),h(i) = (l/c) g ( j ) , for some c E C1 and for a certain value of i and a certain value of j ( i , j = 1,..., n),” then the relation so defined is an equivalence relation.

-

6.8. Prove that the “equality” relations in (6.16), (6.17), and (6.18) are equivalence relations in 8,x .-.x gn

.

6.9. Prove that if S , and S, are dense vector subspaces of &, and X , , respectively, then S , 0, S, is dense in X I 0, H,, and therefore also in 2, 0$Y, . 6.10. Suppose that V, and V, are dense vector subspaces of the Hilbert spaces X , and 2,, respectively. Prove that a linear mapping f Hfof onto Y, , which preserves the inner product, can be extended onto iV2. uniquely to a unitary transformation of &‘,

<

7. The Two-Body Bound-State Problem

with a Spherically Symmetric Potential 7.1. Two PARTICLES INTERACTING VIA A SPHERICALLY SYMMETRIC POTENTIAL When a system consists of only two particles of different kinds, the states are represented by wave functions $(rl , r, , t), + ( t ) E L ~ ( W in ),

11. Measure Theory and Hilbert Spaces of Functions

148

the six configuration space variables r, and r 2 . The time-independent Schroedinger equation is in this case

where, on physical grounds, only potentials depending on r, - r2 exclusively are considered" (see $5). After going over to new coordinates, specified by the relative position vector r and the center of mass position vector R,

r =r2-r,,

we get as a special case of (5.18) H,

=

h2

--A 2M,

h2

-A

R-2m,

+ V(r),

where M , and m, are, respectively, the total mass and the reduced mass of the system: Mo = m, m 2 , (7.4)

+

m, =

m1m2

m1+ m2

*

If we seek solutions of (7.1) of the form

we get as a special case of (5.20) and (5.21)

* In phenomenological treatments of nuclear forces, velocity-dependent potentials are also considered. In that case the more general requirement of translational invariance is always imposed.

149

7. Two-Body Problem: Spherically Symmetric Potential

The equation (7.5) is uninteresting to us, because it is related, as we already know from $5, only to the motion of the center of mass of the two particles. Concerning (7.6), in practice we are most often faced with a spherically symmetric or central potential, i.e., a potential V(r)

=

v0(r), r

=

IrI

=

dx2

+ yz +

x2,

depending only on the distance r of the two particles. This considerably simplifies the problem. We shall treat this problem in the rest of this section in a detailed manner, because it provides a fair sample of the main methods used in solving the eigenvalue problem of the time-independent Schroedinger equation. I t should be noted that in attacking this problem we will find that it is necessary to employ most of the main theorems obtained thus far.

7.2. THEEQUATION OF MOTION IN SPHERICAL COORDINATES Let us write (7.6) in the form (7.7)

Wr)

+ (2m0/fi2)(E

W) = 0,

- vo(y))

where we have simplified the notation by leaving out the subscript b in the internal bound-energy eigenvalue Eb . It is natural to try to solve (7.7) by going over to the spherical coordinates r, 8, +, (7.8)

x = r cos 4 sin 8,

y

= r sin

4 sin 8,

x

=r

cos 8,

in terms of which the Laplacian A becomes (see Exercise 7.1) equal to

almost everywhere, i.e., with the exception of the z axis, which is a set of zero Lebesgue measure. If we express the functionf(x, y , z ) , f ~ L ~ ( [ Win3 terms ), of the spherical coordinates r, 8,4, (7.9)

f&, 8, 4) =f ( r sin 8 cos 4, r sin 0 sin 4, r cos O),

we note that j p ( r , 8,+) is defined on the closed subset 52, of R3, (7.10)

.n, = {(r, e,4): o G 7 < +OO, o G e <

T,

o G9 G 24,

150

11. Measure Theory and Hilbert Spaces of Functions

and satisfies the periodicity condition (7.11) It

fPP,

is well known

that

690) = fP@, 8224.

the

Jacobian

of

the transformation

(x, Y , 4 + (I 4,4) is ](Y,

8, +) = r2 sin 8.

Thus, if we introduce the Lebesgue-Stieltjes measure (7.12)

pp( B ) =

I

r2 sin 0 dr dB d+,

B

on the Bore1 subsets of Q,

B E a&,

, then by Theorem 4.9 we have

i.e., f, EL~(Q, ,p,). It is easy to see that, in fact, f+f, is an isometric linear mapping of L2(Q)into L2(Qp, p,). Furthermore, the range of this mapping is L2(Q, , p,), because if fP(r, 8,4) E L ( ~ ) (, pq) Q ~ satisfies the periodicity conditions (7.1 l), then there is anfEL2(Q) which is mapped into fp ; but since the sets ((r, 6,#): 0 r < co, 6 = T,0 4 2~} and ((1, 8,+): 0 r < co,0 8 T , = 2 ~ obviously } are of pp measure zero, every equivalence classf, E L2(Qp, p,) contains a function which satisfies (7.10). Thus, we can state Lemma 7.1.

<

+

< + < < +

< <

e,

Lemma 7.1. The Hilbert spaces L2(R3)and L2(Qp, ) are isomorphic, and the mapping j-f, defined by (7.9) is a unitary transformation of L2(R3)onto L2(Qp, pup). We can now try to solve (7.7) by the method of separation of variables. We look for solutions of (7.7) of the form (7.13)

'U(4 = R(r) W,$).

Thus, we arrive at the system of equations

We note that (7.14) is an eigenvalue equation for A, in which the potential does not enter. Therefore, we are able to investigate (7.14)

7. Two-Body Problem: Spherically Symmetric Potential

151

without knowing anything more specific about our two-body problem, except for the assumption that the potential is spherically symmetric, i.e., it depends only on r. On the other hand, the equation (7.15) for the radialpart R(r) of the wave function contains Vo(r),and its solution can be approached only when the potential is specified. We shall investigate at the end of this section the solution of (7.15) for the Coulomb potential Vo(r) = c/r, where c is a constant. Consider the sets [0, f

~ C)R1, = [O, T] x [O, 27r] c

Qr = Qa

R2,

and the Lebesgue-Stieltjes measures pr on airand paon gisdefined by pr(B)

(7.16)

=

pS(B)=

J

r2dr,

BE%r,

J

sin t9 dt9 d+,

B E Bis.

B

B

If Y(r), Y eL2(R3), is square integrable in the Riemann sense and if it can be written in the form (7.13), then we have

I R(r) Y(8,+)I2 I R(r)I2r2dr

= 0

=

J Qr

1"

r2 sin 0 dr dt9 d+

I Y(0,+)I2 sin 0 dt9 d$

0

0

I R ( Y ) dpr(r) I~ J- I ~ ( 0+V , d ~ s ( e41, , ?9

which implies that R E L ~ ( Q, p, r ) and Y E L ~ ( Q,, pa). More generally, by using Theorems 2.4 and 6.9 we can infer the following result.

a$,p, ) is the Cartesian Lemma 7.2. T h e measure space (a,, product of the measure spaces (Q, , air, pr) and (Q, , a&, p,), and therefore the Hilbert space L2(QP,pp) is isomorphic to the tensor product L2(Q,,p r ) @L2(Qa,pa)of the Hilbert spaces L2(Q,,pr) and L2(Q, 9 Pa).

7.3. SPHERICAL HARMONICS ON

THE

UNIT SPHERE

We note that since 8 and 4, (6,4)E Q, , can be considered to provide a system of orthogonal coordinates on the unit sphere in R3, we can always visualize a function Y(O,$)for which (7.17)

y(e,01

=

y(e,2 4

1 52

11. Measure Theory and Hilbert Spaces of Functions

as being defined on the unit sphere. A special class of function of the above type are the spherical harmonics

where Py(u) are the associated Legendre functions of the mth order," which can be given by Rodrigues' formula:

The complete solution of the eigenvalue problem for (7.14) results in the following important theorem. The spherical harmonics (7.18) are eigenfunctions

Theorem 7.1. of (7.14)

and they determine an orthonormal basis Z = { Y ~ , l = O 0 , 1 ,,..., 2 m=--I,-l+l, (7.20)

1: 1""

( Y y ( 0 ,+))* Yy'(O,+)sin 8 d0 d+

..., + l } ,

= 6,,~6,,~ ,

0

in the Hilbert space L2(sZ,,p,). We shall carry out the proof of Theorem 7.1 in a few stages, thus being able to state and prove a few results which are of independent interest. In trying to solve (7.14) by the method of separation of variables, we look for solutions of the form (7.21)

Y(O,+)= @(e)@(C).

Thus, we arrive at the following system of independent differential equations: (7.22) (7.23)

dO( 0)

( A sin2 0

+ p) O(0) = 0.

The associated Legendre functions Pzo(u)of the zero order are called the Legendre polynomials.

153

7. Two-Body Problem: Spherically Symmetric Potential

By applying again Theorems 2.4 and 6.9 we get Lemma 7.3.

Lemma 7.3. The Hilbert space L 2 ( s 2 , , p , ) is isomorphic to Po) 0L2([0,2711, where

W O ,771,

p o ( ~= )

IB

sin e de,

BE

.

7.4. THECOMPLETENESS OF TRIGONOMETRIC FUNCTIONS When the periodicity condition (7.17) is imposed on (7.21), it becomes

@(O)

(7.24)

= @(277).

The only linearly independent solutions of (7.22) satisfying (7.24) are m

eim4,

= 0, +1,

+2,...,

and they do belong to L(2)([0, 271).

Theorem 7.2. T h e family {Gi, : m = 0, i - 1 , f2, ...) of elements of L2([0,271) corresponding to the functions

am(+) = (1/1&)

eim*

is an orthonormal basis in L2([0, 271). The fact that {@, , @*l,...} is an orthonormal system is easy to establish: (Om I

1 am,> = 27c

I

exp[i(m’ - m)+]d+ = S,,

.

0

In order to prove its completeness we need Theorem 7.3.

Theorem 7.3. Iff(+),

+

(7.25)

E

[0,27r], is a continuous periodic function

f ( 0 ) =f(24,

with a piecewise continuous first derivative, then

and the convergence of ul(+), u2(+),...,

tof(+) is uniform in [0,27r].

154

11. Measure Theory and Hilbert Spaces of Functions

Proof. Since f’EL2([0,2~11,f’(4)= df (4)/&, and {@, , ,...} is an orthonormal system, we have by Bessel’s inequality (see Chapter I, Lemma 4.1), (7.27)

On the other hand, integrating by parts we get

Therefore, by (7.27) (7.28)

To prove the uniform convergence of +n

on(+)

=

1

cm@m(d),

cm

= <@m

If>,

m=-n

use the Schwarz-Cauchy inequality in Z2(2v) to obtain

where the last inequality follows from (7.28). Since the series converges, for any t: > 0 we can find an N(c) such that

OD

l/m2

7. Two-Body Problem: Spherically Symmetric Potential

155

for all n > N(E),and therefore the uniform convergence of un(x) follows. In order to show that ul(+), 02(+)... , converges tof(+), note that 1 " exp(im#>f * f ( ~exp( ) - z k ) dT 2m m=-n 0

un(+) = -

=

-!r f ((~ f )exp[im(+ 277

T)])

dT

m=-n

0

where in the last integral we have introduced the new variable w = r - 4. We extend the domain of definition off(+) to the entire real line:

f(+

+ 2774 =f(+),

=

51, 52,... *

The function so extended is periodic (with a period 2rr), everywhere continuous, due to (7.25), and has a piecewise continuous derivative. We can now write

Since we have

j2n sin(n +

-l-

2~

&)w

sin(w/2)

1

dw = -

T

o

(4 + cos w +

we deduce from the above two formulas that

* a -

+ cos nu) dw = 1,

156

11. Measure Theory and Hilbert Spaces of Functions

where the function

is piecewise continuous, since

while gz(w)

=f(+

+

w)

-fb)

is obviously continuous. Thus, both functions g l ( w ) and g 2 ( w ) are integrable and we have (7.30)

On(+)

-f(+) =

1

1

2n

[ein%2(w)

-

0

ig1kJ))

+ e-“””(gz(w)+ ig,(w))ldw

Since on the right-hand side of (7.30) are the Fourier coefficients of [0, 27r]),from Bessel’s inequality we can infer functions belonging to L2( that they converge to zero when n ---t 00. Q.E.D.

+

It is easy to finish the proof of Theorem 7.2 with the help of Theorem 7.3. I f f is an arbitrarily chosen element of L2([0,2nl) and E > 0 is any which can given number, then by Theorem 5.6 there is a g, E %‘;(rWl), be represented by an infinitely many times differentiable function gl(x), x E R1, such that

llfl - g1 I l l 2

+m

=

J

=

/i,(X)*

-m

I fl(4 - g,(x)I2 dx < 3/99

where f1(X)

O<x<277 x 4 [O,2771.

If we introduce the function

which is infinitely many times differentiable in [0, 2771, we have -02

157

7. Two-BodyProblem: Spherically Symmetric Potential

We can approximate g(+) in the mean as closely as desired by a continuous and piecewise differentiable function h(+), q5 E [0,21~1,which is periodic, i.e., h(0) = 4277). For instance

is such a function, for which

if +,, is chosen sufficiently small. Since h(+) satisfies the conditions of Theorem 7.3, for sufficiently large n we shall have

I h(4) -

i

cm

E/d%,

cmQm(+)l<

=
I

m=-n

and consequently

By combining (7.31), (7.32), and (7.33) we deduce that

1 < f + 11 If min -

cmam

11

- g 11

g - 11

+ I/

-

Cmam m--73

1<

€3

which establishes Theorem 7.2.

7.5. THECOMPLETENESS OF LEGENDRE POLYNOMIALS We can use Theorem 7.2 to prove the Weierstrass approximation theorem, which will be used later.

Theorem 7.4 ( Weierstrass). Any complex function f ( x ) , x E I, which is continuous in a closed finite interval I = [a, b] can be uniformly approximated in I by a polynomial P(x) in x; i.e., for any given E > 0 there is a polynomial P(x) with complex coefficients, such that (7.34)

I f ( x ) - P(x)l

< €,

a

< x < b.

Proof. Consider the case when 0 < a < b 0 = x0 < x1 = u

< x2 <

< 21~.Choose a partition

< x,,-~ = b < x,,

= 27~

158

11. Measure Theory and Hilbert Spaces of Functions

of [0, 2n]. T h e function x,,

=0

< x < x1 = a

represents a polygon inscribed in the graph o f f ( . ) by joining the points (xl ,f(q)), ..., ( x n - l , f ( ~ , - ~ with ) ) straight lines, and then joining the ends to the real axis at x,, = 0 and x , = 277. Such functions g(x) can approximate uniformly and arbitrarily well the continuous functionf(x) (recall that a continuous function is uniformly continuous in a closed finite interval). Since the function g(x) in (7.35)satisfies the conditions of Theorem 7.3,it can be approximated uniformly by a trigonometric polynomial n

h(x)

=

C

cm eim s.

m=-n

T h e above trigonometric polynomial can be, in its turn, approximated uniformly and arbitrarily well by polynomials, since due to Taylor’s formula

each einlz can be approximated uniformly in [0,2n]by ordinary polynomials. Thus, the theorem is proved for the case when 0 < a < b < 27r. ‘The general case of [a, b] can be reduced to the above case by mapping linearly [a, b] into [0, 2n3, e.g., by using the mapping x-a 7r xw7r-

b-a

+z,

and by noting that a linear mapping maps polynomials into polynomials. Q.E.D. We note that if (7.34)is true, then

Consequently, from Theorems 5.6 aild 7.4 we can easily infer the following lemma.

159

7. Two-Body Problem: Spherically Symmetric Potential

Lemma 7.4. The vector subspace ( po ,p , ,...) of L 2 ( [ a ,b]), spanned by the elements of L2([a, b]) represented by the polynomials p l ( x ) = x,...,

po(x) = 1,

p,(x)

= x"

,...,

is everywhere dense in L 2 ( [ a ,b]). We can turn now to solving the differential equation (7.23), which in the new variable (7.36) = -cos e, -I G < 1, becomes (with p, = -m2) the associated Legendre equation d (1 - U S ) -&[(l - 242)

9 1

- [h(l

- 242)

- m7 w(u) = 0

for the new function w(u) = @(arccos(-u)).

(7.37)

The above associated Legendre equation (called simply the Legendre equation when m = 0) can be written in the more customary form (7.38)

(1 - u2) w"(u) - 2u w'(u) -

[h - "If] w(u) 1u2

= 0.

We can state the following direct consequence of Theorem 4.9.

Lemma 7.5. T h e mapping 0 -+ w , defined by (7.37), is a unitary transformation of L2([0,771, pB)onto L2([- 1, 13).

+

Theorem 7.5.

T h e Legendre polynomials

1 dz(U2- 1)' 2'1! duz ' satisfy the Legendre equation with h =

(7.39)

(7.40)

P'(U)

1 = 0 , 1 , 2 ,...,

=-

(1 - 242) P'"(U) - 2UP,'(U)

-Z(Z

+ 1):

+ Z(1 + 1) Pz(u)= 0.

The set { P o ,P, ,...} CL2([-1, 11) is an orthogonal basis inL2([-1, with +1 2 (7.41) S,,, ( P r I P z , ) = -1 Pz(u)Pz,(u)du = 21+1 *

+1]),

Proof. T o derive the result that PI(.) satisfies (7.40), we start with the identity (1 - u2)(d/du)(u2- 1)' - 21u(u2 - 1)' = 0.

160

11. Measure Theory and Hilbert Spaces of Functions

By taking, with the help of the Leibniz rule, the 1 above equation we arrive at (7.40). By employing the binomial formula we get

+ 1 derivative of the

from which we derive, after carrying out all the differentiations,

(7.42)

"=

;:1

1 even 1 odd.

The above formula shows that P, is a polynomial of degree 1. Integrating by parts 1 times consecutively and using the fact that dk -(U2-1)~u=*l=0' duk we arrive at the relation

From the above we get (P, I p,) ( P I I P l , ) = 0 for I' < I, i.e.,

J

K=O,l,

=0

for m

=

..., 1 - 1 ,

1,..., 1 - 1. Consequently,

-1

which proves that the system P o , P, ,... consists of mutually orthogonal elements of L2([-1, +1]). It is easy to check (see Exercise 7.2) that (7.41) is true for I = l', i.e., the Legendre polynomials are mutually orthogonal but they are not normalized. From (7.39) it is obvious that po(u) = 1, pl(u) = u,...,pm(u ) = urn can be expressed in terms of Po(u),..., P,(u). Consequently the vector subspace (Po ,p , ,...) of L2([- 1, 11) spanned by p , ,p , ,... is identical

+

161

7. Two-Body Problem: Spherically Symmetric Potential

to the vector subspace ( P o ,P, ,...) spanned by the elements of L2([- 1, + 11) represented by Legendre polynomials. Since according to Lemma 7.4 ( p , , p , ,...) is dense in L2([- 1, l]), we have

+

[ P o ,Pl

,...I

Theorem 7.6.

=

[ p o, P I

,...]

= L2([-l, + 1 ] ) .

Q.E.D.

T h e associated Legendre functions

defined for all positive integral values of m,are solutions of the associated Legendre equation, (7.44) ( 1 - 242)

d2PT(u) dPm(u) m2 - 224 (I(Z J1)--i du2 du m = 1 , 2 , 3,....

+

~

+

For a fixed integer value of m the set { P I , P:+, gonal basis in L2([ - 1, l]), with

+

(7.45) Proof.

( P y I PF)

2

+1

=

Pl"(u) = 0,

, P;+2 ,...} is an ortho(I

+ m)!

J-1 P ~ ( uP) ~ ( udu) = 21+ 1 ( I - m)! &, .

By substituting in the left-hand side of (7.44)

drn -P&) durn we get by repeated use of the Leibniz rule PE"(u)= (1 - u

y 2

Thus, we deduce that (7.44) is true from the fact that (7.40) is true. It can be shown by direct integration that (7.45) is true, i.e., {P,", P;+, ,...} constitutes an orthogonal system in L2([-l, +1]). In order to prove that this system constitutes a basis in L2([- 1, l]), note that for any given function f ( x ) , f E L2([- 1, $-l]), we can write

+

(7.46)

(PT

If)

1

+1

=

-1

drn (1 - ~ 2 ) m / ~ f (u ) Pl(u)du.

durn

From (7.42) we can easily see that po(u)= 1, pl(u) = u,..., p,(u) = u" can be expressed in terms of

11. Measure Theory and Hilbert Spaces of Functions

162

i.e., the vector spaces spanned by these two systems are identical:

(Po

dmPm dmPm,,

9

Pl

=

7-1

(F du",...). 9

Thus, we deduce from (7.46) that the relations (7.47)

(Prlf)

= 0,

1 = m,m

+ 1, ...

U ) have (g I h) = 0 for all imply that for h(u) = (1 - U ~ ) " / ~ ~ ( we g E ( p , ,p , ,...). Since according to Lemma 7.4, (p, ,p, ,...) is dense in L2([- 1, 13) and since obviously h E L2([- 1, 11) when f E L2([- 1, l]), there is a sequence g, ,g, ,... E ( P o ,p , ,...) which converges to h. Hence, if (7.47) is true, we shall have

+

+

(h I h)

=

lim(gk I h )

k+m

+

= 0.

The above relation implies that h = 0, i.e., h(u) = 0 almost everywhere. h(u) = 0 almost everywhere, because Consequently,f(u) = (1 (1 - u2)-m/2is finite and different from zero everywhere in (- 1, l), while the Lebesgue measure of { - 1, l} is zero. Thus, we conclude that (7.47) implies that f = 0; consequently, {PE , P;+, ,...} is according to Theorem 4.6(a) in Chapter I an orthogonal basis in L2([-1, +1]). Q.E.D.

+

+

7.6.

OF THE SPHERICAL HARMONICS COMPLETENESS

Now we are in the position to carry through to the end the proof of Theorem 7.1. Due to Lemma 7.5 we can deduce from Theorems 7.5 and 7.6 that for each fixed nonnegative integer m the functions

determine an orthonormal basis in L2([0,7r], pO).By applying Theorem 6.11 to the orthonormal bases {arn: m = 0, & l ,...} (see Theorem 7.2) 1,...}, m = 0, f l , ..., we deduce that the and {@\"I, 1 = 1 m I, I m I system (07 @ Om, 1 = 0, 1, 2,..., m = -1. --I 1 ,..., + 1 }

+

+

is an orthonormal basis in L2([0,771, pO) @L2([0,2~1).From this result and Lemma 7.3 (see also Theorem 4.8 in Chapter I) we can finally conclude that the spherical harmonics determine an orthonormal system in ~ 2 ( Q , pe). We can easily derive from Theorem 7.1 the following result. 9

163

7. Two-Body Problem: Spherically Symmetric Potential

Theorem 7.7,

T h e set

(0,-2, -6

,..., -l(Z

+ 1),...}

contains all the eigenvalues X of the equation (7.14) belonging to eigenfunctions Y(0,+), (0,+) E In, , having continuous second derivatives in SZ, , and satisfying the following periodicity conditions:

Denote by %?;(In,) the set of all elements of L2(In,,pa) which are represented by such functions Y(0,+). Any eigenfunction Y(0,+), Y E %$(Ins), corresponding to the eigenvalue X = -Z(Z 1) is equal, almost everywhere, to a linear combination of the spherical harmonics

+

Y l Z ( 4b),..., y:z(e,

4).

Proof. I t is easy to check explicitly by integrating by parts that the differential operator" form (7.48)

-

defines a Hermitian operator when applied to twice continuously differentiablefunctions Y(0,+), where Y E %?#2,), Y,*P, 4 ) ( L 2 Y N , 4) dcLs(4 4)

Consequently, we can infer, by the procedure used in proving Theorem 5.3, that any two eigenfunctions of -L2 corresponding to different eigenvalues are orthogonal. If Y(,,(O,+) is an eigenfunction of (7.14) (-L'y(~))(e,

4) = Xfi2Y(de,41,

YQ)EL'&',

9

c~d,

we can expand it, due to Theorem 7.1, in terms of the spherical harmonics m

1

(7.49) * A s we shall see in Section 7.8 of Chapter IV, the operator square of the angular momentum L of the system.

La corresponds to the

164

11. Measure Theory and Hilbert Spaces of Functions

However, according to the above-mentioned property of eigenfunctions corresponding to different eigenvalues

+

# -Z(Z 1). (YZ" I Y(l)>= 0, Consequently, Y(*)can be a nontrivial eigenfunction only if h is equal to -l(Z + 1) for some Z = 0, 1,... . I n that case we are left in (7.49) with the finite vector sum +Z

Y[-t(2+1)1 =

C

cmyZm,

cm

=
Y[-t(z+~])*

m=--I

From the definition of equality of functions in L2 spaces we get 1 YL-l(l+l)l(O, +) = Ern=-[c,Yy(O, +) almost everywhere in SZ, . Q.E.D. This completes the systematic study of the angular dependence of internal energy eigenfunctions Y(r) of two-body quantum mechanical systems interacting via central potentials. We turn now to the study of the complete solution of the eigenvalue problem of Equation (7.7) for the special case of Coulomb (attractive) potentials Ir,(Y)

=

c

-C/Y,

7.7. THETWO-BODYPROBLEM WITH

A

> 0. COULOMB POTENTIAL

When we are dealing with the hydrogen atom, our two-body system consists of an electron carrying one elementary unit of negative charge -e, and a nucleus carrying the positive charge +e. The constant c is in this case equal to e2. We shall denote the positive constant c also in the genera1 case by e2. Since we know that the only possible values of X in (7.14) are )I = -l(Z l), Z = 0,1, 2, ..., the equation (7.15) for the radial part of the wave function assumes the form

+

(7.50)

d2Rz(r)

2 ~R,(Y)

2m0E

dr2 + F T + [ T - p

Z(l

where we introduce the subscript 1 in Rl(r)to indicate that the solutions of (7.50) depend implicitly on the parameter 1. If we look for negative energy solutions, E < 0, we can introduce the new variable (7.51)

and in (7.50) make the substitution (7.52)

W )= PZ exp (- P/2)VdP).

7. Two-Body Problem: Spherically Symmetric Potential

165

After some simple algebra we find that v,(p) has to satisfy the conjuent hypergeometric equation

+ (21 + 2

P -d2vz(p)

(7.53)

dP2

-p

dVdP) - (1 )dP

+ 1 - n ) vz(p) = 0,

We shall attempt to find solutions of the above equation which can be written as a series of the following form:

If we insert the above series in (7.53) and assume that we can differentiate term by term, we get an expansion which has to vanish identically, and therefore the coefficients of this expansion have to be zero. By equating these coefficients to zero we get the relations (7.55)

Y(U

+ 21 + 1) a, = 0 ,

which can be looked upon as recursive relations for determining a , , a2 ,... . We can satisfy the indicial equation (7.55) by choosing v = 0. By solving (7.56), starting with a , = 1, we get as a solution of (7.53) a conjuent hypergeometric function given by the Kummer's series (7.57) F(Z

+1

-

n I21

+ 2 I p)

=

1

Z-n+l).-(Z-nfk)pk + c ((21 + 2 ) ... (21 + k + 1 ) k? k=l

*

It can be verified (see Exercise 7.3) that the above power series has an infinite radius of convergence, and since the nth derivative of a function represented by a Taylor series can be obtained within the radius of convergence by differentiating the series term by term n times, (7.57) is indeed one of the solutions of (7.53). The equation (7.53) is an ordinary linear differential equation of the second order; consequently it has only two linearly independent solutions. A second solution of (7.53), which is linearly independent from (7.56), can be obtained by taking the solution v = -21 - 1 of the indicial equation (7.55). If 1 # 0, 4,1, Q, ..., we can derive from (7.56) the coefficients of the series (7.54) for this second solution. We note that this second solution behaves like wp-2z-1when p + 0.

166

11. Measure Theory and Hilbert Spaces of Functions

Unfortunately, in our case 1 = 0, 1, 2, ..., and (7.56) does not have any solutions because it cannot be satisfied for k = -v = 21 1. However, a second linearly independent solution of (7.53) can still be obtained (see Bateman [1953, Vol. I, Section 6.7.1, p. 261, Eq. (13)]), and its behavior for small p is again ~ p - ~ ~ Consequently, -l. R,(r) for this second solution [see (7.52)] behaves like p-l--l for small p. Thus, for 1 = 1,2,... the corresponding function (7.13) is not integrable in the pp measure (7.12) because of its behavior at p = 0, and therefore these solutions are ruled out. For 1 = 0 this solution provides a squareintegrable function Y(r), but it is still ruled out on the following grounds: if a wave function #,(R)Y(r), with Y(r) behaving like l / r for r -+ 0, would be included in the family of functions on which we apply the Schroedinger operator form H, [see (7.3)-(7.6)], then Hs would not define a Hermitian operator (see Exercise 7.4)-a requirement which constitutes one of the postulates* of quantum mechanics. We return to the solution(7.57) of (7.53).If 1 1 - n 4 (0, - 1, -2,...}, then all the coefficients in the power series (7.57) are different from zero. Therefore, when Y + + w , i.e., for p + +m, the function F(1 1 - n) I 21 2 I p ) in (7.57) obviously diverges faster than any finite power of p, and from the similarity of the Kummer's series to the exponential series for e p , one can suspect that F(1 + 1 - n I 21 2 1 p) diverges about as fast as e p . In fact it can be shown (see Bateman [1953, Vol. 1, Section 6.13.1, p. 2781) that in this case the function F(1 1 - n I 21 2 I p) behaves likep-l-l-%Pfor largep. Consequently

+

+

+

+

+

+

+

has the asymptotic behaviour

which indicates that R,$ L2(Qn, ,pr), i.e., the corresponding function (7.13) is not square integrable in that case. Thus, we are left only with the possibility that 1 1 - n E (0, - 1,..., -n', ...}. When

+

(7.58)

l+l-n=-n',

n'=0,1,2

)...,

all the coefficients in the series (7.57) from the n'th on vanish, and the

* See Chapter IV, $1. We shall analyze this problem more completely in $7 of that chapter.

167

7. Two-Body Problem: Spherically Symmetric Potential

series reduces to a polynomial" of the n'th degree. I n this case the radial function &(y)

= pz exp( - p/2)F(I

(7.59)

2me2 P

=

T

y

n

'

+1-

=I

1 21

+ 2 I p),

+ 1, I + 2,...,

is square integrable in the measure pr . Furthermore, for each fixed I , 1 = 0, 1, 2,..., the set { R n l ,n = 1 I , 1 2,...} is an orthogonal system in L2(Q,9 PA,

+

+

Consequently, the corresponding wave functions

n = 1, 2,...,

I

= 0,

1, 2,..., n

-

m

1,

=

-I, -I

+ I,...,

+I,

constitute an orthonormal system in L2(R3);Yalm(r)is an eigenfunction of the internal energy Schroedinger operator (7.61)

and has as an eigenvalue (7.62)

En =

-

m,e4 1 --

2fi2 n2 '

n

=

1, 2,...,

as can be seen from the definition of n in (7.53). It is intuitively obvious from the above procedure that the above values are the only eigenvalues of Hi corresponding to square-integrable functions-as long as we restrict the domain of definition of Hi by requiring that (7.61) defines a Hermitian linear operator on LZ(R3). A rigorous justification of this statement requires some familiarity with the theory of linear operators on Hilbert spaces, and will be discussed in Section 7.7 of Chapter IV.

* Equal to n'! L:?+'(p),whereL:f+'(p) is an associated Laguerre polynomial,

andl,(p) is a Laguerre polynomial (see Exercise 7.6).

168

11. Measure Theory and Hilbert Spaces of Functions

Thus, we have arrived at the conclusion that the internal energy discrete spectrum of the hydrogen atom is m,e4

s.=i-~,--

m0e4 *fi2

?.*.Y

2n2fi2 _m,e4

5.a.

1-

I n spectroscopy it is customary to denote the eigenfunctions corresponding to the above energy levels, in the above order, by the letters s, p , d, f , g ,... . T h e numbers n, 1, and m, are called, respectively, the principal, the azimuthal, and the magnetic quantum numbers. When an eigenstate of the hydrogen atom is given by writing one of the above letters s, p , d,... preceded by a natural number n, then n denotes the principal quantum number. Thus, 3p refers to any of the eigenstates of (7.61) with energy E , , [obtained by taking n = 3 in (7.62)], which is also an eigenfunction of the angular momentum operator L2 defined essentially by (7.48), with an eigenvalue l(1 l), where 1 = 1. It is worth mentioning that the above terminology has become standard in atomic physics, and is used, with certain modifications, even in nuclear physics. I t is applied not only to the hydrogen atom, but also to other ‘atoms. This application to more complicated atoms with more than one electron in their electronic “shell” is made possible by an approximate model of such an atom. According to this model the complicated many-body interactions between each particular electron and the rest of the electrons are neglected, and each electron is envisaged to move in a Coulomb potential generated by the nucleus and all the rest of the electrons which are closer to the nucleus than the considered electron. This model, though too rough for quantitative analysis, provides an adequate qualitative picture allowing a simple classification of the electron “orbits” (i.e., eigenstates of the Schroedinger operator). According to Pauli’s exclusion principle,” there can be in this model (21 1) electrons in states Yn,, with a principal number at most 2 n. All the electrons in such states are said to belong to the same “electronic shell,” and when there are precisely 2 (21 1) electrons in a shell, it is said that the shell has been “filled out.” I n such a “shell model” of an atom, the ground state? of the atom has the property that no electrons can be found in the nth shell if the ( n - 1)st shell is not filled out.

+

x:i1 +

+

* Pauli’s exclusion principle states that in an atom there are at most two electrons in a state specified by given n, I , and m. t The ground state of a quantum mechanical system is the state in which the internal energy of the system is at its lowest possible value; e.g., the ground state of the hydrogen atom corresponds to the principal quantum number n = 1.

169

7. Two-Body Problem: Spherically Symmetric Potential

The above model leads in a natural way to the periodic table of chemical elements if it is postulated that the number of electrons in the electrically neutral atoms of a chemical element is equal to the atomic number (in the table) of that element. Many of the regularities in the chemical properties of the elements, which were first observed by Mendeleev in the 19th century, can be deduced from the above picture of the ground state of an atom. EXERCISES 7.1. Check that for any twice-differentiable function f (x, y, z ) the identity

1 +1sin20 a p a2

f ( y

sin e cos 4, r sin e sin 4,

y

cos e)

is satisfied, except for the points where 0 = 0 and 0 r = x e , , -a < z < +a. 7.2.

= T,

i.e., for

Show that

by integrating by parts 1 times.

7.3. Prove that Kummer’s series (7.57) converges for every p E C1. 7.4. Denote by 93 the subset of all elements ~EL~(SZ, , pp) [see (7.10) and (7.12)] which can be represented by functionsf(r, 0,$) which

are twice continuously differentiable everywhere in SZ, , except maybe for Y = 0, and such that Af eL2(SZp,pp), where

Show that 9 is a linear subspace of L2(SZp,pp) and that A is a linear operator (see Definition 1.1 in Chapter 111) mapping 9 into L2(SZ, , p,).

11. Measuere Theory and Hilbert Spaces of Functions

170

Prove that this linear operator is not Hermitian, i.e.,

L

f*(y,

6, $ ) ( A m ,694)dPp(Y, 4 4

z

(Af>*(y,6, 4) d . 9 8, +) dPp(Y, 4 4) 0,

for some f,g E 9. 7.5. Find the internal energy spectrum and the corresponding eigenfunctions of a two-particle system, moving in three dimensions and interacting by means of a central square-well potential V(Y) =

<

for 0 Y < L for L < r < +co

Iio> o

by solving the radial equation (7.15) with the above potential. Compare the present solutions with the one-dimensional case treated in $5 of Chapter I. Note. This problem is discussed by Messiah [1962, Chapter IX, Section 11, pp. 359-3611. 7.6.

Prove by using the Leibniz rule for the nth derivative of

pn e-p that the Laguerre polynomials

Ln(P) = nW-n =

n! (1

I 1 I P)

+5

n(n (-1)k

k=l

-

1) .*. (a k!

-

k

+ 1)

can be written as Ln(p) = e o

dn (pne-”). dPn

~

Use the above formula to show that the set { l o ,lI ,...}, where Zn(p) = (1In!)e-oI2Ln(p)

are the Laguerre functions, is an orthonormal system in L2([0, 7.7.

Derive from the results of Exercise 7.6 that

fL , o s n n=O

n!

=

exP[-P-sl(l - 41 1 --s

+ co)).

171

References for Further Study

(the function on the right-hand side is called the generating function of Laguerre polynomials). Use this formula and the orthonormality of { l o ,Zl ,...>to show that

7.8. Prove that the orthonormal system is a basis in L2([0, 00)).

+

{I,, Zl ,...} (see Exercise 7.6)

7.9. Prove that the system {h, , h, ,...} of Hermite functions defined by hn(u) = e-ue/ZHn(u1,

where HJu) are the Hermitian polynomials (see Chapter I, Exercise 5.4), is an orthonormal basis in L2(R1).

References for Further Study On measure theory (551-3) consult Halmos [1950], and Munroe [1953]. A less general but very readable book is the work of Kolmogorov and Fomin [1961], which is also useful for 54. On spaces of square integrable functions (54) see Riesz and Nagy [1955], and von Neumann [1955]. For 55 and 57 consult Landau and Lifshitz [1958], Messiah [1962], and Gottfried [1966]. A general treatment of tensor products of Hilbert spaces can be found in the work of Dixmier [1952].

CHAPTER

111

Theory of Linear Operators in Hilbert Spaces

In Chapters I and I1 we met a large number of linear differential operators and other linear transformations. Linearity and a few other properties (such as boundedness, self-adjointness, etc.) of operators, which can be formulated and studied in a general and abstract manner without having to specify the operators, imply that the operators possessing these properties have many other additional properties. These properties are derived in the theory of linear operators. The object of the next six sections of this chapter is the study of that part of the general theory of linear operators in Hilbert spaces which is of immediate importance in quantum mechanics. The spectral theorem, formulated and proved in $6, is the main result of this theory. A knowledge of the theory of linear operators in Hilbert spaces will enable us to study the structure of quantum mechanics from an abstract and very general point of view. In such a context the formalism of wave mechanics becomes a particular realization of the general formalism,

1 . Linear and Antilinear Operators on Euclidean Spaces

1.1.

LINEAR AND ANTILINEAR TRANSFORMATIONS

A mapping of a vector space Vl into another vector space V2is usually called a transformation of Vl into V2or an operator from Vlinto V 2 . I t is evident from the preceding two chapters that a special class of operators, called linear operators, are of great significance in quantum mechanics. Besides linear operators there is a second class of operators which play a role in quantum mechanics, though a very minor one by comparison with the linear operators. These are the antilinear operators. 172

1. Linear a n d Antilinear Operators on Euclidean Spaces

173

Definition 1.l. A mapping f-f‘=A(f),

f € < ,

f’Ectrz,

of the vector space Vl over the field F into the vector space V2over the same field F is called a linear transformation (linear operator in the case that” Vl = V2)if

4-f+ bg) = -4f1 + W

g )

for all a, b E F and all f,g E Vl . For vector spaces Vl and V2over the field @’ of complex numbers, the mapping

f€%,

f-f”=B(f),

f”Ectrz9

is called an antilinear transformation+ (semilinear transformation) if B(-f

+ bg)

= a*Wf

1 + b*B(g)

foralla,bEC1andallf,geVl. It is customary to denote the image A(f) off under a linear or antilinear transformation simply by Af. We have met many linear transformations in the preceding chapters. For instance, the unitary transformations used in defining the concept of isomorphism of Euclidean spaces (see Chapter I, Definition 2.4) are linear. It is important to realize that the simplest type of linear operator is multiplication by a scalar a from the field F:

f--f,

fey.

We shall denote the above operator simply by a. However, the reader should realize that there is a conceptual distinction between a as an element of the field, and a as an operator! As an example of an antilinear operator on the vector space Vo(R1) of continuous complex-valued functions on R1 consider the operator C of complex conjugation (Cf )(4=f

f UO(~l).

*(XI,

This particular distinction between transformation and operator is adopted by us as very convenient, but it is by no means standard. It seems to be the case.that, depending on the author, either the term “transformation” or the term “operator” is used exclusively. t Note that an antilinear transformation is distinct from a.linear transformation only for complex vector spaces, or, more generally, for vector spaces over fields F in which a * operation (or involution operation) is defined by a mapping a + a*, a, a* E F , obeying the conditions that (a*)* = a, a b +a* b* and a b -+ b*a*. Since the case of complex vector spaces is of sufficient generality for quantum mechanics, we ignore the more general possibilities.

+

+

174

111. Theory of Linear Operators in Hilbert Spaces

We have

+ bg(x))

C(af(x)

=

+ b*g*(x)

.*f*(x)

= a*(W )(XI

+ b*(Cg)(x).

1.2. ALGEBRAIC OPERATIONS WITH LINEAR TRANSFORMATIONS Let A , and A , be linear transformations of V, into V, . Then, as can be easily checked, the transformation A defined by Af

= A,f

+ A,f,

fE5

9

is also a linear transformation, called the sum of the transformations A , and A , and denoted by A, A,. Let now A be a linear transformation of V, into V, , and B a linear transformation of V, into V3. Then it is easily seen that the transformation C of V, into V3, defined by

+

Cf = W f ) , f E

c

9

is also linear; C is called the product of B and A , and is denoted by BA (which is, in general, different from AB). Finally, if A is a linear transformation of the vector space V, into the vector space V, over the same field F , and a E F , the transformation aA defined by

(4f = a(Af),

fE5

9

is again a linear transformation. When we are dealing with the set N of all linear operators on a vector space over a field F , we see from the above definitions that for any B and A B are always A , B E 2I and a E F the operators aA, A defined. These operations in N endow 2I with a certain algebraic structure. Such structures have been systematically studied-very much like the vector structure of vector spaces. Sets with such structures belong to mathematical spaces called associative algebras (to be distinguished from Boolean algebras), or simply algebras.

+

Definition 1.2. An algebra over a field F is a linear space N over that field on which, in addition to the vector operations, an operation of algebraic multzplication is defined. An algebraic multiplication operation is a mapping ( A ,B ) I+ AB,

( A ,B ) E 2l x N,

of 2I x 2I into 2I having the properties (1) ( A B ) C

=

A(BC), associativity,

AB E N,

175

1. Linear and Antilinear Operators on Euclidean Spaces

+

( 2 ) ( A + B)C = AC + BC, C ( A + B ) = C A CB, distributivity, ( 3 ) a(AB) = (aA)B = A(aB), associativity of scalar multiplication, for all A, B, C E 2I, and for each element a from the field F. We say that the algebra 2I has an identity if there is an identity element 1 satisfying 1A = A1 = A for all A E 2I. The algebra 2I is said to be commutative if AB = B A for all A , B E 2I. It is called real or complex if the field on which it is defined is the field of real or complex numbers, respectively. It is easy to see that all the complex n x n matrices with matrix multiplication as the operation of algebraic multiplication form a complex algebra.

Definition 1.3. Two algebras 211 and 212over the same field F are isomorphic if there is a one-to-one mapping

which has the property that for any A , , B,

E

211 and a E F .

where A,, B, E 212are the images of A, , B, E 211 , respectively. Such a mapping is then called an isomorphism between 211 and a,. If 212is an algebra of matrices (with the algebraic operations determined by matrix summation and multiplication) then the isomorphism is called a faithful matrix representation of 211 .

Theorem 1.1. T h e set a(%) of all linear operators defined on a real (complex) finite-dimensional Hilbert space & is a real (complex) algebra with identity under the operations of operator summation, multiplication and multiplication of operators by scalars, with these operations defined, respectively, by the relations '

(1.1)

(A

+B)f

= Af

+ Bf,

(AB1.f= A ( B . ) , ( 4= fa(Af>,

A , B E 2I(W A , B E a(&') A E 2I(X), a E Wl(Cl),

111. Theory of Linear Operators in Hilbert Spaces

176

valid for all f E &. If the dimension of 3 is n, and {el ,..., en} is any given orthonormal basis in 2,the mapping A+

r' ""), *"

an1

( 14

'**

AE'%(X),

ann

aij = (ei I hi),i , j

=

1,..., n

is a faithful representation of a(&) by the real (complex) algebra of all n x n matrices; this representation is called the matrix representation of 91(&) in the basis {el ,..., e,}.

Proof. It is a straightforward task to check that % ( X )is an algebra with identity, if the identity is taken to be the linear operator 1 defined by the identity mapping ft+lf=f,

f € 2 .

The matrix in (1.2) is obviously real if & is a real Hilbert space; for a complex Hilbert space 2,(1.2) is in general complex. If {el ,..., en} is an orthonormal basis, by taking advantage of the properties of an orthonormal basis (see Chapter I, Theorem 4.6), we can write

The above relation shows that the mapping (1.2) is one-to-one, since to every matrix corresponds only one linear operator A satisfying (1.3), n

n

i.e., the image Af under A' of any vector f is completely determined by giving the matrix (1 aij (1 in (1.2). If B E a(&) and 1) bij ( 1 , btj = (ei I Bej), is the image of B under the

1. Linear and Antilinear Operators on Euclidean Spaces

177

mapping (1.2), we get by using the linearity of A and B and the definitions (1.1) A

+B

-

IKei I ( A

+ B ) e d l l = It(% I A% + W l l

+

i.e., A B and AB are mapped by (1.2) into, respectively, the sum and product of the matrices 11 aij 11 and 11 b, 11. Since a A HIl(%

I (4 eJll

= II

a(e,

I Wll = a II ad5

we have established that (1.2) is an isomorphism.

It9

Q.E.D.

We recall that every finite-dimensional Euclidean space is a Hilbert space. In general, in any given n-dimensional vector space V we can easily introduce an inner product, e.g., by choosing a vector basis {f,,...,f,} C Y and writing for every f,g E V , n

( f I g> where a, ,..., a, and b, in the basis {fi ,...,fn},

,..., b,

=

c

9

k=l

are the coefficients off and g, respectively,

Consequently, the above theorem can be applied, de facto, to any finitedimensional vector space. I n fact, Theorem 1.1 can be proved directly for finite-dimensional vector spaces, without the introduction of an inner product (see Exercise 1.4).

111. Theory of Linear Operators in Hilbert Spaces

178

1.3. CONTINUOUS AND BOUNDED TRANSFORMATIONS As the concepts of continuous functions and bounded functions are of great significance in the theory of functions, so are the analogous concepts which can be defined for operators acting on Euclidean spaces, or more generally, on normed spaces.*

Definition 1.4. A transformation f-T(f),

f E 4 ,

TfEN2,

of the normed space Nl into the normed space N2is said to be continuous at fo ,fo E Nl, if for any E > 0 there is an & ( E ) such that

I1 T(f)

-

T(f0)lln < E

for

Ilf -fa

1 1 1

<

w,

where 11 h Ili denotes the norm of h E N6, i = 1,2. T h e transformation T is said to be continuous if it is continuous at all points fo E Nl.

Definition 1.5. A set S in a normed space A” is bounded if there is a constant C such that 11 f 11 C for all f~ S. A transformation T of Nl into N2is called a bounded transformation if it maps each bounded set in Nlinto a bounded set in N2. T h e requirement that a linear transformation T be bounded is obviously equivalent to the requirement that there is a constant C such that

<

II Tf 112 < c llf Ill for all f~ Nlsince 11 T(af)l12 = I a I 11 Tf 1, for any a E C1. If A is a linear operator on a normed space N, then the supremum

is called the bound of the operator A ; obviously 11 A 11 2 0. Note that if A is bounded, then its bound 11 A 11 is finite. Vice versa, if 11 A 11 < +CQ then 11 Afll c 11 A 11 for all llfll c, i.e., A is bounded. For linear transformations we have an intimate connection between the concepts of boundedness and continuity. This connection is stated and proved in the second of the following two theorems.

<

<

Theorem 1.2. If the linear or antilinear transformation T of the normed space Nl into the normed space N2is continuous at one point fo of Ml , then it is continuous everywhere in Nl .

* These concepts are defined in their most general form for topological spaces. The present definition is, however, sufficiently general for our purposes.

179

1. Linear and Antilinear Operators on Euclidean Spaces

Proof. If T is continuous at foE Jv;, then for any 8 ( ~ such ) that

II T(f -fo)112

(1.6)

=

II Tf

-

Tfo 112

E

> 0 there is a

<E

for all /If - fo1, < S ( E ) . For arbitrary go E M, we obtain

It Tg - Tgo I12 = I1 T(g - gJIl2 < for all I/ g - go (I,

< 8 ( ~ by ) taking g - go = f - fo in (1.6). Q.E.D.

Theorem 1.3, A linear or antilinear transformation T of the normed space .Vlinto the normed space N2 is continuous if and only if it is bounded. Proof. If

(1 7'11 < f a , and

0, denotes the zero vector in

N,, then

for all /If 1, < 8(r) = E 11 T /Id1. Consequently, T is continuous at the origin 0, E Nl, and by the preceding theorem, it is continuous everywhere in .Vl. Conversely, if 7' is assumed to be continuous at 0, , there is such a 6, > 0 that 1) Tg )I2 1 for all 1) g 1, 8, . For any f E .Vl we have that satisfies )I g 1 , = 8 , and consequently g = a0IIf

<

80

Thus, if llflll

lIflL111

<

Tfllz

=

/I T(80 II f ll;l.fh

< 1.

< c for any given c > 0, then

i.e., T is bounded.

Q.E.D.

1.4. EXAMPLES OF BOUNDEDAND UNBOUNDED OPERATORS The study of operators on finite-dimensional Euclidean spaces is considerably simplified owing to the fact stated in Theorem 1.4.

Theorem 1.4. Any linear transformations of a finite-dimensional Euclidean space 4 into another Euclidean space g2is bounded. Proof.

Let the subindices i

=

1 , 2 in (. I

.)i

refer to the inner

180

111. Theory of Linear Operators in Hilbert Spaces

products in g1 and g2,respectively. If dim 6 orthonormal basis in cYl , we have for any f E cYl

n and el

=

,..., e,

is an

(1.7) n

C (f I ei)l(Tei I Tej>z(ej I f i.

1

>1

.

j=l

Since according to the Schwarz-Cauchy inequality I(% If>lI

< I1 ei

1 1 1

llflll

= llflll

7

we get from (1.7)

< C lif/ll

T h e above relation shows that 11 Tfllz implies that T is bounded. Q.E.D.

for all f E 1 ; , which

T h e above theorem does not apply to infinite-dimensional spaces. In fact, most of the operators of interest in quantum physics are unbounded. For instance, consider the space Vi([wl) of all continuous functions f ( x ) , x E W, of compact support. The operator A ( i l f ) ( x ) =).(fI.

is obviously linear, but it is not bounded. T o see that, take a function f o ( x ) 2 0 with support inside [0, 11 which does not vanish identically. For each a E C1 the function f , ( x ) = f ( x - a), has its support inside [a, a I], and f , ( x ) E Wi(R1). Since (1 fa !I == l ! fo 11 = const, while

+

--m

we see that (I A

)I = +a.

EXERCISES 1.1. Let ao(x),..., a,(x), x E C1,n 3 1, be continuous functions. Show that the mapping

1. Linear and Antilinear Operators on Euclidean Spaces

181

is a linear transformation of the linear space Vn(rW1) (of n-times continuously differentiable functions) into Vo(W).

Remark. T h e above linear transformation can be characterized by the differential operator form dn an(.%) dxn

+ ...

and the specification of its domain of definition Vs.(rW1). 1.2, Let I E P be a finite interval, and let K(x,y ) be a continuous function on I x I. Prove that the mapping (Lebesgue integration)

defines a linear operator on LZ(1). 1.3.

Prove that the linear operator defined in Exercise 1.2 is bounded.

1.4. Let V be a complex n-dimensional vector space, and f , ,...,f , a vector basis in V . Assign to each linear operator on V the complex matrix 11 aik 11, defined by expanding Afk , K = 1,..., n, in the vector basisf, ,...,fn n

Aflc

=

c

aikfi

*

i=l

Show that the mapping A H [ ] uikII is an isomorphism between the algebra of all linear operators on the vector space V and the algebra of all n x n complex matrices. 1.5.

Prove that the linear operator

defined on the space %r(W)of infinitely many times continuously differentiable functions of compact support, is unbounded. 1.6. Denote by %(N) the algebra of all bounded linear operators acting on a normed space JV (such as the Banach space L2(52, p) of all p-integrable functions on 52). Show that the operator bound 11 A 11, A E 9I(N),defined by (1.5), has all the properties of a norm (see Chapter I, Definition 2.2) on %(N).

111. Theory of Linear Operators in Hilbert Spaces

182

2. Linear Operators in Hilbert Spaces 2.1.

LINEAR FUNCTIONALS ON NORMED SPACES

A functional on a vector space V is a real- or complex-valued function defined on that space. As such, a real or complex functional can be thought of as a transformation of the vector space V into the vector space (Rl) or (el)of real or complex numbers, respectively. Hence, we can introduce the fundamental concepts of continuity, linearity, etc. by specializing the corresponding concepts, defined for transformations in general in $1, to the present special case of transformations.

Definition 2.1.

A functional 4,

4:fw4(f),

fE M ,

d(f) E

defined on the normed space N is continuous at the point f o E N if for any E > 0 there is a S ( E ) such that I +(f)-4( fo)l < E for all 11f -f o 11 < a(€). The functional 4 is said to be continuous if it is continuous at all points of Jfr; 4 is said to be bounded if it maps every bounded set of JV into a bounded set in R1 or C1,respectively, i.e., if for every N > 0 there is a constant C , such that I + ( f ) l C, for all f E N such that 11 f 11 N. An important special class of functionals on vector spaces are the ones which possess linearity properties, e.g., the functionals

<

<

4o(f) = f ( O ) ,

fEgo(q,

+(f)= Jh) dx, Definition 2.2. The functional vector space V is linear if

4

4(f + g) = +(f)+ 4(g),

f e P([% bl).

defined on the real or complex

4 w

=

df),

for all f , g E V and all scalars a from R1 or el,respectively. and (el)as normed spaces by We can think of the vector space (W) taking the absolute value of a number to be its norm. By specializing Theorems 1.2 and 1.3 to the case of functionals, we can state the following theorem.

Theorem 2.1. (a) If the linear functional 4, f E N , defined in the normed space N is continuous at one point, then it is continuous everywhere in N ; (b) The linear functional 4 is continuous if and only if it is bounded.

183

2. Linear Operators in Hilbert Spaces

2.2. THEDUALOF

A

HILBERT SPACE

The family of all continuous linear functionals defined on a topological vector space" V is called the dual (or conjugate) space of V , and is usually denoted by V ' .T h e dual space V ' is also a linear space, and plays an important role in the theory of topological vector spaces. By specializing the above statements to normed spaces we get Theorem 2.2.

Theorem 2.2. Let N be a real (complex) normed space, and denote by N'the dual of N , i.e., the set of all continuous real (complex) linear functionals on N.If the sum + +2 of two functionals and +2 on A'is defined by

+,

($1

I$,

+ 42)(f) Mf)+ Mf), =

and the product a+, of a functional defined by

+ on N with a scalar a E Rl(@l) is

( M f )= Mf), then &"I is a real (complex) vector space with the above operations as vector operations. Moreover, N'becomes a normed space if (1 * II is taken to be II $11 = SUP I 4 (f)l for II (f)II 1. The proof of this theorem is trivial and it consists in noting that +2 and a+ are continuous linear functionals if ,+2 are continuous linear functionals. For the special case of normed spaces which are Hilbert spaces we have a very close relation between the space and its dual, as reflected in Theorems 2.3 and 2.4. T o prove the first of these theorems we need the following lemma.

<

+,+

+,+,

*Lemma 2.1. If M is a closed linear subspace of a Hilbert space S and f is a vector in 3, there is a vector f E M such that f" = f -f ' is orthogonal to M. Proof.

Denote by d the distance between f and M, i.e., d

=

inf Ilf

QEM

-

g 11.

= 0, there must be a sequenceg, ,g, ,... E M such that 11 f - g, [I --+ 0 when n --f co;since M is closed, we have in this case f E M and the lemma is proved by taking f ' = f and f" = 0.

If d

* The concept of topology provides the most general means of introducing neighborhoods, and therefore the notion of convergence in an arbitrary set. See, e.g., Treves [1967] on the theory of topological vector spaces.

111. Theory of Linear Operators in Hilbert Spaces

184

Assume that d

> 0, and let g, ,g, ,... E M be the sequence for which d

=

lim Ilf

- g,

n-i m

11.

It is easy to check by using the algebraic properties of the inner product that

4 Ilf

- g9n

+ t Ilf

112

1,

- g,

=

Ilf - 8(gm + gn)1I2+ II Hgm - gn)1I2.

If we let above m, n + co and note that

Ilf

+

- Q(gm

+ g,)1I2 2 d2

because &(g, gn)E M, we get 1) g , - g, II -+ 0. Thus, gl , g, ,... is a Cauchy sequence, which has a limitf‘ E M, since M is closed. Moreover, we have

Ilf -f‘ II = ! i Ilf

-gn

II

= d.

I n order to show thatf” 1M, it is sufficient to prove that for all normalized h E M. By means of the identity (h

If”

- (h

If”)

h) =0

we get

Ilf” 1,

-

(h

If”)

h 112

+ ll(h If”)h 112,

1) 2 d because ( h 1f”)h E M, we have d2 = Ilf” 112 = Ilf” - I2 3 d 2 II2, i.e., I(h If’’)l = 0. Q.E.D.

and since

Ilf”

= Ilf”

(f”I h ) = 0

-

( h 1f”)h

+

Theorem 2.3. (Riesx’ theorem). To every continuous linear funcon a Hilbert space X corresponds a unique vector g E X, such

tional that

+

4(f) = ( g If)

(2.1)

for all f E X .

Proof.

Denote by M, the null space of +: M,

=

{f:+(f) =0 , f E W .

M, is a closed linear subspace of X ; its linearity is evident, while the fact that M, is closed follows from the continuity of 4: if fi,f, ,... E M, and if f = limn+mfn , then

W) = W and since limn+ao+(f - f,)

n )

=

+ 4(f

-f n >

=

C(f

0, we have + ( f )

- fn),

=

0, i.e., f E M,

.

185

2. Linear Operators in Hilbert Spaces

If M, = X , the theorem is established by taking g = 0. Therefore, assume that M, # X . Then there is at least one nonzero vector h 4 M, Since M, is closed, we can use Lemma 2.1 to state that h can be written as the sum of two vectors g' and g", where g" 1M, while g' E M, . The vector g" must be different from the zero vector because otherwise we would have h =g' E M Q . Thus, +(g") # 0, and the vector f - (+(f)/+(g"))g" exists. This vector belongs to M, because

.

Since g" IM,

, we have

for any f E X . From the above relation we get

we obtain (2.1) by setting

In order to establish that g is unique, assume that f l f ) = (g, for all f. Then we have ( g - g1 If

> = (g If)

by choosing f = g - g, we get (g

- (g, - g,

If)

If) = 0;

I g - g,)

= 0, i.e.,

g = gl . Q.E.D.

Theorem 2.4. Let 2 be a Hilbert space and #' its dual. There is a one-to-one antilinear" transformation (2.2)

++g*,

+ € Z , g,EH,

of X" onto &' such that for each

+

E Hf

+(f1 = (gd If> for all f E &'. In the case that X is real, #" is real too, and there is no distinction between linear and antilinear transformations of X' into X.

186

111. Theory of Linear Operators in Hilbert Spaces

ProoJ According to Riesz' theorem, the above mapping is indeed a one-to-one mapping of 2'' into 2'. Since for any g E S the functional 4 defined by

+(f)= ( g If> is continuous, this mapping is onto 2'. The antilinearity of this mapping is evident:

M f ) + M f ) = 4g4, If>+ Kg4, If> =

(a*g+,,

+ b*gd, If>.

Q.E.D.

The above theorem justifies the notation introduced by Dirac, which has become very popular in physical literature. In this notation a vector! of a Hilbert space @ is denoted by I f ) , and called a ket vector. The functional 4 E i@', which under the mapping (2.2) has f as an image is denoted by (f I and called a bra vector. The inner product (f 1 g ) of two vectors f,g E 2' can be looked upon as a product of the bra (f I and the ket I g). The above notation neither presents any real advantages, nor does it enrich the techniques when dealing with Hilbert spaces. * Consequently, it will not be adopted in this book.

2.3. ADJOINTS OF LINEAR OPERATORS IN HILBERT SPACES Riesz' theorem enables us to prove the existence of the adjoint of any bounded operator acting on a Hilbert space. I n order to give a sufficiently general definition of the adjoint of an operator, we enlarge our definition of operators on Hilbert spaces to operators in Hilbert spaces, by dropping the requirement that an operator be defined on the entire Hilbert space.

Definition 2.3. A linear operator A in a Hilbert space @ is a linear transformation of a linear subspace g Aof 2' into 2';g Ais called the domain of dejinition of A and its image under A

W A = {Af:f E QA} is the range of A. If another linear operator B defined on g B 3 g A coincides with A on V, , i.e., Bf = Af for all f E g A, then B is called an extension of A (or A a restriction of B), and we write A C B. *; There have been, however, proposals to incorporate this notation in the theory of rigged and extended Hilbert spaces. Then a ket is an element of .@ and a bra is an element of a larger scale containing SE" (see, e.g., the articles by Antoine [1969], Prugov&ki [1973b]).

187

2. Linear Operators in Hilbert Spaces

Theorem 2.5. Let A be an operator on Z , with domain 9,. Denote by 9,* the set of all vectors g E 2 which are such that for each g there is one and only one vector g* which satisfies the equation = ,

(2.3)

f E g A

9

for all f E 9, . T h e mapping g+g* = A *g,

gEgA*,

is a linear operator, called the adjoint of A, and it exists, i.e., 9,* # 0 if and only if Q A is dense in 2. Proof. In the case that 9,is dense in 2 and g,g* are two vectors satisfying (2.3), g* is uniquely determined by g; namely, if g,* also satisfies (g I -4.) =

for all f E 9, , then
If>= - = 0

for all f E 9, , and since 9,is dense in Z , we must have g* - g,* = 0. Thus, if (2.3) has any solutions, we can define an operator A*, with a domain gA* , consisting of all g for which there is a g* such that g and g* satisfy (2.3)) and mapping g into g*. The operator A* is defined at least for g = 0, since then we can take g* = 0. Moreover, if g, ,g, E BA*, then
+ bgz) If>

+

k 2 I Af> I Af) + b* + b* = ,

=

<%

= a*<&

+

which proves that ag, + bg, E gA* and A* is linear. Assume now that g Ais not dense in Z . Then its closure g, is different from Z , and there is in Z a vector h $ g , . According to h", h' 1g Awhere h' # 0. We have Lemma 2.1 we can write h = h' then

+

If>= 0 = (0 I Af> This shows that when 6,# Z , we can take in (2.3) (h'

for all f~ Q A . g* = h'. Hence, we have at least two distinct solutions, 0 and h', belonging to 0. Thus, A* does not exist, because (2.3) can never have a unique g*

188

111. Theory of Linear Operators in Hilbert Spaces

for anyg, since if for someg there is a g * satisfying (2.3), theng* also satisfy (2.3). Q.E.D.

+ h‘ will

An operator A might not have an adjoint A*, but there m i g h still be many operators B satisfying the relations

for all f E g Aand g E g B .Two such operators A and B are said to be adjoint to one another. We note that if the adjoint A* of A exists, and A and B are adjoint to one another, then necessarily B C A*, i.e., A* is a maximal extension of all the operators adjoint to A.

2.4. BOUNDEDLINEAR OPERATORS IN HILBERTSPACES If A is bounded linear operator which has an adjoint A*, then from Theorems 2.6 and 2.7 it is easy to see that A* is also bounded, and that gA* = s.

Theorem 2.6 (The extension principle). A bounded linear transformation T from the Hilbert space Z1 into theHilbert space Z 2 defined , on a dense domain g T ( g T= Zl), has a unique continuous linear extension to the entire Hilbert space Z1 . Proof. Let f be any given vector in S1 . Since GT = Zl , there is a sequence f l ,f 2 ,... E g Tconverging to f . T h e sequence Tfl , T f 2 ,... is a Cauchy sequence, since according to Definition 1.4 there is a constant C such that

II Tfm - Vn 112 B Cllfm - f n

111

*

Hence, the limit of Tfl , Tf2 ,... exists. Moreover, this limit is independent of the chosen sequence converging to f because iff; ,f i ,... E g Talso converges to f , then limn+m11 T f , - Tf,’ 112= 0 , since

by choosing for Thus, we can define a transformation T’ for all f E Z1 each vector f a sequence f l ,f 2 ,... E g Tconverging to f and setting T’fto be the limit in the Hilbert space norm of Tfl , Tf2 ,..., lim (1 T’f - Tfn/I2 n-r w

= 0.

189

2. Linear Operators in Hilbert Spaces

This transformation T', with gT, = S1 , is evidently linear: T'(uf

+ bg) = s-lim T(uf, + bg,) n-, = s-lim(aTf, + bTg,) n-, m

m

+ ST'g,

= aT'f

where g, ,g, ,... E g T is any sequence converging to g, and "s-lim" . denotes the limit in the norm of S, The fact that T is bounded follows from the inequality

I1 T'fllz < I1 Yf- Tf,112 + I1 Tfn 112

< II T ' f - Tfn + c llfn I12

Ill

Y

since when we let n -+ co,11 T'f - Tf,, 11 -+ 0, while I]f , Ill It is evident that T' is the only continuous extension of space if,. Q.E.D.

II f Ill . T to the entire

-+

Theorem 2.7. If A is a bounded linear operator defined on the entire Hilbert space #, its adjoint A* is also a bounded linear operator defined everywhere in z?, and 1) A 1) = 1) A* I). Proof. For a fixedg E S t h e functional4 defined by+(f) = (g I A f ) , X , is obviously linear. Moreover, 4 is also continuous, as can be seen by using the continuity of A (see Theorem 1.3)

JE

lim

d(f 1 = ( g I ,;?;go Af) = ( g I Afo) = +(fo).

lif-f~ii-+o

Consequently, we can apply to +(f) Riesz' theorem (Theorem 2.3) to deduce the existence of a unique g*, satisfying

I =

+(f) =
By Theorem 2.5, the mapping g w g " = A * g,

g E x ,

defines the adjoint of A as a linear operator, for which we see that gA*= if. From the inequality

I l = l(g I Af>l

< II A II llfll

we get by setting f = A*g

II A*g II

< I1 A II Ilg II.

llg II

190

111. Theory of Linear Operators in Hilbert Spaces

< 11 A 11.

The above shows that A* is bounded and (1 A* 11 the roles of A and A* we arrive at 11 A 11 11 A* Q.E.D. 11 A 11 = 11 A* 11.

<

2.5.

By reversing

11, thus proving that

DIRACNOTATION FOR LINEAR OPERATORS

It is interesting to take a look at the above theorem from the point of view of Dirac's terminology. T o realize the nontrivial implications of this terminology, consider the more general case of a bounded linear operator A defined everywhere on a normed space N . If is an element then the functional +l defined by of the dual N',

+

is again a continuous linear functional, i.e., +l E N ' ,as can be easily proved by using the linearity and continuity of and A (A is bounded and therefore continuous, by Theorem 1.3),

+

Consequently, the mapping $ H4 1

= A1+,

+,+I E

M',

is a transformation of N ' into itself. Furthermore, it is easily seen that A1 is a linear operator on N ' : (A"41

+W Z l ) ( f )

+ b+Z)(Af) M A f ) + WZ(Af)= @A1+1(f)+ bA1+2(f).

= (dl =

Thus, every bounded linear operator A on N induces a linear operator Al on N ' . In case of a Hilbert space S,to every element EX' corresponds, according to Theorem 2.4, a unique vector g4 E S.We

+

+

have seen that in Dirac notation one denotes by the bra symbol (gd I, so that +(f)= (g4 .)fI Moreover, in this notation one writes A'+ = (gd

I 4

agreeing to denote the above introduced operator Al on 2' by the same letter A, but introducing the clause that when A stands, defucto, for the operator Al, then it acts on the bra vector from the right. In this notation one writes

W f ) = (A1$)(f)

= (gd

I A If).

191

2. Linear Operators in Hilbert Spaces

According to Theorem 2.7, we have

Mf) = , which implies that, using Dirac notation, (A*& 1

2.6. CLOSEDOPERATORS AND

=

(gm I A.

GRAPHOF

THE

AN

OPERATOR

In the case that a linear operator A is unbounded and has an adjoint A*, we can deduce from the preceding theorem that A* is also = 2' [so that (A*)* = A** exists]. In fact, if unbounded when A* were bounded, then A** would also be bounded, and since obviously A C A**, this would imply that A is bounded. However, A* has always the property of being closed, which is a somewhat weaker property then boundedness.

a,*

Definition 2.4. The linear operator A in the Hilbert space S is closed if it has the property that whenever f l ,f 2 ,... E g Aconverges to a vector f E S and Afl , Af2 ,... also converges to some vector h E 3?,then f E 9, and h = Af. Theorem 2.8. The adjoint A* of a linear operator A in S is closed. Proof. Let g, ,g , ,... E gA*converge to g E S and assume that A*gl, A*g2 ,... converges to a vector h. Then for any f E 9,

which

the definition of A* implies that g E 9,* and that h = A*g.

Q.E.D. The concept of a closed operator can be readily grasped in terms of the graph of an operator. The graph of an operator is a generalization of the concept of the graph of a real function f ( x ) , x E R1, which, we recall, consists of all the points ( x , ~ ( x ) )in (R2) = (R1)@ (Rl) corresponding to x from the domain of definition off(x).

Definition 2.5. The graph G , of an operator A acting in the Hilbert space S is the subset GA

= {(f,A f ) : f E B A )

of the direct sum S @ S (see Chapter 11, Theorems 6.1 and 6.3). We note that A is closed if and only if whenever

(fl

9

&I),

(f2

Y

Af2),-**

GA

192

111. Theory of Linear Operators in Hilbert Spaces

is a convergent sequence in 2 @ 2, then its limit (f,g) lies in G, , i.e., G, is a closed subset of 2 02 if and only if A is a closed operator. In Chapters I and I1 we encountered many operators A which were Hermitian, i.e., for which (AfI g) = (f I Ag) for all f,g E gA. These operators were in addition linear. I n mathematical literature such an operator is called symmetric.

Definition 2.6. The linear operator A acting in a Hilbert space is symmetric if (AfI g) = (f I Ag) for all f,g E gA. A symmetric operator A is called self-adjoint if A = A*, i.e., A* C A and A C A*. We note that a self-adjoint operator is always closed (because by Theorem 2.8 A* is closed) and densely defined (since A* exists). All the Hermitian operators which have been introduced while studying different physical problems were symmetric. For instance, we have seen that the Schroedinger operator H , defined in Chapter 11, 55, is linear, Hermitian, and defined-under reasonable requirements on the potential (see Chapter 11, Theorem 5S-on the dense subspace %‘Z([W3n) (see Chapter 11, Theorem 5.6). It is an unfortunate feature of unbounded linear operators defined on linear subspaces of a Hilbert space that they do not have closed extensions to the entire Hilbert space (see Section 3.5), and for this reason we had to give particular attention in Definition 2.3 to the domain of definition of an operator. This fact makes the study of unbounded operators particularly difficult, especially since it prevents simple definitions of quite fundamental algebraic operations with such operators. For instance, if A and B are two operators for which gA# 2 and g B# Z , then we can define A B only on gAn g B ,

+

(A+B)f=Af+Bf,

f E g A n g B *

Since it might happen that 9,n g B= {0} even when both 9,and g B are everywhere dense in 2,it follows that A B might not be defined at all except on the zero vector 0. Similarly, in order to define AB, we must assume that Bf €9” whenever we apply A B to some f E g B ,

+

( A B ) f = A(Bf),

f

gB

,

Bf

gA

*

Only in case of multiplication of an operator A by a scalar a can it be stated simply

( a 4 f = 44%

f E 9.4

*

193

2. Linear Operators in Hilbert Spaces

Despite these difficulties we shall have to study unbounded operators in some detail since practically all of the operators which appear in physics are unbounded, such as the position and momentum operators or the Schroedinger operator (see Exercise 2.4).

2.7. NONEXISTENCE OF UNBOUNDED EVERYWHERE-DEFINED SELF-ADJOINT OPERATORS We shall proceed to prove that no symmetric unbounded operator has a self-adjoint extension to the entire Hilbert space. For this proof we need some preliminary results.

*Lemma 2,2. If +1 ,+2 ,... is a sequence of continuous linear functionals on Z, which is bounded for each fixed f E Z, i.e.,

I dn(f)l G

d2,... are uniformly S, = {h: (1 h 1) \< I}, i.e.,

then

I Cn(g)l < c,

Cf

f E z,

,

bounded within the unit sphere

#1,

gE

s, ,

It =

1,2,...,

or equivalently

I dn(f)l

< c Ilf

= 1, 2,...,

11,

for all f E Z. Proof. We shall show first that q$(f), + 2 ( f ) , . . . is uniformly bounded inside at least one sphere {h: 11 h - h,, )I Y } in &. Assume that the above statement were not true! Then we would have I +n,(fl)I > I for at least some integer n1 and some fl E Z . By the continuity of +,J f ), we also have

<

on a sphere S(l)C X with the centre at fl . Since +1( f ), +2( f ), ... are not there must be an uniformly bounded in any sphere, including S1), fi E 5"l)such that I & ( f2)l > 2 for some integer n2 . Due to the continuity of+nl(f 1, I +n,(f)l > 2, f E s(2), for some sphere S2) with its center at f 2 . By continuing this reasoning, we establish the existence of a sequence S(1)3 23'2) 3 -.-of spheres in which I d W k ( f ) I > k, f S ( k ) ,

194

111. Theory of Linear Operators in Hilbert Spaces

for some n k , K = 1,2, ... . Since 2 is complete, Stk) contains at least one element f, , for which we have according to the above construction I +n,(fo)l > k, k = 1,2,... * m

Thus, $,(f0), $,(fO),...is not a bounded sequence, contrary to the assumption of the theorem. This means that if , $, ,... are bounded for each f E 2, then 4, ,... are uniformly bounded on at least one sphere So = {h: 11 h - ho )I \< r} in H,

< co,

I+,(f)l

fES,,

n

=

1,2 ,... .

Consequently, we get by exploiting the linearity of $,(f)

Since for g E S, we have 11 g )I

< 1 and therefore rg - h, E So so that

I +n@g - h0)l

<

l+,(g)l

*Theorem 2.9. QA

=

Y

*.

, 6, ,... are uniformly bounded on S,

we have established that

of definition

< co

:

Q.E.D.

The adjoint A* of a linear operator A with domain Z is a bounded operator on 2.

Proof. Since g A= Z, A* exists by Theorem 2.5. Assume that the theorem is not true for A, i.e., that A* is not bounded. Then there must be a sequence h, , h, ,... E gA* such that 11 h, II,II h, I),... is bounded while 11 A*h, 11 -+ +GO for n -+ CO. Since 11 h, II,ll h, 11, ... is bounded, we have

+

(2.4)

lim II A*g, /I

n+m

=

+a,

for the normalized vectors g, = (1 h, The functionals $, defined by +n(f) =

(g,

11-l

I Af),

IIg,

h,

II

=

1,

. f E 2,

are obviously linear and defined everywhere on Z because g A= 9. Moreover, these functionals are bounded,

I +i(.f)I

=

I(A*g,

I f > l < II A*& I1 Ilfll,

2. Linear Operators in Hilbert Spaces

195

and therefore continuous (see Theorem 2.1). Since in addition the sequence dl 4%,... is bounded for every fixed f E 3?) )

I Mf)l = I
< IIgn II II 4-11= I1 AflL

we can apply Lemma 2.2. According to this lemma the functionals ,d2,... have a common bound for all f E 3?,

I +n(f)l

< C Ilfll,

But then we get by setting f

I bn(A*gn)l

=

71

= 1,

2,e.e

A*gn

= II A*gn

11'

< CII A*gn IL

which implies that

II A*gn I1 < C, contradicting (2.4).

71 ===

1, 29***,

Q.E.D.

From Theorem 2.9 we can immediately deduce that the following theorem is true.

Theorem 2.10 (Hellinger and Toeplitz). Any symmetric operator A defined on the entire Hilbert space & is bounded. The above theorem is evidently true, since A A* and the fact that = iZ? imply that A = A*, while on the other hand A* is bounded by Theorem 2.9. We note that in the Hellinger-Toeplitz theorem we have that A = A*, and therefore A is closed by Theorem 2.8. I n $3 we shall generalize this theorem to any closed operator defined on a dense domain.

EXERCISES 2.1.

Show that the functional

is linear but it is not continuous in the norm

2.2.

Show that the functional

111. Theory of Linear Operators in Hilbert Spaces

196

is linear and continuous in L2(R1), and find a g(x) EL(,#W) such that

2.3. Let A = 11 a i k 11 be the operator acting on following manner:

1) ck 11 E Z2(n) in

the

n

A (1 ci 11

= 11 bi 11,

bi =

%kCk

*

k=l

Show that the above operator is linear and bounded, and that its adjoint is represented by the Hermitian conjugate matrix (1 aZc11,

c u;ck . n

A* 11 ci 11

= 11 bi'

1,

bi' =

k=l

2.4.

Prove that the operator

is an unbounded linear operator on L2(R3). 2.5. Let (f 1 g ) , f , g E X , be a bilinear form on the Hilbert space &' which is bounded, i.e.,

I(f I g)l

< c llfll II g 11.

Prove that there is a bounded linear operator A on 2,with 11 A 11 \< C and g A= X', such that (fig) = (Aflg). Show that (f1 g) is Hermitian if and only if A*

E

A.

Prove that if A * exists and A is any number, then (AA)* and exist (in A A, A stands for the operator which multiplies each vector by the number A) and 2.6.

(A

+

+ A)*

@A)* = A*A*,

(A

+ A)*

= A*

+ A*.

2.7. Show that if A and B are bounded operators with g A= BB = S, then (A + B)* = A* + B* and (AB)* = B*A*. 2.8.

Prove that if A* and B* exist, then ( A

2.9.

Show that if A* exists and A

aA+B = 2,and (AB)* 2 B*A* if gAB =X, B* C A*.

C B,

+ B)* 2 A* + B* if

then B* also exists and

197

3. Projection Operators

2.10. Quote the theorems in this and the preceding sections which show that every Hermitian linear operator on a finite-dimensional Hilbert space is self-adjoint.

3. Orthogonal Projection Operators 3.1. PROJECTORS ONTO CLOSEDSUBSPACES OF A HILBERT SPACE The simplest type of operator, which is of essential significance in the spectral theory of self-adjoint operators as well as in quantum mechanical problems, is the orthogonal projection operator. As the name suggests, this concept of a projection operator acting on a Hilbert space 2 is a generalization of the notion of an orthogonal projection of a vector r in (W)onto a line or a plane in (R3). For instance, if we have a line through the origin of W [i.e., a one-dimensional subspace of (R3)], M

= {te: -GO

< t < +GO}

determined by the unit vector e, (1 e I( = 1, recall that the orthogonal projection of a vector r onto this line is the vector

-

rM= ( e r) e .

Theorem 3.1. Let M be a closed linear subspace of the Hilbert space Z. Denote by M I (or 2 0M) the linear space (see Exercise 3.1) of all vectors orthogonal to M: M* = {h: ( h I g) = 0, g E M}. Then each vectorf (3.1)

E2

can be written uniquely as a sum

f =f'+f'',

f'EM,

f"EML,

and the mapping (34

f+f'=EM(f),

fEs,

f l E M ,

is a linear operator defined on the entire 2, called the projector (or orthogonal projection operator) onto M. Proof. The possibility of the decomposition (3.1) for each f E 2 is guaranteed by Lemma 2.1. I n order to prove the uniqueness of this decomposition, assume that

f = fl'

+

fl",

fi'

E

M,

fl"

E

111. Theory of Linear Operators in Hilbert Spaces

198

Then we have f ‘ -fl’

f

=

- ( f ” -fi”).

Since, on the other hand, f ’ - fl‘ E M, f “ - fl“E M I , and therefore -fit 1f ” - f1”,we get

( f ’ -f1’If’

4f’ -f1’If” -f1”> = 0, and f”- fl” = 0. Thus, the uniqueness of =

-f1’>

which implies f ’ - fl’ = 0 f’and f for each f E 2 is established, and consequently (3.2) defines a I’

mapping of X into M. T o prove that this mapping is a linear operator, note that if g

= g’

af

+ bg

+ g”,

g” E M I ,

g’E M,

we have

+ bg‘) + (d”+ bg”),

= (a?’

where, due to the linearity of M and M I , uf’

+ bg’ E M

and

uf”

+ bg” E MI.

Since the decomposition (3.1) of any vector in 2 is unique, we get

+ bg)

==

af‘ f bgl

which proves the linearity of EM

.

Q.E.D.

(af

+ bg)’

= EM(Uf

= aEM(f)

+ bEM(g),

If M is a closed linear subspace of a Hilbert space 2 and EM is the projector on M, then f’= EMf is called the projection of the vector f E 2 onto the subspace M. The following theorem provides a simple recipe for computing projections in practically important cases.

Theorem 3.2. If M is a separable closed linear subspace of a (not necessarily separable) Hilbert space 2, and if {el, e2 ,...>is an orthonormal basis in M, then (3.3)

where EMis the projector on M. Proof. In order to prove that EMf is given by (3.3) when {el , e2 ,...} is a (countable) orthonormal basis in M, note that g’

=

x k

(ek

If>

ek

199

3. Projection Operators

exists and g' E M (Chapter I, Theorem 4.6) for all f E X. Furthermore, g" = f - g' E M* because for any h E M we get
=

(f' I h,

-

z(f' I

ek)(ek

I h,

= O,

k

where in deriving the third expression we used the fact that (f"lh) =(f"Iel) =(f"Ie,)

=

..- = 0 ,

and then Parseval's relation (4.15) of Chapter I was applied to the orthonormal -basis {el, e2 ,...>in M. Thus, we have f

= g'

+ g",

g" E MI,

g' E M,

and therefore, due to the uniqueness of the decomposition (3.1),

g f = f' = EMf. Q.E.D. An illustration of the above theorem is provided by the special case of a one-dimensional space M. If e is a normalized vector from M, then we have EMf

(e

If)

e-

In Dirac's notation the projector EM is denoted by I e) (e I, thus providing a handy mnemonic device for writing" the above formula in the ket formalism: (3.4)

=

(1

e>(e

I> I f >

=

I e>(e

If>

= (e

If>e.

It is immediately noticeable that in defining the projection on a closed subspace M, the spaces M and M* play a symmetric role. In fact, we see from (3.1) that

(3.5)

EM + E M ,

=

1.

M* is called the orthogonal complement of M in S. We say that X is the orthogonal sum of the subspaces M and MJ-, and express this statement symbolically by writing Jf? = M @ M I . More generally, we define the orthogonal sum of subspaces as follows. *In mathematidal literature, instead of writing

I e><e I, one would write

<e

I *>

e.

200

111. Theory of Linear Operators in Hilbert Spaces

Definition 3.1. Let 9be a family of mutually orthogonal closed linear subspaces of a Hilbert space X , i.e., if MI,M, E 9,then M, 1M, . T h e closed linear subspace

0M

[If:f. u

=

MI]

MES

M E 9

uMs9

spanned by M is called the orthogonal sum of the subspaces M E 9. I n the case where 9can be written as a finite or infinite sequence {M,, M, ,...} of closed subspaces of 2, their orthogonal sum is also denoted by

0

Mk =

MI @ M , @ * * * .

k

I t is significant to note the similarity between the notation for the direct sum of Hilbert spaces and the orthogonal sum of subspaces of a certain Hilbert space. I t will soon be evident that the direct sum of the closed linear subspaces M, , M, ,... (which are themselves Hilbert spaces) is unitarily equivalent to the orthogonal sum of MI,M, ,... .

3.2. ALGEBRAIC PROPERTIES OF PROJECTORS

Theorem 3.3. (a) A linear operator E defined on the entire Hilbert space X is a projector if and only if it is self-adjoint, (3.6)

E

= E*,

E

=

and idempotent, (3.7)

E’.

(b) The closed linear subspace M on which a projector E defined on 2 projects is identical with the set of all vectors f E i%? satisfying the equation (3.8)

Ef

=f

.

(c) 11 E /I = 1 for any projector E which is different from the zero operator 0, i.e., which projects on a subspace M # (0). Proof. Let EM be the projector on the closed linear subspace M of Z . For arbitrary f , g E X we get by using the familiar decomposition f=f’+f”,

g=g’+g“,

f’,g’EM,

f“,gnEML,

that ( f I EMg>

=

(f’

= (EMf

+f” I g’> = (f’1 g’> = (f’I g’ f g”> I g>,

201

3. Projection Operators

which proves that EM* = EM for any f E A? <EMf

I g>

=(f'

. Furthermore, we see from the above that

I g')

I EMg)

= <EMf

=

I g>

for all g E X , which implies that EMf = EMY,i.e., EM = EM2. Note that in particular EMf = f when f E M! Assume now that a linear operator E, with g E= X , is such that E = E* and E = E2. Denote by R the set of all vectors f E A? for which Ef

=f

.

R is a linear subspace of A?,since iff, g

+

E ( u ~ bg) = aEf

E

R, then

+ bEg = af + bg

for all scalars a, b. Moreover, R is closed; namely, if the sequence ,... E R converges to some vector f E X , we get

fi ,fa

by using the fact that E is bounded by Theorem 2.10, and therefore continuous (Theorem 1.3). Since fl ,f 2 ,... can have only f as their limit, we conclude that Ef = f, i.e., f E R. In order to establish that E is the projector on R, decompose each vector g E &' in the following way: g = Eg

But Eg

E

+ ( g - Eg).

R, as we can see by using the relation E2 = E, E(Eg)

= E2g = Eg.

On the other hand, g - Eg G RI,since for any f E R (g-Eg1.f) = < g - E g I E f )

=<E(g--g)If)

=<Eg-EE2gIf) =o,

which shows that E = ER . The above considerations establish the points (a) and (b) of the theorem. Since by (3.1)

I l f 11' 2 I l f ' 1 '

=

11 E M f [I2,

we have 11 EM 11 f 1. If EM # 0, there must be at least one vector f l # 0 which belongs to M and therefore satisfies (3.8). Thus, 11 EMf l 11 = II fl 11, which shows that 11 EM (1 = 1. Q.E.D.

111. Theory of Linear Operators in Hilbert Spaces

202

Theorem 3.3 provides a very useful criterion for verifying whether a linear operator is a projector, without having to find the closed subspace in which it projects (see Exercise 3.2). It will be used so frequently in the following that no explicit mention will be made of the fact that the theorem has been applied while carrying out a certain step.

Theorem 3.4. Two closed subspaces M and N of X are orthogonal if and only if EMEN = ENEM = 0.

Proof. Since EM

=

EM*,we have

(3.9)

for all f,g E X . If M 1N, as we have EMf E M and ENg E N for all f,g~&', we get from (3.9) that (f I EMENg)= 0. Conversely, if EMEN = 0 and f,g are any vectors from M and N, respectively, due to EMf = f and ENg = g we obtain from (3.9)f 1g. Since EMEN = 0 implies that (see Exercise 2.7)

EMEN

= 0 = (EMEN)* = ENEM,

we see that the statement EMEN 3.3.

=

0 is equivalent to ENEM

=

0. Q.E.D.

PARTIAL ORDERING OF PROJECTORS

<

If M and N are two subspaces and M C N, we shall write EM EN, The following theorem gives us two additional criteria for this relation to hold.

Theorem 3,5.

The following three statements about the projectors

EM and EN defined on 2 are equivalent: (a) M C N, i.e., EM <EN ; (b) EMEN = E N E M = EM ; I/ E M f 11 \< 11 E N f 11 for allfE z *

Proof. If M C N, then EMf E N for any f E z,and consequently, by (3.81, EN(EMf)

for all f E 3. Hence, EM

11 EMf 11 since 11 EM

11

=

=

=

EMf

ENEM , which implies

11 EMENfll

< 11 E M

I/ 11 E N f 11

< / / ENfI/

< 1 according to Theorem 3.3(c). Thus, (b:

implies (c).

203

3. Projection Operators

Finally, to prove that (c) implies (a), note for any closed subspace MI we can infer from

Ilfl? thatfe

M, (i.e., f

=

=

+ 11 EMlLfl12

/I EMlf112

E M l f )if and only if

llfll If (c) is true, then whenever f

llfll

= 11

=

E

11 E M l f l l *

M, we have

EMfIl

< 11

Combining the above with the fact that we deduce that

11 f l l

=

ENflI*

(1 ENf (1

< I(f (1 for any f

EX

,

11 E N f I I .

This means, according to the aforesaid, that f E N, thus proving that M C N . Q.E.D.

<

The reason for the particular notation used in designating the relation between any two projectors satisfying the conditions of the preceding theorem is that this relation obviously introduces a partial ordering in the set of all projectors acting on #.

Definition 3.2.

A set S is said to be partially ordered if a relation

5 < 17 is defined for some pairs f , r) of elements of S, which satisfies the following conditions: (1)

6
(2) if f (3) if f

< <

for all f 7 and r) r ) and r )

E

S;

< f , then f = 7; < 5, then f < 5.

Other examples of partial ordering of a set are provided by the “equal or smaller” relation holding between real numbers, and by the relation C of set inclusion holding between the sets in a family of sets.

<

3.4.

PROJECTORS ONTO OF SUBSPACES

INTERSECTIONS AND ORTHOGONAL SUMS

Theorem 3.6. The product E M E N of two projectors EM and in AY is a projector if and only if EM and E N commute, (3.10)

EMEN

=ENEM.

E N

204

III. Theory of Linear Operators in Hilbert Spaces

If this is the case, the set L = M n N is a closed linear subspace of &' and EL

= EMEN.

Proof. If the linear operator EMEN is a projector, by Theorem 3.3, it must be Hermitian, and consequently EMEN = (EMEN)* = EN*EM*

= ENEM.

Conversely, if EM and EN commute, then EMEN is Hermitian, (EMEN)*

= EN*EM* = ENEM = EMEN,

and idempotent, (EMEN)' = EM(ENEM)EN

= EMzEN2 = EMEN,

i.e., EM EN is a projector. T o prove the second part of the theorem, assume that (3.10) is true, and denote by L the closed subspace of i%? onto which EMEN projects, so that EL = EMEN. Then we have

II E ~ E ~ l f< l l II E ~ f l l ,

II E J l l

=

II ELfll

= It E t ~ E ~ f l l II E N ~ I I ,

<

<

<

for all f E 2, i.e., according to Theorem 3.5 EL EM and E L EN, and consequently L C M n N. Conversely, if f E M n N, then ELf = &(EN f) = EMf = f and therefore f E L, i.e., L 3 M n N; thus, L = M n N. Q.E.D.

Theorem 3.7.

The sum

of a finite number of projectors defined on 8 is a projector if and only if (3.1 1)

EMiEM, = 0

i.e., if and only if MI, M,

for i # j ; i , j = 1,...,n,

,..., M,

are mutually orthogonal; in that case

205

3. Projection Operators

Proof. The linear operator A

= EMl

+ + EMn

is Hermitian, A = A*, since it is a sum of bounded Hermitian operators, A*

= EA,

+ + ER, = EMl + + EMn = A . *.*

* * a

Thus, A is a projector if and only if A whenever (3.1 1) is satisfied, since then

=

A2.This will be the case

Conversely, assume that A is a projector. Then for i # j (3.14)

( f IEMf)

+( f I

EMjf)

< =

(fI k-1

( f I Af>

< ( f If),

where in deriving the first inequality we have used the fact that for any projector


=


=

(Ef I E f ) 2 0,

f

€

2

9

and in deriving the second inequality we employed the property that

1) A 11 < 1 if A is a projector. From (3.14) we obtain

11 EM,f 11' If we set above f

=

+ 11 &,f

11' d I l f 112-

EMjf, we get

which implies EM,EM, f = 0 for all f E S,i.e., that M, ,..., M, are mutually orthogonal (see Theorem 3.4). To prove (3.12), assume that (3.1 1) is satisfied, so that A is a projector. Denote by M the space on which A projects, i.e., A = EM . I f f € M, then f

= A f = &,f

+ + EM,f€ M I @ ..*

which shows that (3.15)

M C M, @ * * - @ M,.

@ Mn,

206

111. Theory of Linear Operators in Hilbert Spaces

Suppose thatg E M, @ @ Mn . By Definition 3.1,g is in the closure of the linear manifold spanned by M, U U M, . This means that there n is a sequence g, ,g, ,... E ((Jk=, Mk) k

= g(1' k +

... + gp),

gL1)E M, ,...,gp) E Mn ,

,..., M,

converging in the norm tog. From the mutual orthogonality of M, we get n

llgk -g, 11'

1 (gk

( f ) - g;f)

=

1 g;)

c I[g;) n

=

- g;.4))

- (i)

g1 II2

.4=1

i,j=l

by using that fact that g t ) - g\') E Mi for any k, 1 = 1, 2, ... . Thus, for any fixed i = 1,..., n, IIg(k)-gy)II dllgk-glll*O

for K , z - t o 3 ,

so that g:", gy),... is a Cauchy sequence in M, . Since Mi is closed, this sequence has a limit g('). Thus, we can write our arbitrarily chosen element g from M, @ * * . @ M, in the form g

= g'l'

Moreover, since Mi

+ ... + gcn),

1Mi

g(l)E M, ,...,g(n)E M,

.

for i # j , we have Enig(i)= 8.. t d (j).

It follows that n

n

(3.16)

for the arbitrarily chosen g E M, @ @ Mn . From the above relation (3.16) we deduce that g E M, thus proving that @ M,.

M 3 M, @

(3.17)

The relations (3.15) and (3.17) combined together imply (3.13).

Q.E.D.

Theorem 3.8. The difference EM- EN of two projectors is a projector if and only if N C M; in that case EM - EN

where M (3.18)

=

EMGN

9

0N is the orthogonal complement of M

0N

= (f:f E

N with respect to M:

M , f l N}.

207

3. Projection Operators

Proof. Suppose that EM- EN is a projector, and let L be the space on which it projects: EM - EN = EL. Thus, EM is a projector which satisfies the relation EM = EL

+EN,

In this case we have according to Theorem 3.7 that ELEN= 0 and M=L@N. Conversely, if N C M, then by Theorem 3.5 EN

= EMEN = ENEM.

This implies that the evidently self-adjoint operator EM - EN is idempotent, (EM - EN)^ = EM - EMEN - ENEM + EN = EM - E N , i.e., EM - EN is a projector. Q.E.D. We shall now generalize Theorem 3.7 to the case of a direct sum of a countable infinity of closed subspaces.

Theorem 3.9. If M, , M2,... is a sequence of mutually orthogonal linear subspaces of a Hilbert space Z , and M is their orthogonal sum M

=

M i @ M2@

*.*,

then

for every f E S,

Proof. By applying Theorem 3.7 to EM, derive

i 11

k=l

EMkf112

=

i

k=l

(EM,flf>

=

+ + EMn we

easily

(i

k=l E M J l f )

= ( E M ~ Q . . . w IG~llf1I2, I~)

which shows that the series the sequence (3.20)

C,"=, 11 EM,f 112

converges. Consequently,

111. Theory of Linear Operators in Hilbert Spaces

208

is a Cauchy sequence, as can be seen from the relation

If we denote the limit of the sequence (3.20) by f the operator Af = f' is obviously linear and defined everywhere in A?. Moreover, we have I,

which shows that A = A* and A 2 = A. Thus, A is a projector. Denote by M the linear subspace of A? on which A projects. Iff E M, then f = Af = EMlf f &,f *.* E Mi@ Mz@ . * * ,

+

i.e., we must have M C M, @ M, @ . * * . If we assume that M # M, @ M, @ -.-, there would be a vector which but not to M . Since M, @ M, @ is a closed belongs to M, @ M, @ linear subspace of a Hilbert space and as such is a Hilbert space itself, we could deduce from Lemma 2.1 the existence of a nonzero vector g E M, @ M, @ ... which is orthogonal to M. This implies that g 1M, , k = 1, 2, ..., because we obviously have 11 EM,fII 11 Afll for all f E i@', and therefore M, @ M, @ C M . On the other hand, by Definition 3.1 of orthogonal sums, g is the limit in the norm of linear combinations of vectors from M, , M, ,..., i.e., we have vectors glk E M, ,..., g; E M,, K = 1 , 2,..., such that

<

lim II g - (glk k-m

Consequently, sinceg

+ + g i ) It

= 0.

1M , , M, ,..., we have

(g lg) =

Fz ( g I + + g ; ) = 0, g,k

**.

which shows that g = 0, thus contradicting the assumption that M # M, @ M, @ ..-. Q.E.D.

209

3. Projection Operators

Theorem 3.10. If EM, , EM, ,... is a sequence of projectors defined on 2 which is monotonically increasing

EM^ < EM,<

***,

or monotonically decreasing

,

Enl 2 E M ,2 then we have for any f E i%?

lim II(EM - EMn)fll = 0, n+m

where EM is the projector onto (see Chapter 11, Definition 1.2) M = lim M n . n+m

Proof. In the case of a monotonically increasing sequence, apply Theorem 3.9 to the mutually orthogonal projectors (take M, = {0}) EM,,M,-, = E M , - J%I~-~, k = 1,2,-. to deduce that

where

u m

m

M = @ (Mk

0Mk-1)

=

k=l

Mk.

k=l

The case of a monotonically decreasing sequence can be reduced to the above case by considering the projectors EM: = 1 - EMn * Q.E.D.

*3.5. APPENDIX:EXTENSIONS AND ADJOINTS OF

CLOSEDLINEAR OPERATORS

By using the concept of orthogonal complement, we can prove the following theorem which then easily yields the generalization of the Hellinger-Toeplitz theorem (Theorem 2.10) to any closed operators.

*Theorem 3.1 1. If A is a closed linear operator in A? and A* exists, then the domain of A* is dense in 2,and A** = A. is not dense in #, there is a nonzero vector h orthogonal Proof. If 9A* . But then we also have to gA*, i.e., (h I g ) = 0 for all g E gA* ( 0 I A*g)

+ ( h I -g>

= 0.

210

111. Theory of Linear Operators in Hilbert Spaces

In terms of the inner product ( * (3.21)

I

in X

( ( 0 , h ) I (A*& -g)>,

02,the above relation yields =0

for all g E 9,*. According to Theorem 2.5, which contains the definition of A*, the set (3.22)

{(A*g,-g): g E 9,*}c .@ 02%

is the set of all the points (g*, -g) E Z 0S satisfying (2.3). This set consists of all the points in Z 0Z which satisfy the relation (3.23)

z= 0

for all f~ g A ,i.e., which are orthogonal to the graph G, of A. The graph G, is a linear subset of S 0X because A is linear, and moreover, it is closed since A is closed. Thus, the set (3.22) is the orthogonal complement G; of the closed subspace G, of S 0Z.By (3.21), (0, h) is orthogonal to G i and consequently, (0, h) E G , . This implies that h = AO, i.e., h = 0, thus establishing that 9,* is dense in S. Since B,* is dense in 8, (A*)* = A** exists by Theorem 2.5. The graph of A** consists of all the points ( f , f * ) of Z @ Z satisfying (A*glf>

=

(gIf*>,

g E g A * ,

i.e., for which (3.24)

<(A*&-g) I

(f,f*)>2

= 0.

This implies that G,** = (GA)l = G, and consequently G,** which means that A = A**. Q.E.D.

=

G,,

Let A be a closed linear operator with B, = S. By Theorem 2.9 A* is a bounded linear operator with 9,* = X , and again by the same theorem, A** is also a bounded linear operator with 9,,* = X . On the other hand, by Theorem 3.11, A** = A, and therefore we can state Theorem 3.12.

Theorem 3.12 (The closed-graph theorem). All closed linear operators defined on the entire Hilbert space 2 are bounded. I t is also easy to deduce from Theorem 3.11 the following useful result. *Theorem 3.13. The adjoint A* of the linear operator A has a dense domain of definition in Z if and only if A has a closed linear extension; in that case A** is the smallest closed linear extension of A, i.e., any closed linear extension of A is also an extension of A**.

21 1

3. Projection Operators

Proof. According to Theorem 3.1 1, if A is closed, then a,,= X and A** = A, so that Theorem 3.13 stands. Conversely, if a,*= 2' then A** exists. From the relation
If>

=
1 Af),

gEgA*

3

we obviously get A** 2 A. Moreover, if B is a closed extension of A, then its graph G, is closed in 2' @ X and it contains G, . However, as can easily be seen from (3.23) and (3.24), we always have GA** = (Gi)I. This implies that GA*.is the closure of G, ,i.e., GA**C G, and, therefore,

A** C B. Q.E.D.

EXERCISES 3.1. Let S be a nonempty subset of a Hilbert space X . Prove that the set S A of all vectors f in X which are orthogonal to S

s1 ={f:

=O,gES)

is a closed linear subspace of X .

3.2. Use Theorem 3.3 to show that the linear operator A, acting on the two-dimensional space E2(2),which is represented (see Theorem 1.1) in some orthonormal basis of P(2) by the matrix 1 1 y ( - i

' 3

9

is a projector.

3.3. Show that any projector E on an n-dimensional Hilbert space X can be represented by a matrix of the form

1. 1

0

0 1

0 0.

in a suitably chosen orthonormal basis {el ,..., en], where the number of 1's in the diagonal (the rank of E) is equal to the dimension of the space on which E projects.

212

111. Theory of Linear Operators in Hilbert Spaces

3.4. Prove that any 2 x 2 Hermitian matrix either is a multiple = A*) of a matrix E representing a projector on Z2(2), or it can be written uniquely as AIEl A2E2,where El and E2 represent projectors on Z2(2), and A , , A, are real numbers.

AE ( A

+

4. Isometric and Unitary Transformations 4.1.

ISOMETRIC TRANSFORMATIONS IN BETWEEN HILBERT SPACES

We encountered unitary transformations in Chapter I (Definition 2.4) when defining the concept of isomorphism of two Euclidean spaces. The concept of isometric transformation presents a straightforward generalization of the concept of a unitary transformation, and since it will be of fundamental significance in scattering theory, we shall define it now.

Definition 4.1. A linear transformation T of a Hilbert space Zl into another Hilbert space S2(with inner products (. I -)1 and (-I .)2, respectively) which preserves the inner product, i.e.,

for all f,g E Zl , is called an isometric transformation. If in addition the range g Tof such a transformation is the entire space S2, i.e., 9,= Z 2 , then the transformation is said to be unitary. We have the following weaker condition of isometry. into Z 2is an Theorem 4.1. The linear transformation T of Zl isometric operator if

for all f E X I

Proof. T o prove the above statement, first substitute f (4.1) to derive

+ g for f in

213

4. Isometric and Unitary Transformations

+ ig to obtain < T ( f + 4!)I T ( f + igD2 = II Tf + iz + II Tg = ( f + ig I f + ig>l =Ilf + i ( f I g ) , - i
then f

112,

111,

>1

112,

11l2~

After multiplying the above equation by --i and adding it to the one preceding it, we arrive at the result ( T f I Tg>2 = (f I g>1

for all f , g E Sl . Q.E.D. It is easy to see that an isometric operator, mapping S into S, is necessarily unitary if S is finite dimensional (see Exercise 4.1). However, in the case of linear operators on infinite-dimensional Hilbert spaces these two concepts do not coincide. For instance, if {el , e2 ,...} is an orthonormal basis in the Hilbert space S, then the shift operator =

<ek \k=l

If>

ek

1

If>

=

ek+Z

k=l

is obviously isometric, but not unitary, for any fixed I = 1, 2,.... The relation that characterizes a unitary operator is U* = U-l, where U-l is the inverse of U . The definition of the inverse T-l of a linear transformation T is contained in the following statement.

Theorem 4.2. Let T be a linear transformation with domain BT and range 9,. T h e inverse mapping T-l of 9,into BTexists if and only if the zero vector 0, of g Tis the only vector mapped by T into the zero vector" 0, of g T .If T-' exists, then T-l is a linear transformation with = 9,and range Wrl = g T . domain gTb1 Proof. For the existence of the inverse transformation T-l of the linear transformation (44

f+f'=

Tf?

f E g T ,

f t E g T ,

it is necessary and sufficient that f # g (i.e., f - g # 0,) implies Tf # Tg [i.e., T (f - g ) # O,]. T h e existence of the inverse of (4.2) is thus obviously equivalent to the requirement that Tf = 0, has only the solution f = O1 . Naturally, 0, = O1 if 9,and W Abelong to the same vector space.

214

111. Theory of Linear Operators in Hilbert Spaces

If the inverse transformation T-l is defined on the linear set a*,it is obviously linear, as can be easily seen in the following way: if we have f = T-Y' and g = T-lg', f ', g' E 9T-l = a T it , follows that f ' = Tf and g' = Tg, and consequently for any scalars a, b, af '

i-e., af

+ bg'

= aTf

+ bTg = T(af + bg),

+ bg = T-'(U..' + bg').

AND 4.2. UNITARYOPERATORS OF ORTHONORMAL BASIS

THE

Q.E.D.

CHANGE

Theorem 4.3. A linear operator U , defined on the entire Hilbert space Z , is unitary if and only if

u*u = uu* = 1,

(4.3)

i.e., if and only if U-l

=

U*.

Proof. A unitary operator U is obviously bounded ( 1 1 UII = 1) and consequently its adjoint U exists and is defined everywhere in Z . Since

(f

lg) = (UfI ug> = ( f

I u*ug>

is true for all f,g E Z , we conclude that U*U 9,= Z we deduce that in the relation

=

1. Moreover, from

Uf assumes all the values in Z when f takes on all the values in Z . This implies that UU*g = g for all g E Z , i.e., UU* = 1. Conversely, if (4.3) is true for some linear operator U, we have U-l = U* and therefore

Furthermore, (4.3) also implies that U is isometric: ( U f I ug>

=

( f I u*ug>= < f Ig).

Thus, we have proved that U is unitary.

Q.E.D.

One of the most important features of unitary operators is that they perform the transformation of one orthonormal basis into another.

215

4, Isometric and Unitary Transformations

Theorem 4.4. (a) If {el, e2 ,...} is an orthonormal basis in the Hilbert space 2,then {Ue, , Ue, ,...} is also an orthonormal basis. (b) Conversely, if {el , e, ,...} and {el', e2', ...} are two orthonormal bases, there is a unique unitary operator U such that f?k' = Uek , k = 1, 2, ... . Proof. (a) The set {Ue, , Ue, ,...} is orthonormal, since due to the unitarity of U ( Ue, I Ue,) = ( U*Ue, I e,) = (ei I e,) = S,,

.

To see that { Ue, , Ue, ,...} is also complete, assume that for some f E 2 1,2,...

( f I Uek) = 0,

k

=

(U*f I e k ) = 0,

k

= 1,2,

.

Then we also have

The above relations imply that U*f {el, e, ,...}. Since UU* = 1, f

=

=

... .

0, due to the completeness of

U ( U * f ) = 0.

(b) Given {el, e, ,...} and {el', e,' on all f E X as follows:

,...}, we can define an operator U,acting

(4.4)

The series on the right-hand side converges in the norm since the sequence n

hn =

C (ek I f ) ek',

n

= 1,2,

...,

k=l

is a Cauchy sequence; this statement is a direct consequence of the relation m

11 hm

-

1'

= k=n+l

and the convergence of the series

I(ek I

f >Iz

216

111. Theory of Linear Operators in Hilbert Spaces

It is trivial to check that the operator U defined by (4.4)is linear. Furthermore, we have for any f,g E 2, (4.5) m

=

c

<ei

i.j=1

If>*M e i I g> = .

Hence, the operator U is isometric. From its definition (4.4) it is obvious that the range 9, of U is the entire Hilbert space 2.Therefore, U is a unitary operator. Q.E.D.

4.3. THEFOURIER-PLANCHEREL TRANSFORM The Fourier transform-which is the best known of all integral transforms and is frequently used in quantum mechanics-can be looked upon as being essentially the restriction of a unitary operator to some linear submanifold of an L2([Wn) space. In order to prove this unitarity property of the Fourier transform, we need a few auxiliary lemmas and theorems.

Lemma 4.1 (The Riemann-Lebesgue lemma). If f (x) is a complex function which is Lebesgue integrable on the interval I, I C OX1, then lim

A-m

11

f ( x ) sin Ax dx

=

lim A-?w

I I

f ( x ) cos Ax dx

= 0.

Proof. We shall prove the lemma only for the sine integral, since an analogous proof applies to the cosine integral. Assume that I is a finite interval, for instance that I = [a, b]. Take a function g(x) which is differentiable on [a, b], and for which (see Exercise 4.2)

If@> -&)I

dx

< 42-

Integrating by parts we derive /lg(x) sin Ax dx = -

g(b) cos Xb - g(u) cos hu h

Consequently, for sufficiently large values of A,

I

I l g ( x ) sin Ax dx

I<

42,

217

4, Isometric and Unitary Transformations

and therefore for such A,

I

/ : f ( x ) sin Ax dx

I
/:g(x)sinAxdx

I + s:

If(x) -g(x)J

I s i n k I dx < c,

which proves the theorem. The case of infinite I can be reduced to the above case by chosing a and b, a < b, so that In that case

and the previous considerations apply.

Q.E.D.

Lemma 4.2. Iff(x) is once continuously differentiable and Lebesgue integrable on the interval I in Rn, and if 0 E Rn is an internal point of I, then (4.7)

1 lim -

sink, J,f ( u + x ) x1

..*

r\+rnnr"

sinhx,

-dnx

xn

=f(~).

Proof. We shall give the proof for the case n = 1. The general case can be easily reduced to this case (see Exercise 4.3). Since the origin of R1 is an interior point of I, we can choose a > 0 so that [-a, +a] C I. Then

where by Lemma 4.1 the second integral approaches zero as h --t The function h(x) =

f

f(u

+ x ) -m, X

f

+

00.

0

f'(u), x =0 is a continuous function of x because f ( x ) is continuously differentiable. Therefore, after writing

111. Theory of Linear Operators in Hilbert Spaces

218

we can apply Lemma 4.1 to the second integral on the right-hand side of (4.9) to infer that this integral vanishes in the limit A -+ 00. Since sin Ax dx 1imJ +a -I*m -a x

=

lim A*m

1

+A=

sin t -

+*

sin t

dt = m,

we obtain (4.7) by combining (4.8) and (4.9) and then letting A + 00.

Q.E.D. Theorem 4.5. If f ( x ) E W1(Rm) is Lebesgue integrable on Rm, then its Fourier transform* J(p) = (27ffi)-"/2 /,,f(x)

(4.10)

e - ( i l f r ) p . x dmx

is bounded and continuous on Rm, and if DA

[-A,

=

[-A,

+A]

x

-.. x

+XI,

(4.1 1)

f(x) = (2~rfi)-"/~lim A+m

JD,

f(p)

e ( i 1 f r ) X . pdmp.

If, in addition,f(x) is of compact support in Rm, then (4.12) Proof. The integral on the right-hand side of (4.10) is uniformly convergent in p due to the fact that f ( x ) is integrable and

If ( x ) e--(ilh)p.z I < If ( x ) l . Hence, f(p) exists, it is continuous, and by Lemma 4.1 it is bounded. In order to derive (4.11), first apply Fubini's theorem (Chapter 11, Theorem 3.13) to invert the order of integration, and then Lemma 4.2 to obtain upon setting = f i - l I 2 x , p' = fi-lI2p, h' = fi-lI2A, XI

* The

definition found in mathematical literature corresponds to the choice fi = 1.

219

4. Isometric and Unitary Transformations

-

s,

I f(p)12 dmp. Q.E.D.

Theorem 4.5 constitutes one of the most important results in the theory of the Fourier transform. We are interested, however, in the Fourier transform as an operator on a L2(Rm)space. In that context the following result is of greatest interest to us.

Theorem 4.6. There is a unique linear and bounded transformation U , of L2(Rm)into itself, which maps every element f E Ug(Rm) into

f=

UFf,

f g%c(RBm), fEL2([Wm),

where f(p) is related to f ( x ) by the Fourier transform (4.10); U , is a unitary operator, called the Fourier-Plancherel transform on Rm. Proof. The integral operator defined by (4.10) and having Uc(Rm)as (; Rn) is dense in L2(Rm) a domain of definition is obviously linear. Since % (Chapter 11, Theorem 5.6), and since by the relation (4.12) in Theorem 4.5 the present operator is isometric on V;(Rm), it has according to Theorem 2.6 a unique bounded linear extension to L2(Rm).This extension U , is isometric (see Exercise 4.4). In order to prove that U , is unitary, i.e., that 9?+ = L2(Rm),it is sufficient to show that it maps a dense set onto another dense set. This last property follows from Theorem 4.5; namely, we easily see from (4.10) and (4.1 1) that the dense set consisting of all of functionsf(x) E Um(Rm) for which x$ @ ; f ( x ) is bounded on Rm for any . . ,...,j, = 0, 1,x k2,... is mapped onto itself by U , . Q.E.D. il ,..., 2, , "4.4.

CAYLEYTRANSFORMS OF SYMMETRIC OPERATORS

An important category of isometric operators are the Cayley transforms of symmetric operators A, (4.13) v = ( A - i)(A $1,

+

which play a key role in the theory of self-adjoint operators-as see later in this chapter.

we shall

220

111. Theory of Linear Operators in Hilbert Spaces

*Theorem 4,7. If A is a (closed) symmetric operator in a Hilbert space X , its Cayley transform V = (A - i)(A i)-l exists as a linear operator in X , and it is an isometric transformation of the closed subspace 9A+i onto the (closed) subspace

+

The existence of the operators (A f i)-l follows from the

Proof. relation (4.14)

ll(A

fi>fllZ= (Afl Af> =F i ( f I Af> fi+ = 11 A f l l z + llf112, f E g A

and Theorem 4.2, since according to the above relation (A & i ) f = 0 impliesf = 0. We shall prove now that the ranges 9tAfiof the two operators A f i are closed linear subspaces of X if A is closed. Since WA*dare the ranges of linear operators, they are linear. T o show that 9A+a and WAi are closed sets, we first derive from (4.14) that

llfll

< ll(A fi)fll,

f E B A

+

+

Thus, if the sequencef,‘ = (A i ) f l ,fz‘= ( A i)fi ,... converges to somef‘ E X , the sequence f, ,fz ,... also converges to somefE X . Since A i is obviously closed when A is closed (see Definition 2.4), we must have f E g Aand f’E 9A+i . A similar argument establishes that 9A4 is closed too. and its range is g A ,while The operator (A i)-l is defined on A - i is defined on gA and its range is 9 A - iConsequently, . V = (A - i)(A i)-l maps WA+i onto W A - i . T o establish that V is isometric, note that (4.14) yields

+

+

+

- i, g

If we insert above g

=

(A ll(A

for all f~

11

=

+ i, g 1,

g

gA

*

+ i)-’jf,E WA+i , we get -

i)(A

+ iYfIl = llfll

, i.e., V is isometric. Q.E.D.

In the above Theorem the term “closed” is optional, but it should be noted that each densely defined symmetric operator A has at least one closed extension, namely A**; we see immediately that A** 2 A by looking at the relation (A*fIg)=
fE9A*,

gEBAr

and recalling that since gA* 3 gAand 9,is dense in 2,gA*itself is dense in X, and therefore A** exists and is closed (see Theorem 2.8).

221

4. Isometric and Unitary Transformations

Moreover, as we shall see later, A** C A*, and since obviously A* C A***, it follows that A** is symmetric.

*Theorem 4.8. If the Cayley transform V of the linear operator A exists, then 9, = W1-"and (4.15)

A

= i(1

+ V)(1 - V)-1.

Proof. According to Definition (4.13) of V, 9"consists of all vectors g for which there is an f E g Asuch that (4.16)

g = (-4

+i)f.

When V is applied to the above g, we get Vg

(4.17)

= (A-i)f.

By adding and then subtracting (4.16) and (4.17) we obtain (4.18) (1 - V ) g = Zf. (1 + V ) g = 2 4 , The second of the above relations shows that (1 - V)-l exists; namely, if (1 - V)g = 0, the aforementioned relation implies that f = 0, which due to (4.16) implies in its turn that g = 0, thus establishing, in accordance with Theorem 4.2, the existence of (1 - V)-l. Hence, we can write g = 2i(l - V ) - l f , which in conjunction with the first of the relations (4.18) yields Af=i(l

+ V)(1 - V)-'f,

f E g A *

Moreover, from the second of the relations (4.18) we see that = B A , so that the domains of definition of A and i(1 V)(l - V)-l are identical, i.e., (4.15) is true. Q.E.D.

9(1+)-1

+

The following theorem is of great practical importance in establishing the self-adjointness of a symmetric operator.

*Theorem 4.9. The symmetric and densely defined operator A is self-adjoint if and only if its Cayley transform is a unitary operator. Proof. Assume that A = A*. We shall prove that the domain of definition W,+i of V is dense in 2. Assume that h 1W,, , i.e., that (h I (A i ) f ) = 0 for all f E g,, , or, equivalently, that (h I (A i ) f ) = (0 If) for all f E g A. According to Theorem 2.5 we have h E 9(,,+(p and

+

+

( A + i ) * h =(A* - i ) h = ( A - i ) h

=o.

222

111. Theory of Linear Operators in Hilbert Spaces

I t follows from (4.14) that ( A f i)h = 0 only if h = 0. Thus, RA+iis dense in X , and since WA+iis closed (see Theorem 4.7), we must have g v= wA,i = 2. It can be proved in a completely analogous manner that BV= W,+ = X . T h e identities 9A*i = X establish, in conjunction with Theorem 4.7, that V is unitary when A = A*. Assume now that V is unitary and A* exists. We shall prove that in this case A = A*, where A* exists since gA*is dense in X . Take g E gA* and set g* = A*g. Then (g

I

If>

= (g*

for allfE g A. According to (4.15), for eachfE g Athere is an h E 9" = A? such thatf = (1 - V)h and Af = i(1 V)h. Hence

+

i(g I(1

for all h

EX

+

, or

+ i(g

i(g I h)

Vh)

=

Ih)

(g*

-

(g*

I Vh).

Due to the unitarity of V, ( g I h ) = (vg I


and

vh)

and the preceding relation yields (-ig

- ivg - Vg* + g *

1 Vh) = 0

for all h E 2.Since V is unitary, Vh assumes all the values in X when h takes on all the values in 8, and therefore i(1

+ V)g

-

(1 - V ) g * = 0.

Using the above relation we obtain (1

which shows that g

g

-

=

E

i(1

V)-

-

ig*

2

=g,

g A; the same relation yields

+ V ) g-ig* 2

which shows that g* = Ag. Thus, A* A* = A. Q.E.D.

=g*,

C A, and

since A is symmetric,

223

4. Isometric and Unitary Transformations

We have already seen in Theorem 4.7 that &'A+i and 99A-i are closed linear subspaces of A? when A is a closed symmetric operator. Let us denote by m and n the dimensions of the ortogonal complements of W A 4 and @ ' a + i , respectively, m

=

dim W i - i ,

n

=

dim &';+dm

Naturally, m and n can assume any nonnegative integer values, including 03 when the corresponding space is infinite dimensional. Moreover, if X is not separable, we have to distinguish between infinite dimensions of separable and nonseparable subspaces. The two numbers m and n are called the deficiency indices of A , and the are the deficiency subspaces of A . According two subspaces 9i-i and to Theorem 4.9, A is self-adjoint if and only if m = n = 0. If the deficiency indices of A are equal but not zero, then we can easily extend the Cayley transform V of A to a unitary operator Vl on X . This can be achieved by choosing any two orthonormal bases {el', egr,...} and {el", e2",...} in a;-, and @i+i, and extending V to &'i+i by writing Vlek" = ekr.Since this extended Vl is unitary, the operator

+

A,

= i(1

+ VJ(1

-

V,)-1

is self-adjoint by Theorem 4.9, and it obviously provides an extension of A. Hence, if m = n > 0, the closed symmetric operator A will have many self-adjoint extensions. If, however, m # n, then A has no selfadjoint extensions. As a matter of fact, if A, were a self-adjoint extension of A, then the Cayley transform V , of A, would be unitary, and it would be an extension of the Cayley transform V of A. Since obviously V maps @iFi isometrically onto 9i+i, this would imply that m = n, which contradicts the assumption. I n summary, we can state the following theorem (see also Lemma 7.1 in Chapter IV).

Theorem 4.10. A closed symmetric operator A has self-adjoint extensions if and only if its deficiency indices m = dim9?i-d and n = dim @i+i are equal; moreover, A = A* if and only if rn = n = 0. Theorems 4.7-4.10, and in particular Theorem 4.10, indicate that the self-adjointness or essential self-adjointness of a Hermitian operator in an infinite-dimensional Hilbert space is by no means a "natural" feature of such an operator. On the contrary, there are many more symmetric operators which have no self-adjoint extensions than symmetric operators which possess self-adjoint extensions; namely, to any isometry which is not the restriction of a unitary operator there corresponds a closed symmetric operator without self-adjoint extensions. T h e reader should

224

111. Theory of Linear Operators in Hilbert Spaces

keep this in mind when trying to decide within the context of some quantum mechanical theory whether or not some obviously Hermitian differential operator represents a self-adjoint operator.

4.5. SELF-ADJOINTNESS OF POSITION AND MOMENTUM OPERATORS IN WAVEMECHANICS As an easy application of the results obtained in this section, we can prove the self-adjointness of the j-coordinate position operator Qj and the canonically conjugate momentum operator Pj . We define Qj as the operator with domain gQj consisting of elements f of L2(Rm),representable by functions f ( x l ,..., xm) which are such that xjf(xl ,..., xm) are Lebesgue square integrable, and acting on f E gQ{ in the following way:

(Qjf )(XI

(4.19)

9-7

xm) = % f ( x i

,.**)

f E gQj

xm),

9

The above operator is obviously symmetric (see Exercise 4.5). T o prove that Qj* = Q j , we will show that the Cayley transform (Q, - i)(Qj i)-l is unitary (see Theorem 4.9). we can write The ranges of Qj f i are L2(Rm);e.g., if g €L2([Wm),

+

g(x1 ) - * vx m ) = (xi

+ i) 4 x 1

,*-*)

xm),

where obviously h E gQ, . Moreover, the operator (Qj - i)(Qj isometric,

+

i)-1

is

and since its domain and range are (see Theorem 4.7) @Q,+.i and respectively, this operator is unitary. Hence, Qj is self-adjoint. We define P, by (4.20)

Pj = U;lQjUF,

Qp9 =

UklgQj,

where UF is the Fourier-Plancherel transform introduced in Theorem 4.6; in (4.20) we used a very useful notation, according to which if T is any transformation mapping the set S, C g Tonto S , we write (4.21)

S,

=

TSl

= {Tf: fE

Sl}.

225

4, Isometric and Unitary Transformations

The self-adjointness of Pi follows from the unitarity of U, and the following lemma.

Lemma 4.3. If A is a self-adjoint operator in 8, and U is a unitary then the operator operator on S, B

=

UAU-',

9~= U ~ A

is self-adjoint.

Proof. Since obviously .GB = S because gA= 8 (see Exercise 4.6), it follows that B* exists. We must have B* 2 UA*U-l = B due to the following relation: ( f I Bg)

=

( f 1 UA*U-lg)

=

(U-lfl A*U-lg)

=

(AU-lfI U-'g)

=

(UAU-lflg)

= (Bf

Ig),

f y g € UBA.

By inverting the roles of A and B, and noting that U-l is unitary, we get A* 1 U-'B*U, which implies that B* C UA*U-l = UAU-l = B . Hence, B = B*. Q.E.D.

As a result of the above considerations we can state Theorem 4.1 1.

Theorem 4.1 1. The wave mechanical canonical position and momentum operators Qjand P, , defined by (4.19) and (4.20),are selfadjoint. In order to see that (4.20) is nothing but a precise definition of the symbol -ifi(a/8xj), note that for f belonging to the set U;l%;(IWm) [which according to Exercise 4.6 is dense in L2(Rm)since %,$P)is dense] we have p5f = UF1Qj.L where ( Q i J ) ( P l ,-..,Pm)= P 5 f h ,..-,Pm)= g(Pl

Y.*.Y

Pmh

Hence (f'jf)(xl , * * * *

xm) = (Ui'g)(xl ,.**, x m ) = (27rh)-m/2

p j f ( p l ,...,p m )e(ilfi)'"'x dmP

111. Theory of Linear Operators in Hilbert Spaces

226

where the interchanging of the order of differentiation and integration in carrying out the third step in the above derivatioii is easily deducible from standard theorems of the calculus, when the fact thatfE %‘:(OXm) is taken into account. EXERCISES 4.1. Prove that an isometric operator V defined on a finitedimensional Hilbert space 2 is necessarily unitary. 4.2. Prove that if f ( x ) is Lebesgue integrable on [a, b ] , there is for any given E > 0 an infinitely many times differentiable function g ( x ) E q m ( [ a ,61) such that

4.3.

Prove in detail Lemma 4.2 for any integer n.

4.4. Prove that if A and B are bounded linear operators satisfying A C B , and if g Ais dense in Q B , then B is isometric when A is isometric. 4.5. Show in detail that the operator Qj defined in the subsection 4.5 [see (4.19)] is linear and Hermitian, and that Qj*2 Qj . 4.6. Show that under a unitary transformation the image of a dense set is also a dense set.

5 , Spectral Measures 5.1. THEPOINTSPECTRUM OF

A

SELF-ADJOINT OPERATOR

We encountered the eigenvalue problem when determining the bound-energy spectrum of a quantum mechanical system from the wave mechanics point of view. In this section we shall formulate the eigenvalue problem for symmetric operators in general.

Definition 5.1. A complex number h is said to be an eigenvalue (characteristic value) of the linear operator A, with domain of definition BA, if there is at least one nonzero vector f E Q A satisfying the eigenvalue equation (5.1)

Af

=

Af.

Each vector f E g Asatisfying the above equation is called an eigenvector (or characteristic vector) of A with eigenvalue A. All the eigenvectors with

227

5. Spectral Measures

the same eigenvalue h form a linear subspace MA, which is usually referred to as the characteristic subspace corresponding to the eigenvalue A. If MA is infinite dimensional, then MA is not necessarily closed. However, it is easy to see that the characteristic subspaces of closed linear operators are always closed.

Theorem 5.1. If the linear operator A is symmetric, all its eigenvalues are real and the characteristic subspaces corresponding to different eigenvalues are mutually orthogonal. Specialized versions of Theorem 5.1 were proved in Chapter I (Theorem 5.1) and Chapter I1 (Theorem 5.3). The proof in the general case proceeds along the same lines. T o derive that an eigenvalue h is real if A is symmetric, choose any nonzero eigenvector f and use its symmetry property to write

Xf I f >

=

(f I Af>

=

= (Af If>= = A*,

thus showing that h = A*. The orthogonality of two characteristic subspaces MA1 and M A S , corresponding to two different eigenvalues A, and A, , follows from the orthogonality of any two respective eigenvectors fiE MA1 and f, E MAz :

Akfl If,>= Qlfl Ifi>= (Afl If,>

=

(fl I Afz>

=

(fr I A,fz>

=

Mfl If,>.

The set S{ of all eigenvalues of an operator A is called its point spectrum (see also Definition 3.3 in Chapter V). 5.2.

SPECTRAL RESOLUTION OF SELF-ADJOINT OPERATORS PUREPOINTSPECTRUM

WITH

Consider a closed symmetric operator A for which the closed linear space spanned by its eigenvectors is the entire Hilbert space 2‘ (an operator with a so-called pure point spectrum), i.e., for which, by Definition 3.1,

iF= @ M A , AsS;

where :S denotes the point spectrum of A. We assume that % is separable, so that S; is necessarily at most countable, and hence it can be written in the form sp” = {Al, A, ,...}. By using Theorem 3.9 we get

228

111. Theory of Linear Operators in Hilbert Spaces

If we have for s ,me f

then the sequence f l ,f i

,...,

(5.3) converges to f , while A f l , A f 2 ,..., n

II

(5.4)

is a Cauchy sequence, due to (5.2). Since i%? is complete, this sequence has a limit g. Therefore, A being closed, we must have that f E g Aand

Vice versa, iff E B A ,(5.2) must be satisfied. T o see that, note that the restriction of A to MA1@ @ MAn is a bounded operator and therefore, A being closed, it must be defined everywhere on MA1 @ -.-@ MAn. Hence, the vector f n , defined in (5.3), belongs to g A .Moreover, A obviously maps MA1 @ @ MAn into itself, and therefore (Afn

I A ( f -fn)>

=

(AYn

If

-fn>

= 0.

By using this relation we immediately get

which shows, in conjunction with (5.4), that (5.2) must be satisfied if f E 9 A .

The above arguments can be applied to A*, A* 2 A, to derive that gA* consists of all vectors f satisfying (5.2), so that gA*= g,,. Hence, we have proved Theorem 5.2.

Theorem 5.2. If the closed linear manifold spanned by all the eigenvectors of a closed symmetric operator A acting on a separable Hilbert space i%? is identical with i%?, then A is self-adjoint, and 9A consists of a the vectors f satisfying (5.2).

229

5. Spectral Measures

In the considered case, the complete solution of the eigenvalue problem is accomplished when the characteristic subspaces MA,, MAS ,... are found, ,... have been specified. or equivalently, when the projectors EM EMAS A1 ’ However, this formulation of the eigenvalue problem is not sufficiently general, because it does not cover the cases of self-adjoint operators whose eigenvectors do not span i%? (such as the general case of the Schroedinger operator), or do not even have any eigenvalues at all corresponding to Hilbert space eigenvectors (such as the position and momentum operators in quantum mechanics). Fortunately, one can solve the most general case of self-adjoint operators after rephrasing the above offered solution of the eigenvalue problem for operators A with pure point spectrum in the following way: the knowledge of the projectors E M , EMA2 ,... on the characteristic subspaces MA1,MA,,... of A is A! equivalent to the knowledge of the projector-valued set function (5.5)

E(B) =

1

EMA,

BE@,

AEBnS;

for all Bore1 sets B on the real line. The set function E(B),B E al,whose assumed values are projectors, has certain properties which are common to a class of projector-valued set functions which are known as spectral measures. Moreover, these properties, which we shall formulate and study later in this section, make E(B) unique in the sense of E(B) being the only spectral measure that can be associated with A. When it is thus generalized, the eigenvalue problem becomes the problem of determining the spectral measure of A, and as such it can be applied to any self-adjoint operator-as we shall see in 96.

5.3. WEAK,STRONG, AND UNIFORMOPERATOR LIMITS The above approach to the problem necessitates a generalization of the concept of measure which would also apply to operator-valued set functions. A careful analysis of the concept of limit of sequences of operators is essential for that. As seen in the following definitign, we can distinguish three main concepts of limits for operator-valued functions.

Definition 5.2. Let A(t) denote an operator-valued function on some subset S of R1, assigning to each t E S a bounded operator A(t) acting on a Hilbert space 2.We can define A to be a limit of A(t) when t tends to to (where to is a real number or kco) in the following three ways: (a) If for all f,g E 2, (5.6)

(f I 49 = ;iT(f I 4 9 g h

230

111. Theory of Linear Operators in Hilbert Spaces

A is the weak limit of A ( t ) when t -+t o , and we write A

A(t)

= w-lim t+tO

-

A(t)

or

t+t0

A.

(b) If for allf € 2

gI I ( 4 )

(5.7)

- 4fll = 0,

A is the strong limit of A ( t ) for t -+ to : A

= s-lim t+tO

A(t)

A ( t )---+A.

or

t+tO

(c) Finally, if 11 A - A(t)ll -+ 0 when t limit of A ( t ) and write A

A(t)

= u-lim t+to

+t o ,

we call A the uniform

A(t)

or

A.

These three definitions of limits are related in the sense that a uniform limit is also a strong limit, and a strong limit is also a weak limit, though the converses of these statements are by no means true. Thus, from the inequality ll(A - 4t))fll < It A - 4)llllfll we derive that when A = u-limt+lo A ( t ) , then (5.7) is also true. By means of the Schwarz-Cauchy inequality we get

I(f l ( A - A(t))g)l < llfll ll(A - 4

) ) g /I,

which shows that (5.7) implies (5.6). As an example of strong convergence which does not imply uniform convergence, consider a sequence E l , E, ,... of mutually orthogonal nonzero projectors. According to Theorem 3.9, there is such a projector E that n

E

However, the sequence

=

Ek .

s-lim n+m

k=l

ZP2),..., n

E(") =

Ek ,

n

=

1,2,...,

k= 1

is not uniformly convergent, since it does not satisfy the necessary condition (see Exercise 5.3) of uniform convergence:

11 E(n+l)- E(n)ll+O

as n-+

00;

231

5. Spectral Measures

in fact, since En+,# 0, we have

1) E(n+l)- E'")))= 1) En+l1)

=

1.

Definition 5.2 suggests that there are two different kinds of convergence of sequences of vectors in a Euclidean space.

Definition 5.3. Iff(t) is a vector-valued function assigning a vector f ( t ) from the Euclidean space d to each value t from the set S, S C W , we callf a weak limit off(t) for t -+to ,

I f ) = lim,,,o(g I f ( t ) ) for all g E &; we say that f is a strong limit off(t) for t + to , if (g

f = s-limf(t) t+tO

xf,

f(t)

or

if limbtoII f -f(t)ll = 0. From the inequality Kg I f

- f ( W < llg 1I Ilf -f(9IL

we conclude that iff is a stronglimit off(t), then it is also a weaklimit. The converse is by no means generally true. It has to be emphasised that we have been considering until now only limits in the norm for sequences of vectors. T h e concept of the limit in the norm is obviously identical with the concept of strong limit.

5.4. SPECTRALMEASURES AND COMPLEX MEASURES We are now ready to define the concept of a spectral measure. T h e reader should review Definition 2.1 in Chapter I1 of an ordinary measure, and note the formal analogies.

Definition 5.4.

A spectral measure E ( S ) on the measurable space

(S, d)is a projector-valued function which assigns a projector E(S)to each element S of the Boolean u algebra d in such a manner that (1) E ( 3 ) = 1;

(2) E

(CJ

k=l

c E(Sk),if S, , s,,..., is any sequence m

sk) =

k=l

from d. In the last relation above, we define m

n

of disjoint sets

111. Theory of Linear Operators in Hilbert Spaces

232

We agree, in general, that given an infinite sequence E l , E, ,... of projectors, the sum of their infinite series shall always mean, if not otherwise stated, the strong limit of the partial sums:

If we choose in the relation (5.8)

E

(pl

c m

sk) =

si n sj

E(Sk),

= %

for i # j ,

k=l

S 2 -- S 3 -= 0 we get E ( 0 ) = 0. By taking above instead just S3 -- S 4 -- -.. = 0,we obtain E (S , u S,) = E(S,) E(S,) if S, n S, = 0 , so that E(S,)E(S,) = 0 by Theorem 3.7. More generally, setting B = R n S, S, = R - B, S, = S - B for R, S E d,we get E(R)E(S)= [E(S,) E(B)][E(S2) E(B)] = E2(B) = E(B) since S, n S, = S,n B = S, n B = o.Hence we can state the following theorem.

+

+

+

Theorem 5.3. If E ( S ) is a spectral measure on (X, d), then E( o ) = 0 and E(R)E(S)= E(R n S) whenever R, S E d. The concept of a spectral measure can be related to that of a measure most conveniently by introducing the concept of a complex measure. For our purposes the following definition of a complex measure is most adequate, though it is not the customary one.

Definition 5.5. A complex measure p ( S ) on the measurable space (3,d )is a finite linear combination, with complex coefficients c1 ,..., c,

c Y

P(S) =

CuPdS)

u=l

of a given number v of Jinite measures pl(S),..., py(S) on (%, d). A measurable functionf(0, E X , is integrable on R E d with respect to p if and only if it is integrable on R with respect to each of the measures pl ,..., pv ; in that case, by definition, (5.9)

With the above definition of integration most of the results of Chapter 11, $3 on integration can be easily generalized directly to complex measures. Moreover, it is easy to see that the above integral (5.9) is

233

5, Spectral Measures

independent of the way the complex measure p is written as measures. That is, if

”

(5.10)

P(S) =

c

c

a

sum of

V’

~ u P u ( S >=

CB’PB’(S)¶

B=1

u=l

any function f ( e ) integrable on R with respect to pl ,..., py is also integrable with respect to pl’, ..., pLr and (5.1 1)

The simplest way to verify (5.11) is by noting that forf(() = xs((), S C R, (5.1 1) becomes (5.10) and is therefore certainly true. Consequently, (5.11) is also true when f(() is a simple function. Following the usual procedure, indicated by Definitions 3.4 and 3.5 of Chapter 11, one can arrive at the verification of (5.11) for the most general case whenf(8) is an extended complex-valued function on R. The reader can easily convince himself that a complex measure is a complex-valued o-additive set function on the measurable space (3,d). It is worthwhile to mention that due to a fundamental theorem of measure theory” the converse is also true: any finite complex-valued a-additive set function on a measurable space (3, d )is a complex measure (see Theorem 5.8).

*Theorem 5.4. A projector-valued set function E ( S ) , S E d, defined on the measurable space (%, d )and such that E ( X ) = 1, is a spectral measure if and only if for any two given vectors f , g P f m

(5.12)

is a complex measure on Proof.

(5.13)

=

(f I E ( S )g),

S E ,rQ>

(X, d).

From the easily verifiable relation

(f

containing on the right-hand side only set functions of the form (h I E(S)h), which are obviously measures, we immediately deduce that P,,~(S) is a complex measure.

* Hahn’s theorem: see Theorem 5.7 in

Appendix 5.6.

111. Theory of Linear Operators in Hilbert Spaces

234

Assume now that, conversely, (5.12) is a complex measure for any two given f,g E 2. This implies that P ~ , ~ (isS )u additive, i.e., for any sequence S, , S, ,... of disjoint sets from d

(5.14) (f I E(S1 U S2 U *.*)g) = pr,p(S1U S2 U m

By taking in the above relation S,

E(S1 u S2)

* a * )

00

=

S,

=

=W I )

= 0 , we

derive

+ E(S2)

for any two disjoint S, , S, E: d.This implies, according to Theorem 3.7, that E(Sd E ( S J

=0

for any disjoint sets S, , S, E d.Thus, when, in general, the sets in S, , S , ,... E d are disjoint,

C E(Sn) = s-lim c E(S,) n+ m k=l

n=l

exists by Theorem 3.9, and due to (5.14) we have

This establishes that E ( S ) is a spectral measure.

Q.E.D.

In defining the concept of a spectral measure E ( S ) we required that S should assume values from a Boolean u algebra instead of an algebra. This requirement does not entail any loss of generality, as can be seen from the following analogue of Theorem 2.3 in Chapter 11.

*Theorem 5.5. Let Eo(S) be a projector-valued set function defined on a Boolean algebra d oof subsets of a set %, Suppose that E o ( X ) = 1 and EO

Lu

sk

1

m

=

EO(sk)

k l

for any sequence S , , S2 ,... of disjoint sets from d o ,for which S, E d o .In that case there is a unique spectral measure E(S),

u;=,

235

5. Spectral Measures

S E d ,defined on the Boolean cr algebra d generated by do, which is an extension of Eo(S),i.e., Eo(S) = E ( S ) for all S E do. Proof, T o prove the theorem, we shall follow the lead suggested by the proof of Theorem 2.3 in Chapter 11, and introduce for each subset R of % the equivalent of an infimum of the family of all projectors (5.15)

u m

f

Eo(sk),

k=l

s k 3

R,

Sk

E

4 , si n s k

= 0

for i # k.

k=l

This “infimum” E(R)can be defined as the projector onto the closed subspace M(R) which is the intersection of all subspaces on which the projectors (5.15) project (see Exercise 5.7). I n this way, we have defined a projector-valued set function on the family of all subsets R of 3, which obviously has the property that

(f 1 E ( R ) f ) = inf

f(I

f

I

EO(Sk)f):

k=l

fi

Sk

R,

sk

k-1

4

I

*

According to Chapter 11, Theorem 2.3, ( f I E ( S ) f ) , S E d ,is a measure on d.Hence, as we see from (5.13), it follows that ( f I E ( S ) g ) , S E d ,is a complex measure on a2 for any vectors f and g . Thus, E ( S ) , S E d , is by Theorem 5.4 a projector-valued measure on d , which evidently coincides with Eo(S)for S E d o . Q.E.D.

5.5. SPECTRAL FUNCTIONS A concept very closely related to that of spectral measures, and also very useful when formulating the spectral theorem in 96, is that of spectral function. We shall give a definition of spectral function in Rn, rather than limiting ourselves to the more conventional special case of n = 1. on Rn Definition 5.6. A projector-valued function E,,= EAI,...,,,, is called a spectral function (or spectral family, or decomposition [resolution] of the identity) if it satisfies the following requirements:

< E,,, whenever A < A‘,

(1) E,,

(2) E,, = E,,, (3) E-, where A’

A,

<

A1’)

=

=

A ‘ 4 ,,‘>A

s-l-l~ E,, = 0

3 A for A

*..,A,

s-lim E,,,,

< A,‘.

=

(A,

and

E+m = s-lim E,,= 1, A-bfm

,..., A,)

and A’

=

(Al‘ ,...,A,’)

means that

111. Theory of Linear Operators in Hilbert Spaces

236

T o every spectral measure E(B), B e a n , on

*Theorem 5.6.

(W",an)corresponds a unique spectral function E, ,X E R", satisfying (5.16)

and, conversely, to each spectral function on Rn corresponds one and only one spectral measure on (UP, 93"") satisfying (5.16).

Proof, If E(B), B E an,is a spectral measure, the projector-valued function E, , h E Rn, defined by (5.16) satisfies the first requirement of Definition 5.6 since for A' A,

>

+

E,* = E(z,*) = E(ZA) E(I,, - 1,) 2 E(I,) = E, ,

while the other two requirements of Definition 5.5 are satisfied by EA due to the continuity from above of the spectral measure E(B) (see Exercise 5.8); for instance, s-lim EA, = s-lim E(I,,) = E(I,) = E, A ' X A

S

A'>A A'*,

4

.

T o prove the converse, define a projector-valued set function E(B) on the Bore1 algebra 9?;in the following way. For intervals I, defined in (5.16), the set function E(I,) is defined by (5.16). Assuming now that the E(B) is defined for two sets B , and B , , write E(B, u B,) E(B, n B,) E(B,')

+ E(B,)

=

E(B,)

=

E(B1)E(B,),

= 1

-

if B, n B,

= 0,

E(B,).

T h e reader can easily convince himself that by the above indicated operations one can extend the domain of definition of E(B) from the intervals I, to the family Inof all intervals in W", and then to and that the derived function E(B), B E gt, is u additive on 9?:. Since

at,

E(B), B E gt, satisfies the condition of Theorem 5.5. Therefore, it has a unique extension to gn, and this extension is a spectral measure. Q.E.D. *5.6 APPENDIX:SIGNEDMEASURES

T h e real and imaginary parts of a complex measure are u additive but, in general, not nonnegative set functions. These set functions belong to the class of signed measures.

231

5. Spectral Measures

Definition 5.7. A countably additive set function p(S) on a measurable space (X, d),for which p( 0 ) = 0, is called a signed measure if it assumes values only from the set R1 u { co} or only from the set R1

+

u {-Go}.

+

The reason for not allowing a signed measure to assume co as well as - co values is to avoid the ambiguities inherent in the addition of 0 ~ and -co. We recall that, in general, a measure was not a special case of a complex measure since a complex measure is, by definition, finite. Hence, only the class of finite measures is a subset of the family of complex measures. However, we note that all measures are certainly special cases of signed measures. One could ask the question whether a signed measure could be always written as a linear combination of measures. The following theorem provides an affirmative answer to this question.

+

*Theorem 5.7 (Hahn's theorem). If p is a signed measure on the measurable space (3,d),then there are two measurable disjoint sets S ( + ) and Sc-1 whose union is X, such that p(S n S(+))2 0 and p(S n S(-1) 0 for all S E d .

<

Proof. It is sufficient to consider only the case when p(S) assumes values only from R' u {+co}. Denote by F ( - )the family of all sets R such that p ( R n S) 0 for all S E d .If

<

(5.17)

y =

choose a sequence R, , R,

inf p ( R ) ,

RESF(--)

,... E F ( - )such that y = lim p(&). k++m

The sets S,

=

(JF=l

R, also belong to F(-)since

for all S ~ dMoreover, . we also have

)

111. Theory of Linear Operators in Hilbert Spaces

238 and consequently

y = lim k+m

p(Sk).

The set

u m

S(-) =

m

Sk

uR

--

k=l

k

k=l

< 0 for any S

also belongs to 9(-) since p ( S n S(-)) seen from the relation

E ~as, can be

m

Setting S

=

X,we also get p(S(-)) =

Liym

p(Sk)

= y.

>!

We claim now that p(S n S ( + ) ) for all S E d if S(+)= X - S-). Assume to the contrary that p(S n S(+))< 0 for some E d . The set SA+)= S n S(+)could not belong to 9(-), since then we would and have $6” u S(-)E F(-) p(Si+) u S ( - ) )= P(Si+))

+ p(S(-)) < Y,

which contradicts (5.17). Hence, SA+)has measurable subsets of positive measure p. Let n, be the smallest positive integer for which there is a measurable subset Sl+)of Sh+)such that p(S:+)) > l / n , . Since p(S;+’ - S‘+’) 1 = p(Si+))- p(S:+’)

< p(Si+)) < 0,

we can apply the same argument to Sl;“ - Si+)to infer the existence of a smallest positive integer n2 such that SA+)- S:+)contains a measurable set Si+)with p(Si+)) 2 1/n2. We can continue with this procedure, thus obtaining the sets S;+),Si+),... . Then

and consequently

For arbitrary S E d we have

(5.18)

p

( S n (s;+) -

u s;+)))<

2 -1

k=l

1

n, - 1‘

239

5. Spectral Measures

In fact, if that were not so for some 1 = KO , then nk, would not fulfill the conditions of its definition since we could choose n;, < nk, so that

and replace Sil) by

T h e sequence n, , n2 ,... diverges to infinity since

and the measures of Sl;“ and Si+)- (J;=l Si+)are nonpositive and therefore finite. By combining this result with (5.18) we obtain m

k=l

for all S E&’. This would imply that Si+)- (Jkm_l SL+)is a set which belongs to %(-I, is of negative measure, and is disjoint from S(-).Hence, we would have

(SC-) u (S‘+’ 0

which contradicts (5.17).

-

pi+)))

< CL(S(-))= Y,

Q.E.D.

It is easy to see that if p(S) is a signed measure, then the set functions p(++’(S) = p(S n S ( + ) ) and p(-)(S) = -p(S (7 S+)) are measures. These measures are called the upper variation and the lower variation of p, respectively. According to the above theorem, we can always write p ( S ) = pL‘+’(S)- p ( - ) ( S ) ,

sE d .

Using this result we arrive at the following conclusion.

Theorem 5.8. Any countably additive set function p(S) on a measure space (%, d )for which p( 0 ) = 0 and which assumes values from C1is a complex measure (in the sense of Definition 5.5). In order to establish the above theorem we only have to note that Re p(S) and I m p(S) are signed measures and therefore they can be written as the difference of their respective upper and lower variations.

240

111. Theory of Linear Operators in Hilbert Spaces

EXERCISES 5.1. Show that in an n-dimensional Hilbert space 2 the eigenvalue problem Af = A f can be reformulated as a matrix eigenvalue problem

c n

=

ASflk

Aik

9

=

<ei

I Aek),

xi

= <ei

I f >,

k=l

in some orthonormal basis {el

,..., e,}.

5.2. Show that every linear Hermitian operator A acting on an n-dimensional Hilbert space 2 has a pure point spectrum containing at most n points.

5.3. Prove that if A,, A, ,..., is a uniformly convergent sequence of operators on a?' then for each E there is an N ( E such ) that 1) A, - A, (1 < E for all m, n > N(E).

5.4. Let {el, e, ,...} be an orthonormal basis in an infinite-dimensional Hilbert space 2.Denote by I ek)(ek I the projector (3.4) onto ek . Show that on account of Theorem 4.6(b) in Chapter I, n

I

5.5.

= s-lim n+ m

W

C I ek>(ek 1 = 1I ek)(ek I. k=l

k=l

Show that when s-lim,,to A(t) exists, then lim 11 A(t)fll = 11 s-lim A(t)fll t+to t+tO

for all f E X .

5.6. Show that if A = s-limt+CA(t) and B = s-lim,,,o B(t), then s-lim,,,o A(t)B ( t ) exists and is equal to AB in the case that 11 A(t)ll < c for all t.

5.7. Show that the intersection of all the subspaces in any given family of closed linear subspaces of a Hilbert space a?' is a closed linear subspace of 2. 5.8. Prove that any spectral measure E(B), B E d,is continuousfrom above and from below, i.e., if B = limn+mB, , then E(B) = s-lim E(Bn), n+ m

where B , 3 B, 3

5.9.

3 B and B , C B, C

Prove that 11 A* 11

=

11 A (I.

C B, respectively.

6. The Spectral Theorem for Unitary and Self-Adjoint Operators

5.10, Show that if A = u-lim,,,o A , and B A

= u-lim,+lo B,

241

, then

+ B = u-lim (A, + B,), t4,

AB

=d

i m A,B,,

A*

= u-lim A:.

t+tO t-t,

5.11. Explain why any infinite orthonormal sequence e, , e2 ,... in a Hilbert space A? is weakly convergent to 0, but it does not converge strongly to any vector. *6. The Spectral Theorem for Unitary and Self-Adjoint Operators

6.1. SPECTRAL DECOMPOSITION OF A UNITARY OPERATOR We shall first state the spectral theorem for unitary operators, and later on proceed to prove it in a few stages.

*Theorem 6.1. To every unitary operator U on a Hilbert space A? corresponds a unique spectral function E, having the properties that E, = 0, E,, = 1, which is such that n

U

= u-lim n-, m r+O

1 eiAL(E,,- E,,-,), k=l

where the limit is taken over finer and finer partitions of [ 0 , 2 r ] : (6.2)

0 = & < A, < -.. < An = 27r.

Since uniform convergence implies weak convergence, we can write

for any two vectors f,g E 2. It should be noted that since EA1

< EAe for A, < A2 , we have

111. Theory of Linear Operators in Hilbert Spaces

242

Thus, the support of E(B) is within the set [0, 2 ~ 1 ix., , if B n [ 0 , 2 ~ ]= la.

E(B) = 0

Consequently, the integration in (6.3) can be extended over R1. The formula (6.1) is customarily written in the symbolic form U

eudE, ,

= R'

and as such is known as the spectral decomposition of U. The basic idea of the construction of the spectral function En consists in defining E , as the strong limit of monotonically decreasing sequences of polynomials in U and U-l. We define the yet unfamiliar concept of a monotonic sequence of bounded operators so as to obtain a generalization of the already familiar case for projectors (see Theorem 3.10). 6.2.

MONOTONIC SEQUENCES OF LINEAR OPERATORS

Definition 6.1. If A is a symmetric operator in A? and c is a real number, we write A c or A c if, respectively, ( f I A f ) c( f 1 f) or ( f I A f ) 2 c( f I f ) for all f E g A. In that case A is said to be bounded from above or below, respectively; A is said to be positive if A 0. If A and B are two symmetric operators on A? and A - B 0, then we write A 2 B . It is very easy to check that the above relation is a partial-ordering relation (see Exercise 6.2).

<

>

<

>

>

Theorem 6.2. If A, , A, ,... is a sequence of symmetric operators A, - * * ) or monoon A?,which is monotonically increasing (i.e., A, tonically decreasing (i.e., A, A, ...), and bounded, respectively, from above (i.e., A, c for all k and some constant c) or from below (i.e., A, 2 c), then A, , A , ,... converges strongly to a limit A , which is a symmetric operator on A?.

<

<

>

>

<

<

<

. Since the operators A,, Proof. Consider the case A, A, A, ,... are symmetric and defined everywhere on A?,they are bounded (see Theorem 2.10). We shall assume that all the operators A , , A, ,... are positive; the general case B, B, --- can be easily reduced to this case by setting A, = B, II B, II:

+

(f I A k f )

=

< <

(f I B k f )

+
Bl

I l f > k (f I Blf>

+ 11

Bl

I/ (f If) 2 0.

T o prove the strong convergence of A,f , A, f , ..., note that A, - A j 2 0

243

6. The Spectral Theorem for Unitary and Self-Adjoint Operators

if i > j . For any positive symmetric operator A we have the generalized Schwarz-Cauchy inequality (see Exercise 6.1): l
(6.6)

mI2< (fl A f ) ( g IAg).

By inserting above A = A, - A j and g (6.7)

=

(A, - A3)f,we get

((4- Ai)fl(Ai- A m 2 s
< <

Let c be an upper bound of the sequence A, A, (f I A , f ) c Ilfl12 for all i = 1, 2, ... . From the inequality

<

, so that

< f l (Ai - A , ) f ) d I2

=

A,

< c2 IlflIZI1g IlZ,

The above inequality implies that 11 A, - A j 11 (6.7) we obtain

(6.8) Since

I

- Aj

I - ~4 f 1 1 4 G c3 I I ~ I I ZIW ~


,

f,g E 2.

< c. Using this result in

~- (fi f M>I. )

.**

s c llf1I2,

the above sequence of numbers is convergent, so that the term I(f I A i f ) - (f I Ajf)l can be made arbitrarily small for sufficientiy large i and j . In the light of this remark, (6.8) yields the strong convergence of A,f, A,f, ... . The linearity and Hermiticity of A, where Af

=

s-lim Anf, n+ m

is easily derivable from the corresponding properties of each of the operators A, ,A, ,... (see Exercise 6.3). Q.E.D.

6.3. CONSTRUCTION OF SPECTRAL FAMILIES FOR UNITARY OPERATORS

Assign to every trigonometric polynomial

k=-n

111. Theory of Linear Operators in Hilbert Spaces

244 the operator p( U), where (6.10)

Then we can state the following lemma.

Lemma 6.1.

The mapping

P( U )

(6.1 1)

-+

of the family of all trigonometric polynomials (6.9) into the family of linear operators on if has the following properties: (6.12)

alpl(eio)

+ %Pz(eiq)

(6.13)

pl(eiw)p2(e”)

(6.14)

(p(e””)*

Moreover, if p(e”) 2 0 for all y

E

-

aIpl(u)

+ azpz(U),

PI( U )P 2 (

U),

+

(P(U))*.

W, then p ( U ) is a positive operator.

Proof. The verification of the relations (6.12)-(6.14) is straightforward algebra and is left to the reader (Exercise 6.4). In order to prove that p ( U ) 0 if p(eiq) >, 0, we use a lemma by FejCr and Riesz (see Exercise 6.5) which states that if p(eiW) 0, then there is another trigonometric polynomial q(eiq) such that

>

p(e”)

=

1 q(eiq)I2 = [ q ( e i ~ ) ]q(eiq). *

Using (6.13) we get


which shows that p ( U )

> 0.

(4(U)fI d U ) f ) 2 0,

Q.E.D.

We can extend the mapping (6.11) to the family Spof all functions ~ ( y )y ,E W , which are the limits

(6.15)

w(v) = lim p,(eiq) n+m

of monotonically decreasing sequences (6.16)

pl(eiv) >, p,(ei@)

**.

0

of positive trigonometric polynomials. This result constitutes the following lemma.

6. The Spectral Theorem for Unitary and Self-Adjoint Operators

245

Lemma 6.2. To each function w ( y ) E Fpcorresponds a symmetric positive operator A, on S?

which is independent of the chosen monotonic sequence (6.16). The mapping (6.17)

is such that (6.18)

Wl(4

wz(d

+

AW1WB

= A,014u,*

Proof. Due to the last statement in Lemma 6.1, we have for any sequence (6.16) that p,(U) 2 pz(U) 2

... 2 0.

Hence, by Theorem 6.2, the above sequence has a strong limit A,which is a symmetric operator on &'. Moreover, A 2 0,

(f I Af) = ?I;_mm (f I P n ( U ) f ) 2 0. In order to prove that the limit A is independent of the chosen sequence (6.16), take two such monotonically decreasing sequences (pkl)(eiq)}and {fi~.)(eiq)}, for which lim p:)(eiq) n-rm

=

lim pr)(ete).

n-1-

The above assumption implies (see Exercise 6.6) that for every integer n there is some pfJei") such that

piy(eiQ)< pF)(eiw) + l / n .

By using again the last statement of Lemma 6.1 we infer that

p t ( U ) < p:'(u) which implies that s-lim pkl)(U ) n+m

+ 1In,

< s-lim p r ) (U). n-m

By reversing in the above argument the roles of pa)(ei@) and pA2)(eiq)we can deduce that s-lim pLz)(U) < s-lim pi1)(U), n- m

n+m

which implies (see Exercise 6.2) that the above two limits are equal.

246

111. Theory of Linear Operators in Hilbert Spaces

Finally, the property (6.18) of the mapping (6.17) follows from (6.13) by using the result of Exercise 5.6 on strong limits. Q.E.D. Consider the following family of periodic functions for h for h

nA(P)

n,+(p)=

1 (6*19)nA(') = 10

1

when 2nn when 2nn

nA(v)defined

by

< 0, > 27r,

< tp < 2nn + h + h < P < 2(n + 1) 0 < h < 27ry = 0,f l y f2,... . T,

72

We shall first show that the operators EA assigned to the above functions by the mapping (6.17) are projectors, and later on we shall prove that these projectors constitute the spectral family of Theorem 6.1.

Lemma 6.3. The functions (6.19) belong to the family Pp. The operator EA assigned to IJA(v)by the mapping (6.17) is a projector in &'. Proof. I t is easy to construct" for any given function IJ,+(y)a monotonically decreasing sequence UdP)

> uz(P') > ...

of once continuously differentiable and periodic functions un(p)) = un(P

Un(P)

+ 2kn)y

k

converging to nA(v).Choose a sequence 6, (6.20)

0 < an

<8

=

f l y f2y.**y

, 6, ,... --+0, where

[min{un(~)- u n - l ( ~ )u, n + l ( ~ ) un(~)>I*

According to Chapter 11, Theorem 7.3, there can be found for each un(v)satisfying the above imposed conditions a trigonometric polynomial

pn(eiq)for which

I un(P) - pn(eiV)I < a n

9

P E R1*

Due to the conditions (6.20), we have that p,(eiV) obviously

> p,(eio) >

. a * ,

and

limpn(eiq) = iim un(p) = n,(~). n+ w

n-m

This proves that n,(~p) E Pp.

* See the construction of the functions h,(x) in the course of the proof of Theorem 5.6 in Chapter 11.

6. The Spectral Theorem for Unitary and Self-Adjoint Operators

247

If EA denotes the symmetric operator in 8 assigned to ITA(v),by (6.18) we shall obtain EA2 = EA, due to the fact that nA2(P) = nA(P)*

Q.E.D.

Consequently, EA is a projector.

6.4. UNIQUENESS OF THE SPECTRAL FAMILY OF A UNITARY OPERATOR We have to prove now that the projectors En defined in Lemma 6.3 constitute a spectral family. From the obvious relation nA(F)

<

nA(P),

nA'(P))

"3

and (6.17) we get (6.21)

EAEAj

h

= EA

< A'.

Thus, in order to establish that EA , h E [wl,is a spectral family, we are left with the task of demonstrating that EA = EA+,

(6.22)

=

s-lim EA'.

A '++O

First, we note (as can be easily and explicitly proven in the same fashion as in Lemma 6.3) that a monotonically decreasing sequence {pn(ei@)} of trigonometric polynomials can be found which is such that

I,isnm pn(ef")= nA(ef@), and in addition pn(eim) >, nA+l/n('P),

71

= 1, 2,****

Consequently pn(u) 2

EA+l/n

9

and since EA \< EA+l/n, EA

< s-lim EA+1/, = s-lim pn(U ) = EA , n+m

n-w

i.e.,

s-lim EA+l/, = EA . n-w

As any sequence hl , A,

,...

--t

containing only the numbers h we see that (6.22) is true.

h

+ 0 can be majorized by a sequence

+ l/n, n = 1,2,..., and converging to A,

248

111. Theory of Linear Operators in Hilbert Spaces

T o establish now that EA, h E R1, of Lemma 6.3 is the desired spectral family, note that the definition (6.19) of 17,(~p) yields immediately that (6.4) is satisfied. Thus, we have to verify that (6.1) is true. If we keep the notation of (6.1) and (6.2), we see that for q~ Ak

< <

Since the above holds for k = l,..., n, we have (6.23)

for any ~p E [O, 27~1.From the last statement in Lemma 6.1 about the mapping (6.17), alias (6.1 l), it can be easily derived that (6.23) implies

Thus, we have

for all f E &', and consequently

which proves (6.1). T o establish the uniqueness of the spectral decomposition of U, assume that E,' is a spectral function satisfying (6.1) and (6.4). Because of the pairwise orthogonality of El, - E;,-, for different values of K, we easily derive (using the results of Exercise 5.10) that for any nonnegative integer m

6. The Spectral Theorem for Unitary and Self-Adjoint Operators

249

i.e., in symbolic notation,

I,

eimw dE,'

= Urn,

m

= 0,

f1,... .

Due to linearity of the above integration procedure, we easily establish that for all trigonometric polynomials p(e") we have

for any f, g E X . With the help of Theorem 3.10, Chapter 11, which remains valid for complex measures, we can generalize the above formula to any function w ( y ) E 9--

by approximating w ( y ) by a sequence (6.16) and then going to the limit. Thus, in particular, for w(p)) = n A ( y )we get

i.e., after carrying out the above integration,

Since E,' = 0, and the above identity is true for any f,g E A?, we get En' = EA for all h E R1.This establishes the uniqueness of the spectral decomposition of U and completes the proof of Theorem 6.1.

6.5. SPECTRAL DECOMPOSITION OF A SELF-ADJOINT OPERATOR The spectral theorem for self-adjoint operators on a Hilbert space can now be stated and proved.

111. Theory of Linear Operators in Hilbert Spaces

250

Theorem 6.3. (The spectral theorem). T o each self-adjoint operator A in a Hilbert space X corresponds a unique spectral function E , , X E R1, such that the domain BAof A consists of all the vectors f for which

and for any f E g Aand g E L%' (6.25)

Proof. We shall derive the existence and uniqueness of the spectral function of A from the existence and uniqueness of the spectral function of the Cayley transform V of A. Inasmuch as A is self-adjoint, its Cayley transform V is a unitary operator (see Theorem 4.9), which, therefore, has a spectral decomposition V = JLoSzn, et@dEm. According to Theorem 4.8, iff e g A we have (6.26)

Af

= i(1

+ V )h,

h

=

(1 - V)-lf.

Consequently, for any g E X we get by applying in the last step of the following derivation the result of Exercise 6.7:

where the complex measure P ~ , ~ ( BB ZE ) , L P , is defined as

6. The Spectral Theorem for Unitary and Self-Adjoint Operators

25 1

On the other hand, by making use of the fact that f = (1 - V)h and E(B)V = V&B) (see Exercise 6.8), we easily derive (6.27)
= J B (1 - ei')

1

[o,znl

ei,

d(g I E,E(B) h )

= P&h(B),

by noting that by (5.16) and Theorem 5.3 &,E(B)

= e ( B n [0,

TI),

B C [0,2~].

Thus, we obtain

where in the last step we had introduced the new variable h = -cot +p,

T E (0,274,

and in terms of it the new spectral function EA

= B(-z arccot A ) =

gw

*

ForfE g Awe get from (6.27)

< f I E ( B ) f ) = JB (1

-e i')

&f I E,h)

= JB

(1 - eiw) d(h 1 I?&*

=

1 1 - eiw ID d(h I E,h) B

=

4 B

and consequently (see Exercise 6.7)

sin2 $p d(h I E,h),

252

111. Theory of Linear Operators in Hilbert Spaces

The above consideration proves the existence of a spectral function En satisfying (6.24) and (6.25) for allf E g A. We now want to show that there is only one such spectral function. Suppose E,’, X E R1, is a spectral function satisfying (6.24) and (6.25) for allf E gA. It is easy to see (see also Exercise 6.9) that the operator A’ defined on the set (6.28) as the operator for which

A’f satisfies

for all g E &, is linear. By the original requirement, we have A’ 2 A and consequently A’*_C A* = A (see Exercise 2.9). On the other hand, A’ is symmetric, since for g E gA, ,

Thus we also have A’ C A’*C A, i.e., A = A‘. Now we can reverse the earlier procedure of going over from the spectral function of I/ to the spectral function of A, and derive that ED!

=

ELt*@

is the spectral function of the Cayley transform of A. Due to the uniqueness of @‘,% (see Theorem 6.1), it follows that & = I?@‘, i.e., E, = EA’. The above argument establishes at the same time that (6.28) coincides with the domain of definitiongAof A. Q.E.D. The spectral function E, and the corresponding spectral measure E(B) satisfying the conditions of Theorem 6.3 are said to belong to the operator A. In that case it is customary to write symbolically (6.29)

A

= JRl

h dE,

.

The integral on the right-hand side is usually referred to as the spectral decomposition of A. The integration in (6.25) extends effectively only over the spectrum S A of the operator A, where the spectrum can be defined as follows.

6. T h e Spectral Theorem for Unitary and Self-Adjoint Operators

253

Definition 6.2. A point h E R1 belongs to the spectrum S A of the self-adjoint operator A with the spectral measure E(B) if E(I) is nonzero for every open interval I containing A. If h $ SA, then E(I,) = 0 for some open interval I, containing A. Hence, all the points in I,, do not belong to the spectrum. Thus, all the points not belonging to the spectrum constitute an open set, and therefore the spectrum is always a closed set. Definition 6.3. A point h E S A is said to belong to the point spectrum if the spectral measure of A is different from zero at A, i.e., E({h}) # 0. The set :S = S A - S< is called the continuous spectrum of A.

Si of A

One of the most important properties of the spectrum of a selfadjoint operator A is that it is the support of the spectral measure of A, lee.,

(6.30)

EA(SA)

=

1.

In order to establish the above relation, let us write m

SA=

ns,,

SJSJ.-,

k=l

+

where S, is defined as the union of all intervals [n/2k,( n 1)/2k]which contain at least one point of the spectrum of A. Now, if a closed interval I does not contain any point of the spectrum then EA(I) = 0. In fact, if that were not so, we could split the interva€ in two halves and then at least on one of the halves the spectral measure would be different from zero. By continuing the process we would obtain a monotonic sequence of intervals which shrink to a point in I, and that point would obviously belong to the spectrum. Using this result, we can immediately infer that EA(SA)= 1 and consequently, due to the continuity from above of spectral measures (see Exercise 5.Q we conclude that

i.e., (6.30) is true.

6.6. THESPECTRAL THEOREM FOR BOUNDEDSELF-ADJOINT OPERATORS We conclude this section by showing that for bounded self-adjoint operators the integral in (6.29) can be proven to converge uniformly, as was the case with the similar relation (6.1) for unitary operators. T o prove this statement we need the following lemma.

254

111. Theory of Linear Operators in Hilbert Spaces

Lemma 6.4. If A is a bounded self-adjoint operator on &', its spectrum SA lies within the interval [-I[ A (I, +I/ A 111, and at least one of the two numbers --)I A I( and +(I A (1 belongs to the spectrum of A. Proof. If A E S A and if E ( B ) , B E S, is the spectral measure of A, then E(I,) # 0, I, = ( A - E, A E) for any E > 0. Hence, there is a nonzero f . E 2 for which E ( l , ) f , = f , . Consequently (see Exercise 6.10)

+

<

so that I I A I - E I 11 A 11 for any of the above E > 0, i.e., 11 A (I. This establishes that SA C [-I/ A 11, 11 + A 111. If 01 = sup{I A I: A E SA>, then by (6.30) and Exercise 6.10

II Afll'

=

1

d II JWll' <'a llf1I2,

[-u.+ul

IXI

<

f E 9,4= 2,

<

i.e., 11 A 11 \< 01. On the other hand, 01 )I A I( since S A C [-I1 A 11, f l l A 111. Hence 01 = 11 A 11, and since S A is closed we must have that either 01 E S A or -a E S A , or both. Q.E.D. We are now ready to prove Theorem 6.4.

Theorem 6.4. Let - 1 1 A 11 = A,, < A, < < A, = +(I A 11 be a partition of the interval [-(I A (I, +]I A 111 and Al', ..., A', any points for which Ak-, A', < A,, K = 1,..., n. Then in the limit of finer and finer partitions

<

(6.31)

w =

...n I A,

max

k=l..

- A,.-l

I,

where EA is the spectral function of A. We shall only sketch the proof of this theorem. By following the procedure used in proving Lemma 6.1, it can be easily established that the mapping p ( x ) +p(A) of the set of polynomials in x E R1 into the set of linear operators in 2 has the following properties:

Pl(4

+P z ( 4

Pl(4 P z ( 4

P*(4 p(A)

0

-+

P I ( 4

+P z ( 4

P l ( 4P 2 ( 4

[P(A)I*, if p ( x ) 2 0.

255

6. The Spectral Theorem for Unitary and Self-Adjoint Operators

Then, by using the method employed in proving Lemma 6.2 one can show that the characteristic function x,,(x) of the closed interval In = [-I] A 11, A] is on the interval [-I/ A I), +I1 A 111 the limit of a monotonic sequence p l ( x ) 2 p 2 ( x ) 2 of positive polynomials. Combining these two results one can easily show that EA s-limpn(A) - n++m

is a spectral function with EllAll= 1. From the construction of this spectral function it follows that, given any partition -11 A 11 = A, < A, < -..< A, = +[I A 11 of the interval [-I1 A II, +II A 111, we have

A summation in k

G hk(EAk

- 'A,-,)

- 'Ak-1)

'k-l('Ak

- 'Ak-1).

= 1,..., n obviously yields

n

n

Furthermore, from the inequality n

n

c

- 'Ak-1)

k-1

-

c

hk-l(EAk

- 'Ak-1)

k=l n

=

c

- 'k-l)('Ak

- 'Ak-1)

k=l

c n

<

(EAk

- 'Ak-1)

=

k-1

we see that both sums in (6.32) have to converge to A in the limit w Hence, (6.31) is established. From (6.31) we easily deduce that


=

I

d(f

+ 0.

I

for allf, g E 2.On account of the uniqueness of the spectral decomposition of A, we conclude that EAis the spectral function of A. This completes the proof of the theorem.

EXERCISES 6.1.

Prove the generalized Schwarz-Cauchy inequality

G (f I Af >
II2

111. Theory of Linear Operators in Hilbert Spaces

256

A

6.2. Show that if A and B are two symmetric operators on €3 and B 2 A implies A = B.

>

H,then

6.3. Show that if A,, A, ,... is a sequence of self-adjoint operators on fl and s-liQ-,m A, f = f' for each f E 2,then A(f ) =f' is also a self-adjoint operator. 6.4.

Verify explicitly that Lemma 6.1 is true.

6.5. Prove the lemma of Fej& and Riesz, which states the following: if p(ei") is a positive trigonometric polynomial, p(eiw) 2 0, there is a trigonometric polynomial q(e") such that p(ei") = I q(ei')I2. 6.6. Assume that ul(x),u,(x) ,..., and w,(x), v 2 ( x ),..., are two monotonically decreasing sequences of continuous functions on the finite interval I, for which limn+mu,(x) = limn-,mvn(x). Prove that for each integer n there is some other integer K, such that

<

Uh&)

Wn(4

+ lln.

6.7. Show that iff (S)g(() andg(f) are p-integrable complex functions on R, and pl(S) denotes the complex measure P~(s= )

J

MO,

S

s c R,

then f ( f ) is p, integrable on R and

I f ( 04-43 R

=

J f ( t ) g ( t )d P ( 0 R

6.8. Show that if U is a unitary operator and E(B), B E S,is its spectral measure, then UE(B) = E(B)U. 6.9.

Show that if E, , h E R1, is a spectral function, the set

is a linear subspace of A?. (Hint: 4 1) E,( f 6.10.

Prove that if A

=

+ g)l12 < 11 E,f 1, + 11 E g

112).

A*,then for f ~9~

II Afl12

=

w

/I2 WI

w>.

References for Further Study Consult any text on functional analysis, such as Akhiezer and Glazman [1961], Kato [1966], Riesz and Sz. Nagy [1955], Schechter [1971], Yosida [1974].

CHAPTER

Iv

The Axiomatic Structure of Quantum Mechanics

In the quantum mechanical problems encountered thus far we have had to limit ourselves basically to the special version of quantum mechanics known as wave mechanics, in which we could characterize the state as a wave function. T h e knowledge of the theory of linear operators in Hilbert spaces acquired in Chapter I11 will enable us to study the structure of quantum mechanics from an abstract and very general point of view, in which the formalism of wave mechanics becomes a particular realization of the general formalism. We start this chapter with a section on the theory of measurement in quantum mechanics, which investigates the relation of the basic theoretical constructs to experimental material. We proceed then to explore the basic mathematical concepts of quantum mechanics, formulating in the process the fundamental axioms which these constructs have to obey. This entire exposition will lead in a natural way to specialized versions of quantum mechanics, such as the already familiar case of wave mechanics.

1. Basic Concepts in the Quantum Theory of Measurement 1.1. OBSERVABLES AND STATES IN QUANTUM MECHANICS The theory of linear operators in Hilbert space developed in Chapter I11 has equipped us with all the necessary tools for a very general formulation of quantum mechanics. As we saw in the Introduction, the basic categories of primitive concepts of quantum mechanics are, as in the case of classical mechanics, 257

IV. The Axiomatic Structure of Quantum Mechanics

258

those of state and observable. In order to be able to give a mathematical description of these objects, we associate in classical mechanics with any given system a phase space; e.g., with an n-particle system we can associate a 6n-dimensional vector space. For the same reason, we associate a Hilbert space with any given system which we intend to describe quantum mechanically. For instance, as we have seen in 95 of Chapter 11, if the system consists of n particles, which are of different kinds, without spin and moving in three dimensions, then in wave mechanics (which is a special version of quantum mechanics) we associate with that system the Hilbert space L2(W"). Let us assume now that we are dealing with a particular quantum mechanical problem in which a certain system has been specified . ' is (e.g., a hydrogen atom) and with which a certain Hilbert space % associated [e.g.,L2( UP)]. I t is then postulated that to each observable corresponds in the formalism a unique self-adjoint operator acting in &' -though the converse is by no means required, i.e., there might be selfadjoint operators in @ . which do not correspond* to any observables. As an example of operators corresponding to observables related to a two-particle system (e.g., the hydrogen atom) described wave mechanically in LZ(W), consider the self-adjoint operators Qp),&?), Q;*), Pp),PjY),Py),i = 1, 2, defined as follows. QP) is the self-adjoint operator [see (4.19) of Chapter 1111 with a domain of definition

where ( 1.2)

x

= (x1 ,Y1,z19 x2

which acts on a (1.3)

J/I

, Y z , 4, d6x

= dX1 ..* d % ,

E ~ Pin)the following way: (QZ!"'+)(x)= xi$(x).

An analogous definition holds for QP)and QP). Py)is the self-adjoint operator [see (4.20) of Chapter 1111 with a domain of definition

In the early days of quantum mechanics there was a tendency to assume that each self-adjoint operator represents an observable (see von Neumann [1955]). However, the existence of so-called superselection rules (which will be considered later in this chapter) indicated for the first time that this requirement lacks experimental support.

1. Basic Concepts in the Quantum Theory of Measurement

259

P?) acts on any (G E W;( UP)C Sy)(R6) according to the formula

PjY)and Pp) are defined analogously. The operators QF), Q:v), Q f ) correspond then to observables which from the experimental point of view represent measurements of the x, y, and z coordinate, respectively, of the ith particle in a Cartesian inertial reference frame. Similarly, Pr),P!v), Pjs) correspond to the measurements of the x, y , and x components, respectively, of the momentum of the ith particle with respect to the same frame." As another example of an observable for the same system, consider the operator H' with domain of definition g H , = Yc( W), which maps a 4 ~ 9into~ '

For a very wide class of suitable potentials V ( x ) ,the above operator H' is essentially self-adjoint, i.e., its adjoint (H')* 2 H' is a self-adjoint operator. Its self-adjoint extension (H')*, customarily denoted by H , is called the Hamiltonian of the system, and is an observable corresponding to measurements of the total energy of the system. In general, for any given system, there are correspondence rules specqying which self-adjoint operator corresponds to each particular observable related to the system. These rules are part of the set of correspondence rules relating the formalism to the experimental framework. We shall meet some of the most important of these rules, which are concerned with the most fundamental observables, such as energy, position, momentum, angular momentum, spin, and charge, as we proceed in our study of quantum mechanics. Let us turn now to the concept of state. As a generalization of the consideration in $5 of Chapter I and $5 of Chapter 11, it is postulated, in general, that a state' is specified by a vector-valued function Y(t), See Exercises 1.5 and 1.6 for the spectral measures of these operators.

'We are adopting here the Schroedinger picture of quantum mechanics. In $3 we shall

encounter other physically equivalent pictures.

260

IV. The Axiomatic Structure of Quantum Mechanics

Y(t)E S, of the time parameter t; Y(t)is never zero, i.e., 11 Y(t)ll > 0 for all t E W. We have, again, as in the aforementioned sections, that if Y(t)describes a state, then cY(t),for any nonzero constant c, describes the same state. The reason for this assumption lies in the nature of Born's correspondence rule, which is the fundamental principle on which the accepted interpretation of the quantum mechanical formalism lies. We shall now focus our attention on this rule.

1.2.

THECONCEPTOF COMPATIBLE OBSERVABLES

Let A be a self-adjoint operator corresponding to an observable (also referred to as A), and denote by E(B), B E @,the spectral measure of A

=I

+m

A

XdE,.

--m

Consider a system which is known to be at time t in a state described by the normalized vector Y(t).In keeping with a suggestion first forwarded by Born [1926a, b] for the case when A is a position observable [see (5.2) of Chapter 111, we interpret the measure ( 1.8)

PA,(t,(B)= (F(4I

'u(9>,

B E a,,

as a probability measure in the following way: a measurement to determine the value of A, carried out on the system at the time t, has the probability P&t,,(B)to have as outcome a value within the set B. A generalization of this interpretation can be applied to any finite set of compatible observables. This necessitates an understanding of the concept of compatibility, which is empirically rooted and to which we give the following meaning.

Definition 1.I. The observables A, ,...,A, related to a certain system are compatible if, in principle, arbitrarily accurate measurements of the simultaneous values of these observables can be carried out; i.e., if for any prepared a E S from the set S C [Wn of feasible (cf. Definition 5.2) simultaneous values of A, ,...,A,, and for any open confidence. interval I containing a, the confidence probability p of actual readings falling within I can be brought arbitrarily close to one by a suitable design of apparatus for the simultaneous measurement of A, ,..., A,, . Denote the self-adjoint operators corresponding to the observables A, ,...,A, also by A, ,..., A, . It is a basic feature of all quantum mechanSee Sections 2.49 and 5.21 of Dietrich [1973].

261

1. Basic Concepts in the Quantum Theory of Measurement

ical theories that if the n observables A, ,..., A, are compatible, then the corresponding operators must commute with each other. Note, however, that in general A, ,..., A, can be unbounded operators which are not defined on the entire Hilbert space 2 and, therefore, for which we do not have an a priori guarantee that the commutator [ A i ,Ak]= AIA, - A,A,

is defined on a domain which is dense in 2.We bypass this difficulty by introducing the following concept of commutativity.

Definition 1.2. Two self-adjoint operators A, and A, are said to commute if their respective spectral measures E(')(B) and B 2 ) ( B ) commute, i.e., (1.9)

[E")(B,),E'2)(B2)]= 0

for all B, , B, E B'l. In the case when A, and A, are bounded self-adjoint operators with 9A1 = g A ,= 2, (1.9) is a sufficient and necessary condition (see Exercises 1.1 and 1.2) for [A, , A,] = 0.

1.3. BORN'S CORRBPONDENCE RULE FOR DETERMINATIVE MEASUREMENTS The above considerations show that when A, ,..., A, are self-adjoint operators corresponding to compatible observables and E(l)(B), ..., E(")(B) are their respective spectral measures, then E(l)(B,), ...,E(n)(B,)commute, and consequently the operator E(B, x

(1.10)

***

x B,)

= E")(B,)

* a *

E(")(B,)

is a projector (see Chapter 111, Theorem 3.6). If Y(t) is a normalized vector, )I Y(t)ll = 1, representing the state of a system at the time t, then the generalization (1.11)

Z'$;;.*"n(B,

x

x B,)

I

= (Y((t) E"'(B1)

E(")(B,)Y(t)>

of (1.8) is interpreted as being the probability that a measurement designed to determine the simultaneous values A, respectively, will yield a result A, E B,

,..., A,

E B,

,

B,

when carried out on the system at time t.

,..., A,

,..., B, E 9,

of

A, ,..., A,,

IV. The Axiomatic Structure of Quantum Mechanics

262

We can generalize the above statement further by noting that the set function (l.ll), defined by (1.11) only on the Borel sets of Rn which belong to (a1), = x ..- x 9Y1,can be extended to a measure PA,t;i"*An(B),B €9, on the family anof all the Borel sets in W". Moreover, by Theorem 5.5 in Chapter 111, the projector-valued set function (1.10) can be extended to a spectral measure EA1***.*An(B) defined for all B E an,and such that PA'.*.*mAn(B) Y(t) = (Y(t)I EA'.***mAn(B) Y(t)).

This suggests the following interpretation, which applies to measurements which are designed to determine (i.e., determinative measurements; see the Introduction) the simultaneous values of any finite number of given compatible* observables (see Section 8.9 for incompatible ones).

Definition 1.3 (Born's correspondence rule for determinative measurements). Let A , ,..., A, be n commuting self-adjoint operators acting in #, with spectral measures EA1(B),..., EAn(B), respectively, and representing n compatible observables. Denote by EAI..

(1.12)

...An (B),

BEW",

the unique spectral measure which satisfies the equalities (1.13)

....""(Ill x

EA~.

for all choices of B ,

*.-

x B,)

- - - EAn(B,)

= EA1(B,)

,..., B , E al. If the vector-valued Y(t)€X,

--co

function Y(t)

< t < +a,

represents an already prepared state of a given system, then the measure (1.14)

pAi ,....An Y(t)

( B ) = 11 Y(t)ll-2(Y(t) I EA1*...*An ( B ) WO)

is a probability measure: for each B E the number PA;,;j.*Jn(B)is the theoretically predicted probability that a measurement carried out at time t to determine the simultaneous values A, ,..., A, of A, ,..., A,,, respectively, will yield a result within B, i.e., (A, ,..., A,) E B at t . The first attempts of treating incompatible observables were made in the 1960s (She and Hefner [1966], PrugoveEki [1966. 19671, Park and Margenau [1968]). A consistent extrapolation of Born's correspondence rule turns out to be indeed feasible (PrugoveEki [1976], Ali and PrugoveEki [1977], Schroeck [1978]), and in fact quite essential for relativistic quantum mechanics (PrugoveEki [1978d, 1981a-c]).

1. Basic Concepts in the Quantum Theory of Measurement

263

According to the above interpretation

J

-m

represents the mean value (also called expectation value) of the observable A for the system in the state Y(t) at time t . If A is an unbounded operator, then (1.15) is not necessarily defined for all vectors Y(t)E &', since the integral on the right-hand side might diverge for some such vectors. However, if Y(t)E g A, then by the spectral theorem (1.16)

<

>

and since I h I A2 for I X I 1, the integral in (1.15) converges and (A)F(t)exists. As an example of compatible observables, take the six position observable Q?),..., i = 1, 2, defined by (1.1) and (1.3). T h e spectral measure (1.12) is (see Exercises 1.4 and 1.5)

and the probability of discovering the first particle at r1 and the second particle at r2, where x = (rl, rZ)E B, is given according to (1.14) by (1.17)

This agrees with (5.2) of Chapter 11. The six operators PF),..., i = 1,2, representing momentum and defined by (1.4) and (1.6), also correspond to compatible observables. In this case we have (see also Exercise 1.6) (1.18)

(Ep1*p2(B)+)(x, t ) = ( q 3 X B . $I)(x, t>.

Thus, in terms of the Fourier-Plancherel transform $(p ) of #(x), (1.19)

The operators QF)and P p ) do not commute. This feature corresponds to the experimental fact that the position and momentum are incompatible observables.

264

IV. The Axiomatic Structure of Quantum Mechanics

1.4. BORN'SCORRESPONDENCE RULE FOR PREPARATORY MEASUREMENTS T h e above correspondence rule (Definition 1.3) and discussion of the process of measurement is related to the subject of determinative measurement. In this case, we have a system about which it is already known to be in a state described by the vector-valued function !P(t). Thus, we have a situation in which, on the one hand, the correspondence rule in Definition 1.3 predicts the probability that different observables of the system have certain values at any given time t , while, on the other hand, a procedure of measurement (such as those described in the Introduction for position and momentum) enable us to experimentally determine the values of these same observables. These considerations do not tell us, however, how to make sure that a system is indeed in a state Y ( t ) . This last question involves preparatory measurements or preparation-of-state procedures. Let us assume that we have (for the considered system) n compatible observables represented by the operators A, ,..., A , , which are complete with respect to some simultaneous eigenvector Yo, (1.20)

AIYo= $')Yo,..., A,Y,

= *'Yo,

i.e., for which Yo is, up to a multiplicative constant, the only vector satisfying the n eigenvalue equations in (1.20) with the respective eigenvalues hio),..., A;'). According to Definition 1.1 of compatible observables, there must be not only determinative but also preparatory procedures which would enable us to prepare the system so as to have at some time to a value of A, in the E neighborhood of hio). If h ~ o ' , . . . , h ~are o ) not accumulation points of the spectra of A, ,..., A, , respectively, then for some co there are in addition to hio)no other points in the spectrum of A, which lie in the co neighborhood of hio). Thus, there should be, in principle, a sufficiently accurate (€,-accurate) preparatory measurement which would tell us that the considered system had at time to the value h(kO)for the observable A,, k = I , ..., n. Then the accepted framework of quantum mechanics says that the state Y ( t ) , - m < t < +m, of the system immediately after the preparatory measurement is such that (1.21)

W O ) =

Yo 9

where Yo satisfies (1.20). As we shall see in 93.2, the dynamical law of quantum mechanics assures us that in this case there is only one state Y ( t ) satisfying (1.21).

1. Basic Concepts in the Quantum Theory of Measurement

265

An ideal preparatory measurement, like that above, is rarely the case in practice. Usually we can establish that the simultaneous values A, ,..., A, of the n compatible observables A, ,..., A, are within an n-dimensional interval I, or, more generally, within a Bore1 set B E L3" at some instant t o . For instance, we have seen in the Introduction that the experimental arrangement on Fig. 2 does not offer us an absolutely precise knowledge of the momentum of a charged particle which emerges from the apparatus in the interaction region, but it prepares a range A of momenta. We see that we cannot limit ourselves to the very idealized case treated by (1.20). Instead, we should be able to cope with the most general imaginable case. T o arrive at such a generalization of the case treated by (1.20), we shall recast (1.20) in a form which yields itself to straightforward generalization. The vector Yo satisfying (1.20) can be characterized as the unique vector (up to a multiplicative constant) satisfying (1.22)

EAI.

..., A , ({A?'} x

*.*

x

{At'})

Yo= Y o ,

where the above projector is given by the formula (1.13) with B, = {A:')}, ..., B, = {A:)}. Thus, by Theorem 3.3 in Chapter 111, Yois the unique vector (up to a multiplicative constant) in 3 ' for which (1.23)

The characterization of Yo by (1.22), or equivalently by (1.23), can be applied to the problem of characterizing the state of a particle prepared by the apparatus described in Fig. 2 (see the Introduction) to have at time to its momentum within A. We achieve this generalization by introducing the projector EPl(A), which, for instance, in the case of the two-particle system considered earlier, can be written as EP'(A) = EP1*Pa(A X R'),

with EP1*'a given by (1.18). According to (1.22) we can state that this two-particle system, with the first particle emerging from the aperture O3(see Fig. 2) at time t o , can be afterwards in any state Y(t)which at to satisfies the relation (1.24)

266

IV. The Axiomatic Structure of Quantum Mechanics

It should be noted that the outlined preparatory measurement obviously does not single out a unique state Y(t),but rather a whole set of states. This state of affairs is by no means peculiar to quantum mechanics, but is shared by other physical theories. It is due to the everpresent “errors” of measurement. Thus, when trying to determine” in classical physics the state r(t), -GO < t < +GO,of a particle of mass m moving in a known force field by measuring its momentum p at some instant t o , we determine, in fact, only some range A of momenta, since “errors” of measurement are always present. After this measurement the particle could be in any of the states r(t) for which (1.25)

mr(t,) E A .

A simultaneous measurement of position of the particle, which would locate it at to within the space region A’, would reduce the selectable set of states to those which in addition to (1.25) also satisfy the relation r(to)E A‘.

However, even this measurement will not, strictly speaking, single out a unique state r(t) since in practice A and A’ are never one-point sets. If we return to the quantum mechanical case, it is interesting to note that the experimental arrangement in Fig. 2 prepares not only a range A of momentum values of the particle, but also a range A’ of the position coordinates; namely, if a particle has emerged in the interaction region during the time interval to to to A t while the aperture 0, was open, the particle must have been somewhere within the confines of that aperture at the instant t o . However, the accepted interpretation of quantum mechanics deals only with the measurements of compatible observables, and cannot take advantage of the additional information that is derived from simultaneous measurements of incompatible observables. We get the general form of this interpretation for the case of preparatory measurements by straightforward generalization from the special case of momentum measurement discussed above.

+

Definition 1.4 (Born’s correspondence rule for preparatory measurements). Suppose that a preparatory measurement of n compatible observables, represented by the self-adjoint operators A, ,..., A, , has been carried out on the system at time t o , and that this measurement establishes that the simultaneous values A, ,..., A, of these observables *Since the disturbances of macroscopic systems caused by the impact of photons used in the measurement process are negligible, the distinction between preparatory and determinative measurement is of no practical consequence in classical mechanics.

1. Basic Concepts in the Quantum Theory of Measurement

267

are within the set B E at the instant to . Then, after the instant t o , the system can only be in a state Y(t)satisfying at tothe relation P$,;;;.Jn(B) = 11 Y(to)ll-yY(to) I EA’..***“n(B)Y(to))= 1.

(1.26)

According to the above, a preparatory measurement assigns to the system the entire family of all states satisfying (1.26). Since, in general, this family contains more than one state, the theory will not predict a for the outcome within B’ E am of a sharp probality P&;;j*.*Am’(B’) determinative simultaneous measurement of A,‘, ...,A,,,’ at time t, ( t , > to),but rather a whole range of such values. Statistical considerations (to be mentioned in $8 in connection with the “statistical operator”) can replace this range by a representative “mean” value.

1.5. THESTOCHASTIC NATURE OF OF

THE

QUANTUM THEORY

MEASUREMENT

As we see from Definitions 1.3 and 1.4, the only link between the formalism and the physical reality is via the probability measures (1.14). An experiment consists in first having the system submitted to a preparatory procedure at some instant to and then carrying out a determinative measurement at some later instant t , . T h e Born correspondence rule in Definition 1.4 enables us to specify a set of states Y(t)as being the states in which the system can “find itself” after the preparatory procedure. We note that even in the most ideal case we cannot assign to the system at time to a single vector Yo, but rather an entire family (ly, Y = cY0 , c E C1,c # 0} of vectors. This explains the reason for the statement that if Y(t)represents a state, then c Y ( t ) , I c I > 0, represents the same state. From Born’s correspondence rule for determinative measurements (Definition 1.3) we note that the quantum mechanical theory, as opposed to the classical theory, does not predict the outcome of a single measurement carried out at some instant t , > t o . T h e prediction is statistical, and as such it refers to a large ensemble of identical measurements rather than a single measurement. This means that the preparatory procedure should be carried out at the time to on a large sample of N identical and independent systems. Then the theory predicts the frequency with which a determinative simultaneous measurement of certain compatible observables A, ,..., A, , carried out at time t, will have an outcome within the set B; i.e., if out of the N systems a number v(B) will have at t values within B for A, ,..., A,, then, if the theory is correct, we must have the following approximate relationship, (1.27)

IV. The Axiomatic Structure of Quantum Mechanic8

268

with an accuracy which increases with an increasing number N of system in the ensemble. It is conceivable that in some exceptional" cases a measurement on the set A, ,..., A, of compatible observables, which is complete with respect to Yoand satisfies (1.20), can serve not only to determine whether the values of A, ,..., A, are A:)', ..., A;'), but also as a preparatory procedure which leaves the system with the values A!'), ..., A)'; for A, ,..., A,, respectively-at least for a short time after the measurement. If the system was prior to this measurement in a state Y ( t ) , then after the measurement at time t , it will be in a state P(t)for which Y ( t o ) = Yo. In this case, the probability (1.28)

.....

pA Wto) i A, ({A?)}

x

*.-

x

{At)})= I(Yo I Y(to))12

can be called the transition probability of having the interference introduced by the process of measurement cause the transition of the system from the state Y ( t ) to the state Y ' ( t ) which is equal to Yoat t o . For this reason the expression /('PII Y2)12 for two normalized vectors representing two states at some instant is sometimes called a transition probability.

EXERCISES 1.1. Assume that A, , i = I, 2, are symmetric operators on 2 = gA1 = g A , with spectral measures P i ) ( B ) ,i = 1,2. Show that if [E(l)(B,),~ ? ( ~ ) ( h= ~ )0] for all B, , B, E gl, then [A, , A,] = 0. 1.2. Show that if A,, k = 1, 2, are commuting bounded symmetric operators on X , then their spectral measures commute.

Note. A corresponding statement is by no means true when A,, A, are unbounded operators, i.e., when QAl # &? and QA2 # 2.In fact, it has been shown by Nelson [1959] that there are self-adjoint operators with a common dense domain which they leave invariant and on which they even commute (being essentially self-adjoint; see $7) but possessing noncommuting spectral measures (see also Exercise 5.4). 1.3. Suppose p(x) 2 0 is a Lebesgue integrable function on R". Introduce the Lebesgue-Stieltjes measure p,(B) = JB p ( x ) dnx, B E 9'. That this is not always so can be seen from the extreme case when the system itselfk destroyed by the determinative measurement, e.g., by becoming a part of the detector (for instance, when the detector is a photographic plate).

269

2. Functions of Compatible Observables

Show that a Bore1 measurable functionf(x), x E Rn,is pDintegrable on Rn if and only iff(x) p(x) is Lebesgue integrable on Rn,and in this case#

1.4. Use the result of Exercise 1.3 to establish that the spectral measure (E(k)(B)#)(x)= xB(xk)#(x), x E Rn, belongs to the operator (Qk$)(x)= x k #(x), x = (x, ,..., xn), which has the domain of definition g o k = {#: Jwn I xk#(x)12 dnx < +a}.

The spectral measures of Q1,..., Qn are (Eor(Bk)#)(x)= Prove that the spectral measure E ( B ) defined by (E(B)#)(x)= xB(x) #(x) coincides with Eol.***om(B).

1.5.

XB,(xk)

#(x).

1.6. Prove that if U is a unitary operator and A is a self-adjoint operator with the spectral measure EA(B), then the spectral measure of the self-adjoint operator A, = U-lAU is EA1(B)= U-'EA(B)U.

2. Functions of Compatible Observables 2.1. FUNDAMENTAL AND NONFUNDAMENTAL OBSERVABLES In $1 we had postulated that in quantum mechanics an observable is represented by a self-adjoint operator in a Hilbert space and that a state is represented by a vector-valued function Y(t),t E R1. Thus, as the most basic requirement for a quantum mechanical description of a system we have the following.

Axiom 0. To any quantum mechanical system belongs at least one complex Hilbert space i f in which the quantum mechanical theory of that system can be formulated. If h is a normalized vector in the complex Hilbert space S, the set h = {ah: a E @I, I a I

= 1)

is called a ray in 2.We note that if Yois a normalized vector representing a state at an instant to , then for any complex number a, I a I = 1, aYo is another normalized vector representing the same state. These This result can be deduced also from Radon-Nikodym's theorem (see Halmos [1950]) once it is observed that p(x) equals the Radon-Nikodym derivative dp,,(x)/d"x.

270

IV. The Axiomatic Structure of Quantum Mechanics

remarks justify the frequently made statement that “a state of a quantum system is (at any given instant) a ray in the Hilbert space.” In analyzing the concept of observables we have demanded until now that an observable play the dual role of being related to a symbol of the formalism (a self-adjoint operator, in case of quantum mechanics) and at the same time be anchored directly in the experiment by being related to some empirical procedure (or procedures) for measuring it. We shall call observables satisfying both these conditions fundamental observables, to distinguish them as a subfamily of the wider family of observables in general, which will be introduced next. In order to realize the necessity of enlarging the concept of observables, consider the concept of a function of observables in classical mechanics. Take, for instance, a one-particle system. If we know the position r and momentum p of the particle at some instant, then we can compute the values of other observables-such as angular momentum r x p, kinetic energy” p2/2m, potential energy V(r)-since all these other observables are functions of r and p. More generally, if we take any real-valued function F(r, p) in the six variables of r, p, we can think of it as an observable, because an indirect measurement of F(r, p) can be carried out by measuring the position and momentum of the particle and then computing F(r, p). The above argument indicates that we can generalize the concept of an observable in quantum mechanics by introducing functions of one or more already given fundamental observables. However, while in classical physics all observables are assumed to be compatible, this is not the case in quantum physics. Thus, in quantum mechanics we shall be able to give an adequate definition only of functions of a single observable or, at most, of a set of compatible observables.

2.2. BOUNDED FUNCTIONS OF COMMUTING SELF-ADJOINT OPERATORS The mathematical basis of the concept of a function of observables is contained mainly in Theorems 2.1 and 2.3 of the present section.

Theorem 2.1. Let E”i(B),..., E”-(B) be the spectral measures of n commuting self-adjoint operators A, ,..., A, in X . Denote by EA1.e. .A* (B), BE*, the spectral measure on which satisfies the relation

*

The mass m and the force field characterizing the problem are assumed to be given.

27 1

2. Functions of Compatible Observables

for all B, ,..., B, E gl. If F(X, ,..., A,) is a bounded Bore1 measurable function on Rn,

there is a unique bounded linear operator A satisfying the relation

for all vectors f,g E i%?. This operator A, which is conventionally denoted M. by F ( A , ,..., An),is bounded and 11 F ( A , ,..., A,)ll

<

Proof.

Since the complex measure pf,s(B)= (f I E"l**"*An(B)g), B E any

is equal to a sum [see (5.13) of Chapter 1111 of finite measures, any bounded measurable function on Rn is integrable (see Chapter 11, Theorem 3.9). Hence, the functional

is defined for all f,g E &'. This functional is obviously a bilinear form. Let us show that (f I g ) is a bounded bilinear form. Construct a sequence s1 ,s2 ,... of complex simple functions on Rn

Bik n Bi,k = 0

for which (see Exercise 2.1)

and for any f,g E S

for i # i',

IV. The Axiomatic Structure of Quantum Mechanics

272

An application of (2.5) and of the Schwarz-Cauchy inequality, first in &‘ and then in Z2(nk),yields

< M llfll Ilg 11. The above inequality implies, when combined with (2.6),that

I(f I g)l

< M llfll

II g II.

Thus, ( f I g ) is a bounded bilinear form, and therefore by a straightforward application of the Riesz theorem (Chapter 111, Theorem 2.3; see Chapter 111, Exercise 2.5), we infer the existence of a linear operator A for which
and 11 A

= (fig)

(1 \< M. Q.E.D.

An operator A for which a Bore1 measurable, bounded’ function and commuting self-adjoint operators A , ,..., A, can be found such that A = F(Al ,..., A,) is said to be a function of the operators A , ,..., A,. It is important to note that for two different functions F,(Al ,..., A,) and F2(A, ,..., A,) we shall have F,(A, ,..., A,) =

F(A, ,...,A,)

The case of unbounded functions is going to be treated in Section 2.4.

273

2. Functions of Compatible Observables

F,(A, ,...,A,) in the case that these two functions coincide when A, E S A 1 ,..., A, E SAn (see Exercise 2.2). Thus, we can defineF(A, ,..., A,) even in the case when the function F(Al ,..., A,) is defined only on the set SA1 x x S A n , and is bounded and Borel measurable on SA1 x x S A n . We can do that by extending F(Al ,..., A,) to a bounded this can be done, for instance, by settingF(A, ,..., A,) = 0 function on UP; when (A1 ,..., A,) 4 S A 1 x --.x S A n . In the above considerations we have allowed F(Al ,..., A,) to be a complex function. We shall establish that F(A,,..., A,) is self-adjoint whenF(A, ,...,A,) is real by proving Theorems 2.2 and 2.3.

Theorem 2.2. Let {A, ,..., A,} be a set of commuting self-adjoint operators. If F(A, ,..., A,) is a complex-valued, bounded, Borel measurable function on IWn and F*(A, ,..., A,) is its complex conjugate, then F*(A, ,..., A,) is the adjoint of F(A,,..., An). Proof. According to Theorem 2.1

‘The same theorem applied to F*(Al ,..., A,) yields (g I F*(Al ,..., A , ) f )

=

/

F*(h) d(g

I E:’”..oA”f.

R”

Thus, we get

for allf, g E A?. This shows that F*(A, ,..., A,) = [F(A,,..., A,)]*. Q.E.D.

Theorem 2.3. If F(A, ,..., A,) is a real bounded Borel measurable P and A, ,..., A, are n commuting self-adjoint operators, function on U

274

IV. The Axiomatic Structure of Quantum Mechanics

then the operator A = F(A, ,..., A,) measure P ( B ) satisfies the relation

is self-adjoint and its spectral

EA(B) = EA1.**-.An

(F-YB))

(2.7)

for all B E 9,.

Proof. If F(A, ,..., &) = F*(A, ,..., A,), preceding theorem [F(A, ,..., A,)]*

= F*(A,

we have according to the

,..., A,)

= F(Al

,..., A,).

Thus, the operator A = F ( A , ,..., A,) is self-adjoint. The projector-valued set function (2.8)

E(F)(B)= EA1***** W-’(B))

is obviously a spectral measure on @. We shall prove that for all f,g (2.9)

(f I 4) =

I

P

A4

f I EYk),

and that consequently the spectral measure E(F)(B)defined in (2.8) is indeed the spectral measure of A. From (2.3) we see that the relation (2.9) is equivalent to the relation

In the case whenF(A, ,..., A,) = xBl(A, ,...,A,), B, E@, an easy computation shows that (2.10) reduces to the relation (2.8), which is true by definition. Hence, we can immediately conclude that (2.10) is also true when F(A, ,..., A,) is a simple function. From this result, (2.10) can be established straightforwardly first for F(A, ,...,A,) nonnegative, and afterward for any real function F(A, ,..., &), by employing Definitions 3.4 and 3.5 in Chapter I1 for integration on measure spaces. Q.E.D.

2.3.

ALGEBRAS OF COMPATIBLE OBSERVABLES

Suppose that the self-adjoint operators A, ,..., A, represent compatible observables. If we carry out at some instant t a simultaneous measurement of these observables, we will determine, ideally, a measure P$(j;**An(B)-in the case that the system was in the state Y(t)at that instant. Thus, we can also compute the measure (2.11)

A

pY&q = P”,’,.;;-~~(F-~(B)).

275

2. Functions of Compatible Observables

It is then natural to say that the measure (2.11) provides the distribution of values of the observable represented by the operator = F(Al ,..., A,). This ruling is in agreement with (1.14), since by Theorem 2.3 we find that P&(,,(B),defined by (2.1 l), satisfies indeed the following relation:

A

We shall say that A represents a bounded observable which is a function of A, ,..., A , . The family of all such observables has an algebraic structure, which can be established on the basis of Theorem 2.4.

Theorem 2.4. If Fi(Al ,..., A,), i = 1, 2, are two bounded Borel measurable functions on UP and A, ,..., An are commuting self-adjoint operators, then

+

Proof. Since Fl and F2 are bounded functions, their sum Fl F, and product F, F, are also bounded. I n addition, F, F, and F, F, are Borel measurable according to Theorem 3.5 of Chapter I1 so that the operators

+

A

=P-1

+ FZ)(Al ,-..,4,

B

= (Fl

*FZ)(Al

7***,

An),

are defined by Theorem 2.1. We easily obtain (2.12) by using the definition of A as a function of A, ,..., A , to derive that for all f,g E 2,

In order to prove (2.13), write first

276

IV. The Axiomatic Structure of Quantum Mechanics

obviously vanishes on sets B, lying outside B. Consequently (see Chapter 111, Exercise 6.7)

which, when combined with (2.14), shows that (2.13) is true. Q.E.D. We note that albeit in general we cannot add or multiply unbounded operators without paying close attention to domain questions (cf. Chapter 111, §2.6), we do not run into similar problems with bounded functions of self-adjoint operators A, ,..., A, representing compatible observables. Denote by ,..., A,) the family of all bounded selfadjoint functions of A, ,...,A, . If a is a real number andF(A, ,..., A,) E Ob(A, ,..., An), then obviously also (uF)(A,,..., A,) E Ob(A1 ,...,A,), Moreover, if F,(A, ,..., A,), F,(A, ,...,A,) E Ob(A1 ,..., A,), then on account of Theorem 2.4,

Thus, &+,(A, ,..., A,) is a real algebra, usually referred to as the algebra generated by the observables A , ,..., A,. Furthermore, Ob(A1 ,...,A,) is obviously a commutative algebra.

277

2. Functions of Compatible Observables

In studying the relation between theory and experiment, it is often convenient to consider instead of the original observables A, ,..., A, the algebra @,(A, ,..., A,). The reason for this is that, while some of the operators A, ,..., A, will be, generally speaking, unbounded and therefore not defined on the entire Hilbert space, the operators in @,(A, ,..., A,) are bounded and defined on the entire Hilbert space. On the other hand, @,(A, ,..., A,) supplies all the physical information that A, ,...,A,do. The reason for this is that this information is contained exclusively in the expressions (2.15)

(Y(u(t) I E A l * . * * * A n( B )'u(t>>,

where EA1*..*pAn(B) are bounded functions of A,

,..., A,,

as can be seen by taking forF(A, ,..., A,) in (2.3) the characteristic function xB(h, ,..., A,) of the set B E an. If A is an operator representing an observable and Y(t)~9~, then the expression

is called the expectation value of A at time t . In $1 we saw [see (1.15)] that in the case of a determinative measurement of A at time t , (2.17) is the mean value of the statistical sample obtained as an outcome of that measurement if the system was in the state Y(t)at time t . According to (2.15) and (2.16) we always have

This prompts us to state that a knowledge of the expectation values of all self-adjoint bounded functions of the observables A, ,..., A, is equivalent to a knowledge of the statistical distribution of the simultaneous value of A , ,..., A, . Hence we shall call such functions also observables.

*2.4. UNBOUNDED FUNCTIONS OF COMMUTING SELF-ADJOINT OPERATORS We shall study now the general concept of a function F(A,,..., A,) of A, ,..., A, for the case when F(A), h E R", is any Bore1 measurable function on Rn. n self-adjoint commuting operators

278

IV. The Axiomatic Structure of Quantum Mechanics

Theorem 2.5. Let A, ,..., A, be n commuting self-adjoint operators B E &In, the spectral measure defined by (2.1). in 2, and EA1*-**Jn(B), If F(h), h E Rn, is a Bore1 measurable function, then there is a unique linear operator A which satisfies (2.19)

Iw”F(X)

( g I Af) =

d(g I E?9..**Anf)

for all g E &. This operator is defined on the domain (2.20)

9~=

1) 1., I

F(A)I2d 11 E?”“’””fll2

<

A is usually denoted byF(A, ,..., A,). Proof. Denote by R the set of all vectors f E 2 for which the linear functional v,(g)

=

w”

F*(h) d(E?*.***An f lg)

exists for all g E 2 and is continuous. For each such f E R there is by the theorem of Riesz (Chapter 111, Theorem 2.3) a vector h such that (h I g )

=

v m

or equivalently, and with the notation h = A( f ), ( g 1 A(!))

=

Iw”

F(h) d ( g I EA1*..***”f).

The set R is linear, because if f, ,f, E R, we can derive for anyg E X , by using the rules of integration,

i.e., a, f,

+ a2f, E R; the above relation can also be written as

2. Functions of Compatible Observables

279

In order to show that R coincides with the set g Adefined in (2.20),let us show first that f E R implies that f~ g A. Taking in (2.19) g = Af, we get

11 Af [I2

=

/

-S n ,

F(h)d(Af I E P * - - . * A ” f ) P”

F(A)d(E,A’**.**A”f I Af )*,

and since we have (see the proof of Theorem 2.4)

we obtainfE g A(see Chapter 111, Exercise 6.7):

Conversely,let us assume that the above relation holds for some f E Z, and let us show that f E R.Choose a sequence of simple functions $,(A), s2(A), ... such that (see Exercise 2.1)

If we write s&)

in the form (2.4), we can derive for any g E &‘

280

IV. The Axiomatic Structure of Quantum Mechanics

T h e expressions on the left-hand side of (2.22) constitute a convergent sequence since they are monotonically increasing for k = 1, 2,..., while the right-hand side is bounded by

As this observation stays true if Re aik is replaced in (2.22) by Im aik , we arrive at the existence of the limit

This argument establishes, incidentally, the following inequality:

Since the above argument stands for any g E X , it follows that f E R. Q.E.D. I t should be noted that, in general, the domain of definition of F(A, ,..., A,) is not dense in X . However, in the case that this domain is dense and F(h) is real, it follows from Theorem 2.6 that F(A, ,..., A,) is self-adjoint.

*Theorem 2.6. If A, ,..., A, commute and A a dense domain of definition, then A * = F*(A, function F*(h) is the complex conjugate of F(h).

= F(A,

,..., A,),

,..., A,)

has where the

Proof. Consider the operator D, which is defined as that operator which satisfies

(f I Dg)

=

J'

F*(h) d(f

I EP*.*..An g>

R"

for all f E X . I t is obvious from Theorem 2.5, especially from (2.20), that the domain g Dof D is identical to g A. If we take f,g E g A= g D , we get

(f I Dg)

=

(IF(h) d(g I EP**..*An f >)* R"

=(g

I a>* =(4lg>,

28 1

2. Functions of Compatible Observables

which shows that D C A*. We shall prove that D = A* by showing that CgD. Assume that h € B A *and h* = A*h, so that the relation

gA*

is valid for all f ~ function

5 . Consider 9 ~ the bounded and Bore1 measurable

According to Theorem 2.1, the operator A,* bounded. We easily see that A,*h E g A,

= F,*(A,

,..., A,)

is

282

IV. The Axiomatic Structure of Quantum Mechanics

From (2.23) and the above equation we get

II &*h 112

=
I &*h> d II h* II II

II,

which implies that

(1 IF,(h)12 a”

d 11 E,A’**-”nh112)

1/2

= 11 A,*h

11

< 11 h* (I.

If we let M + +a,then the left-hand side in the above relation stays bounded, and since lim F,(h)

M+m

= F(h),

the integral

exists and is finite (see Chapter 11, Theorem 3.10). Thus, h E 9,, , which C g D. Q.E.D. establishes that gAI We can generalize the concept of a function of compatible observables ,..., A, by including all real Bore1 measurable functions F(h) which are such that the set (2.20) is dense in 2‘.Theorem 2.6 guarantees that the operator F ( A , ,..., A,) representing such a function will be selfadjoint, and therefore we can call it observable. As a generalization of Theorem 2.4 to the unbounded case we prove Theorem 2.7.

A,

Theorem 2.7. If F,(A, ,..., A,), i = 1,2, are any two functions of the commuting self-adjoint operators A, ,..., A, , then (2.24)

+3‘2(A1

9...7

An) C (FI

+F d A 1

5.-7

An),

4%) c (Fl -F2)(A1 An). any function F(A,,..., A,) and any integer m 2 0 we have

(2.25)

For

Fl(A1 ~ * * An) . ~

(2.26)

F l ( 4 ,..*, An)F2(A1

[W,

Y’**Y

Y*..7

)**a>

An)]” = (F”)(Al

4.

Proof. It is easy to see that 9~~ n 9~~ C~

F ~ + F *

since whenever Fl(h) and F2(X)are square integrable with respect to a measure p,(B) = 11 E”’--.”n(B)f 112,

283

2. Functions of Compatible Observables

+

the function (Fl F2)(h) is also square integrable (see Chapter 11, Theorem 4.2). If we introduce the functions

F,(A,

,.-a,

A,)f

+ F2(Al

,.**,

&)f

=P I

t F2)(-41

¶**+,

A9J.f.

We turn now to proving (2.25). By the already familiar procedure of dealing with spectral-resolution integrals, we get

I F,(h)I2 d 1) E:‘B-*-AnF2(A1 ,..., An)f/12

=

1,. IFl(h)F2(h)12 (1 E,A’”.a-A,f!)2. d,,

The above relation shows that whenever F,(Al ,..., A,) f belongs to the domain of definition ofFl(Al ,...,A,), so thatFl(Al ,...,A,)F,(A, ,..., A,) f is defined, then f belongs to the domain of (Fl * F,)(Al ,..., A,). If we take such a vector f, and apply Theorem 2.4 to FIMl(Al,...,A,) and F2Mz(A1,..., A,) to obtain (2.27)

FIMl(Al

)*--,

,..*)

An)FzM2(Ai

= (FIM1 * F 2 M 2 ) ( A 1

Y*-*)

&)f,

we get by taking the strong limits, first for M , -+ co [note that the operator FIMl(Al,..., A,) is bounded and therefore continuous] and then for Ml ---f co,

F l ( 4 ,..-,A,)FZ(Al

Thus, (2.25) is established.

,.a*,

A,)f

= (Fl *F,)(Al ¶**.,

A,).f.

IV. The Axiomatic Structure of Quantum Mechanics

284

From (2.25) it follows directly that [F(Al

I...,

4 1 " c (Frn)(Al,..., A,)

for any m = 0,1, 2, ... . However, when m 2 2, the square integrability of (F(A))" for some measure pI implies the square integrability of (F(A))rn-l,since I F(X)12(m-1) I F(A)I2" at any point X where I F(X)I 2 1. This means that QFm C .CBP,-, . Hence, by taking in (2.27) F, = F and F2 -- Fm-1, we get for f E gP C gP-l in the limit M2 --+ 00

<

+

FM1(A1,..*,A,)Fm-'(A1

,**.,

An)f= (%,)(A1

,***,

An).$

By letting M l -+ +a we arrive at the result that FFm-'f = Fmf and that gFF,,,-, 3 .CBF,,, , i.e., FFm-l 2 Fm. Thus, by induction in m [F(A1 ,***, An)]" 2 (Frn)(A1).**s An), i.e., (2.26) holds.

Q.E.D.

T h e relation (2.26) indicates that the definition of functions of observables given in Theorem 2.5 coincides with the algebraic definition whenever the algebraic definition is feasible. I n fact, the function of n commuting operators A , ,..., A, corresponding to a polynomial of n real variables A, ,..., An is equal to the operator obtained by replacing A, by A , for i = I , ..., n everywhere in the polynomial. I n particular, if

p(A) = a$

+ un-lXn-l + + a,A + a, **.

and A* = A , and if p ( A ) is defined by

it is easy to establish that

by using the linearity property of the integral in (2.28) and the relation

(2.26) with F(X) = A. EXERCISES 2.1. Prove that if F(A), A E Rn,is Bore1 measurable on Rn and p(B), B E gnJ is a complex measure, then there is a sequence of simple func-

285

3. The Schroedinger, Heisenberg, and Interaction Pictures

tions that satisfy (2.21) with sign ReF(h) = sign Re Sk(h), sign Im = sign Im Sk(h), and F(h) is p integrable if and only if

F(h)

exists, in which case the above limit is equal to

2.2. Let A, ,..., A, be n commuting self-adjoint operators and F(h), h E R”, a function which vanishes on S A 1 x x SAn. Show that F ( A , ,..., A,) = 0. 2.3. The proof of Theorem 2.5 is independent of Theorem 2.1. Derive Theorem 2.1 from Theorem 2.5. 2.4. Show that in general the symbol C cannot be replaced in (2.24) and (2.25) by the symbol =. 2.5. Show that any two bounded functions F,(A, F2(Al,..., A,) of A, ,..., A, commute.

,..., A,)

and

2.6. Prove that to each spectral function E,, , A E R1, corresponds a unique self-adjoint operator A such that

forallgE&andforallfE9A. 2.7. Show that if Q1 ,..., Q, are the self-adjoint multiplication operators (Qk#)(xl ,..., x,) = Xk#(x1 ,..., x,) in L2(Rn), then F(Ql ,...,Q,) coincides with the operator A defined by (A+)(%

Y...,

x,)

= F(x1 Y...,

BA = {+: F 3.

3.1.

*

xn)

+(x1

,-**,

+ EL’(R~)}.

xn),

The Schroedinger, Heisenberg, and Interaction Pictures THEGENERAL FORMOF

THE

SCHROEDINGER EQUATION

We have seen in $$l and 2 that in order to “describe” a system quantum mechanically, we have to associate with it a Hilbert space (42, Axiom 0). The observables are represented by self-adjoint operators in X , while the physical states are vector-valued functions Y(t)in the time variable t E W. Clearly, not every vector-valued function can be a state, because in that case-as can be easily seen from the Born corre~

286

IV. The Axiomatic Structure of Quantum Mechanics

spondence rules in $l-the theory would lack predictive power. We must require that a function Y(t) which represents a state should satisfy a dynamical law (see t h e Introduction). This law is given in quantum mechanics by an equation of motion, called the Schroedinger equation. We can arrive at the general form of this equation by generalizing the Schroedinger equations of Chapter I, $5 and Chapter 11, $5. T h e essential feature of those special cases was the existence of an operator H, which in those cases was the Schroedinger operator [essentially given by the Schroedinger operator form in (5.8) of Chapter 111 and it was also related to the total energy observable. More generally, we postulate for each system which can be described quantum mechanically, the existence of a self-adjoint operator H, commonly called the Humiltoniun of the system. This operator is assumed to play a double role in the theory. On one hand, it represents the total energy observable, and it is therefore related to measurements of the energy of the entire system. On the other hand, it plays a key role in the, equation of motion; namely, as a generalization of (5.5) in Chapter 11, we postulate that a state satisfies the Schroedinger equation

There are, however, two mathematical questions which a careful look at the above equation immediately brings to mind. The first question is concerned with the meaning of the derivative (3.1). With some foresight (see also Exercise 3.1), we answer that question by defining 1 At

+ At) - Y(t)).

d W ) = s-lim - (Y(t dt

At+o

The second question is related to the fact that since H is usually an unbounded operator, we cannot find a solution of (3.1) for every state Yo prepared at some instant t, since we have to require that YoE gH, while gH # X . We can solve this last difficulty by formulating the dynamical law by other means than the equation of motion (3.1).

3.2.

THEEVOLUTION OPERATOR

Consider the function expf-(i/fi) h(t - to)), which is continuous (and therefore Bore1 measurable) as well as bounded for h E R1. Due to these properties, we can define a bounded operator (34

w, to)

= exp[-(iP)

H(t - to)],

287

3. The Schroedinger, Heisenberg, and Interaction Pictures

customarily called the time-evolution operator. I n accordance with Theorem 2.1, U(t, to) is the operator satisfying (3.3)

for all f, g E 8,where EA is the spectral function of H, H

(3.4)

=

5,

h dEA

We shall prove a theorem from which the main properties of U(t, to) can be derived immediately. In proving that theorem we need two lemmas that will find many later applications.

*Lemma 3.1. (Lebesgue's dominated, or bounded, convergence theorem). Let fl(t),f2(t),... be a sequence of nonnegative functions defined on the measurable set R of a measure space (.%, d , p) and integrable on R. Suppose that limn+mfn(t) = f(t) almost everywhere. If there is a function g(5) integrable on R and such that 1 f,(t) I g(t) for all n = 1, 2, ..., thenf(6) is integrable on R and

<

Proof.

By Theorem 3.4 in Chapter 11, f ( 8 ) is measurable so that gn(5) = (Ifk(5)

-f(5)1

: k = 71,

71

+ l,...},

n

= 1,

2,...,

<

are also measurable (see Chapter 11, Exercise 3.3). Since g,(t) 2g(t), the functions g&) are integrable on R (see Chapter 11, Theorem 3.9). Furthermore, lima+mga(t) = 0 almost everywhere on R. Since gl(t) 2 g&) 2 ..., we can apply Theorem 3.10 of Chapter I1 to derive

We get the desired result by observing that ) ( , f I

-f(t)

I

< g,(t).

Q.E.D.

*Lemma 3.2 (Fatou's lemma). If fi(<),f2(t),..,is a sequence of nonnegative functions, integrable on the subset R of measure space (3, d ,p) and converging almost everywhere to a functionf(t), and if

288

IV. The Axiomatic Structure of Quantum Mechanics

for all n, thenf(5) is integrable on R and

Proof. Consider the functions

which are integrable on R (see Chapter 11, Exercise 3.3) and note that f ( 5 ) = limn+cog,([) almost everywhere in R. By applying Theorem 3.10 in Chapter I1 to the sequenceg,(e), g2(5),... we infer thatf(5) is integrable on R and

, ( Since g

5)
which establishes that M is a bound for the integral off(.$) over R. Q.E.D.

Theorem 3.1. valued function (3.5)

If A is a self-adjoint operator in A?, the operatorUt

t

= eiAt,

E

R1,

has the following properties: (a) Uo = 1; (b) U , is a unitary operator on A? for all t E W; (c) Ul1Ut2= Url+t2for all t, , t , E W1; (d) i f f E B A , iAf

= s-lim t+O

~

Utf - f t

and, conversely, the above strong limit exists only iff E B. ~

Proof. (a) If EAis the spectral function of A

3. The Schroedinger, Heisenberg, and Interaction Pictures

289

then by Theorem 2.1 ( f I utg)

(3.7)

When we insert above t

=

=

< f I uOg>

I,

eiAt

d ( f I EAg).

0, we obtain

IRl

d ( f I EAg)

=

=

( f Ig>

for all f, g E X', which proves that U , = 1. (b) According to Theorem 4.3 in Chapter 111, U , is unitary if it satisfies the relations

ut*ut= utut*= 1.

(3.8)

The above relation can be established immediately by noting that =

eiAt(eiAt)*

1

and using Theorems 2.2 and 2.4. (c) By using Theorem 2.4 we get ( f I UtlUt,g>

=

s,

eiAtleiAtz d( f

=

IR1d(f

=

< f I Ut,+t,g>

ei~(t1+t2)

for all f, g E X , which proves that U1,U1,= UL,+,,. (d) The domain of definition of the operator ( l / t ) ( U t- 1) - iA

(3.9)

is 9.. since , U, is defined everywhere on X . T h e operator (3.9) is a function of A, corresponding to the complex function Ft(X) = ( l / t ) ( e i A t 1) - ih.

(3.10)

For f E BAwe have by Theorems 2.5 and 2.7 (3.11)

I/ 7 1

(Ut - 1 ) f

I

-

iAf

1)

2

( f ( U t- 1) - i A ) *1 ( p t- 1) - iA

=

(f

=

IR1 I (eiht 1)

1

-

- ih

I2

d ( f I EAf ) .

IV. The Axiomatic Structure of Quantum Mechanics

290

Since we have by the mean value theorem of the differential calculus (l/t)(eiAt - 1) = ih(cos X8,t

+ i sin XB,t),

0 < 81,2 < 1,

we get for all t # 0 and X E R1,

I F,(X)I2< I h l2 I cos h8,t

+ i sin X8,t

- 1 l2

< 4 I h 12.

Thus, I F,(X)l2is a function majorized for t # 0 by the function 4 I h 12, which is integrable in the measure p,(B) = (f I E(B)f ), and for which limt+,, I FXX)lz= 0. Consequently, by Lemma 3.1 we have

and we can conclude from (3.11) that (3.6) is true. Assume now that

exists for some g E &'. Due to Theorems 2.1, 2.2, and 2.4 we can write (3.13)

1 l l p t

2

-

1k/I= (g

I

1 jdut

- I)*(Ut -

lk)

The existence of the strong limit (3.12) implies (see Chapter 111, Exercise 5.5) the existence of a constant M such that 11(1/t)(u, - 1)f 112

<M

for all t E R'. Since in (3.13)

we get by applying Fatou's lemma (Lemma 3.2) that X2 is integrable with respect to the measure p&B) = ( g I E(B)g). According to the spectral theorem in Chapter 111, 96,this implies that g E g A. Q.E.D. Theorem 3.1 shows that the operator U(t, to) in (3.2) is unitary. Furthermore, from (3.6) it follows that iff E BH then for any t E R1 (3.14)

s-lim r+q

U(t, to)- 1 2 - to

i f =-Hf. A

3. The Schroedinger, Heisenberg, and Interaction Pictures

291

The operator U(t,to)obviously leaves BHinvariant, i.e., iff E g Hthen , U(t, t , ) f ~BH, for all t , to E Rf. This can be seen by applying the spectral theorem, and noting that [E, , U(t, to)] = 0, to obtain

From Theorem 3.lc we easily get

w ,to) W

(3.15)

O

Y

21)

=

w ,tl).

Combining (3.14) and (3.15), we derive that for any f E B ~ d ifi - U(t, to)f = ifi s-lim U(t dt At-0 = ifi

s-lim

U(t

Af+O

+ At, to) - W t , to) f At

+ A t , t ) - 1 U(t,to)f At

= HU(t, to)f.

Thus, if YoE g H, then

Yf) = U(4 t o ) yo satisfies the Schroedinger equation (3.1), and since U ( t o ,to) = 1, it also satisfies the initial condition Y ( t o )= U ( t 0 ,to) Yo= Yo.

We shall see in a later section that Y(t)is, as a matter of fact, the only vector-valued function of t satisfying these conditions and the requirement that 11 Y(t)ll = (1 Yo11 for all t E [wl.

3.3. THESCHROEDINGER PICTURE The formalism of quantum mechanics espoused in the last two sections has the striking feature that a state is represented by a vector-valued function Y(t)in the time variable t , while the observables are represented by self-adjoint operators which do not depend on t. This formalism for quantum mechanics is called the Schroedinger picture. The statement that in the Schroedinger picture the observables are represented by operators which do not depend on t has to be accepted with one qualification; namely, there are cases when the Hamiltonian of

292

IV. The Axiomatic S t r u c t u r e of Quantum Mechanics

the system depends on t, i.e., is given by H(t ) . Such instances will occur when a system G interacts with another system GI,and we know how the interaction term in the Hamiltonian depends on t. I n such a case it is very often impractical to study the behavior of the compound system G Gl, when, for instance, Glis very large" and complicated. We shall summarize the characteristic features of the Schroedinger picture in the following axioms, in such a way as to include the possibility of the most general case when the Hamiltonian is time dependent.

+

Axiom S1. A Hamiltonian H ( t ) is assigned to every quantum mechanical system; H ( t ) is an operator-valued function of the time parameter t. For each value o f t E R1, H ( t ) is a self-adjoint operator in the Hilbert space X associated with the system (see $2, Axiom 0) and H(t) represents the total energy of the system. Axiom S2. All the observables of the system that are not functions of the energy are represented by self-adjoint operators in S? which do not depend on t. Two or more observables are compatible if and only if their corresponding operators commute. Axiom S3. There is a unitary operator-valued function U(t,to), t, to E R1, called the time-evolution operator, which is such that

w o

(3.16) (3.17)

U(t, tl)

Y

to) =

1,

Wl , to) = w , to),

to E R1,

t, tl ,to E R1,

and is strongly continuous in t, to E R1, and for all f~ 9H(1) ,

Axiom S4. A state of the system is represented by vector-valued functions Y(t)which satisfy the relation

W)= w ,to) W o ) ,

II W 0 ) l l > 0,

for all t , to E R1. If c is a nonzero complex number, Y(t) and cY(t) represent the same state.

* Such a case occurs when-6, is an electromagnet producing a magnetic field which varies with time. It is true that in principle we could study the system B GI by considering the electromagnet itself to be a many-body system consisting of many atoms, electrons, ions, etc., but such an approach would be hopeless to pursue from the computational point of view. Instead, we could measure the microscopic electromagnetic field produced by the electromagnet, and treat the problem by introducing in the Hamiltonian of the system B a term which is a function of that field, and represents, to a good approximation, the interaction of the system and the electromagnet.

+

3. The Schroedinger, Heisenberg, and Interaction Pictures

293

We have seen that the conditions (3.16)-(3.18) are satisfied by the operator-valued function (3.2) when H ( t ) is the same for all t E R1. A certain generalization of (3.2) can be obtained when H ( t ) depends on time in some regular manner, but the considerations become quite involved if the operators H ( t , ) and H(t,) do not commute for t, # t , . In the time-independent case it is easy to see that (see Exercise 2.5) (3.19)

%t)W

=

(W) I En(B) 'u(t)> 9

II'Y(t)ll = 1,

is time independent. The physical meaning of this result is that the probability of measuring a certain value of the energy of the system in state Y(t)is the same at all times, i.e., that the energy is conserved. Naturally, if H ( t ) is intrinsically time dependent, this statement does not stay true in general. Physically, this is due to the interaction of the system with an external source in the course of which energy can be transferred to or from the source (see also Section 7.9).

3.4. THEHEISENBERG PICTUREAND PHYSICAL EQUIVALENCE OF FORMALISMS Let A be a bounded operator representing an observable, and let (3.20)

be the expectation value of that observable in the state (3.21)

Y(t)= U(t,O)P,

P = Y(0).

As we have seen in $2, the link between the quantum mechanical formalism and experiment is contained solely in the expectation values (3.20) of all possible bounded observables, so that two formalisms in which all expectation values are the same are physically equivalent. Since we can write (3.20) in the form

(3.23)

act) = u-ys,0) AU(t, O),

we could describe the same physical state of the system by giving the vector and?' the operator-valued functions A(t)in (3.23) for all bounded observables of the system. This formalism of quantum mechanics is known as the Heisenberg picture.

294

IV. The Axiomatic Structure of Quantum Mechanics

If D is an unbounded operator representing an observable, the operators B ( t ) = v - y t , 0 ) DU(t, 0 )

are self-adjoint for all t E R1 because U(t,0) is unitary (see Chapter 111, Lemma 4.3), and represent the same observable in the Heisenberg picture as D did in the Schroedinger picture. If the Hamiltonian H of the system is time independent, we have

since e-cr/fi)Hr leaves g Hinvariant and commutes with H. Thus, in that case Z?(t) will be time independent (in physicists’ language, the energy is conserved) and given by the same operator as in Schroedinger picture. In the more general case of time-dependent Hamiltonian H(t), we have in the Heisenberg picture (3.24)

A ( t ) = v - y t , 0 ) H ( t ) U(t,0).

In the Schroedinger picture the dynamical law was the Schroedinger equation (3.1), which had to be satisfied by the state Y(t).Analogously, in the Heisenberg picture the dynamical law is embodied in an equation of motion, which has to be satisfied by any operator function A(t) representing an observable. Since the link between the representations of states and observables in the two pictures is given by (3.21), (3.23), and (3.24), we can derive the equations of motion in the Heisenberg picture from the equations of motion in the Schroedinger picture.

Theorem 3.2. Let a(t)represent an observable in the Heisenberg picture, and assume that the same observable is represented by a bounded operator A in the Schroedinger picture. Iff and A(t)f are in the domain of definition of A(t) for all t E R1, then A(t)f satisfies the Heisenberg equation (3.25)

d&t) iti -f = [A(t),&(t)]$ dt

Proof. By definition, (3.26)

dA( t ) f = s-lim dt At-0

A(t

+ At) - A(t)fAt

3. The Schroedinger, Heisenberg, and Interaction Pictures

295

Straightforward algebra yields (3.27)

act + A t ) - Act) At - u-yt -

+ A t , 0 )A U(t + A t , 0 ) u-yt,0 ) A U(t,0) At U(t + A t , 0)- U(t,0) = u-yt + A t , 0)A -

At

+ u-v + 4A0)t - U-V,0 )A qt,0). WhenfE9fi(t),we get from (3.18) (3.28) ifi S;!.? U(t A t , 0) - U(t,0)f

+

At

By using the unitarity of U-l(t

+ i(;

u-yt

=

H ( t ) U(2, 0 ) f .

+ A t , 0) we obtain

+ A t , 0) - ;u-yt,0))A H ( t ) U(t,0)f 1 .

The first term on the right-hand side of (3.29) converges to zero due to (3.28) and the fact that A is bounded and therefore continuous (see Chapter 111, Theorem 1.3), while the second term converges to zero since, due to (3.17) and (3.16), (3.30)

for any t, , t, (3.31)

U-’(t, Y

E

tz) =

W z

Y

tl)

UP, and therefore

+ A t , 0 ) - u-yt,0)) = s-lim(U(0, t + At) U(0,t)) = 0.

d&( u-yt

-

At+O

Thus, we see that (3.32)

+

s-lim U-l(t At, 0)A At40

i

U(t

+ At, 0)- U(t,0)f At

i

= - - u-yt,0 ) A H ( t ) U(t,0)f = - ti Act) f l ( t ) f .

fi

IV. The Axiomatic Structure of Quantum Mechanics

296

If AU(t, 0 ) f E gH( 1) , or equivalently, a(t)fE 9 g (r) , we derive by employing (3.18) and (3.30) (3.33) = s-lim At+O

U(0, t

+ A t ) - U(0,t ) A U(tY 0)f At

Now we can obtain (3.25) by applying (3.27) to f and using (3.32) and (3.33) when taking the strong limits for A t +-0. Q.E.D. T h e Heisenberg picture is a formulation of quantum mechanics which is physically equivalent" to that of the Schroedinger picture. Consequently, its main features are derivable from Axioms Sl-S4. In order to formulate the dynamics in the Heisenberg picture independently of the Schroedinger picture, we introduce the unitary operators

O(t) = U(t,0).

(3.34)

For o(t)f~ 9?H(1) , i.e., w h e n f g 9?;(1) , (3.17) and (3.18) yield

ih s-lim At+O

O(t + A t ) - O(t)f At

= H ( t ) O(t)f =

O(t)R(t)f.

Furthermore, the relation (3.23) can be recast in the form (3.35)

A(t) = O-l(t) A(0) O(t).

The above relations do not contain any direct reference to the Schroedinger picture. Thus, we can now characterize the Heisenberg picture as the formalism fulfilling the following Axioms Hl-H4.

Axiom H1. A Hamiltonian k(t) is assigned to every quantum mechanical system. A(t) is an operator-valued function of the time parameter t . For each value of t E W, A(t) is a self-adjoint operator in the Hilbert space Z' associated with the system. k(t)represents the total energy of the system at time t .

* The physical equivalence of these two pictures was established by Schroedinger [1926b] only after both these pictures had been independently developed by Heisenberg [1925], and Schroedinger [1926a], in the matrix and wave mechanics versions, respectively.

3. The Schroedinger, Heisenberg, and Interaction Pictures

297

Axiom H2. A state of the system is represented by a time-independent nonzero vector !?' E 2;if c # 0, c!?' represents the same state. Axiom H3. There is a unitary and strongly continuous operatorvalued function ??(t) such that O(0)

(3.36) and such that for allf

1,

E 9fi( 1)

ih s-lim

(3.37)

=

At-0

O(t + A t ) - O(t)f At

=

O(t)A(t)f.

Axiom H4, An observable is represented by an operator-valued function a(t),where for each t E R', A^(t)is a self-adjoint operator in %. Moreover, A(t) satisfies the equation

Act) = P ( t ) A(0)U ( t ) .

(3.38)

3.5. THEFORMALISM OF MATRIX MECHANICS The Heisenberg picture and the Heisenberg equation (3.25) were first introduced by Heisenberg in the matrix mechanics version of quantum mechanics. We can obtain the matrix formulation of quantum mechanics for any particular system by choosing in the Hilbert space 2 assigned to that system an orthonormal basis* { e l , e2 ,...} which is contained in the domain of definition gA( !) of all operator functions A(t)representing observables. T o each A(t) we assign an infinite matrix 11 A i k ( t ) l l (see also Exercise 3.2), where

A,,(t) = <ei I Act) ek),

i, k

=

To a state represented by the normalized vector one-column matrix 1) c k I), where c,=(e,lPf/>,

1,2 ,... .

9, we assign an infinite

K = I , ,.... ~

11 c k 1) E 12( co) and that (A(t)9li= <ei I A(t)Y)=
Note that

m

=

c <&)

ei

I e k x e k 13)

k=l

k=l

8 is in practice infinite dimensional and separable.

k=l

298

IV. The Axiomatic Structure of Quantum Mechanics

Thus, 11 f&(t)ll can be considered to be an operator in Z2(co), which acts on the elements of Z2( co)according to the rules of matrix multiplication. T h e Heisenberg matrix picture can be useful when the Hamiltonian I? of the system is time independent and the elements {el , e2 ,...} can be chosen to be eigenvectors of A, i.e., if we can write ek = u k where

(an example of such a system is the quantum mechanical harmonic oscillator; see Exercise 3.3). I n that case is represented by the matrix with components

o(t)

3.6. THEINTERACTION PICTURE T h e physical equivalence of the Schroedinger and Heisenberg pictures is based on the fact that the physically significant quantities, i.e., the expectation values (3.20), are the same in both pictures. When the Hamiltonian is time independent, the Schroedinger picture is characterized by the property that the vectors representing states are time dependent, while the operators representing observables are time independent; in the Heisenberg picture the opposite is true regarding time dependence. It is clear that one can invent a large number of “intermediate” pictures “between” these two pictures, which would all preserve the expectation values, but in which the time dependence will be partially assigned to vectors representing states, and partially to operators representing observables. T h e most significant example of such an intermediate picture is the interaction picture, which is useful when the Hamiltonian H ( t ) of the system can be written in the form

where H , is the “free” Hamiltonian of the system (usually) describing a system without interactions; for instance, in the case of the n-particle

3. The Schroedinger, Heisenberg, and Interaction Pictures

299

system of Chapter 11, $5, H , is the self-adjoint operator defined by the differential operator form

while V is the multiplication operator defined by the potential

W, ,...)r,J

In order to arrive at the interaction representation for Hamiltonians

(3.39),we write the expectation value (3.20) in the form (3.40)

(4y ( t ) = (W) I 44W)>,

where

(3.42)

= e(i/fi)Hot TJ,

(t 3)P;

A(t)and p(t)are taken to represent in the interaction picture the observable and the state which were represented in the Schroedinger picture by A and Y(t)= U(t,O)Y, respectively. I n the case where V ( t ) is time independent, (3.42) becomes (3.43)

qt)=

,(i/fi)H0t e-(i/fi)Htp

If we apply Theorem 3.2 to (3.41), we obtain that for a n y f E a H 0 (3.44)

the above relation is the Heisenberg equation for the “free” system. Let us introduce the operator-valued function

The vector function p(t)representing a state in the interaction picture satisfies a Schroedinger equation with the Hamiltonian H,(t), (3.46)

dp(t) iti -= H,(t) dt

if P(t)E aHl(t) (see Exercise 3.4).

P(t)

300

IV. The Axiomatic Structure of Quantum Mechanics

EXERCISES 3.1. In formulating the Schroedinger equation in wave mechanics (see Chapter 11, $5) for a state of n particles given by a wave function $(x, t), x E W n , we have been using the derivative i,hL(x,t) = a,$(x, t ) , ifilClt(X9t ) = (HslCl)(x, t),

instead of the strong limit of (3.1). Show that

in the case when $ r ( ~ t) is , continuous in t and Lebesgue square integrable in x.

3.2. Assume that {el, e2 ,...} is an orthonormal basis in the infinitedimensional Hilbert space A@,and that A , B ,... are all the linear operators in 2, which are such that their domains of definition as well as the domains of definition of their adjoints contain the set {el, e2 ,...}. Assign to each of the operator A , B , ... the infinite matrices

I/ A**I/?II 4, I/,...,

i, k

=

1, 2,..*,

where Aik = (ei I At?,), etc. Show that if e l , e2 ,... are in the domains Of A B and A B , then ( A B)ik = A,, Bik and (AB),, =

+

x:i"= A1 ijBjk

+

+

*

3.3. Use the results of Chapter I, Exercise 5.4 to compute the matrix elements &k(t) and pik(t) of the harmonic-oscillator position operators x(t)and momentum operators p ( t ) in the basis {Yl , Y2,...} of energy eigenfunctions. 3.4. Prove that the vector function p(t) representing a state in the interaction picture and satisfying the relation q ( t ) = e(i/fi)HoiU (t 0) also satisfies the equation )

m,

iti- dy(t) - H,(t)!P(t) dt

if p(t)E gH0 n 9H(t) and H,(t) is given by (3.45). 3.5& Show that if A is a self-adjoint operator in 2, and if E([-a, + a ] ) f = f for some real 01 > 0, then eiAY

=

s-Iim n++m

(iAt)k C (iAt)k klf = C f. k!

k=O

k=O

4. State Vectors and Observables of Compound Systems

301

3.6. A well-known phenomenon in wave mechanics is the evanescence of wave packets. For example, if H , is the kinetic energy Schroedinger operator in L2(R3),given by (H,$)"(p) = (p2/2m)$(p), then it can be proved" that

if B, is any bounded Bore1 set in R3. Prove that for any open bounded set B, the expansion

is incorrect [though (HOn$)(r) = ( -(fi2/2m)A)n$(r) is well defined, since #(r) is infinitely many times differentiable and of bounded support] because it contradicts the evanescence property of e-(i'fi)HolI#.

4. State Vectors and Observables of Compound Systems 4.1. SUPERSELECTION RULESAND STATEVECTORS Until now we have been talking mainly in terms of the Schroedinger picture of quantum mechanics, in which the observables are represented by time-independent self-adjoint operators, while the states are vectorvalued functions oft. I n 93 we also encountered, however, the Heisenberg picture, in which the observables are represented by time-dependent operator-valued functions, while the states are time-independent vectors, as well as other pictures of quantum mechanics. I n each of these pictures we can define the concept of state vector as a vectorw hch€i'! can represent, at least at one instant t , some state of the system. T h e natural question to ask is whether all the vectors in the Hilbert space i V assigned to the system (92, Axiom 0) are state vectors. It is immediately clear from the last section that the zero vector 0 can never be a state vector, Inasmuch as we do not adopt the requirement that a state vector be normalized (a request which is a matter of mere convenience), we can say that it was generally believed until the early 1950s that all the other vectors in S are state vectors in any quantum mechanical theory. A paper by Wick et al. [1952] dispelled that belief by pointing out the existence of superselection rules in quantum field theories.

* See Exercise 7.7

in Chapter V.

302

IV. The Axiomatic Structure of Quantum Mechanics

The presently known superselection rules are those of charge, baryon number, and univalence. These superselection rules state essentially that the Hilbert space X is a (in general infinite) direct sum of so-called coherent subspaces M, , M, ,..., and that only the nonzero vectors belonging to these subspaces are state vectors. However, these superselection rules are primarily of significance only in theories treating systems with a variable number of particles, for instance in field theories. Thus, the charge superselection rule states that if Yland Y,are eigenvectors of the charge operator with eigenvalues el and e, , respectively, where el # e 2 , then alY1 a2Y2is not a state vector if a, # 0 and a, # 0. Since we are primarily concerned with nonrelativistic quantum mechanics, in which the nature of the particles and their number does not change, the existence of the above-mentioned superselection rules will not be of any immediate concern to us.

+

4.2. THEHILBERTSPACE

OF COMPOUND SYSTEMS

The main task of quantum mechanics is to study the interaction of different parts of a system. Nonrelativistic quantum mechanics deals with the case when the nature of these parts remains unchanged during the entire duration of the experiment, so that each one of these parts can be thought of as a stable “particle.”+ T h e question which naturally arises is whether we know how to construct the formalism for a more complex system if we know which are the formalisms of the constituent particles. In the case that a system 6 consists of n diferent other systems (e.g., particles”) 6, ,..., 6, with which the Hilbert spaces Sl,...,Xn are associated, respectively, then the rule is to associate with G the Hilbert tensor product X = X, 0 .--0X, . Let us assume, in addition, that at a certain instant t we know that the n systems 6, ,..., 6, are in states represented at that instant by the state vectors Y,E % ,..., ‘, Y, E X, , respectively. Then quantum mechanics rules that the entire system 6 is at that instant in a state represented by the state vector Y, 0 0Y, . Since in this case we can talk about the state vectors of the individual systems G1,..., 6, separately, we say that 6, ,..., G, move independently at time t . However, in general, the state of G will 66

* It should be noted that the use of the word “particle” does not preclude the possibility of an intricate internal structure. Thus, an a-particle consists of two protons and two neutrons, a deuteron consists of one proton and one neutron, etc. In fact, there is a tendency in modem elementary-particle physics to treat all particles on an equal footing, and talk about the transformation or decomposition of one particle into other particles rather than of breaking up of a particle into constituent parts.

303

4. State Vectors and Observables of Compound Systems

not be representable at later times by a tensor product of n vectors from S,,..., 2, , respectively. Due to the interaction of G, ,..., G,, the state vectors representing the state of 6 at later times can be vectors of a more general form in 2. 4.3.

TENSOR PRODUCTS OF LINEAR OPERATORS

Let the operator A, be an observable of the system Gk. In the case where Gk is an independent part of the system G, we can certainly measure A, . Hence, this question arises: which mathematical entity related to the Hilbert space S = S,@ -.-@ S, represents this observable. In order to answer this question, we have to define the following concept.

Definition 4.1. Let A, ,..., A, be n bounded linear operators acting on the Hilbert spaces 2,,..., 2, , respectively. The tensor product A, @ ... @ A, of these n operators is that bounded linear operator on .X, @ @ S, which acts on a vector f, @ *.. @ f, , f, E X ,,..., f, E S,, in the following way: - a -

The above relation (4.1) determines the operator A, @ ... @ A, on @ f, , and therefore, due to the the set of all vectors of the form f, @ presupposed linearity of A, 0 .--@ A, ,on the linear manifold spanned by all such vectors. Since this linear manifold is dense in S,@ ... S, and the operator defined by (4.1) in this manifold is bounded (see Exercise 4.1), it has a unique extension to S, @ @ S,. Consequently, the above definition is consistent. Coming back now to the earlier posed question of how to represent in #= @ ..- @ 2, an observable of Gk representable in z k by the bounded operator A,, we see that the natural candidate is

---

Ak

(4.2)

=

1@

@ 1 @ A, @ 1 @

*.*

@ 1.

The fact that this is the right choice can be established by verifying that the physically meaningful quantities, i.e., the expectation values (2.17), are the same for A, when Gk is an isolated system in the state Yk at t , or when Gk is at the instant t an independent part of G, which is at t in the state y, @ -.. @ yk @ -.. @ y, ; namely, if we take the normalized state vectors Y, ,..., Y, , then (Yl @

-.-@ YnI (1 @

=V l

I Y1)l

*.*

W

***

k

@ A, @

***

@ 1) Y1@

I A k Y I K ) k *.. W n I Yn)

=W

@ Yn) k

IAPdk *

IV. The Axiomatic Structure of Quantum Mechanics

304

We note that in Definition 4.1 we defined A, @ @ A, only in the case when A, ,..., A, are bounded. I n case of unbounded operators difficulties appear with the domain of definition of A, ,..., A,. If the domains of definition of A, ,..., A, are gA1 ,..., g A nrespectively, , then @ A, uniquely by (4.1) at least on the algebraic we can define A, @ 0, gagA, . However, if A, ,..., A, were selftensor product gA1 adjoint unbounded operators, then the A, @ -.-@ A, so defined might not be self-adjoint when defined on such a domain. When we are trying to define the operator i&which would be a generalization of (4.2) to the case when A, is unbounded, we can circumvent the above-mentioned difficulty by considering the spectral measure EAk(B)of A,. Since

is also a spectral measure of projectors in % (see Exercise 4.5) we can define & as the self-adjoint operator with the spectral measure (4.3) (see Exercise 2.6). 4.4.

THEOBSERVABLES OF A SYSTEM OF DISTINCT PARTICLES

Let 0,

)...,0,

be the sets of observables corresponding to the systems

G, )...,G,, respectively, and denote by 8, ,..., 8, the same observables represented by operators in % = 0 0Zn, such as the operators

(4.2) in the case of bounded observables. Naturally, it is true that the set 0 of observables of the compound system G contains 8, u u 8,. However, in general it contains additional observables. One way of obtaining additional observables is by taking functions of compatible observables from 8, u u dn . For instance, let Hk be the operator in %k representing the energy observable of the system G,, i.e., the Hamiltonian of 6 k , and let EHk(B) be its spectral measure. Denote by 8, the energy operator of 6, in %, i.e., the self-adjoint operator with a spectral measure built from EHk(B) according to (4.3). It is very easy to check that the spectral measures of Hiand Aicommute, i.e., that Ri and are compatible for any i, j = 1,..., n. Hence, according to Theorem 2.7, the operator 8, 8, is selfadjoint. This operator obviously represents the total energy of the system 6 in the case when 6, ,..., 6, are independent, since it gives us the probability of obtaining energy values El En for 6 when the probabilities for the energies El ,..., En of 6,,..., Gn , respectively, are known.

ffi

+ +

+ +

4. State Vectors and Observables of Compound Systems

305

The operator H which represents the total energy of the system B = {G, ,..., Gn} is usually taken to be of the form

H 2 ii, + * * *

+ Rn+ v,

where V is a self-adjoint operator referred to as the interaction part of H. Indeed, the presence of V reflects, from the physical point of view, the existence of interaction between the n parts 6, ,..., 6, of G. Let us illustrate the above consideration with the case of n different particles without spin, which was treated in $5 of Chapter I1 in the context of wave mechanics. In that case # was taken to be L2(R3%),and we can take Sl = = Xn = L2(R3). The space L2(R3n) is isomorphic to L2(R3) @ @ L 2 ( R 3 ) (n factors), and according to Theorem 6.9 of Chapter 11, the mapping (4.4)

+1(r1) 0 * * *

-

01Cln(rn>

+1(4

**-

1Cln(rn)

induces a unitary transformation U between these two Hilbert spaces. The operator Rk is in this case given essentially by the differential operator form -(fi2/2mk) Ak , and therefore Eln is essentially -((fi2/2m1) A , -.- (fi2/2mn)An), and represents in this case the kinetic energy of the system 6. The interaction term V is determined by the potential V(r, ,..., rn),

a, + +

+ +

(W)(rl ,..., r,)

=

Wl ,..., r,) #(r,...,rn).

We note that H corresponds to the Schroedinger operator form (5.8) in Chapter 11.

4.5. SYMMETRIC AND ANTISYMMETRIC TENSOR PRODUCTS OF HILBERTSPACES We have emphasized many times that our considerations on compound systems did not apply, up to this point, to systems containing identical particles. A widening of the scope of our considerations can be achieved only after the introduction of some additional mathematical material on tensor products.

Definition 4.2. The closed linear subspace of the Hilbert space = Sn, spanned by all the vectors 2 = .X; @ -..@ #%, Sl= of the form

306

IV. The Axiomatic Structure of Quantum Mechanics

where the sum is taken over all the permutations (k, ,..., k,) of (1,..., n), is called the symmetric tensor product of Xl )...)Xn and it is denoted by )

%0

s *..

or

~ S X ,

X?".

Similarly, in the case of the set of all vectors in &' of the form

(4.7)

if (K, ,..., k,) if (k, ,..., K,)

; :1

,...,A,)

n(K1

=

is even is odd,

the closed linear subspace spanned by this set is called the antisymmetric tensor product of X, ,..., X, , and it is denoted by

% @A

@*znor

X?*,.

The factor ( r ~ ! ) - l /in ~ (4.5) and (4.6j has been introduced for the sake of convenience in dealing with orthonormal bases (see Exercise 4.2). In case that f, ,...,fn are orthogonal to each other and normalized in Sl , then due to this factor, the symmetric and antisymmetric tensor products of these vectors will be also normalized. If we take the inner product of a vector (4.5) with a vector (4.6) we get zero. This implies that XPsn is orthogonal to SPA"(see Exercise 4.3)) when these spaces are treated as subspaces of 210". If A, ,..., A, are bounded linear operators on X, , then A, 0 0 A, will not leave, in general, XPsnor &PA" invariant. We can define, however, the symmetric tensor product (4.8)

A, 0 s

0 . .

C

0 s A, = (k,

.....k,)

A,,

0 0A,,

which leaves S y A "and XPSn invariant, and which can be considered therefore as defining operators on these respective spaces, by restricting the domain of definition of this operator to these spaces. 4.6.

THECONNECTION BETWEEN SPINAND STATISTICS

Let us consider now a system 6 of n identical particles of spin u. We recall from the Introduction that the spin projection in some direction can assume only integer and half-integer values. If X, is the Hilbert space associated with each particle in this system, then not every nonzero

307

4. State Vectors and Observables of Compound Systems

vector in Xd" is a state vector. It is an assumption of quantum mechanics, which is consistent with the known experimental material, that the Hilbert space X of state vectors associated with 6 is the symmetric tensor product in the case that the particles have integer spin, and the antisymmetric tensor product SPA",in the case that the particles have half-integer spin. The above assumption is based, primarily, on the statistics of gases. A given macroscopic volume of gas can be considered to be a system G with a large number n of identicle particles which are, in the first approximation, independent. Well-known statistical consideration, starting with a basic assumption of equal a priori probability for each particle to have its energy within a certain range, lead to a certain distribution of particles (energy distribution) along the energy spectrum. This procedure involves counting the number of all possible states of the system which are compatible with a certain energy and some other exterior conditions. It has been observed that the assumption that the state Y of the system (e.g., gas) 6 in which particle G k is in the state Ykand particle 6, in the state Y , is identical to the state of 6 in which 6, is in the state !PI and 6, in the state y k leads to a correct energy distribution in the case of particles of integer spin. This assumption is usually referred to as the principle of indistinguishability of identical particles in quantum mechanics, and the statistical considerations based on it are referred to as the BoseEinstein statistics. It reflects the fact that the symmetric tensor product Y10 s Yndoes not change under a permutation of y k and y l . In the case of a gas consisting of particles of half-integer spin, it has been observed that one must add to the assumption of indistinguishability of particles the assumption that no two particles can be in the same state. This last assumption is referred to as the Pauli exclusion principle (which we have met already in Chapter 11, $7) and the statistics based on it is called the Fermi-Dirac statistics. The Pauli exclusion principle reflects the mathematical feature of the antisymmetric tensor product Yl@A * - . @ A Y n that it is obviously equal to the zero vector if two vectors Ykand Y , in it are identical for some K # 1. The indistinguishability of identical particles is reflected in the further natural assumption that those observables A of the system 6 which are built from observables of individual free particles, are symmetric tensor products (4.8) of these. Thus, in measuring the expectation values ( ~ 4 ) ~ of some particle, we cannot say whether it refers to the first, second, ..., or nth particle, since obviously for any such A

os

(A)...oSykOS...OSyrOS... =

(A)...oSrlBs...QS%~s...

9

(A)...O~ykO~...O~yll~~... - (A)...B~rlB~...B~ySIB~...

308

IV. The Axiomatic Structure of Quantum Mechanics

Let E(B), B E @ , be the spectral measure of some observable of a single particle from our system of identical particles. Then 1

n! - E(B) 0 s

1.-

0 s E(B)

(n factors)

is a projector, as follows from the fact that it is self-adjoint (see Exercise 4.4) and idempotent (see Exercise 4.5), (E(B) @

a * *

JT(B))~ = (E(B))' @

@ (E(B))2= E(B) @

**.

... @ E(B).

Similarly, by using the definition (4.8)and Exercise 4.5, it is easy to check that (4.9)

~p,) oS-.&q~,),

B~ n B~ =

o

for i + j ,

is self-adjoint and idempotent, and therefore it is a projector. More generally, if in the family (4.10)

B11

,-.*,B,nl ,**., B R,*.., ~ B,nk

n,

9

+ + "*

n k = 71,

of n Bore1 sets, any two sets with the same first index are identical while any two sets with different first indices are disjoint, then (4.1 1)

1 n,!

*'*

n,!

[E(B,,) os. - *

os[E(BK*)os

***

QS

E(Blnl)]@S

.**

OSE(BkTIk)l

is a projector (see Exercise 4.6). Thus, the expectation value of (4.1 1) for some state Y(t)in the Schroedinger picture gives us the probability of having ni particles within the sets B,, = = Bini for each i = 1,..., R. However, if B, ,..., B, are not pairwise disjoint, then the operator (4.9) is not, in general, a projector, and its expectation value does not yield the probability of finding at some instant one particle in B, , one particle in B, , etc. This probability is given by the expectation value of a projector which is the sum of projectors of the form (4.11), obtained by writing B, @ @ B, as a union of disjoint sets of the form (4.10) and adding the operators (4.1 1) corresponding to these sets.

4.7. SPINAND STATISTICS FOR

THE

BODY PROBLEM

As an illustration of the above consideration on the connection between spin and statistics we shall formulate the wave mechanics for n identical particles of spin 0.Wave mechanics postulates that a state vector will be represented in this case by a wavefunction (4.12)

#(rl ,s1 ,..., rn ,s,),

rk E R3,

sk = -a, -u

+ 1,..., u,

309

4. State Vectors and Observables of Compound Systems

which is Lebesgue square integrable and symmetric (4.13)

$(..., ri ,si ,..., ri ,si ,...)

..., ri ,sj ,..., ri ,si ,...)

= $(

if u is an integer, or antisymmetric (4.14)

$(..., ri , si ,..., ri ,sj ,...)

..., rj ,sj ,..., ri ,si

= - $(

,...)

if u is half-integer. It is easy to see that the Hilbert spaces of functions (4.13) and (4.14) in which the inner product is taken to be

c .*. c +O

(4.1 5)

=

+O

s1=-0

SRSnj*(ri ,s1 ,.-, rn

sn)

S"=-O

x g(r1 s1 ,..., r,, sn) drl

drn

I

are unitarily equivalent under the unitary transformation induced by the mapping (4.16)

$(rl 9 4 0 ... 0$(rn 9 4

+

to the spaces &'psnand (4.17)

~1)

* a .

$(rn

9

~ n )

#PA", respectively, where (see Exercise 4.7)

& = L2(R3)@

@L2(!R3)

(20

+ 1 terms).

The Hamiltonian of the system is usually taken to be of the form

H I Ts

+ V,

where Tsis the kinetic energy operator essentially given (see Theorem 7.3) by (--k2/2m)(d1 An). Thus, Ts is already symmetric with respect to the n particles. If V is the potential energy given by a potential V(r?, s1 ,..., r, , sn), then the principle of indistinguishability of identical particles requires that this function is symmetric under any permutation of the indices 1,..., n. We shall now summarize the main conclusions reached in this section in the following axioms.

+ +

Axiom PI+ Let G by a system consisting of K * n particles G;'),..., Gnl (1),..., GP), ..., G e , where the particles carrying the same superscript are of the same kind, while those carrying different superscripts belong to distinct species. If &+) is the Hilbert space associated with each one of the particles Gp),..., Glt'J , v = 1,..., K , then the Hilbert space for 6 is (4.18)

310

IV. The Axiomatic Structure of Quantum Mechanics

where 8 should be taken to be the symmetric tensor product if the corresponding systems are of integer spin, or the antisymmetric tensor product if the corresponding systems are of half-integral spin.

Axiom P2. I n the absence of superselection rules, every nonzero vector in 2' is a state vector. Axiom P3. If at some instant t the particles Gill, ..., GF: are free ..., P : ) , respectively, then the state of G at that and in the states ?Pi1), instant is represented in the Schroedinger picture by (4.19)

(Y.y @

-..@ Y2)@

-**

0 (Yy@

-.a

@ Yk)).

representing in the SchroeAxiom P4. If E is a projector in 9(") dinger picture an observable of the particle G?) when that system is free, then the same observable is represented in 8 by the projector

ny factors

where the operator in brackets acts in ( 2 f ( y ) ) @ n y . It is easy to derive from (4.20) the expression for any projector related to measurements on the system 6, when the constituent parts of 6 are independent (see Exercise 4.8). In particular, we can arrive at the expression (4.11) for measurements on systems of identical particles. EXERCISES 4.1. Prove that the linear operator A, @ ..-@ A, on the separable Hilbert space 2fl @ .-.0 2', has the bound (1 A, (1 .-.I(A , 11. 4.2. Show that if e l , e2 ,... is an orthonormal basis in HI, then {ekl @a *.- 0" ek, : K, ,..., k, = 1, 2,...} is an orthogonal basis in Hpsn, whereas {ekl @ A @* ekn : K, ,..., K, = 1, 2,...} is an orthonormal basis in Spa. 4.3. Prove that the subspaces Sf"" and Hf'*'" of 2'd" are mutually orthogonal. 4.4. Show that if A, ,..., A, are self-adjoint bounded operators on 2', ,..., 2', , respectively, then A, @ ... @ A, is also self-adjoint and

bounded.

311

5. Complete Sets of Observables

4.5. Prove that for any bounded linear operators A, B, ,..., B, , and any scalar a

,..., A,

and

Show that the first two of the above relations are true also for symmetric tensor products, while the third one is not true, in general, for symmetric tensor products. 4.6. Use (4.8) and the results of Exercises 4.4 and 4.5 to prove that (4.11) is a projector.

4.7. Prove that the spaces Loz(IWan)syrn and Lo2(Wn)Bnti consisting, respectively, of all the symmetric (4.13) and antisymmetric (4.14) functions which are Lebesgue square integrable in the variables rl ,..., r, are unitarily equivalent to S p s nand respectively [with Sl given in (4.17)], under the linear transformation induced by (4.16). 4.8. If EcY)(B)is the spectral measure of some observable of the system Bp) = .--= Bt: (see Axiom Pl), use Axiom P4 to build the projector whose expectation value gives the probability of observing simultaneously one particle in BPI, one in B r ) , etc. for all Y = 1,...,K, when the particles in B are independent. 4.9. Let U, ,..., U , be unitary transformations of the Hilbert spaces S,',..., S ,'onto the Hilbert spaces 2," ,...,s," Define . U , @ @ U, as that linear transformation of 2, = S,' @ @ 2,' into 2, = 2: @ @ 2,"which maps fl @ @ fn into U,f , @ 0 Unfn

..-

...

...

(see Definition 4.1). Prove that U, @ .-.@ U, is a unitary transformation of 8,onto 2,.

"5.

Complete Sets of Observables

5.1.

THECONCEPT OF

A

COMPLETE SETOF OPERATORS

The concept of a complete set of observables was first introduced by Dirac [1930] and it has remained in extensive use ever since. The original formulation is, however, merely heuristic in the general case, and it becomes rigorous only in the special case of a set of self-adjoint

312

IV. The Axiomatic Structure of Quantum Mechanics

operators A, ,..., A, acting in a separable Hilbert space Z? and having pure point spectra (which must be countable since 2 is separable),

sA&= {@), A:'),...},

K

=

1,..., n.

Under these circumstances {A, ,..., A,) is called a complete set of operators if the following statements are true for some S C Rn:

---

(a) T o each n-tuple (A, ,..., A,) E S C SA1 x x S A n can be assigned a vector YAl An from the common domain of definition of A, ,..., A, which satisfies

(b) The family of all vectors (5.2)

YA

is an orthonormal basis in

7

(A1

)...,A,)

E

s,

2.

Note that the completeness of the set {A, ,..., A,) implies that the operators A, ,..., A, have a common dense domain of definition containing the linear manifold spanned by all the vectors YAl,....An, and that they commute with one another on that domain of definition. Another fact which has to be borne in mind is that, due to BoseEinstein or Fermi-Dirac statistics, not all vectors (5.2) represent physical states, but only those vectors which are appropriately symmetrized or antisymmetrized with respect to the indices A, ,..., A, referring to the same observables of identical particles of integer or half-integer spin, respectively. In the presence of identical particles, the Hilbirt space of physical states will not be 9, but rather a closed subspace Z of 9, constructed in accordance with Axiom P1 in $4, in which the appropriately symmetrized vectors in (5.2) will constitute an orthonormal basis. The operators A, ,..., A, will not, in general, leave &' invariant, (Consider, e.g., the spin operators of identical particles in wave mechanics.) However, appropriately symmetrized polynomials p ( A , ,..., A,) in A, ,..., A, leave &' invariant.

Definition 5.1. Let 6 be a system consisting of a finite number of particles, and A, ,..., A, obvervables of single particles from that system. We say that the function f(xl ,..., x,), x1 ,..., x, E R1, is appropriately symmetrized with respect to {A, ,..., A,} if it is symmetric (antisymmetric) f(..., Xt ,..., x* ,...) = j-f(..., x,

,..., xi ,...)

313

5. Complete Sets of Observables

with respect to any two of its variables xi and x i , whenever A, and A, represent the same observable quantity (e.g., energy, spin projection, momentum component) of two identical particles of integer (halfinteger) spin. Since the set (5.2) is an orthonormal basis in A?,we can expand any vector Y E 2 in terms of that basis,

(5.3)

y

= (A'..

c..

+(A1

,...,u FA

'....*An

9

.An)ES

(5.4)

where we have (5.5)

Denote by Z*(S), where S C SA1 x *.. x S A n , the space of all functions $(Al ,..., A,) defined on S and satisfying (5.5). It is easy to see that Z2(S) is a separable Hilbert space and that the one-to-one mapping (5.6)

P !

-

Wl ,..., hJ

is a unitary mapping of H onto Z2(S). On the other hand, the space 1"s) is isomorphic to the space e2(Rn, p ) of all properly symmetrized and antisymmetrized (in the case where some of the particles in the system are identical) complex functions $(Al ,..., A,) which are square integrable on Rn in the measure p

(5.9)

s c S"'

x

**.

x

SAn.

In other words, for an arbitrary Borel set B in Rn, p (B) is equal to the number of the points in the set B n S. The elements of the Hilbert space x2(RBn,p ) are equivalence classes of almost everywhere equal functions, i.e., the square-integrable (in the measure p) functions $l(x) and y&(x), x E U P , represent the same element of L2(R", p ) if = $2(A) for all A E S. The wave functions yr(x) are

IV. The Axiomatic Structure of Quantum Mechanicr

314

such that if x1 and x2 are variables corresponding to the same observables of two identical particles, then #(...,x,

,..., x,

,...)

=

&#( ..., X*

,..., x1 ,...),

where the plus sign has to be taken in the case that the particles have integer spin, and the minus sign for particles of half-integral spin. Thus, e2(Rn, p ) is, in general, a closed linear subspace of the space L2(Rn, p ) of all functions on 08% which are square integrable in the measure p. In the case where all particles are different,L2(Rn,p) is identical to e2(Rn, p). The mapping

d W),

d ( 4 4-P W),

(5.10)

4 EL2(Rn,P ) ,

which is such that #(A) = #(A) for all h E S, is a unitary transformation of

P ( S ) onto e2(Rn, p). Thus, (5.6) and (5.10) determine a unitary transformation (5.11)

of

Yt+#(x)

=

UY,

# ELyR",p),

Y€E,

z onto D(w,p).

If we denote, in general, by g Athe domain of definition of an operator A , then we have that for Y E gAk

where Ak#(h) E Z2(S). Thus, Y E g Aif and only if $(x) = U!P is from the domain of definition gAk, of the operator A,' acting on L2(Rn,p ) as follows: (5.12)

(Ak'#)(4= X

k W ,

4E9Ak5

gAk. obviously consists of all # €L2(Rn,p ) for which

xk

#(x) is square

integrable, i.e., (5.13)

gAl;,

=

I#:

# EL2(Rn,P),

Xk2

PI"

I #(X)Iz

~ P ( X )

< +a/.

Moreover, we obviously have (5.14)

A;

=

UAkU-l.

The existence of a unitary transformation (5.1 1) of Z onto e2(Rn, p) which is such that the operators Ax',.,.,An', defined in (5.14), satisfy (5.12) and (5.13) is evidently a necessary and sufficient condition for

315

5. Complete Sets of Observables

,..., A,, to be complete in the sense of the definition given at the beginning of this section. This approach to the concept of a complete set of self-adjoint operators A, ,..., A, with discrete spectra leads to a straightforward and natural generalization to the case of self-adjoint operators A, ,..., A,, with arbitrary spectra.

A,

Definition 5.2. The self-adjoint operators A, ,...,A,, acting in the Hilbert space 2, constitute a complete set of operators in the Hilbert space &', which is a subspace of 2, if the following three conditions are met: (a) There is a measure p(B) in the Bore1 sets B of the n-dimensional Euclidean space Ra with support S C S"I x ... x S"*, where S"1, ..., SAa are the spectra ,of A, ,..., A,, , respectively. (b) There is a unitary transformation U of 9 onto the Hilbert space L*(Rn, p), such that the operators A;

=

UAJJ-l,

k = 1,...,n,

are the multiplication operators (A; Jl)(x) =

Wm,

9 E 9,;>

with domains of definition 9"k, = 19:

I

R"

Xk2

I Wl*444 < +a,9 €L2(Rn,P I ] .

(c) U maps X onto the subspace e2(R,, p) of L2(Rn, p ) consisting of all appropriately symmetrized functions from L2(Rn, p).

If the first two of the above requirements are fulfilled, the Hilbert space L2(1Wn,p) is called a spectral representation space of the operators A, ,..., A, , and All,..., A,' the spectral representation (or canonical form) of the operators A, ,..., A,. If A, is an operator such that {A,} is complete (i.e., n = 1 in the above definition) then A, is said to be an operator with a simple spectrum.

SETSOF OPERATORS 5.2. CYCLICVECTORSAND COMPLETE The significance of the concept of a complete set of operators in quantum mechanics is reflected in the following assumption, which is ordinarily tacitly made in nonrelativistic quantum mechanics.

Axiom C. For every quantum mechanical system 6 there is a finite set {A, ,..., A,} of operators in 2, which is complete in the Hilbert

316

IV. The Axiomatic Structure of Quantum Mechanics

space 2, 2 C A?, associated with that system, and such that, in the absence of superselection rules, every function F(Al ,..., A,) which leaves 2 invariant and, when restricted to 2,is a self-adjoint operator in 2,represents an observable of 6. T o give an example of a complete set of operators of the kind mentioned in the above axiom, consider a system 6 of n particles without spin. We have seen in the preceding section that if all the particles in the system are different, then in wave mechanics we associate with 6 the Hilbert space L2(R3,).In this case 2 = i%? = L2(IW3,)and the position operators (XiW.1

Y...,

r,)

(YdWl

7-..9

r,) = Yi+@l

(Zi#)(rl ,..., r,)

= xi$(.,

= zi+(rl

¶***¶

9.e.9

r,), rn),

,..., r,),

i = ly...,n,

constitute a complete set of operators, which in this case represent observables. Thus, in this example L2(R3,) is the spectral representation space of the position observables and the above introduced operators X , , Yi, 2, , i = 1,..., n, are the canonical forms of these observables. In case that some of the particles in the system are identical, we have seen in the preceding section that L2(IW3,)is not the Hilbert space associated with 6; instead, the space L2(R3,) of appropriately symmetrized Lebesgue square-integrable functions is the space X' associated with 6. In this case, the operators X i , Yi, 2, , i = 1,..., n, acting inL2(R3") = are still a complete set of operators, but they do not represent observables since they do not leave 2 = L2(R3,) invariant. However, any operator A = F(Xl , Yl , 2, ,..., X , , Y , , 2,) (AW-1

,.**¶

r,) = Wl

,.-a)

r,)

Wl

Y...,

rn),

where F(rl ,..., r,) is a real appropriately symmetrized and bounded Borel-measurable function, will represent an observable of 6; for instance, the projectors (4.1 1) and (4.20) are such functions. It has become customary to refer to any set {A, ,...,A,} obeying Axiom C as a complete set of observables even when, strictly speaking, only their appropriately symmetrized functions are observables. Additional examples of complete sets of observables will be encountered in 97, when the formalism of wave mechanics will be derived from the general quantum mechanical formalism espoused in this chapter (see also Sections 4.1-4.4 in Chapter V). We shall relate the completeness property of A, ,..., A, to the existence of a cyclic vector for A, ,..., A,.

317

5. Complete Sets of Observables

Definition 5.3. The vector Yo of the Hilbert space 2' is a cyclic oector in 2' with respect to the commuting operators A, ,..., A, in the Hilbert space 9, where 9 3 2', if the following three conditions are fulfilled:

. (a) YoE BAln n gAn (b) For any integers R, ,..., R,

=

0, 1, 2,...,

(c) The linear manifold

spanned by all vectors of the formp(A, ,..., A,) Yo,corresponding to all choices of appropriately symmetrized polynomials p in A, ,...,A,, is everywhere dense in 2'. In the remainder of this section as well as the next section, we shall gradually establish the following main theorem.

Theorem 5.1. The set of self-adjoint operators A, ,..., A, acting in the separable Hilbert space A? and with range in $ 3 S is complete if and only if there is a vector !PoE Xwhich is cyclic with respect to A, ,..., A,, and A, ,...,A, commute on the dense linear manifold 9(Y0;A, ,..., A,) generated by applying to Yoall possible polynomial forms p(A, ,..., A,) which are appropriately symmetrized. To prove that the completeness of A, ,..., A, on a separable 2' implies the existence of a cyclic vector Yo for A, ,..., A, we first verify Lemmas 5.1 and 5.2.

Lemma 5.1. If L2(Rn, p) is unitarily equivalent to a separable then the measure p is u finite. Hilbert space S, Proof. If p is not u finite, then there is an uncountable family F of mutually disjoint measurable sets of nonzero measure. Denote by xs(x) the characteristic function of the set S. The functions

IV. The Axiomatic Structure of Quantum Mechanics

318

constitute an orthonormal system in L2(R", p), since for Sl , S, E 9, we have if S, # S,

Under a unitary transformation of L2(1Wn;p) onto X , the images of all elements from 9 would consitute an uncountable orthonormal system in 8.This is impossible if is separable. Q.E.D.

.#

*Lemma 5.2. Let p be a u-finite measure and h(x) a continuous positive function representing an element of L2(W,p). The function (5.17)

fo(x)

= exp(-a(l

xl

+ .-.+ I x,

I

I))+),

a

> 0,

represents an element of L2(R", p), and the family of all functions

(5.18)

P(X)fO("),

obtained by letting p ( x ) range over all polynomials in x E U P is dense in L2(R",p). Proof. Since we have

the functions (5.18), which include the function fo(x) itself, are square integrable because

I p(x)f0(x)l2 s const I W12. Due to the u finiteness of p, the family of all continuous functions on Rn which represent elements from L2(R",p) and are of compact support is dense in L2(Rn, p) (see Exercise 5.1). Hence, it is sufficient to show that any continuous function g(x) of compact support and representing an element of L2(Rn, p) can be approximated arbitrarily well in the norm

II * II,

by a function of the form (5.18).

319

5. Complete Sets of Observables

We carry out the change of variables

where w( y) is some analytic function on R1, mapping R' in a one-to-one 1 for all manner into a finite interval I and such that I dnw/dyn I n = 0, 1, 2, ... (such a function cah be built from exponential functions). Under this transformation Rt" is mapped onto the n-dimensional rectangle

<

R

=I

x

..-x I .

Since g(x) is of compact support, the continuous function fo(x) is bounded from below on the support of g(x), and therefore the function g(x)lfo(x) is integrable and square integrable on Rn. The function g l . - ' ( ~ ) ] l f ~ [ p - ~ ( uwhere )], p-'(u) is the inverse of p(x), is continuous and of compact support R. Hence, according to the Weierstrass approximation theorem (see Chapter 11, Theorem 7.4), we can approximate that function arbitrarily well on R by a polynomial, i.e., we can find a polynomial

for which

If we denote by pl the measure in R resulting after the change of variable x + u = &), we have

By expanding each w(xk) according to the Taylor's formula to the order k, we obtain

320

IV. The Axiomatic Structure of Quantum Mechanics

where pk(x) is a polynomial of the order n(k - l), and y k ( x ) is the remainder. A simple computation shows that this remainder can be majorized as follows:

where the coefficients c, are constants built from the coefficients of q(u), and can be chosen to be independent of k, due to the requirement that I dnw/dyn I 1. By employing (5.19) we get

<

By taking advantage of the relation

which is a direct consequence of the triangle inequality whenever , f 2 fo E L2(Rn, p), we obtain by using (5.21)

fi .fo

-

[Jw" Ifo(412 I

1/2

P k ( 4 - dP(X)l12

4441

for sufficiently large values of k when 01 > 1. Combining the above relation and (5.20), we arrive (by using the triangle inequality) at the inequality

which is the desired result when a

> 1.

321

5. Complete Sets of Observables

The case with 0 < a < 1 can be reduced to the preceding case by substituting the variables xk by the new variables xk' = 2axk. Q.E.D. As a direct consequence of Lemmas 5.1 and 5.2 we get the following theorem upon appropriate symmetrization of f o ( x ) in (5.17).

Theorem 5.2. If the Hilbert space z(Rn, p) is separable, it contains a vector which is cyclic with respect to the canonical operators A,', ..., A,' defined in (5.12).

5.3. THECONSTRUCTION OF SPECTRAL REPRESENTATION SPACES We shall prove now that if there is a cyclic vector for the self-adjoint operators A, ,..., A,, then {A, ,..., A,) is a complete set. The proof will be obtained by constructing the representation space e 2 ( R n ,p).

*Lemma 5.3. Assume t h a t f e i@ is a cyclic vector with respect to the self-adjoint operators A, ,..., A, in the separable Hilbert space 2, and that p is the measure satisfying the relation

for all polynomials p ( x ) . Then there is a unique unitary operator U , mapping L2(R,, p) onto i@,such that (5.24)

U(X, . p ) = p(A, ,...)A,)

EA'*...*A"(I)

for all appropriately symmetrized polynomials p ( x ) and all finite intervals

I in Rn.

Proof. Note first that every polynomial p(x) is square integrable on R" with respect to the measure ,usince

Define an operator U,, from e 2 ( R n ,p) into i@ by

322

IV. The Axiomatic Structure of Quantum Mechanics

where p ( x ) is any appropriately symmetrized polynomial. The so-defined operator U, is isometric,

in deriving the second step we have made use of the fact that ,..., A,) is a polynomial in A, ,..., A,. The measure p is finite and, therefore, the family of all continuous functions in E2(Rn,p) of compact support is dense in e2(Rn, p) (see Exercise 5.1). Since, according to the Weierstrass theorem, each such function can be approximated uniformly and arbitrarily well by a function of the form x,(x) p(x), where I is a finite interval and p ( x ) is an appropriately symmetrized polynomial, it follows that the family of all such functions is dense in 12( Rn, p). Thus, the domain of definition of the operator U, can be extended to E2(Rn, p) in a unique manner, and the operator U obtained in this way is isometric. Moreover, since f is cyclic with respect to A, ,..., A,, the family of all vectors p ( A , ,..., A,) f is dense in A?, and, therefore, the family of all vectors

p*(A, ,..., A,)p(A,

EA‘*--A~(1)p(Al,..., A,)f,

is also dense in X . Hence, the range of Uo is dense in X , i.e., the range of U has to be 2.Thus, U is unitary. Q.E.D. of intervals which are Note that if we take a sequence I , C I , C such that I , u lau = Rn, then x I n ( x ) p ( x )converges strongly to p ( x ) and EA1*.*.>An(I)p(Al ,..., A,) f converges strongly to p(A, ,..., A,) f. Since

and U is bounded and therefore continuous, we get in the limit UP

= P(A1 7*.’,

4.f.

By using Lemma 5.3 we can easily prove Theorem 5.3.

Theorem 5.3. Let f €9be a cyclic vector with respect to the self-adjoint operators A, ,..., A, in the Hilbert space 9, let p be the

323

5. Complete Sets of Observables

finite measure defined in (5.23) and U the unitary operator mapping Lz(W", p ) onto 2 and satisfying (5.24). Then the self-adjoint operators U-'AkU,

k

,..., n,

=1

in Lz(Rn, p ) are the (p-almost everywhere) multiplication operators $

(U-lAkU$)(x) = X k $('k),

g"-lA,lJ

9

and therefore for any of the operators A in Theorem 2.6, U$)(X)

=F

$ E qpAU *

( 4 #(XI,

Proof. We shall prove the theorem by showing that U-'A,U coincides on functions of the form xI(x)p ( x ) (I is a finite interval) with the operator A,', (Ak'$)(x)

#

= xk

gAk'

3

with domain

Since x k p ( x ) is a polynomial when p ( x ) is a polynomial, we have [u-'Aku(p ' x,)](x)

=

[U-lAkp(Al )...)A,) EA""**An(l)](x)

= xkp(x)

Now, U-lA,U is self-adjoint because A , is self-adjoint and u is unitary, while A,' is known to be self-adjoint. Thus, the operators U-lAkU and A; must be identical because they coincide on the set of all functions h(x) = xI(x)p(x),which is dense inL2(Rn, p). I n fact, since ( U-lAkU)* = U-'A,U and t,h is any vector in 9u-lAku , we have (U-lA,u$ 1 h )

= ($

1 U-'AkUh)

=

=

s,"

$*(X)

Smn

x&X)

(x,$(x))*

dp(X)

h(4 444

for all h(x) = p ( x ) x,(x), which implies that (U-lA,U#)(x) = xkt,b(x); the representation of U-lAU follows in a similar manner. Q.E.D.

324

IV. The Axiomatic Structure of Quantum Mechanics

It is evident that Theorems 5.2 and 5.3 imply Theorem 5.1. It should be noted that a complete set of operators can have many spectral representation spaces corresponding to different measures p. For instance, if we take the case of the single operator X in L2(R1),

( X W 4 =xN4, which has a simple spectrum, then rx'is an example of a cyclic vector for this operator, since the family of all functions p ( x ) c x e l 2 [where p ( x ) is a polynomial] is dense in L2(R1) (see Chapter 11, Exercise 7.9). For a polynomial p(x), x E R1, we shall have

where dp(x) = ec2*' dx.

Thus, Le(R1,p) will be another spectral representation space of X .

5.4. CYCLICITY AND MAXIMALITY We have seen in the previous section that in the case that n identical particles are present, it can happen that the self-adjoint operators A, ,..., A, corresponding to observables of individual particles will not, in general, represent observables of the entire n-particle system. Instead, appropriately symmetrized functions of A, ,..., A, will represent observables, provided we are working in the Schroedinger picture. A special case of such functions will be the projectors Ezl*-.**zn(B) corresponding to Bore1 sets B whose characteristic functions xB(x) are appropriately symmetrized. It is therefore convenient to express the property of fo being cyclic in terms of projection operators. The next two theorems are related to this problem, and lead to the important Theorem 5.6.

Theorem 5.4. If fo is a cyclic vector in 2 with respect to the selfadjoint commuting operators A, ,..., A, in #, then the linear manifold

spanned by all vectors E$1**.***,f0 , X E UP,is dense in 2.

325

5. Complete Sets of Observables

Proof. Assume that the set (5.25) is not dense in 2.Then there exists a nonzero vector g which is orthogonal to all the vectors in this set, i.e., (g

I E:',...*An fo) = 0,

A € R".

Using Theorem 2.7 [see also (2.28)], we can write for any polynomial P(A1 >...> An) (g I P(A, 9 . * * > A,)fo) =

Then the identity (g

w"

P ( 4 4 g I E:'"".AYo>.

I Ef1*...vA,fO)= 0 implies that (g I P(A1 ) . * * Y 4 f o ) = 0

for any polynomial p(A, ,..., A,), which contradicts the assumption that

fo is cyclic with respect to A, ,..., A,, i.e., that the set of all vectors p ( A , ,..., A,) fo , corresponding to all polynomials p(A), is dense in 3. Q.E.D The following result represents a converse, in a limited sense, of the above theorem.

"Theorem 5.5. If go is a vector in the Hilbert space cyclic with respect to { E ? v . . . J n : A E Rn} so that (5.26)

(E:'-**'Ango:

2 which

is

h € R),

is dense in 2, then any vector fo of the form (5.27)

f o = exp[--or(l A,

I

+ + I A, 1)Igo ***

9

01

> 0,

is a cyclic vector with respect to the self-adjoint commuting operators

A, ,...,A,

.

Proof. T h e vector fo in (5.27) is in the domain of definition of any polynomialp(A, ,..., A,), sincep(A, ,..., A,) exp[-a(l A 1 I ... I An I)], A, ,...,A, E R1, is a bounded function and therefore

+ +

If fo were not cyclic with respect to A, ,..., A, , then the linear manifold (5.28)

(E,A'*****"nexp[--ol(l A, I

+ + I A, I)] go:h E R")

IV. The Axiomatic Structure of Quantum Mechanics

326

would not be dense in A?, i.e., there would be a nonzero vector

f orthogonal to (5.28), (f

I E,A’..*** ”n exp[-a(l A, I + * * *

+ I An I)] go) = 0,

h E R”.

This implies that (5.29)

I

Rn

P(A) exp[-a(l& I =

1.. p(h)

+ .*. + I

An

I)] d(f I E,A’”‘’’””gJ

d ( f I EP*****4 exp[-a(l A1 I

+ + I An I)] go> **.

= 0.

Since the measure

(5.30)

p ( B ) = (f I EA’**.**An(B)go),B E P,

is finite, we can take in Lemma 5.2 h(h) = 1 and conclude that the set of all functions (5.31)

P(h) exp[-a(l

A1

I

+ .*. +

is dense in L2(Rn, p). Thus, due to the fact that any characteristic function is square integrable in the measure (5.30),it can be approximated arbitrarily well in the mean in the measure (5.30) by a function of the form (5.31). By taking such an approximation to the limit and using (5.29) we obtain

x&) of a Bore1 set B

This implies thatfis orthogonal to the set (5.26). But this is impossible if

f # 0 and (5.26) is dense in H. Q.E.D. In relation to the most general case encompassed in Definition 5.3, Theorems 5.4 and 5.5 refer to the case when H = 9. The formulation of the equivalent of these theorems in the more general case when 2 is a nontrivial closed subspace of A? can be achieved by replacing A? by A? in the above two theorems, and substituting the vectors in (5.25) and (5.26) by their projections onto 2. This means, de fucto, replacing the projector-valued function Ef;.-.*.-;ein the A1 ,...,An E R1 by an appropriately symmetrized operator-valued function

327

5. Complete Sets of Observables

where the above sum is carried out only over those permutations of (A, ,...,A,J which permute the same observable of identical particles. The alterations in the proofs of Theorems 5.4 and 5.5, which are necessary when we are dealing with this more general case, are very easy to carry out; they are left to the reader. The following theorem states the important maximality property of the algebra generated by a complete set of observables.

Theorem 5.6. If the bounded operator A, defined on the Hilbert space 2,commutes with the operators A, ,..., A,, where {A, ,..., A,} is a complete set of operators, then A is a function of A, ,..., A, . Proof. Since {A, ,..., A,} is a complete set, there is at least one vector f o E 2 which is cyclic with respect to A, ,..., A, . According to Theorem 5.4, the linear manifold (5.26) is dense in sequence f , , f a ,...

9.Consequently,

there is a

which converges strongly to A f , . If we introduce the function F ~ ( A= ) &)

e p -)):A

+ -..+ &Je(x

- A$J),

where

then we observe that f , = F,(A, 11j-i -fm

112 = JRn

,..., A,) f o . Since

I~ t ( h) Fm(h)12 d II E > . * . * * ~ 112 Y~

and f , ,f i ,... converges, we deduce that F,(A), F2(A),... is a sequence of Bore1 measurable functions which is fundamental in the mean with f o 112, B E 9,. According respect to the measure p0(B) = 11 EA1s*.**A,(B) to the Riesz-Fischer theorem (Chapter 11, Theorem 4.4), Fl(A),F2(A), ... converges in the mean, with respect to the measure po , to a po-squareintegrable function F(A). Hence, by using Theorem 2.5, we arrive at the conclusion that

Since A commutes with E(1, ...,Ef;, it commutes with any operator

328

IV. The Axiomatic Structure of Quantum Mechanics

I t is easy to check that Df,E 9F(Al,...,Am) , and ADfo = DAfo = DF(Al ,..., A,) f o

= F(A,

,..., A,)

Dfo .

Thus, A andF(A, ,..., A,) are operators which coincide on the set of all vectors Df,, i.e., on the dense linear manifold (5.26). Now, A is bounded and g A= X , while F(A,,..., A,) is closed, since by Theorem 2.6, F ( A , ,..., A,) also has to be the adjoint of the densely defined operator F*(A, ,..., A,). Hence, we conclude* that A = F(A,,..., A,). Q.E.D.

EXERCISES 5.1. Prove that if p is a o-finite measure, then the family of all infinitely many-times differentiable functions which are of compact support and belonging to e2(Rn,p ) is dense in ez(Rm,p). 5.2. Use Lemma 5.2 to deduce that the set of all functionsp(x) e-BlSl, /3 > 0, where p ( x ) are polynomials in x E W, is dense in L2(R1). 5.3. Let {A!'),..., Aiy},..., {A:"),..., A;:)} be complete sets of observables of the systems 6, ,..., 6, , respectively, and let these operators act ,..., 9,(which have as subspaces the Hilbert spaces in the spaces 9, X, ,...,Srn associated with 6, ,..., 6,). Denote by A:")the self-adjoint @ 1 [see operator with the spectral measure 1 0 @ EP)(B) @ (4.3)], where EP)(B) is the spectral measure of A:). Show that {Ail),..., ,..., ,..., A;:)} is a complete set of observables of the system 6 = {G, ,..., G,}.

5.4. Show by making use of the definition of a spectral measure, that if [E(l)(B,),E(2)(B,)]= 0, the set S of all vectors f satisfying E(1)(B,)E(2)(B,)f = f,where B , , B , are any bounded Borel sets in R1, is dense in 3'. If A, and A, are the self-adjoint operators corresponding to E(l)(B)and E(,)(B),respectively, prove that S C gAlA8 n gAaal and that A,A,f = A,A,f for all f E S. 5.5. Show that if A is any self-adjoint operator and E(B), B ELF, is its spectral measure, then for any bounded Borel set R we have BAE(R) = X , =9E(R)A= g A and , A E ( R ) f = E(R)Af for all f E g A,i.e., AE(R) is the extension of E(R)A to the entire Hilbert space.

* In other words, A necessarily belongs to the Abelian algebra generated by A, ,...,A,, and therefore {A, ,..., Am}is complete if and only if that algebra is maximally Abelian. This observation is especially significant when dealing with infinite sets of commuting observables (de Dormale and Gautrin [1975]).

329

6. Canonical Commutation Relations

6. Canonical Commutation Relations 6.1.

THEEMPIRICALSIGNIFICANCE OF

COMMUTATION RELATIONS

One of the most striking features of the position and momentum observables defined in (1.1)-( I .6) (where, in general, we can take i = 1,..., n ) is that all position operators as well as all momentum operators commute among themselves, while this is not true for each pair consisting of a position operator and a momentum operator. As a matter of fact, it is very easy to see that a position operator corresponding to the position coordinate of a certain particle in the system commutes with any momentum operator except the one belonging to the same momentum component of the same particle; for instance, QY) commutes with Piv) and P f ) , i = 1 ,..., n, as well as with Pjz) for i = 2 ,..., n, but we have

(6.1)

(5)

( 8 1

(d

(5)

PI - PI

(5)

8 1

)#

= ifi#,

# Eg~,(a)p,(4

This can be easily verified by taking for

i,h

9p,(z)~,(4

-

differentiable functions:

The commutation relations satisfied by the position and momentum operators in wave mechanics are called canonical commutation relations. In general, given 2n self-adjoint operators Q1,..., Qn , Pl ,..., Pn in a Hilbert space 2,we say that they satisfy canonical commutation relations in which Qkis canonically conjugate to Pk, k = 1,..., n, if (6.2)

(QkPk

- p k Q k ) f = ifif,

k

,..., n,

= 1

for any f belonging to the domain of definition of [Qk

9

P k l =QkPk

-PkQk

3

while any other two operators from {Ql ,..., Qn , Pl ,..., P,) commute. In the first section of this chapter we identified commutativity of operators representing observables with compatibility of those observables. The empirical meaning of compatibility was given in Definition 1.1. From the discussion in $1 we must draw the conclusion that canonically conjugate observables are incompatible, i.e., that there is reason to believe that there is a lower limit to the accuracy with which simultaneous measurements of such observables can be carried out on the same system. Such is the case with canonically conjugate position and

330

IV. The Axiomatic Structure of Quantum Mechanics

momentum observables, as revealed by an extensive analysis of different “gedunken” experiments (see Messiah [1962, Vol. I]). It is important to realize that the commutation relations satisfied by canonically conjugate observables also imply that the outcomes of many measurements of those observables on systems in the same state possess certain statistical features. This conclusion can be drawn from the following result.

Lemma 6.1. I f f is a vector in the domains of definition of the operators A,,, [A,, A,], and A,,, where A, and A, are self-adjoint operators in 2,then


(6.3)

Proof.

2 tl(f I [A, Azlf>I2. 9

Since for any real value of h

(f I AlY) - Xf I w,, A2I.f) = (f I = =

[A,, -iA(A,A,

+ w I A2Y) - A,A,) + P A , 2 ] f )

(f I (A1+ i

W ( A 1 --i&)f) <(A,-iAAz)fl ( A , - - i W ) f )

> 0,

we deduce that Im (f I i [A, , A J f ) = 0, and the discriminant of the above polynomial of second degree in h has to be nonpositive. Thus, we have

I A,lf)12 and (6.3) follows immediately.

-

4 ( f I A l Y X f I A2Y>

< 0,

Q.E.D.

Let us take a normalized state vector ?Pfrom the domain of Qk2,Pka, and [Qk, Pk].Since [ A , , A,]Y = iw, where

Ai

= Qk

-

(p I Q k y ) ,

A,

=p

k

- (y I pky>,

we get, by applying Lemma 6.1, (6-4)

(y I ( Q k - (y I Q k y ) ) ’ Y ) < ~ 1 (pic - (p I p k y ) ) ’ y ) 2 h2/4-

The quantities uy(Qk)and oy(Pk),where og2(Qk)

=

I

R

(6-5)

(A - (y I !&y>)z d 11 E f k y l 1 2= (y I (&k -
33 1

6. Canonical Commutation Relations

are the standard deviations of the probability measures 11 EQk(B)Y [I2 and respectively. Thus, inequality (6.4), which can also be written in the form ~ v ( Q duv(Pk) Z fi/Z

11 EPk(B) Y 112,

indicates that we cannot find states for which the standard deviations of canonically conjugate position and momentum are both arbitrarily small. The standard deviations u y ( Q k ) and uY,(Pk)can be obtained experimentally by preparing an ensemble of systems in the same state, and then carrying out at the same instant, when the systems are in a state represented by Y, a measurement of Qk on the systems in one part of the ensemble, and a measurement of Pk on the remaining systems in the ensemble. This procedure, which is implicit in the theory of measurement of $1, enables us to compute 11 EQk(B)Y 112 and 11 EPk(B)Y ]I2, from which uv(Qk)and uv(Pk)can be computed.

6.2. REPRESENTATIONS OF CANONICAL COMMUTATION RELATIONS If Q1,...,Qn , Pl ,..., Pn are self-adjoint operators satisfying canonical commutation relations in which Qk and Pkare canonical conjugate pairs, we write, symbolically, (6.6) [Qi ,Qk] = [Pj ,P,]

= 0,

[ Q j ,Pk] = i i i S j k ,

j,k

,..., n.

=1

Any set of 2n operators satisfying the above relations is called a representation of the canonical commutation relations. If U is a unitary operator, it is easy to see that the self-adjoint operators (6.7)

Qj’ = UQ,U-l,

Pj’ = UP,U-l,

j

=1

,...,n,

constitute another representation of the canonical commutation relation

(6.6). In fact, the spectral measures of Qk’ and Pk’ are E@=‘(B)= U E Q ~ ( Bu)-1,

E ~ ~ ‘ (= B )U E ~ ~ (u-1, B)

and it is therefore easily verified that [EQ”(B1),Epk’(B,)]= UIEQj(B1),Epk(B,)]U-’

= 0,

j # K,

and that Q i ,..., Qd, as well as PI‘,..., Pn’,commute among themselves. Moreover, for f E 9Q,ppjt-pjtQ,t - u 9 Q i p i - p , Q j we have [Qj‘,

Pj’]f

=

U[Qj, P j ] U-lf

= iiiUU-lf = ifif.

332

IV. The Axiomatic Structure of Quantum Mechanics

Two representations {Ql,..., P,} and {Ql’, ...,P,’}of (6.6) which satisfy (6.7) are said to be unitarily equivalent. The question arises whether all representations of (6.6) are unitarily equivalent. The answer to this question has physical implications, since unitarily equivalent representations are physically indistinguishable. As a matter of fact, assume that there are two quantum mechanical theories F and F’ describing the same system. If the observables of the two theories are related in a one-to-one fashion, A t , A’,

AEY,

A’EY’,

and there is a unitary transformation of # onto #‘ such that A’ = UAU-’,

then, by setting Y = UY,we map all the states of one theory onto the set of all states of the other theory in a one-to-one fashion. The only physically important quantities, namely the expectation values of bounded observables, (A‘)p, = (!”’ I A’!”’)’ =

=

( W I UAU-lUY)‘


are the same in both theories, and therefore F and F’ are physically indistinguishable. From the mathematical point of view, the question of whether all representations of (6.6) are unitarily equivalent is a version of a uniqueness problem, which consists in finding out the manner in which all solutions of (6.6) are related. We already know that there are solutions to the problem, one of them being the Schroedinger representation

(Qd)(%xn) = x k ,.**,

#(XI >---,

(6.8) (Pk+)(Xl

,**a,

x,)

=

-2

%A

a

-#(x1 8%

,-.a,

xn),

in which # = L2(Rn). Due to difficulties with the domains of definition of unbounded operators, which do not coincide with the entire Hilbert space, the solution of the problem posed above becomes feasible only after the the problem is rephrased in a technically different fashion. This is achieved by the following heuristic procedure.

333

6. Canonical Commutation Relations

Assume that P and Q is a canonically conjugate pair [Q, PI

(6.9)

= iti.

If P and Q are treated like bounded operators and (6.9) like a relation valid on the entire Hilbert space, one obtains by induction (see Exercise 6.2, and for rigorous results see Putnam [1967]), [Q,P n ] = itinPn-l.

(6.10)

Setting, formally (see Exercises 3.5 and 3.6),

and using (6.10), one derives in the same vein

Multiplication of the above relation from the left by ciUp yields e-'"Q

eipu = Q - ?iu.

By taking the nth power of the above equation, one obtains e-iPuQn

Consequently, for er

E

eiPu

= (Q - tiu)".

[wl m

e-iPu e i Q ~eiPu

=

c ciPU -eipu n!

(iVQ)"

n=O

Finally, multiplying the above relation from the left by eim, one gets the Weyl relation, (6.1 1)

,iQv eiPu

= e-iKuv ,iPu eiQv,

u,

E R1.

The above relation involves only bounded operators defined on the entire Hilbert space. It is advantageous to adopt as a starting point in our study of representations of canonical commutation relations the Weyl

IV. The Axiomatic Structure of Quantum Mechanics

334

relation (6.11), which has a precise meaning, instead of the symbolic relation (6.9). It will be shown that commutation relations, such as [Q1,Q2] = 0, can also be replaced by relations -- e i Q * e~i Q 1 , ~

eiQlu ei O a V

u, w E R1,

involving only bounded operators. Such considerations necessitate an understanding of the basic properties of operator families eitA, t E R1, where A = A*. 6.3.

OPERATORS ONE-PARAMETER ABELIANGROUPSOF UNITARY

The basic properties of the family (6.12)

U , = eiAt,

tE

R1,

are given in Theorem 3.1. I t follows immediately from this theorem that U , = 1, U;’ = U,* = U-, , and UflUfa= UfaUfl . These three features of the operator family U , , t E R1, characterize it as an Abelian group of unitary operators. Moreover, this family is weakly continuous in t, i.e. (see Exercise 6.3) w-lim U , = U,,.

(6.13)

t+t,

It is a significant result, which constitutes one of the fundamental theorems of functional analysis, called Stone’s theorem, that every continuous one-parameter group of operators is of the form (6.12). The key word in this statement is “continuous.” The following lemma contributes to the understanding of continuity in functions U(t), t E R1, where U(t) are unitary operators.

Lemma 6.2.

f

.

If

f = w-limn+mfn

Il f , 11

and

< Ilfll,

then

= s-limn+mfn

Proof. Since (f Ifn)

Ilf

-fn

+.

11’

(f If)

=

llf.112,we have

= Ilfllz - 2 Re =

-

Ilfll’

+ llfn 11’

+ + llfn (Iz, En

where en - P O when n -P +a.On the other hand, so that 0

Thus,

Ilf - f, 11

+.

< Ilf

0 when n -+

-fn

1’

+co.

<

en

*

Q.E.D.

[If, [I2 - Ilfl12 < 0,

335

6. Canonical Commutation Relations

Assume that U(t) are unitary operators for all t

E

R1, and

U(to)= w-lim U(t), t+t,

i.e., U(t)fconverges weakly to U(t,)ffor anyfE

II W)fII = llfll

=

&. Since

I1 W 0 ) f I L

we can apply the above lemma and infer that U(t)fconverges strongly to U(t,)ffor anyfE &, i.e., U(t,)

= s-lim t+t,

U(t).

Thus, in the case of one-parameter families of unitary operators, weak continuity implies strong continuity.

*Theorem 6.1 (Stone's theorem). T o every weakly continuous, one-parameter family U(t),t E R1, of unitary operators on a Hilbert space &, obeying

Wl + t 2 ) = Wl) Wz),

tl ,t, E R',

corresponds a unique self-adjoint operator A

such that U(t) = e i A l for all t

= JRl

E

X dEAA

R1.

Proof. We shall prove the theorem by building the spectral function EAAof A from the operators U(t),t E R1. Consider first the special case of families Ul(t)which are periodic in t, with the period 27r, i.e., Ul(t 27r) = Ul(t).We shall prove that there is a family of mutually orthogonal projectors E n , n = 0,f l , f 2 , ... with a sum equal to the unit operator,

+

2

*m

(6.14)

I*

E,

= s-lim

2 E,

= 1,

r++m n=-r

n=-m

such that Ul(t)can be expanded in the following series: +m

(6.15)

Ul(t) =

1

,=-W

+T

eintE, = s-lim 1-00

1 eintEn. n=-r

IV. The Axiomatic Structure of Quantum Mechanics

336

It is easy to verify that

are bounded bilinear forms infandg. It follows from Riesz’ theorem (see Chapter 111, Exercise 2.5) that there is a unique bounded linear operator satisfying (f I Eng) = ( f I g)n for all f,g E S.In the symbolic notation

I

Zn

En = (1/27r)

(6.16)

e-int lJl(t)

dt

0

it becomes obvious that Enwould be the Fourier coefficient of Ul(t)if Ul(t)were an ordinary function. The operator En is self-adjoint, since

+

27r and taking where the last step was carried out by setting t , = t, into account that U,(tl 27r) = Ul(tl). Moreover, using again the periodicity of Ul(t), we obtain (cf. Chapter V, Theorem 3.4)

+

(6.17)

vl(t)E,

= (1/24

s””

e-intlUl(t

0

= (1/27r) eint

J

t+an e-ints

+ tl) dtl ~ ~ dt, ( t ~ )

t

I

27I

= (1/27r) eint

0

e-intaUv,(t3) dt3 = eintEn

,

337

6. Canonical Commutation Relations

and consequently

j

=I?

j

an

EmEn= (1/2?r)

e-""t1(t)

En dt

=

2ff

(1/2?r)

0

e-i(m-n)tEndt 0

for m + = nn.

Thus, we have established that Eo , E,, ,... are mutually orthogonal projectors. Hence, it follows from Theorem 3.9 in Chapter I11 that

is also a projector. It is obvious that 1 - E is orthogonal to all projectors E n . Consequently (6.18)

(1/27r)

s"" 0

e-int( f I Ul(t)(1 - E)g) dt

= (f

I En(l - E ) g )

= 0,

n = 0, i-l,...

for all f,g E H.Since (f I Ul(t)(l- E ) g ) is a continuous periodic function in t , of period 27r, and has vanishing Fourier coefficients (6.18) on the interval [0, 27r], it follows that (f I Ul(t)(l - E ) g ) = 0 for all t E W1 and all f,g E H.Hence, Ul(0)(l - E) = (1 - E) = 0, i.e., E = 1. Thus, (6.14)is established. Multplying (6.14)from the left by U(t) and using (6.17),we obtain (see also Chapter 111, Exercise 5.6)

This establishes that (6.15)is true when U ( t )is periodic in t. Let us now turn our attention to the general case, when U(t) is not periodic. Introduce the one-parameter family (6.19)

U1(t)= U(t)(U(27T))--t'2n,

where the power in t is defined in terms of the spectral decomposition of the unitary operator U(27r) [see (6.5)of Chapter 1111, 2n

U(2m) =

eiA d & , 0

IV. The Axiomatic Structure of Quantum Mechanics

338 by the expression

( U ( ~ P )= )$

s""

eiAtdBA.

0

We obviously have U,(2.n) = 1. Since U(t) commutes with U(2w), U(t) U(2P) = U(t 3. 2P) = U(27r

it also commutes with U1(tl)U&)

=

8, , and

+ t ) = U(27r)U(t),

consequently with (U(27r))-1I2".Hence

U(t1)up2)( U(2r))-t1/2n(U(27r))--tz12a= U1(tl

+ t2),

In addition, when t + to , the weak convergence of U(t)to U(to)and the / ~(U(27~))-~0/~" " (see Exercise 6.4) uniform convergence of ( U ( 2 7 ~ ) ) - ~to imply that Ul(to)= w-lim Ul(t). t+tO

The above results establish that Ul(t)is a periodic weakly continuous one-parameter group, with period 23r, which can be therefore expanded in the series (6.15). Consequently, for any f,g E X (6.20)

(f I W ) g > = (f I ~ , ( t ) ( w 4 ) t / 2 " g )

Introduce the spectral function (see Exercise 6.5) (6.21)

339

6. Canonical Commutation Relations

where N(A) is the greatest integer satisfying N(h) (6.20) becomes

J

< A. In terms of E,A,

-m

Furthermore, E,R is uniquely determined by U(t). In fact, if I is any finite closed interval, its characteristic function is the pointwise limit of a uniformly bounded sequence of generalized trigonometric polynomials (see Exercise 6.6)

Setting

and using (6.22) to obtain

we conclude with the help of Lemma3.1 that the above expression a,and consequently approaches zero when n .+

+

EA(I) = s-lim pn[U(t)]. n-m

Thus, U(t) determines EA(I) completely, and in turn EA(I)determines (see Chapter 111, Theorem 5.6) EA(B)for any B E S. Q.E.D.

6.4. REPRESENTATIONS OF WEYLRELATIONS We are now ready to pose the problem of representations of the commutation relations (6.6) in terms of one-parameter continuous families eaQkf, e i p x t , K = 1,..., n. This reformulation of (6.6) is made possible by Theorems 6.2 and 6.3, which will be proved next.

Theorem 6.2. Two self-adjoint operators A, and A, in the Hilbert space % commute (in the sense of Definition 1.2) if and only if (6.23)

for all t, ,t, E R1.

eiAltl eiA,t,

= eiA,ta eiAltl

340

IV. The Axiomatic Structure of Quantum Mechanics

Proof. According to Definition 1.2, A, and A, are said to commute if and only if (6.24)

[EA1(B1)7 E A a ( B 2 ) 1

= 07

for all B, ,B, €a1. If (6.24) is true, we can apply Theorem 2.4 to infer (6.23): e i A l t eiA,t

= C ( A , + A , ) t = ei(A,+A,)t

= i A , t eiA,t.

Conversely, assume that (6.23) is true. T o prove that (6.24) is then also true, we resort to the construction of Ef1 and Efp carried out in the proof of Stone's theorem. First, we observe that [UF)(t1),uy(t2)] = 0,

t,

7

t , E R1,

where Uik)(t)is defined by (6.19) in terms of V k ) ( t )= &Axt, k = 1,2. This implies that all the corresponding projectors ELk), n = 0, &-1, ..., in (6.16) and the spectral functions of U'k)(2~)commute with each other for k = 1,2. From this we deduce that the spectral functions E$ and E$ commute. Q.E.D. We have seen that (6.11) can be derived from (6.9) only in a heuristic manner (but see also Putnam [1967], $§4.6-4.11). However, (6.9) is a rigorous consequence of (6.1 l), as stated in the following theorem. Theorem 6.3. If A, and A, are two self-adjoint operators in 2 satisfying (6.25)

e i A l t l eiA2t2

= e-it,t2 eiA2t2 e i A l t ,

for all t , , t , E R1, then (6.26)

AlAJ

= if

+

A2Alf

for any f E ~ A 1 A 2 - - A , A , = g A , A , n g A 2 A l Proof. Using Theorem 3.1, Exercise 5.6 in Chapter 111, and the fact that e i A 1 6 1 is a bounded operator, we derive from (6.25)

341

6. Canonical Commutation Relations

Since A J E gA1 because f E gAIA, , we can again employ (3.6) to deduce (note that the existence of the third limit follows from that of the preceding two limits!): -AIAzf

1 tl

.

= s-lim - (e'Aiti t1-0

1 t,

= - d i m -it, t1-0

=

-if

l)(iAzf)

eiAitif

1 + s-lim(iAz) (e%% tp0 tl

1 + s-lim(iA,) - (e%ti tp0 tl

- 1)f

- 1)f.

The above relation implies that for all g E BA, (6.27) ( g I AlAZf) = i(g = i
If>+ p

p z g I (-i/t1)(e'"ltl

If>+ (4 IAlf)

= i(g

-l)f>

If>+ ( g I AZAlf),

where the last step is valid due to the assumption that f E 9 A , A l , i.e., A, f E gA8 . Since g assumes all values in gAa , which has to be dense in 2 because A,* exists, we infer that (6.26) follows from (6.27). Q.E.D. The above two theorems indicate that the commutation relations (6.6) are equivalent to the following set of Weyl relations: [eq(iPj.j>, exp(iP,u,)l (6.28)

= lexp(iQj0j), e x ~ ( i Q , ~ d= l 09

exp(iP,uj) exp(iQ,w,) = exp(ifi8,,urw,) exp(iQ,o,) exp(iPjuj), 241

,...,24, ,01 ,..., 0, E w .

However, strictly speaking, we know only that (6.6) follows from (6.28), where the precise meaning of [Qk, Pk] = ifi is given by (6.2). In any case, whenever we say from now on that Q1,..., Q m, Pl ,..., P, are a representation of the canonical commutation relations we mean, according to an established tradition, that the W'eyl relations (6.28) are satisfied. The representations of canonical commutation relations are classified in two classes. The irreducible representations are those which leave no nontrivial closed linear subspace invariant, i.e., for which there is no closed linear subspace M # {0}, M # X , such that exp(iP,u,)f, exp(iQ,o,)fE M,

k = 1 ,..., n,

for all f E M. Any representation which is not irreducible is called reducible [an example is provided by (8.76) in Section 8.101. The following theorem was proved rigorously for the first time by von Neumann [19313.

IV. The Axiomatic Structure of Quantum Mechanics

342

Theorem 6.4 (won Neurnann’s theorem). All irreducible representations of the canonical commutation relations (6.28) on a separable Hilbert space are unitarily equivalent. Any reducible representation Qk ,P k ,k = 1,..., n, on a separable Hilbert space 3? is the direct sum of a finite or infinite sequence of irreducible representations, i.e., 3? is the direct sum of a sequence M, , M, ,... of mutually orthogonal closed linear subspaces and exp(iPf)u)EM,,

exp(iP,u) =

exp(iQp)w)EM,,

exp(iQkw) = U

U

where Q?),...,Qk), PP),..., Pg) are irreducible representations in Mu. It will be shown at the end of this section that the Schroedinger representation, in which Q k and P k are given by the operators (4.19) and (4.20) of Chapter 111, is an irreducible representation in L2(R”). This establishes the existence of representations of (6.28) for n = 1, 2, ... . In quantum field theory, or in any quantum mechanical theory of systems in which the number of particles can vary and become arbitrarily large, one is concerned with representations of the canonical commutation relations for an infinite number of degrees of freedom, i.e., for an infinite sequence of operators Q, ,PI ,Q 2 , P, ,... . It is worth mentioning -though this case lies beyond the scope of the present book-that von Neumann’s theorem cannot be generalized to this case. In fact, there is an infinite number of irreducible unitarily nonequivalent representations of an infinite number of canonical commutation relations [Garding and Wightman, 1954a, b].

*6.5 APPENDIX: PROOF OF

VON NEUMANN’S THEOREM

We shall prove in detail von Neumann’s theorem only for the case n = 1, i.e., when we are dealing with two canonically conjugate operators Q and P which satisfy the Weyl relation (6.29)

eiPu e i Q w

= eifiuu

eiQ~ eiPu,

u,

R1.

This case contains all the essential features of the more general case. The lengthy proof of this theorem, which occupies the rest of this section, can be ignored at the first reading. Let us introduce the two-parameter family S(u, w )

(6.30)

= e - [ ( d / Z ) f i ~ ~ ] e i P ~e i Q V

of unitary operators. It is easy to verify that (6.31) S(u1,

S(uz

02)

+

= ex~[(i/2 fi(u1~z ) - ~ ~ 2 S(UI ) l UZ 9

w1

+

w2)

343

6. Canonical Commutation Relations

by using (6.29). Taking in the above relation u = u1 = -u2 and and noting that S(0,O) = 1, we obtain the identity

w = v1 = - w 2 ,

(6.32)

S(-u, -v)

=

S*(u, w )

=

s-yu, w),

u, 0 E

R'.

If p(u, w ) is a function which is Lebesgue integrable on W,the integral

(fI

d

o

=

I

P(U,

w)
I S(%4 B ) du dv

W=

is defined for any f,g E i?F since (f I S(u, v)g> is continuous and bounded in u, w E R1 (see Chapter 11, Theorem 3.9). I t is easy to check that (f I g), is a bounded bilinear form, and consequently it follows from Riesz' theorem (see Chapter 111, Exercise 2.5) that there is a unique bounded linear operator A, satisfying (f 1 A,g) = (f I g), . The function p(u, v) is called the kernel of A, . It is straightforward to verify by direct calculation, employing the relation (6.33)

(f I A,g> =

1

w8

P@, a ) ( f I

s(4.)g>

du dv,

that the following statements are true: (6.34) A,*

= A,,

p(u,) = p*(-u, -).,

A,,+,,

= A,,

+

4 , s

where p, p1 , and p2 are any functions which are Lebesgue integrable on R2. Moreover, we easily deduce that A,,,*

= A,

9

(6.35)

B(% 4 = 1., exp[(Q)

fi(uw2 - 7 4 1 Pl(U - u2 ,w - .2)

by the following procedure:

(f I A,,A,,g)

P2@2

,.2) d% d.2

9

IV. The Axiomatic Structure of Quantum Mechanics

344

in which Fubini’s theorem has been used twice. We need now the following lemma.

Lemma 6.3. A,

=0

if and only if its kernel p(u, v ) vanishes almost

everywhere.

Proof. A,

=

0 implies that S(-u’,- v ’ ) A,S(u’, v ’ ) (6.31) in the process,

=

0 for any

u’, v’ E R’. We obtain, using

Letting u1/27rliiand v‘/2nlti assume integer values, we infer that (6.36)

for any trigonometric polynomial F(u, w ) of period 1. Since p(u, v ) is integrable, and (f I S(u, w ) g ) is integrable and bounded, we can take limits and generalize (6.36) to the case whenF(u, w ) can be any continuous function of period 1 > 0 (see Exercise 6.6). Thus, we deduce (see Exercise 6.8) that for any fixed f,g E H,(f I p(u, v) S(u, w)g) = 0, except on a set R(f,g) of measure zero. If {e,} is a countable orthonormal basis, we conclude that (em I p(u, v ) S(u, v) e,) = 0 except on the set R= R(e, , en), which is of measure zero. Hence, p(u, v ) S(u, v ) = 0 outside R, i.e., p(u, v ) = 0 almost everywhere. Q.E.D.

u,,,

It follows from Lemma 6.3 that the operator A = A,, where v2)], is different from zero. Moreover,

po(u, v ) = exp[-(fi/4)(u2

+

345

6. Canonical Commutation Relations

it is easily verified by using (6.31) and (6.33) that AS(u, w)A has the

kernel

1

+ i(uw2 - wu2) - $(u - u2)2 - i ( w -

I., exp [

[ ti + w2 + u12 + w12)]

= exp - 4 (u2

217 fi

[

= - exp -

fi

(u2

+

w2)]

[

exp -

fi

(u12

+

-

du, dw2

fi

(u32

+ w32)] du, dw,

w12)].

This implies that (6.37)

2n

AS(u, w)A = - exp fi

[-

fi

5 (242

+ +)]A.

In particular, for u = w = 0 we obtain A 2 = (2n/fi)A.In addition, we can derive from (6.33) that A = A*. Thus, (fi/2n)Ais a projector. Denote by M the closed linear subspace of i W onto which (fi/27r)A projects. For any f,g E M we obtain by using (6.31) and (6.37)

Let {el, e2 ,...} be an orthonormal basis in M, and denote by M a the closed linear subspace spanned by all the vectors S(u, w) e, , u, w E R'. We easily deduce from (6.38) that the subspaces Ma are mutually orthogonal .

346

IV. The Axiomatic Structure of Quantum Mechanics

It follows from (6.31) that, for all u, v E W1, S(u, v )f E Mu iff E Mu . Denote by Xothe orthogonal complement of 0, Mu . By virtue of (6.32), for any f E X i = 0, Mu and g E Xo, we have

fI.

I g> = (S(--U, -4f I g> = 0 because S(-u, -v) f E 2‘;for all u, v E W1. On the other hand, it



= <S*(UY

follows from the construction of M u that 0, M u 3 M, where M had been defined to be the subspace onto which (fi/27r)Aprojects. Consequently, A f = 0 for all f E Xo. But, this implies that So= {0}, i.e., X = 0, M, . As a matter of fact, we have just proven that S(u, v ) leaves Xoinvariant, and therefore, if Xo# {0}, we could apply Lemma 6.3 to the Hilbert space Soand to the restriction of S(u, v) to Xo, and deduce that the restriction of A to Xois not zero. It is clear from the construction of M, that S(u, v ) , and therefore eim and d o w , u, v E W’, do not leave any nontrivial subspace of Mu invariant. Hence, eipuEMa and eiQvE are irreducible representations of the ?a canonical commutation relations in M, . Thus, the first part of von Neumann’s theorem is proved. Consider now two irreducible representations P, Q and P’,Q’ in X . In view of the above construction, M and M’ are spanned by the single vectors e, and e,’, respectively, and the sets of vectors f,,, = S(u, v ) e, and f;,, = S’(u, v) e,’ are dense in 2.Introduce the operator U , which is defined on f,,, by Uf,,, = f;,, and then extended to the entire Hilbert space X . It will be shown that U is unitary. By virtue of (6.38)

lf;z,vz>, so that 1) UfU,,11 = 1) f,,, 11. Thus, U can be extended to the entire Hilbert =
space by the extension principle (see Chapter 111, 92), and the result is an isometric operator. Moreover, since the set { f;,,: u, v E Wl} is dense in #, the range of U is X , and U is unitary. It follows from (6.31) that S ( U , v)fu,.v, =

exp[(i/2)

fiw -

~U’)lfu~+u.v*+vY

= exp[i(fi/2)(uvt- vu’)]f:,+u,v,+v

S’(u,

.

Consequently, we obtain the relation u-’S’(U, w )

ufu.*v. = U-’S‘(u, v )fu,,v, =

exp[i(fiP)(uv’- ~~’)lfu~+,,,~+o = S(U, v)fu,.v*

9

6. Canonical Commutation Relations

347

which can be extended straightforwardly to 2,so that S(u, w ) = U-lS'(u, v)U.

Since S(u, 0)

eipu, S(0, v ) = eiQw, etc., we arrive at the result that eiQv = U-lei*'vU. This concludes the proof of von Neumann's theorem. =

etPu = u-leiP'uuand

The proof of the theorem for the more general case of 2n operators proceeds along identical lines, S(u, v ) being replaced now by

Q1,...,Qn, PI,..., P,

w, Y'..,

u,

9

v1 Y...,

0,)

= exp[-(i/2) fi(u,o, X

exp[i(Qivi

+ + u,~,) ] exp[i(P,u, + + P,u,)]

+ + ***

Qnvn)].

In the case where Q and P constitute the Schroedinger representation in L2(R1), we have (see Exercise 6.9) (6.39)

(eiPu+)(x)= +(x

+ tiu),

(eiQv+)(x) = eiow+(x).

Consequently, we easily compute that

(W, .)+)(x)

= exp[i4x

(6.4) (A+)(x) =

I

+ *Wl +(x + w,

exp[-(ti/4)(u2

R2

+ v2)] exp[iw(x + @u)] +(x + tiu) du dw

Hence, the equation A+ = (27r/fi)+ has only one linearly independent i.e., M is one dimensional. This solution, namely $,,(X) = e--(1/2fi)z2, establishes that the Schroedinger representation is irreducible.

EXERCISES 6.1. Verify that the commutator [A,B] = AB - BA of linear operators defined everywhere on Z has the following properties: [A,Bl

=

-[B,AI,

[UA, B] = [A,UB] = U [ A , B ] ,

[A,BC] = B[A, C]

+ [A,BIC.

348

IV. The Axiomatic Structure of Quantum Mechanics

6.2. Derive (6.10) from (6.9) by induction, using the results of Exercise 6. I. 6.3.

Show that s-limf,lo eiAt

6.4.

Show that if U is a unitary operator, ~ - l i m ~ -U, ~ f= ~ Ub.

6.5.

Prove that Ef defined in (6.21) is a spectral function.

=

eiA6o.

6.6. Let f ( A ) , A E R1, be a continuous function of compact support. Prove that there is a sequence of trigonometric polynomials p,(A) = z k a n k exp[(2n/n) &A] which converges pointwise to f(A). Explain also why the same statement is true whenf(A) is the characteristic function of a finite interval. 6.7.

Verify that the relations (6.34) are true.

6.8. Show that if p ( x ) is integrable on R”, and JRnF(x)p(x) dx = 0 for any continuous function F(x) of compact support, then p ( x ) = 0 almost everywhere. 6.9.

Derive relations (6.39) and (6.40) for (Q+)(x) = x+(x), UF’QU, , where U , is the Fourier-Plancherel transform in LZ(R1) (see Chapter 111, Eq. (4.20) and Theorem 4.6).

P

7.

=

The General Formalism of Wave Mechanics

7.1. A DERIVATION OF ONE-PARTICLE WAVEMECHANICS In the preceding sections we outlined a very general framework for quantum mechanics, which is contained in Axioms 0, S (or H), P, and C. In practice, more specific formulations of this formalism are usually required. Such formulations are achieved by making other assumptions in addition to Axioms 0, S, P, and C. These assumptions reflect the peculiarities of the particular class of systems under consideration. The assumptions most frequently made are those leading to the formalism of wave mechanics. It is believed that these assumptions apply in the nonrelativistic domain to all microscopic systems which can be considered to consist of a finite number n of stable particles (i.e., of particles whose nature does not change during the duration of the experiment). The fundamental system in nonrelativistic quantum mechanics is the microscopic quantum-mechanical particle (which we have in mind whenever we talk about a “particle”). The following requirement reflects the basic nature of the concept of a particle.

349

7. The Formalism of Wave Mechanics

Axiom W1. The set { Q ' Z ) , Q(U), Q(Z), S"}of the position observables Q(z),Q ( Y ) , Q(z),and the spin projection S"in a given direction n,constitute a complete set of observables of any one-particle system. The above theoretical assumption reflects an interpretation of certain experimental facts which determine the empirical nature of a particle. In other words, a particle can be defined from the experimental point of view as an empirical object to which measurements of position, momentum, and spin can be related. The observables Q ( Z ) , Q ( Y ) , and Q(z) correspond, naturally, to the measurement of the x, y , and x coordinates, respectively, of the position of the particle in some inertial Cartesian frame. It has become customary to choose the x axis of this frame to point in the direction n in which the spin projection is measured (i.e., the direction H in Fig. 3 in the Introduction). When this is the case, we denote the observable S"by SZ). Axiom W1 implies the existence of a spectral representation space L2(R4,p) of the complete set {Q(Z), Q ( V ) , Q(*), S"},whose elements are represented by p-square-integrable functions

(7.1)

#(%Y , x ,

4

1., I #(%Y ,

x7

4124 4 Y , 4 < +ma s 7

It does not tell us anything, however, about the nature of the measure p. In order to decide which measures p we have to choose, we have to invoke two additional axioms, which will be given next. The following axiom reflects the experimental fact that the spin projection of a particle of spin a in some direction n can assume only the values -u, -0 1,..., +a (see the discussion at the end of the Introduction).

+

Axiom W2. The spin projection observable s" of a one-particle system of spin a has the pure point spectrum (7.2)

s

={-

0,

,..., u}.

-o + l , - u + 2

In the spectral representation spaceL2(R4, p) of {Q(Z), Q ( Y ) , Q(z),S"},the operator S"has the spectral representation

(so#)(% Y , 2,s)

Y , 2,s)

= S#(%

so that

(ES"(B)#)(x, Y , z,s,

= x B ( s ) #(%,

Y , x7

$1,

Take a function of the form #(x7 Y , % s, = xB1(x7

Y7

z, xBa(s),

IV. The Axiomatic Structure of Quantum Mechanics

350

where B, and B, are bounded Borel sets in R3 and R', respectively. Such a function obviously belongs to L2(Iw4,p). If B C B, , we have

II ES"(B)# 112

=

I

P4

x s p ,y , 2) XB(S) dp(x, y, 2,s)

= p(B1

Thus, if B n S,

= 0 , we

get

p(B1 x B )

=

Moreover, Es"({s}) = 1 for s E S,

dB1 x B ) =

(7.3)

x B).

I( ES"(B) # 112

= 0.

. Hence, we can write

c

aEnnS,

Ml

x

{4)Y

where the sum is always finite, since by the assumption in (7.2) S, is a finite set. Relation (7.3) has been established by the above argument for the case of bounded B, E B3and B E @,but it can be extended also to unbounded Borel sets by using the continuity from below of measures to derive for arbitrary B, and B

BP) = B, n {(x,y, 2): -n < x < ny -n < y < n, -n < 2 < n}, =B

n [-n, +n].

I n view of the structure (7.3) of the measure p, we find that the inner product (-I -)in L2(Iw4,p) is of the form

due to the fact that

If we introduce the measures ps(B) = p ( B x {s}),

s =

--(T

,-0

+ 19-Y

U Y

351

7. The Formalism of Wave Mechanics

on g3, we can write

However, without any further assumptions we cannot determine in more detail the measures ps(B), B E g3for s = -u,..., +a. The extra postulate which we need for determining ps follows.

, and Pa) represenAxiom W3. The self-adjoint operators P ) P(v), ting the momentum component observables of a one-particle system commute with the spin operator S".I n addition, the operators Q(%), Q(Y), Q@),P ) P(v), , P(@satisfy canonical commutation relations in which Q(%) and Q(v)and P(v),Q(2) and F a ) are canonically conjugate. In order to be able to take advantage of Axiom W3, we note that (7.4) implies that the mapping $(x, Y , z,4

-+

$s(x,

Y , 4,

s

=

-0

Y

+ I,..., u,

-0

of Lg(IW4, p ) onto @;=-oL2(R3,p.,) establishes a unitary equivalence between these two Hilbert spaces. Since we have

(Q") $)(x, Y , z, 4

= x$(x,

Y , s,4

= x$&,

Y , 4,

and similar relations hold for Q ( v )and Q(2), it follows from von Neumann's theorem (Theorem 6.4) that ps have to be Lebesgue measures on g3, and that for differentiable functions $(x, y, x, s), $ E gP(=) ,

with corresponding relations holding for P(v)and P(&).

7.2.

WAVEMECHANICS OF PARTICLE SYSTEMS

On the basis of the above discussion we have concluded that the spectral representation space of the set {Q(z),Q ( Y ) , Q(z),S"} constituting, according to Axiom W1, a complete set of observables of a single particle system of spin u, is the space L2(R4,p), where p is the measure satisfying p ( B , x {s}) = p?'(B,), p(B1

x B,)

= 0,

s = -u

, -u + l , . . . ,

B, n {-u,...,

+

U}

u,

=

ar,

and p\"(B,) is the Lebesgue measure of B, , B, €a3. We have also seen that a state $ can be represented at any instant by an element of the

352

IV. The Axiomatic Structure of Quantum Mechanics

+

direct sum @,L2(R3) of (20 1)L2(R3) spaces. Wave mechanics is ordinarily formulated in this @,L2(R3)space, in which a state vector is a wave function X,y,sERl,

#(x,y,z,s),

s=-u

, -u + l , . . . ,

+a,

which is Lebesgue square integrable in the x, y, z variables for each fixed value of s from the set S, = {-u, ..., +o}. By means of Axiom P in 94 we can generalize the above results to the n-particle case. Suppose that we are dealing with a system 6 of n different particles of spins O, ,..., o n , respectively. Then according to AxiomP, we can associate with 6 the Hilbert space (7.5)

+

where the kth factor, k = I , ..., n, is a direct sum of (2ak 1) L2(R3) spaces. The linear mapping Ul , which assigns to a vector of the form

(7.6)

(#pil @

* a *

@

@

***

@

(#zn @

@ &)), #!j1

,...,):#

€L2(IW3)

from Xl the vector (7.7)

5

* * a

sl=-ol

(@@

... @#)':

sn=-on

from the Hilbert space

determines a unitary transformation of Xl onto X2(see Exercise 7.1). Moreover, by Theorem 6.9 in Chapter 11, there is a unitary transformation of [L2(R3)]@nonto L2(UPn),which maps +(xl, yl,zl,sl) 0 --.@ $(xn > yn 9 zn > sn). By carrying +(%a 9 yn 9 zn 9 sn) into +(XI 9 ~ 1 21 , 3 $1) out this transformation of each one of the terms [L2(IW3)]@n of the direct sum (7.8) into L2(Rsn)we arrive at a unitary transformation (see Exercise 7.3) of the Hilbert space X2into the Hilbert space (7.9)

The elements of X (called the conjiguration representation space) are, for fixed s1 ,..., s, , square-integrable wave functions (7.10)

y1

%l9 s1

Y..,

9

Yn 9

sn),

sk = -Qk

)*-v

+

uk *

353

7. The Formalism of Wave Mechanics

Thus, U = UIU, is a unitary transformation of Sl onto S mapping a vector of the form (7.6) into (7.1 1)

$(l)(Xl 9 Y1 Y z1 9 s1)

-..$(,)(X,

9

Yn 9

2,

9

4.

The x-coordinate position operator in 3fl for the kth particle is

&W*0

$k(%

,Y k ,zk ,Sk) 0 -)

0(Xk$k(Xk ,YK ,Zk ,s,)) 0... for $3?) and $.3’: It is very easy to

=

**-

9

and corresponding expressions hold check (first for vectors in @ represented by functions of the form (7.7), and then generalize the result) that the x-coordinate position operator in &‘ of the kth particle Qf) =

k = 1,..., n,

U&?)U-’,

is given by (cf. Definition 5.2 of spectral representation spaces)

(Q%’$)(*.*,xk

3

Y k

zk

sk

,*.*)

== xk$(-**, x k , Y k zk

9

sk

,...).

Naturally, similar expressions hold for QP) and Q:). An analogous consideration easily yields that the momentum operator Pp)canonically conjugate to Qp) satisfies the relation

when P p ) is acting on differentiable functions from the domain of definition of P p ) . The above considerations cover the case of n-particle systems containing no identical particles. In the more general case when identical particles are present, the Hilbert space 2‘ of the system will be a closed linear subspace of the Hilbert space in (7.9), consisting of all appropriately symmetrized (see Definition 5.1) functions in (7.9); i.e.,the wave functions of the system will be symmetric with respect to position-spin variables (r, s) corresponding to particles of integral spin (commonly called bosom), and antisymmetric with respect to variables corresponding to particles of half-integer spin (commonly called fermions).* The entire discussion of this section can be duplicated with the complete set (see Exercise 7.5) {P(s),P ( g ) ,P5), S”} instead of {Q(%), Q(g), Q(E),Sn}.The procedure leads then to the same Hilbert space (7.9), except that now the state vectors of this space are wave functions

J ( P K Ply),P?’, s1 ,...,P,(+),P,(Y) ,p , ,&A (2)

it

See also Exercise 7.4.

ss = -0s

,.**,

Uk

2

354

IV. The Axiomatic Structure of Quantum Mechanics

in momentum variables (so that as the spectral representation space Z of {P, , S,” : k = I , ..., n} is called the momentum representation space) and the momentum operators Pp’ have the representation (Pp+)-(...,p p , p p , p p , Sk ,...)

=p y J (

...,p p , p p , p y , Sk ,...),

with corresponding relations holding for P,$”and P f ’ . The almost complete symmetry of the problem under the interchange of position and momentum variables (note that [Q‘”’, P(“)] = -[PS), Q‘“)], etc.) makes it obvious that Q&? is represented now by

when acting on differentiable functions from its domain of definition; similar relations hold for QP) and Qt).

7.3. THESCHROEDINGER OPERATOR I n any picture of quantum mechanics (see 93) we postulate for every quantum mechanical system the existence of a Hamiltonian H, which plays the dual role of determining the dynamics of the system and representing the total energy observable. However, none of the general axioms provides any clue about the more specific form of H. In nonrelativistic quantum mechanics one normally approaches the problem of finding H of a particular system of particles by looking at that system from the classical-mechanics point of view. Thus, one chooses a classical model with a potential V from which all the forces acting between the particles of the system can be derived. The transition to wave mechanics is made by writing the Schroedinger operator €or that system. The hidden assumption contained in this procedure is the following.

Axiom W4. The Hamiltonian H , of a system of n spinless particles is a self-adjoint operator which maps elements a+b E g H egiven , in the configuration representation space by twice everywhere continuously differentiable functions, into

355

7. The Formalism of Wave Mechanics

whenever the above expression represents an element of Z ; Hs is the only self-adjoint operator which maps all such elements IJ E gHsinto elements representable by (7.12). An operator H , satisfying the above conditions is called a Schroedinger operator. For particles with spin, H , contains in general terms which depend on the spin-component operators. Since spin operators are obviously bounded, the results derived later in this section are easily generalized to that case. Sometimes even “velocity-dependent” potentials, containing first derivatives in the position variables, i.e., depending on the momentum operators, are used. Their treatment might require special attention, and the reader interested in such cases is referred to the literature quoted at the end of this chapter. We note that Axiom W4 imposes constraints on the functions which can play the role of potentials. A potential which has too many singularities, or some singularities that are too strong, might not satisfy the second requirement of Axiom W4.

7.4. CLOSURES OF LINEAR OPERATORS The main reason for not treating the Schroedinger operator exclusively as a differential operator is that differential operators given by means of differential operator forms are ill defined, since their domains of definition are ambiguous. However, one hopes that when defining a linear operator on a dense domain by means of the Schroedinger operator form, one will be able to extend that operator uniquely and in a natural way to a self-adjoint operator. This goal is usually achieved by taking the closure of that operator.

Definition 7.1. The linear operator A is the closure of the linear operator A if it is the minimal closed extension of A, i.e., every closed operator C satisfying C 1A also satisfies C 2 A. The existence problem of the closure of an operator is treated in Theorems 7.1 and 7.2.

*Theorem 7.1. If the linear operator A in the Hilbert space Z has an extension which is a closed operator, then it has a minimal extension 2,i.e., its closure exists. Proof. Let A, be some given closed extension of A. Define the operator A in the following manner: the vector f is in the domain of definition of A if and only iff E gAo and for any B > 0 there is a vector g E g Asuch that

Ilf-glJ

(€9

IIAof-4Il

-=E*

356

IV. The Axiomatic Structure of Quantum Mechanics

When f E g A ,we define

Af = Aof. The so-defined operator A is certainly linear, since if f, ,f2 E 9 x and # 0, a, # 0, there are, for any given E > 0, vectors g , ,g2 E g Asuch that

a,

i

=

1,2.

Consequently

+ ‘ddl < which implies that a, f, + a2 Awl+

11 ‘If1

f %fa

- (‘1gl

€7

11 Ao(alf,

f2 E B

a2fA

A

+ ‘2f2)

- A(‘lgl

+

‘d2)ll

<

‘J

and

= Ao(a1f1

+

‘zf2).

Finally, the operator 3 ; s closed. T o see that, note that if f, ,fa ,... E g X converges to f E &?‘ and Af, , Af2,... converges to $, then for any integer n there is a g , E g Asuch that

llfn

- g n II < I/%

1I Aofn

- Agn

II < l/n.

Consequently, if l / n < €12 and

Ilf

-fn

II < 42,

I l f - a n II < 42,

we get by combining the above two inequalities and applying the triangle inequality the following:

Ilf - g n II <

€9

IIJ- Agn II <

6.

This result, in conjugation with the fact that A, is closed (and therefore

3 = A o f )yields that A is closed too. It is easy to infer from the way in which A was constructed that any closed extension of A contains A. Consequently, A is the minimal closed extension of A. Q.E.D.

Theorem 7.1 guarantees the existence of the closure A of a linear operator A only in those cases when it is known that A has a closed extension. However, symmetric operators A with GA= X have A* as an extension, and since A* is closed (see Chapter 111, Theorem 2.8), A exists.

*Theorem 7.2. If A is a symmetric operator densely defined in 2, its closure exists and is identical to A**.

357

7. The Formalism of Wave Mechanics

Proof. Since A** is the adjoint of an operator, it is closed. Moreover, A** 1A, so that A has a closed extension. Consequently, Aexists and A C A**. The relation A C A** implies A* 2 A*** = A* (see Exercise 7.6), and therefore A* is -densely defined and has an adjoint A**. We shall prove that A** = A. Both A and A** are closed linear operators and consequently their respective Cayley transforms V and Vl are closed also (see Chapter 111, Theorem 4.7). Moreover, A C A** implies that V C V , . We recall that the domain of definition g Vof V is closed and identical to 9a+i . Thus, @ . 0g Vconsists of all vectors f E S satisfying

( ( A+ i ) g If> = 0 for allg E gX+( . Since the above relation is identical to

(&If>

=(gIif),

g E g A ,

it follows that f E-2 '0g pif and only if A*f = Zjc. Since A* = A*** = (A**)* (see Exercise 7.6), it follows that 2 0g V= &' 0BVl, i.e., g v=-$Bvl .- Consequently, the relation Y C V , holds only if V = Vl, i.e., A = A**. Using the result A = A**, we obtain from A C A C A** that A* ;2 A* 1A*** = A*. By taking adjoints in the preceding relations we get A** C A C A**. Thus we have arrived at the desired result A** _= 2. Q.E.D.

7.5. THESCHROEDINGER KINETICENERGYOPERATOR By choosing in (7.12) V(xl ,y1 ,z1,..., x, ,y , , zn) = 0 we obtain the Schroedinger operator form for the kinetic energy operator

(7.13)

(Ts#)(%,Yl

x

9

xn Y Yn Y

Zl ) . * * Y

+(x1 9 %

9

z1 Y...)

4

xn ,Yn

9

In order to be able to treat T, as a differential operator in L2(RSn), we require that each t,h E L2(Ran) to which Tsis applied be representable by a twice differentiable function with square integrable second derivatives. An operator T, defined in this fashion would be symmetric but not selfadjoint. It will be shown, however, that such an operator is essentially self-adjoint.

358

IV. The Axiomatic Structure of Quantum Mechanics

Definition 7.2. The symmetric operator A in 3? is said to be essentially self-adjoint if its closure A is self-adjoint. According to Theorem 7.2, A** is identical to the closure A of A. Since (A**)* = A* 3 A** (see Exercise 7.6), we see that A is essentially self-adjoint if and only if A* = A**. *Theorem 7.3. The symmetric operator T o ,defined on the family doof all elements f E L2(R3") representable by functions 1

,..., r,) exp [- 2ti (r12+ + rn2)] corresponding to all choices of polynomials P(r, ,..., r"), and acting on

(7.14)

f(rl

,..., r,)

= P(r,

-**

f E goaccording to

is essentially self-adjoint. Its closure n

To is identical to the operator

.

(7.15)

Proof.

The operator T, is a function n

(7.16)

Ts

= K(P1

,..a,

Pn);

K(P1 ,..., P,)

=

C Pk2/2mk k=l

7

in the sense introduced in Theorem 2.5, of the commuting self-adjoint o$erators Pf), PLY),Pp',K = 1,..., n (see Chapter 111, Theorem 4.11). Therefore, from Theorem 2.7, and in particular (2.26), we deduce that for f e g T S ,T, acts on the Fourier-Plancherel transform 3 of f in the following way (see also Exercise 7.7): (7.17)

Now, g o is invariant under the Fourier-Plancherel transform U, (see Exercise 7.7), i.e., U F g 0= g o ,so that iff E go, thenfis representable by a function of the form P(P, ,-**, Pn) eXPr-(2fi)-l(Pl2

+ + Pn2)1. ***

It follows that T,f is defined by (7.16) when f E go, i.e., goC gTS . In view of the fact that a0is dense in L2(IW3")(see Lemma 5.2), the set

I . The Formalism of Wave Mechanics

359

gTS[consisting of all f E L2(R3,) for which the function on the righthand side of (7.17) is square integrable] is dense in L2(R3,). Thus, T 8 is a real function of self-adjoint operators and a densely defined operator. Therefore, Ts is self-adjoint by Theorem 2.6. It is easy to see that To C T, . As a matter of fact, we have (Tof)(r1 ,.**,

rn)

x exp[(i/fi)(pl rl = (Td)(rl

+ ... + pn r,)l dp1 ... dpn

,..., r,)

since for f E goC 53 of the form (7.14) the order of differentiation and Ts integration can be interchanged by Lemma 3.1 and the mean value theorem of differential calculus. We shall prove now that Ts is the closure of T o .Let us show first that (1 To)gois dense in L2(R3n). Assume that g €L2(R3,) is orthogonal to (1 To)9,,. Since UFBo= go, this assumption implies that

+

(7.18)

+

JD3nH(P1 Pn" I...,

+ K(P1 ,...,Pn)l P(P1

,.**,

+ + pn2)]dp,

x e~p[-(2fi)-~(p,~ ... for all choices of polynomials P(p, (7.19)

k(P1 ,...,Pn) = i(Pl l . . . ? Pn"

+ K(P,

,...,p,). ?*..)

Pn) dpn = 0

The function

Pn)l exp[-(4fi)-l(Pl2

+ ..*+ Pn2)l

is square integrable, and according to (7.18) it is orthogonal to all functions of the form (7.20)

P(p1 ,..., pn) exp[-(4fi)-l(p12

+ + pn2)1.

By Lemma 5.2 the family of all functions (7.20) is dense in L2(R39. Therefore, we must have h(pl ,...,p,) = 0 almost everywhere. Since [I

+ K(P1 ,--*. Pn)] exP[-(4fi)-1(p12 + + Pn2)]> 0, ***

Pi ,...,Pn E R3,

360

IV. The Axiomatic Structure of Quantum Mechanics

we must have f(pl ,..., p,) = 0 almost everywhere, i.e., g = 0, and (1 To)gois dense in L 2 ( R 3 9 . Let us take anyfE gTS . Since (1 To)gois dense inL2(RS”),there is for any E > 0 a vector f,E gosuch that

+

+

lU + Tdf

- (1

+ TO)f€II <

c,

i.e., in view of To C T, , Il(1

+ T s ) ( f -f€)ll<

It is easy to see from the relation

+

I I ( ~ ~ s ) ( -L)IIZ f that since K(p,

= Ran

1[1

+W P ~

P ~ ) U-~xP,)*..,Pn)12 d ~ 1 * dPn *.

,...,p,) 2 0,

I l f -f€ II < Il(1 + Td(f -f€)ll< II Tsf - TOf€I1 = II Ts(f -fJll < 10 T d f -fdl < €9

+

€9

and consequently T, C T o . However, T, is self-adjoint, and therefore closed, and since To C T, , we must have T, = T o . Q.E.D.

7.6. THESCHROEDINGER POTENTIAL ENERGYOPERATOR The operator V representing the potential energy is given in the Schroedinger picture by means of a potential V(r, ,..., r,)

(V~dJl(r1,***,

rn)

=

V(r1

,.**>

rn)

,*.*,

rn),

where V(rl ,..., r,) is a real function. It is quite obvious that V, is a function of QP),Qiv),Q?’, ... in the sense of the definition of a function of commuting operators given in Theorem 2.5 (see also Exercise 2.7): (7.21)

We cannot expect, however, that V , = V,* for any choice of the real function V(rl ,..., rn). The requirements subsequently imposed on V(rl ,..., r,) will enable us to prove next that V , is self-adjoint and, later on, that T , V , is also self-adjoint by virtue of the fact that Vs is bounded relative to T, in the sense of the following definition.

+

Definition 7.3. An operator K is said to be bounded relative to another operator A , or A-bounded, if gK3 gAand if there are constants a, b 2 0 such that 11 Kfll a 11 Afll b l l f l l for all f~ gA. The in-

<

+

361

7. The Formalism of Wave Mechanics

fimum of all values a for which there is some b such that the preceding condition holds true is called the relative bound of K with respect to A, or its A-bound. We note that if K is bounded in the sense of Definition 1.5 in Chapter 111, then it is bounded relative to any operator A with domain gAC g K , and that its A-bound is zero. Indeed, in case of such a bounded operator K we can satisfy the conditions of Definition 7.3 with the choice a = 0 and b = (1 K 11. However, an operator K can have a relative bound zero with respect to another operator A without actually being bounded. Such examples are provided in the next theorem which, in fact, states that V, has T,-bound zero, whereas the conditions on the potentials for which the theorem is valid are sufficiently liberal to allow for the possibility of V, being an unbounded operator.

,..., r,)

*Theorem 7.4. Assume that the potential V(r, written in the form n

V(ri ?..,

rn) =

Vo(ri

p...,

rn)

+ 1 Vij(ri

can be

n

-

Ti)

i,+l itj

+ C Vor(ri), i-1

,..., r,) is measurable and bounded on R3,, while Vgj(r), i , j = 0, 1, ..., n (i < j ) are locally square integrable and bounded at infinity," i.e., there are positive constants R,C such that

where V,,(rl

(7.22)

I Vij(r)l2dr < +a,

I Vij(r)l

< C,

r

> R.

For such a potential, the operator (vSf)(rl

,.-, rn) = V(r1 ,.-, rn)f(r,

rn),

defined on the family gVsof all f €L2(R3%)for which the function V(r, ,..., r,)f(rl ,..., r,) is square integrable, is self-adjoint, and gVs 3 g T SMoreover, . the operator V, is such that for any constant a > 0 there is another constant b > 0 such that

(7.23)

II VsfII

< a II TsfII

+ b llfll

for all f E g T S . It has become customary to denote the class of all functions f(r) having these two properties by L1(R3) Lm(R3),where L1(R3) and Lm(Ra)denote the Banach spaces of functions that are measurable and, respectively, Lebesgue integrable or bounded on [Wa.

+

IV. The Axiomatic Structure of Quantum Mechanics

362

Proof. I t follows from the assumptions on V(r, ,..., r,) that V(r, ,..., rn)f(rl ,..., r,) is square integrable on R3" if f E g o . This conclusion can be easily reached by using Theorem 3.9 in Chapter 11. This theorem can be immediately applied to Vo(rl ,..., rn), which is bounded I Vo(rl ,..., r,)l < b, and therefore

I

(7.24)

Vo(rl ,.-, rn)f(ri ,-., rn)l < b, If(r1 ,-, r,)I.

Its application to Vij(ri - rj) and Voi(ri) is somewhat more roundabout (see Exercise 7.8). Since gois dense in L2(R3")and, by virtue of (7.21)) V , is a real function of the position observables, it follows from Theorem 2.6 that V, is self-adjoint. A glance at (7.24) makes it obvious that V , satisfies (7.23) with a = 0 and b = b, . The proof of (7.23) for Vij is more intricate, and it will be carried out next only for the representative case of V,, . In the new variables

the function

is of the form (7.14) wheneverfe 9, . This is due to the fact that r;"

+ ri2 = r12+ r:.

In addition, the Jacobian of the transformation rk +. rk', k = I,..., n, is equal to one, dr,'

**.

dr,'

= dr,

... dr, ,

so that writing Vi2(r1') = V12(r1- r2),

II V12f1I2

I V ~ ( r 1- r2)f(rl ,..., r,)l2 dr,

= Rn

=

jRn I Vi2(r1')g(r1',...,r,')I2 dr,'

**.

dr, dr,'.

T o be able to estimate the above integral, we split the domain of integration into two parts (7.25)

363

7. The Formalism of Wave Mechanics

where R‘ = 2-lI2R and R is the constant appearing in (7.22). We shall study the two parts separately. According to Fubini’s theorem (Chapter 11, Theorem 3.13)

TI‘ 4R’

I Vi2(rl’)g(r1’, ..., rn’)I2dr,‘

dr,‘

rl’***r; E R ~

If g’(pl’ ,..., pn’) denotes the Fourier transform of g(rl’ ,..., rn’), we have, by (4.12) and Theorem 4.6 in Chapter 111,

+

Moreover, for an arbitrary positive constant E , (1 ~~(p;~)~ and )-’ I g(pl’,..., pn’)l(1 are square integrable in pl’ E W. Hence, we can apply the Schwarz-Cauchy inequality in L2(R3) to these two functions, and thus derive

+

(7.28)

364

IV. The Axiomatic Structure of Quantum Mechanics

in the last step of the above derivation we have used the fact that, in terms of the momentum variables p1 ,..., pn , 1

PI' =

-(PI

-

fl

PA,

1

PZ'

+ rl' - plf + r2' p2', and 2Pl PZ + PZ2) 9 P12 + PZ2 = -(PI

.\/2

- + r2 - pz =

so that r1 p1

PiZ= UP12 -

~ 2 1 ,

*

P,'

~ 3 = ' ~3 v * * ?

consequently

*

Inserting the last inequality of (7.28) in (7.27), and noting that

1",

I2(P;,*.*,P,1)l2 dPl'

--*

dP,'

= Pn

7. The Formalism of Wave Mechanics

365

Since E > 0 can be chosen arbitrarily small, we have arrived at the result that the inequality (7.23) is true for all f E goand any a priori given a > 0, if we choose E such that

(7.29) Take now a vector h E g T sand , let h, ,h, which T,h = s-lim Tshn .

,... E gobe a sequence for

n+m

Since (7.23) has been established for vectors from go, we can write

II vs(hm - hnll < a II T~(hm- hn)Il

+ b /I hm - hn 1,

which implies that V s h l , Vshz ,... is a Cauchy sequence. Hence h'

= s-lim n-m

Vshn

exists, and since V, is self-adjoint, and therefore closed, we have h E gVs and h' = Vsh; hence BTsC g V sFurthermore, .

II V SII ~< II VShn II

+ II V S ( -~ hn)II < a II Tshn II + b II hn II + II V S (-~ hdlj

and in the limit n -+

+

00

we obtain (7.23) for an arbitrary h E gTS.

Q.E.D. The very general type of potential with which Theorem 7.4 deals consists of three distinct parts: Vo , which describes a global interaction in between all n particles; Vij, describing a two-body interaction in between particle i and particle j; and Voi , describing the influence of some sources external to the system (such as the external fields discussed in Section 7.9) onto particle i. As examples of particular practical importance we can mention the case of Coulomb and Yukawa two-body interactions corresponding to the choice

(7.30)

Vo(rl ,..., r,)

= Voi(ri) = 0,

Xii exp( -Kii(ri

v..(r. 13 a -I 3 . =)

- rj))

I T i - ri I

,

i<j,

with K~~ = 0 and qj> 0, respectively, where hij are coupling constants (which in the Coulomb case equal the product eiej of the respective charges of the ith andjth particle-see Chapter 11, Section 7.7).

IV. The Axiomatic Structure of Quantum Mechanics

366

OF THE SCHROEDINGER OPERATOR 7.7. THESELF-ADJOINTNESS

The standard method of proving the self-adjointness of an operator of the form A K, where A is already known to be self-adjoint, is based on a result first proven by Rellich [I9391 (see Lemma 7.2 later in this section), in which the useful notion of operator core figures prominently. D a subset of Definition 7.4. Let A be a closed operator in g A ,and A D the restriction of A to D, so that A D C A. The set D is called a core of A if AD= A.

+

By Theorem 7.3 the set gois a core for the self-adjoint operator T, . As has been illustrated so many time on the differential operator forms we have encountered in preceding sections, such forms define operators on L2 spaces only when applied to suitably chosen sets D of functions. In order that such forms unambiguously lead to self-adjoint operators, it is essential that these sets D represent cores (i.e., that the action of the differential operator form on all elements of D represent an essentially self-adjoint operator). It should be realized, however, that one and the same operator can have many cores. In fact, if we reconsider the proof of Theorem 4.10 in Chapter 111, we see that the argument used there leads to the following more general conclusion. Lemma 7.1. A densely defined symmetric operator is essentially self-adjoint if and only if both its deficiency indices are zero. Thus, as long as for some A = A* a set D C gAis such that the restriction A D of A to D has a Cayley transform whose domain gA,+* and range g'a,-a are dense in X, that set D is going to represent a core for A. For example, we might have used Lemma 5.2 to build dense sets D of functions (5.18) with a choice of fo(r, ,..., r,) that would be different from the exponential function in (7.14) and thus arrive at other" cores for T , , but then the computations of the Fourier transform of such functions needed to establish Theorem 7.3 would not have been as straightforward.

Lemma 7.2. (Rellich's theorem). If A = A * and K is symmetric and relatively bounded with respect to A with an A-bound smaller than 1, then A K is a self-adjoint operator with domain 9A+K = .9,, ; furthermore, A K is essentially self-adjoint on any core of A.

+

+

+

Proof. We shall prove first that A K is self-adjoint by showing that its Cayley transform V is unitary (see Chapter 111, Theorem 4.9), i.e., that %A+K+i = WA+K-i = S.

* The set D = %fb(R3") is a very popular alternative choice (e.g., see Reed and Simon [1975], Sections X.l-X.41).

7. The Formalism of Wave Mechanics

367

Since for any h > 0 the operator h-lA is self-adjoint, we infer from Theorems 4.7 and 4.9 of Chapter I11 that the ranges of the operators A f ih = X(h-lA f i) coincide with S and that (A f ih)-l exist. Moreover, (A f ih)-l g E g AC gK for all g E and in accordance with the Definition 7.3 of relative boundedness, (7.30)

I/ K ( A f i v g II

< a II A(A fih)-lg ll + b ll(A It i w g II,

where we can choose a, 6 2 0 with u < 1, since the A-bound of K is assumed smaller than 1. On the other hand, as in (4.14) of Chapter 111,

f ih)JIIz

(7.31)

so that by setting above f

= =

I/ A f l l z

+

llf1I2,

3

(A f ih)-lg we get

Consequently, chossing in (7.30) at fixed 0 sufficiently large value of A > 0, we obtain

I I WfiPgII

f E9A

< ( a +bP)Ilgll


< Ilgll,

and 6 >, 0 a

gE*.

+

This implies (see Exercise 7.9) that the range of 1 K(A f iA)-l equals %. Since we have seen that the ranges of A f ih equal X, we conclude that the ranges of A

+ K fih = [l + K ( A fih)-l](A fih)

also equal S.By Theorem 4.9 in Chapter I11 this implies that kl(A K) kcelf-adjoint, and therefore so is A K. If D is a core of A then for any f E g Athere must be a sequence fl ,fi ,... E D such that f, -+ f and Af, -+ Af as n -+ co. But then

+

+

II K ( f -fn)ll

< a II A ( f -fn)ll

+ b Ilf

-fn

+

II

-+

0

+

so that the domain of the closure (A K ) Dof A K restricted to D contains the domain of the closure A, = A. Since 9 A + K = g Aand A K is self-adjoint, we conclude that (A K ) D= A K, i.e., D is a core of A K. Q.E.D.

+

+

+

+

Since by Theorem 7.3 the T,-bound of V , is zero, we can immediately apply the above lemma to T, V , , and thus state the following result.

+

Theorem 7.5.

(Kuto's theorem).

The linear operator

HBo

acting

368

IV. The Axiomatic Structure of Quantum Mechanics

on any function of the form (7.14) as follows, (7.32)

(Ha0f)(r1)*..> rn)

is essentially self-adjoint on goif the potential satisfies the conditions of Theorem 7.4, and its closure coincides with the Schroedinger operator H , = T, V , ,which is self-adjoint and has the same domain as T, in Theorem 7.3.

+

zo

Theorem 7.5 was first established in essentially the above form by

T. Kato [1951]. As discussed earlier, there is a great latitude in the choice of core, and a rather popular alternative choice is %,,CO(IW3n). Although Theorem 7.5 is applicable to all the specific cases of interactions treated in the next chapter, it does not cover all cases of physical interest. However, extensive studies of essential self-adjointness of other types of Hamiltonians appearing /In wave mechanics have been carried out by a host of researchers," so that all cases of practical interest in nonrelativistic quantum mechanics have by now received suitable treatment. From a purely mathematical point of view, this field of study is a subdiscipline of the theory of elliptic differential operators, and considering the problem in the abstract one cannot expect to have selfadjointness for arbitrary choices of potentials-such as potentials more singular than r2at the origin. When applied to Schroedinger operators the general theory of elliptic differential operators (see Hormander [19691) leads to other physically important results, such as the following theorem which we state without proof.

'Pheorem 7.6. Any eigenvector t,b of the Schroedinger operator in Theorem 7.5 can be represented by a function +(rl ,..., r,) that has continuous second derivatives in the region where the potential has continuous first derivatives, and it satisfies there the time-independent Schroedinger equation (7.33)

]zl

h2

2m,

+ [A - V(rl ,..., r,)]I +(rl ,...,r,)

= 0.

This theorem assures us that by completely solving (7.33) (as we have done in 97.7 of Chapter I1 for a particle moving in a Coulomb potential), we indeed obtain all the eigenvectors and eigenvalues of the Schroedinger operator H , , i.e., that there are no additional eigenvectors f E gHs

* Stummel [1956], Wienholtz [1958]. Jorgens [1964], Schminke [1972], and many others; see also Sections X2, X4, and X 5 of Reed and Simon [1975]for a review of some of these results.

369

7. The Formalism of Wave Mechanics

that we might not have reached through the differential equation (7.33) because they would not have been representable by differentiable functions.

7.8. THEANGULAR MOMENTUM OPERATORS In classical mechanics angular momentum is next in importance to energy, position, and momentum. This marked significance of angular momentum observables is retained in quantum mechanics. Following the analogy with classical mechanics, we postulate that the components of the angular momentum of a single spinless particle along the x,y, and z axes, respectively, are represented in quantum mechanics by the following operators in L2(R3): L ( z )3 - Q ( v ) p ( z ) - pCv)QCz) L(v)3 - Q(z)p(z) - p(z)Q(v), (7.34) L(z)3 - Q(z)p(v) - p(z)Q(v).

The expressions on the right-hand side of these relations are obviously symmetric operators. Indeed, consider for example L@): Q(z)and P(v), as well as P(z) and QQ(v),are commuting self-adjoint operators, so that QQCs)P(v) and PQ(s)Q(v) are self-adjoint by Theorem 2.6. Hence QQ(s)P(Y) P ) Q ( Y ) is certainly symmetric, but in view of Nelson's counter-example, * we cannot claim immediately that it is self-adjoint, despite the fact that

is well-defined on certain dense linear sets, such as Wb1(R3). A simple way of bypassing this technical difficulty is through the use of Stone's theorem (Theorem 6. I). We consider therefore rotations Rp) of the Cartesian inertial frame of reference by an angle 4 around the 2 axis. For a fixed spatial point, such a rotation gives rise to a change in its coordinates (x, y, z) with respect to the original frame to coordinates

(7.36) x'

= x cos

4 + y sin 4,

y'

=

-x sin 4

+ y cos 4,

z'

= x,

with respect to the new frame. We surmiset that the configuration space wave function +(r) = +(x, y , z ) will be represented in this new frame by $'(r')

= +(r) = +(Rg)-'r'),

R(Z)-' d = RE;

,

See the note following Exercise 1.2 on p. 268. A concise discussion of this counterexample can be found in Section VIII.5 of Reed and Simon [1972]. + Since by Born's correspondence rule (Definition 1.3) only expectation values (1.14) and (1.15) and not state vectors themselves are measurable, a rigorous argument has to be based on group-theoretical approaches to quantum mechanics (see, e.g., Miller [1972], Barut and Rpczka [1977]).

IV. The Axiomatic Structure of Quantum Mechanics

370

where, in general, if R denotes a given rotation of the coordinates system, R-1 stands for the inverse rotation, so that R-IR = RR-I is the identity transformation [in fact, note that a rotation is an isometric transformation in the real Euclidean space (R3) and as such it can be viewed as a unitary operator on (R3)-cf. Chapter 111, Exercise 4.11. These considerations suggest the introduction in L2(R3) of the linear operators (7.37)

(Uy$)(r)

=

wQ9

that are unitary since the transformation (7.36) is obviously linear and one-to-one, and its Jacobian equals one. Furthermore, it is easy to check that U(Z)U(Z)= U(Z) 41 42 d1+d2 ' 4 1 ' 4 2 R1, so that by Stone's theorem there is a unique self-adjoint operator

L(=)for which (7.38)

U t ) = exp[(i/fi)L(z)+],

4 E W.

On the other hand, by Theorem 3.1, (7.39)

(i/h)L(Z)+= s-&[(up'+

- 99/41,

+

E +d

CL2(R9),

and for $(r)E%?b1(R3), the outcome of taking the above limit coincides with (7.35) multiplied by i/fi, as can be easily verified by inserting the expressions (7.36) for R r ) r into (7.37) and explicitly computing the limit by elementary calculus techniques. By also considering rotations around the x and y axes we arrive in a similar manner at definitions of and L(")as self-adjoint operators that are the extensions of the symmetric operators appearing on the right-hand side of the relations in (7.34). The so-obtained operators, however, do not commute, and by (6.6)(see Axiom W3)we easily obtain (7.40)

,

[L(=), L(Y)]C

[I,@),L("] C ifiL("),

L(")]C ifiL(v).

Thus in defining the square of the angular momentum operator by analogy with classical mechanics conventions, (7.41)

L2

+ (L'"')2+ (L(Z))2,

= (L'"')2

we gain run into the technical problem of establishing self-adjointness -as opposed to mere symmetry properties. General (Casimir operator) techniques for dealing with problems of this nature are made available by the group theoretical approach to symmetry problems in quantum mechanics. These techniques are, however, beyond the scope of this book, but fortunately we can handle this particular case by elementary techniques based on the results of

371

7. The Formalism of Wave Mechanics

$7.6 in Chapter 11. Indeed, according to those results, L2(Qp,pp) = ,pr) @ L2(Q8, p.,), and (Chapter 11, Theorem 7.1) the spherical harmonics Y , m constitute an orthonormal basis in L2(Q,,pa). Let US therefore set by definition

L2(Qr

(7.42) ( L 2 W ,8,4)

f z(z + 1) c +Z

= fi2

y,m(e,

m=-1

Z=O

4)

1

+*(y, 01,4') y Z m w 4') d p , 4') ~

0,

whenever the infinite series on the right-hand side of (7.42) converges in the mean to an element of L2(Qp ,pp). T h e resulting operator is selfadjoint (see Chapter 111, Theorem 5.2) and by Theorem 7.7 in Chapter 11 its action on #(r, 8,d) E %b2(Q,) @ Up2(Q,)coincides with that of the differential operator form obtained by inserting (7.34) into (7.41). One of the useful by-products of these considerations is the joint spectral measure EIQI>ILI*L''' of I Q 1, I L I, andL@),which for B = B, x B, x B, E 9, is defined by (7.43)

( E I Q I . I J + ( ~ ) (xB ,B, x B&)(Y, 8,+) = XB1(T)

c c

YZrn(R4)

%ZEB,AmcB,

SI, $*(y, O', 4') yzm(e',4')

dP*(8'9 4')

where, naturally, I E (0,1 ,...}, m E { - I ,..., +I}. Clearly, the existence of this measure extablishes that {I Q 1, I L I,&} constitutes a complete set of observables inL2(Qp ,pp).We note that, strictly speaking, this measure corresponds in fact to {I Q I, L', Lz},where

(7.44)

L'

= I[-1 2

+ (1 + 4L2)1/Z],

but in accordance with a tradition dating back to old quantum theory (see Messiah [ 19621) the variable f i I is thought of as being related directly to 1 L I = (L2)f, albeit the eigenvalues of L2 actually equal f i 2 Z ( Z 1). An even more frequently used complete set of observables is {I P I, I L I, Lz}. If &(k, 8,+) equals the Fourier-Plancherel transform of $(r) expressed in spherical coordinates, then

+

(7.45)

( E I P I . I L I J ( " ) (xB ,B,

s,

x B,)+)"(K,8, 4)

&*(A, 0', 4') Ylm(e',4') sin 8' do' d4'.

312

IV. The Axiomatic Structure of Quantum Mechanics

In the presence of spin the operators (7.34) retain the meaning of orbital angular momentum components, but spin components have to be included to arrive at total angular momentum operators. Naturally, has to be added as seen in the discussion in Section 7.1, in that case SZ) to the sets consisting of I Q I (or I P I), I L I, and L(”)to arrive at complete sets of observables.

* *7.9.

TIME-DEPENDENT HAMILTONIANS

In 93 we had formulated the Schroedinger picture for the general case of time-dependent Hamiltonians H ( t ) . Such time-dependence would be encountered only if the system under consideration is not isolated from its environment, so that energy is being exchanged between the system and some external field sources. Naturally, as mentioned in the footnote on p. 292, by including those sources as part of the system we arrive at a new, enlarged system whose Hamiltonian is not time-dependent, but the resulting mathematical model might be either too difficult to handle exactly, or an appropiate mathematical description might not even be available within the nonrelativistic context studied in this volume. A typical instance of the second kind is provided by a charged particle (electron, proton, ion, etc.) moving in a laser cavity and therefore continuously bombarded by photons. The exact mathematical description of the swarm of photons filling the cavity requires the quantization of the electromagnetic field within the cavity and therefore comes under the heading of (relativistic) quantum field theory. However, if the particle moves at nonrelativistic speeds in relation to the cavity, the time-dependent (nonrelativistic) Hamiltonian (7.46)

H ( t )1(2m)-l [-iti V - (e/c) A(r, t)12

+ eA,(r, t )

for a particle of charge e in an external four-potential (A,, A) related to the electromagnetic field in the cavity might represent a reasonable approximation. Rellich’s theorem (Lemma 7.2) provides a suitable tool for establishing the self-adjointness of (7.46) under reasonable conditions on the functions A,, A(”),A ( V ) , and A@).However, we are still faced with the problem of defining the time-evolution operator (see p. 292) (7.47)

by giving a suitable meaning to the above integration symbol, which in physical literature bears the name of time-ordered integral. Indeed, since H(tJ does not in general commute with H(t,) for t, # t 2 , we

373

7. T h e Formalism of Wave Mechanics

have to exercise caution in preserving the appropriate time-ordering as we (heuristically) combine “infinitesimal” contributions in the following manner, (7.48)

U(t1

+ dt, t o )

= exp[-(i/fi) H(t1)

4W

l

9

to),

to arrive at a sensible definition of U(t, to) which satisfies (3.16H3.18). The clue for a general definition of (7.47) is provided by (3.17) which, if valid, by mathematical induction leads to the conclusion that (7.49)

U(t,t o )

=

W t , tn) U(t, 3

tn-1)

**-

W z

9

tl)

Wl

9

to)

regardless of how small 6 = max{l t

- t, 1, I ti -

I : j = 1,...,n)

is chosen to be. Hence we set, by definition, 1

(7.50)

U(t, to) = d i m n++m

6++0

I7 exp[-(i/fi) .

H(tj-l)(tj - tjW1)]

j=n

where the order j = n, n - 1, ..., 1 in the above operator product is very essential. Under suitable conditions on H ( t ) [such as the existence of a common domain of definition for all H(t)], it can be shown not only that the strong limit (7.50) does exist, but also that Axiom S.3 on p. 292 is indeed satisfied. In Chapter V, we apply the framework of this chapter to quantum scattering problems and we shall concentrate on the case of timeindependent Hamiltonians, for which it is easily seen that (7.50) indeed equals (3.2). Hence we direct the reader interested in details of the general theory of time-ordered integrals (7.47) to Chapter XIV, Section 4, of Yosida [1974], and for details on the general scattering theory for time-dependent Hamiltonians extrapolating that of Chapter V to the articles by PrugoveEki and Tip [1974].

EXERCISES 7.1. Let Sl ,...‘ , Sm‘ and Z1 ,..., ’H ’ ,”be Hilbert spaces. Prove that the mapping of

374

IV. The Axiomatic Structure of Quantum Mechanics

into

7.2. Generalize and prove the statement contained in Exercise 7.1 to the cases when (a) m = +a,n < +a; (b) m < +m, n ’ = +m; (c)m = n = +a. 7.3. Let u k be a unitary transformation of the Hilbert space S k ’ onto the Hilbert space &kIf for K = 1, 2, . Show that the linear transformation U = U, @ U2 @ .-. of Sl = S k ‘ into &?2 = @;=l &k” is unitary, where U is defined by

...

W l Of2

O -)

=

Gfl O &fa 0

fl

**-,

E2

E2

1 ,fi

2

,...

7.4. Prove that the unitary transformation U of the Hilbert space (7.5) onto the Hilbert space (7.9), defined in the text, maps the subspace of all symmetric (antisymmetric) functions from Xl in (7.5) onto the subspace of all symmetric (antisymmetric) functions from &‘ in (7.9); more generally, it maps any subspace of “appropriately” symmetrized functions in (7.5) onto the subspace of functions in (7.9) “appropriately” symmetrized in the same way (see Definition 5.1).

7.5. Derive from Axioms Wl-W3 the result that {P), P(Y),P), S”} is a complete set of observables. 7.6.

Show that if A** exists, then A*** = A*.

7.7.

Show that the Fourier transform of a function

is a function of the form P,(pl

,..., p,)

exp[-l/2fi(p12

+ + pnz]. **.

7.8. Show that if V(r) is locally square integrable and bounded at infinity, then V(r1 - rJf(r1 , 4

7

f(r1 ! r2) = P(r1

9

r2) exp

[-

1 (r12

+ r?)]

is square integrable on R6.

+

7.9. Prove that the range of 1 A, where (1 A (1 < 1, is the entire n Hilbert space &‘, and that (1 A)-l = u-limn-tmCk=O ( - l ) k A k .

+

8. Completely Continuous Operators and Statistical Operators

*8.

Completely Continuous Operators and Statistical Operators

8.1.

COMPLETELY CONTINUOUS OPERATORS

375

A compact set in a complete metric space-in particular, in a Hilbert space-is a set having the property that any infinite sequence of elements from that set has a convergent subsequence. Linear operators on Banach or Hilbert spaces which map bounded sets into compact sets are called completely continuous or compact. T h e following equivalent definition of completely continuous operators emphasizes their main feature which is of importance to us. Definition 8.1. T h e linear operator A on the Hilbert space X is said to be completely continuous (or compact) if for every infinite sequence f l ,f a ,... E X ,such that 11 f , 1) C for all n = I, 2,..., the sequence Afl ,Af2 ,... contains a strongly convergent subsequence. We shall see later that the density operator p describing states in statistical quantum mechanics, as well as the product pA, , where A, is any bounded observable, are completely continuous operators. The theory of completely continuous operators has many applications, some of which we shall encounter in Chapter V. For this reason, the foundations of this theory deserve special attention.

<

Theorem 8.1.

Completely continuous operators have the following

properties: (a) They are bounded. (b) The products A B and B A of a completely continuous operator A and any bounded operator B on X are completely continuous operators. (c) The linear combination alAl a2A2of two completely continuous operators is a completely continuous operator.

+

Proof. (a) Suppose that A is not bounded, i.e., that there is no constant C such that 11 Af 11 C (1 f 11 for all f E X . In that case we can find a sequence f l ,f 2 ,... of normalized vectors, 11 f , 11 = 1, such that (1 Afl 11 , Afi 11 ,... diverges to infinity. But then Afl , Af2 ,... contains no convergent subsequence, which is impossible if A is completely continuous. (b) If f l ,f 2 ,... is any bounded sequence, then Afl , Af2 ,... contains a convergent subsequence Afj, , Ah, ,... . Since B is bounded and therefore continuous (see Chapter 111, Theorem 1.3), BAf,, , BAh, ,... converges, i.e., BA is completely continuous. Further, Bfi ,Bf2 ,... is bounded. Therefore, there must be a subsequence Bfk, ,Bfk, ,... such that

<

376

IV. The Axiomatic Structure of Quantum Mechanics

A(Bfkl),A(Bfk),... is convergent. Consequently, AB is also completely continuous. (c) If g, , g , ,... is any bounded sequence, there is a subsequence gil ,gss ,... such that A,gil ,... is convergent. Moreover, since gil ,gjs ,... is also bounded, it must contain a subsequence gk, ,gk, ,... such that A,gkl, A,gka ,... is convergent. Consequently, the sequence alAlgkl U2A,gkl,... is convergent, which shows that a,A, a,A, is completely continuous. Q.E.D.

+

+

Theorem 8.2. If the linear operator A on 2 is the uniform limit of a sequence A , , A , ,... of completely continuous operators, then A is completely continuous. Proof. Let f, ,f, ,... be any bounded sequence, 11 f, 11 from 2. Construct a family of subsequences {fAO'

=f n } 3

{fy}3

***

3 {f;)} 3

< C, of vectors

*.-

of f, ,fa ,... by the recursive procedure in which, for every i = 0, I, 2,..., {f:+"} is a subsequence of {f:)} such that {Ai,,f:+"} is convergent. Since {f:'} is a subsequence of { f,} and therefore bounded, { fAi+l)} exists, due to the complete continuity of A t + , . From the inequality

11 A f p - A f k( k )11

< I( A f y - A J y I( + I(A , f y - A n f y 11 + (1 A n f y - Af'k) k I( < II A - An II II + Ilfp)II) + II A,(fji) -f;9Il < 2c II A - A, II + It An(fi"' -fK'k')II, we infer that {Aff)} is convergent; namely, for any given 6 2C (1 A - A , (1 < 4 2 for sufficiently large n, while

II A n ( f y )-f;k) l if sufficiently large i, R 2 tt are chosen.

> 0 we have

<42 Q.E.D.

There are many continuous linear operators on an infinite-dimensional Hilbert space X which are not completely continuous. A conspicuous example is the identity operator on i@ (see Exercise 8.1). As a generalization of the operator I f ) (f I, let us define for any two vectors f,g E W . the operator If) ( g I in the following way: (8.1 )

If>(g If'

=
If'>f,

f' E a?.

377

8. Completely Continuous Operators and Statistical Operators

Lemma 8.1. Every operator of the form A

=

1 f ) ( g I is completely

continuous.

Proof. In fact, if fl ,f 2 ,... is a bounded sequence, ( g I f i ) , ( g I f 8 ) ,... is obviously a bounded sequence of numbers. Hence, the set {( g I f n ) > has at least one accumulation point a. Thus, we can choose a subsequence {(g I X,)} converging to a. Consequently s-lim Afk, =f lim ( g n-m

n-m

Ifkn)

= af

exists. Hence, A = I f ) ( g I is completely continuous.

Q.E.D.

It follows from the above result and Theorem 8.l(c) that any operator of Jinite rank, i.e., an operator that can be reduced to the form n

n

(8.2)

'k

I e;>

(ek

I = 1 I e;)

k-1

'k

(ek I

9

k-1

is a completely continuous operator. Moreover, by Theorem 8.2, any operator which is the uniform limit of operators of the form (8.2) is completely continuous. We shall prove that, conversely, any completely continuous operator is such a uniform limit. T o prove this result we need two additional results which will be derived as part of Theorem 8.3 and in Lemma 8.2. It should be evident by now that all linear operators on a finitedimensional Hilbert space &' are completely continuous. In fact, if A is a linear operator on &' and {el ,..., e,} is an orthornormal basis in S', then it is easily seen that A

I hk)

=

<ek

1,

k=l

where h, = Ae, ,k = 1,..., n. In view of Lemma 8.1, we have established that A is completely continuous.

Theorem 8.3. Every symmetric and completely continuous operator A has a pure point spectrum S A = S,; which has no accumulation points, with the possible exception of zero. The characteristic subspace of every nonzero eigenvalue is finite dimensional. Proof. Assume that 01 # 0 is an accumulation point of the point spectrum of A. This would imply that there is an infinite sequence A,, A, ,... of distinct eigenvalues of A such that I A1 I, I A, I,... > 4 I a I. If f l , f 2 ,... are the corresponding unit eigenvectors Afn=hnfn,

n = 1 , 2 ,...,

IV. The Axiomatic Structure of Quantum Mechanics

378

the sequence Af, , Af, ,... has no convergent subsequence since for any m # n we have A, # A,, which implies that f, I f,) = 0 and

<

II Afm

-

Afn

1'

=

I Am 1'

+I

An

1' > a2/2.

Thus, A would not be completely continuous. Hence, St has no nonzero accumulation points. Assume now that the continuous spectrum S< of A is not empty, so that S< contains infinitely many points (see Exercise 8.2). Choose an infinite sequence of nonzero A,', A,', ... E SA, such that 1 A,' I < 1 A,' I < --., and positive numbers r , , r, ,... such that the intervals I,, n = 1, 2,... between the points A', - r, and A', r , are disjoint and do not contain the origin. Since A,', A,', ... are points of the spectrum of A, the projectors EA(I,) are nonzero. Therefore, there are vectorsg, ,g, ,... satisfying

+

E"(In)gn=gn,

IlgnII=1,

fi=l,2,.-*-

Due to the disjointness of the intervals I, , EA(I,) EA(I,) = 0 when m # n. Using the spectral theorem, it is easy to compute that (Agm I Agn) = ( E m

I Pd

I5

I - rn)2

for m # n for m = n.

Thus, we have for m # n

II Agm - Agn 1 '

= I1 Agm

11'

+ I1

&n

11' > 2(1 h' I - r,)',

i.e., Ag, , Ag, ,... contains no convergent subsequence. Hence, if A is completely continuous, its continuous spectrum must be empty. If M is a characteristic space of A corresponding to the eigenvalue A, then Af = Af for all f E M. Consequently, if A # 0, A is a multiple of the identity operator when restricted to M. Therefore, M has to be finite dimensional if A is completely continuous (see Exercise 8.1). Q.E.D.

Lemma 8.2. Every completely continuous operator A on H can be written in the form A = VK,

K = e A ,

where K is a positive-definite completely continuous operator, and V is an isometric operator defined on the range of K.

8. Completely Continuous Operators and Statistical Operators

379

Proof. Since A is completely continuous and A* is continuous (see Chapter 111, Theorem 2.7), A * A is completely continuous. Hence, A*A, and consequently also K = 1 / A A , have a pure point spectrum, which has no accumulation points, with the possible exception of zero (see Theorem 8.3). This implies that K is a completely continuous operator (see Exercise 8.3). Let us define now the operator V on the range W Kof K by setting for every g E g K Vg = Af,

g

= Kf E 9 ? K .

The operator V is uniquely defined for everyg E B K .In fact, ifg = Kf Kfl for f # f l ,we have IIAf--flll2

=(f-f1lA*A(f-f1)) =(f-f1lK2(f-f1))

=

=o,

which'implies that Af = Afl . It is easy to check that V is linear. From the relations

II vg (I2

=


=

(Kfl K f > = 118112,

we infer that V is isometric.

= (f

I K 2 f)

gE9?K,

Q.E.D.

We are now ready to investigate the relation of completely continuous operators in general to operators of the form I f ) (g I.

Theorem 8.4. An operator A on any infinitely dimensional Hilbert space 2 is completely continuous if and only if there are orthonormal systems {el , e2 ,...} and {el', e2',...} in # and positive numbers A, , A2 ,... -+0,such that A can be reduced to the canonical form (8.3)

A

=

c I ekl)

(ek

I

3

k

where, in case that the sum is infinite, we define

and the set {Ak} consists of all the eigenvalues of

a.

Proof. According to Lemma 8.1 and Theorem 8.1(c), an operator of the form

IV. The Axiomatic Structure of Quantum Mechanics

380

is completely continuous. Hence, by Theorem 8.2, any operator which is the uniform limit of operators (8.5) is also completely continuous. Conversely, assume that A is a completely continuous operator. According to Lemma 8.2, A = V K , K = 1 / A A , where K is positive definite and completely continuous. Hence, in view of Theorem 8.3 and Theorem 6.4 in Chapter I11 the spectral resolution of K is

where hl', h2',... are the eigenvalues of K; in addition, if K has infinitely many eigenvalues, the convergence of the series in (8.6) is uniform. By selecting an orthonormal basis in each of the characteristic subspaces Mj (which are finite dimensional by Theorem 8.3), we arrive at an orthonormal system {el , e2 ,...} in X . Let us set Ak = hj' if ek E M i . Assume that we are dealing with the nontrivial case when K has infinitely many nonzero eigenvalues

Since V is isometric, 7L

A

(8.7)

Setting

ek' =

=

VK

= u-lim n-m

C

hk

V I e k ) (ek I

.

k=l

Vek , we can easily check that (V I ek)

<ek

I)f

= V((ek

If> e k )

=

(ek

If) ekl,

i.e., V 1 ek)(ek I = I f?k')(ek I. Inserting this result in (8.7), we arrive at (8.3). Moreover, the isometry of I/ implies that {el', e2',...} is an orthonormal system since {el , e2 ,...} is orthonormal. Q.E.D.

8.2. THETRACE OF

A

LINEAR OPERATOR

In order to be able to study systematically different classes of completely continuous operators, we need the concept of trace.

Definition 8.2. A linear operator A defined on the separable Hilbert space X is said to be of the truce class if the series x k ( e k I Ae,) converges and has the same value in any orthonormal basis {ek} of X . The sum Tr A

=

C (ek I Ae,) k

is called the truce of A.

38 1

8. Completely Continuous Operators and Statistical Operators

We shall investigate the existence of the above trace in the context of future theorems. But first we would like to remark that every linear operator A on an n-dimensional Hilbert space 2f has a trace; namely, if {el ,..., e,} and {elf,..., en’} are any two orthonormal basis, the sum in (8.8) is finite and the same in both bases:

c n

n

<ek

I Aek)

=

k=l

c

<‘?k

I ei’><ei’ I Aej’><ei‘

I ek>

i.j.k=l n

R

n

(e,’

=

I Ae,’).

i=l

We note that in the infinite-dimensional case we cannot immediately establish, using the same method, that the sum in (8.5) does not depend on the basis employed. The reason is that in dealing with multiple infinite sums the interchange of the order of summations is legitimate only if we know that the convergence of the multiple sum is unconditional. We shall see that this is the case when A is positiwe definite,i.e., when (f I Af) 2 0 for all f E H.

Lemma 8.3. If A = K*K, where K is a linear operator defined on the infinite-dimensional separable Hilbert space S, then the series

has the same sum in all orthonormal bases {el ,e2 ,...} on S; thus, if that sum is finite, T r A exists. Moreover, A is bounded and 11 A 11 T r A.

<

Proof. Let {el, e2 ,...} and {elf, e2f,...} be any two orthonormal bases in A?. The double series m

(8.10)

is a series with positive terms; it converges if and only if the three sums

IV. The Axiomatic Structure of Quantum Mechanics

382

in (8.11) are equal (see, e.g., Randolph [1968, p. 162, Theorem 3]),

c (c

(8.1 1)

m

m

i=l

k=l

I Kek)l')

= k=l

(g

I Kek)12)

i=1

According to Parseval's relation [(4.15) of Chapter I], m

c (K*ei' I ek)(ek 1 K*ei') m

I(ei'

I Kek)!'

=

k=l

= 11

K*ei 112,

k=l

c m

m

I(ei'

I Kek)12

=

I=1

(Kek

1 e,'>(ei' I K e k )

= 11 K e k

1 '-

i=1

Hence, from (8.1 I), (8.9) converges if and only if (8.10) converges, and m

m

c

m

c I( Kej' (I2, m

m

=

I(ei' I Kej')lz =

i.j=1

j=1

thus proving that T r A exists. By using the Schwarz-Cauchy inequality, which yields I(Ked I Ke,)I II Ke, II II Kek II, we get

m

m

m

c(

= k=l

<

m

m


1 ei>(ei I Aek>)

11 Aek

=

i=l

119 '

k=l

since the interchanges of the order of summation are permissible on account of the unconditional convergence of a series with positive terms. From the above we obtain (8.12)

II Ae, 11'

< (Tr 4'.

Given any nonzero vector f E X , we can adopt

f/llfll

to be the first

8. Completely Continuous Operators and Statistical Operators

383

element el of an orthonormal series, and, therefore, we have by (8.12)

II Af II < ( T r 4 l l f 11. The above inequality shows that A is bounded and I( A

(1

< Tr A. Q.E.D.

8.3. HILBERT-SCHMIDT OPERATORS In Chapter V we shall encounter integral operators which belong to a special class of completely continuous operators called the HilbertSchmidt class.

Definition 8.3. A completely continuous operator, having the canonical decomposition (8.3), is called a Hilbert-Schmidt operator if (8.13)

Theorem 8.5. A bounded linear operator A defined on the (separable") Hilbert space iP is a Hilbert-Schmidt operator if and only if A*A is of trace class. h o o f . If A is a Hilbert-Schmidt operator, then by (8.3) (8.14)

(ek

I

=

k

c 11

Aek

11'

k

=

c k

In the case where the orthonormal system { e k } is not already a basis, we can complete it to an orthonormal basis in A? by adding to it an orthonormal basis {e,"} in the orthogonal complement of the closed linear subspace [el, e2 ,...I spanned by {ek}.Since Ae," = 0, and therefore also (e: I A*Aei") = 0, we have by virtue of (8.14) and Lemma 8.3 that the trace of A*A exists and

(8.15)

Tr[A*A]

=-

hkZ.

k

Consider now any bounded linear operator K for which Tr(K*K) exists. Since any normalized vector f E A? can be chosen as the first element of an orthonormal basis in A?,we have

We give the proof only for the separable case, but the result is true in general (see Schatten [19601).

IV. The Axiomatic Structure of Quantum Mechanics

384

Thus

+y,

Assume that A is a bounded linear operator with Tr[A*A] < and let us prove that A is a Hilbert-Schmidt operator. Since the proof is trivial when X is finite dimensional, we shall consider only the case of operators A on infinite-dimensional separable Hilbert spaces. Choose an orthonormal basis {el , e2 ,...} in 2, and define the linear operators n

A,

=

I Ae,)(e, I ,

n

=

,... .

1,2

k-1

Thus, 11 A - A , 11 + 0 when n + co. Since A, , A, ,... are completely continuous operators, it follows from Theorem 8.2 that A is also completely continuous. It is easy to reestablish (8.15) by using the earlier procedure, and thus prove that A is a Hilbert-Schmidt operator.

Q.E.D. For any Hilbert-Schmidt operator A, the quantity

exists, and is called the Hilbert-Schmidt norm of A. As a matter of fact, using first the triangle inequality in &? and then the triangle inequality in 12 spaces we obtain (see also Theorem 8.10)

11 %A1

+

ad2

112

=

(1

+

I/(alAl

k

< (F [I1

= I a1

@lAlek

I II A1 112

11

ek

+ 11

+I

a2

112)1'2

aZAZek

111')

I II A2 112

1/2

8. Completely Continuous Operators and Statistical Operators

385

Moreover, according to (8.16),

I1 A II < (TrL4*441)1’2= II A 112 which shows that 11 A = 0 implies A = 0. (8.18)

8.4. THETRACE NORMAND

THE

9

TRACE CLASS

Let us consider now a completely continuous operator A, having the decomposition (8.3), for which (8.19)

If the above series is infinite, its convergence implies that hk -+ 0 when

k + + 00. Consequently, Ak2 < A, for sufficiently large values of k. Hence,

Ckhk2 converges when Ck

converges. This shows that any completely continuous operator A satisfying (8.19) is a Hilbert-Schmidt operator.

Lemma 8.4. If A is a completely continuous operator on the Hilbert space Z having the canonical decomposition (8.3), then (8.20)

where the supremum is taken over all choices of countable orthonormal systems { fi}and {gi}.

Proof. Using (8.3) we get

Applying Bessel’s inequality (see Chapter I, Lemma 4.1) to the orthonormal systems (fi}and {gi}we obtain

c i

(8.21)

I(fi

I e,’>12

< 11

ek’

/I2

=

1,

IV. The Axiomatic Structure of Quantum Mechanics

386

In proving Lemma 8.3, we used theorems on the convergence of double series. In view of (8.21), we can apply those theorems to justify the reversal of the order of summation in the following computation:

,<

c

Ak

k

Since we have

c

I(ei

Aek>l

=

9

k

k

we conclude that (8.20) IS . true.

Q.E.D.

We shall now show that the functional (8.22)

on the class of Hilbert-Schmidt operators satisfying (8.19) is a norm, called the trace norm. Since for every series with positive terms

we obtain

II A 112

(8.23)

< II A Ill

Hence, )I A ) I = 0 implies that )I A For any complex number a,

)I2

=

*

0, and consequently A

Finally, by making use of Lemma 8.4 we obtain

II A,

+

A2

Ill

c < sup c =

SUP

Kfi

a

i

=

II A1 111

I
+ A2)Zi)l I A,g,>I + SUP c I
i

+ I1 A2 Ill

*

= 0.

8. Completely Continuous Operators and Statistical Operators

387

Thus, I( * (II satisfies all the conditions of a norm on the class of HilbertSchmidt operator for which (8.19) is true. Theorem 8.7 states that this class is identical to the trace class of Definition 8.2 when X is a separable Hilbert space. T o prove it, we need the following result.

Theorem 8.6. point spectrum.

Any self-adjoint operator of trace class has a pure

Proof. Any self-adjoint operator A of trace class can be written as a difference of two positive-definite self-adjoint operators A, and A, , A

=

I

A dE,

=

A(+)- A(-)

R'

(8.24) A(+)=

AdE,,

A'-)

Ilo,+d

=

-

1

A dE, 9

(-m, 0 )

both having a finite trace (see Exercise 8.6). Consequently, it is sufficient to prove the theorem for positive-definite operators. Assume that A is positive definite, and choose any A, > 0. Let < A, be n numbers greater than A, such that A, < Az < (8.25)

.'.,

El, f El',

EA, f EAo 9

EAnf E&-'

Since EAk- EAk-l# 0, there are n normalized vectorsf, (EAk

- E,Ak-l)

f k

=f

k

9

'

=

-

,...,f ,

such that

l9***9

According to the spectral theorem we have (8.26)

(fk

I Af k>

=

Iw

A d(fk

I EAfk>

By completing f , ,...,f , to an orthonormal basis f , ,fi ,..., in &+we get, using the positivity of A and (8.26), 7l

Tr A

=

(fk k

I Af k> 2

(fk

I Af k > >, nAO

*

k=l

The above inequality shows that the number of points A,

,..., AR for which

IV. The Axiomatic Structure of Quantum Mechanics

388

(8.25) is true is smaller than T r A/A, , i.e., the spectral function En can change only at a finite number of points in ( A , , +a).Thus, A has exclusively a point spectrum in ( A , , +a),and since A, > 0 can be chosen arbitrarily small, A has a pure point spectrum. Q.E.D.

Theorem 8.7. A completely continuous operator A on the Hilbert space A? is of trace class if and only if it is a Hilbert-Schmidt operator for which T r [ m ]

=

c he <

+oO,

k

where

are the numbers appearing in the decomposition (8.3) of A.

Proof. The positive numbers hk occurring in the decomposition

(8.3) of a Hilbert-Schmidt operator A are the eigenvalues of 4Hence, if A* = A, we must have ek’ = ek when h k E SA, and ek’ = -ek when --hk E SA. In the decomposition A

= A(+)- A(-)

A(*)=

9

c

1 ek>

(ek

1)

*A&

the operators A(*)are positive definite. Hence, we can write A(?=)

d-d-,

and apply Lemma 8.3 to infer that

exists if

x k hk

< + 00.

Consequently Tr A

=

T r A(+)- Tr A(-)

also exists. The more general case of nonsymmetric A can be reduced to the above case by setting A

= A,

+ iA, ,

A,

=

$(A

+ A*),

A,

=

-(i/2)(A

-

A*).

In fact, the operators A, and A, are obviously symmetric HilbertSchmidt operators, for which we get

using the properties of the trace norm 11

*

(I, .

8. Completely Continuous Operators and Statistical Operators

389

Consider now a self-adjoint operator A of trace class. According to Theorem 8.6, A has a pure point spectrum. Hence, (8.27)

A

=

1 hdEAA

=

08'

C

hEA({h}),

A€SA

where the convergence of the above series is uniform if the series is infinite. Each characteristic subspace MA corresponding to a nonzero eigenvalue h E SA has to be finite dimensional; in fact, if that were not true for some A # 0, we would have Tr[EA({h})] = +coo, which would imply that the trace of A does not exist. Choosing an orthonormal basis in each characteristic subspace MA, A E S A , we obtain, by taking the union of all such bases, an orthonormal basis { e l , e2 ,...} in 2'.Since each MAis finite dimensional, (8.27) yields (8.28)

where hk' is the eigenvalue of A for the eigenvector ek . In taking the trace of A, m

m

we can use any orthonormal basis, and therefore any order of el , e, ,... in the above series. Hence, the series Ckm,lhk' converges unconditionally, and consequently it converges absolutely (see Randolph [1968, Section 3.91):

c m

< +a,

I

1.

k=l

It follows from (8.28) that A is a completely continuous operator. Since the convergence of Ckm,lh k implies the convergence of Aka (Ak, h k for sufficiently large k), we easily deduce that A is a HilbertSchmidt operator. If A is any operator of trace class, then obviously A* is also of trace class:

xr=l

<

m

Tr A*

=

c m

(ek

I A*ek)

k=l

=

(ek

I Aek)*

= (Tr A)*.

k=l

+

Hence, A, = &A A*) and A , = - ( i / 2 ) ( A - A*) are self-adjoint operators of trace class, and consequently they are Hilbert-Schmidt operators for which 11 A, [ I 1 < co and I[ A, < co. This implies that

+

+

IV. The Axiomatic Structure of Quantum Mechanics

390

= A, + iA, is also a Hilbert-Schmidt operator. Moreover, since 11 * ( I as defined in (8.22) is a norm,

A

Tr v m ' = /I A

i.e.,

Ck

converges.

11

< I1 A1 + I/ 4 1 1 1

1 1 1

< SW,

Q.E. D.

8.5. STATISTICAL ENSEMBLES AND

THE

PROCESS OF MEASUREMENT

When we were discussing in $1 the concept of preparatory measurement in quantum mechanics, we saw that for n compatible observables represented in the Schroedinger picture by the operators A, ,..., A, it is usually not the case that we can prepare at an instant to n absolutely accurate simultaneous values A!'), ..., A:) for A, ,..., A, . Instead, in general, a preparatory measurement yields the weaker result that the simultaneously prepared values of A, ,..., A, are within an n-dimensional Borel set B. Therefore, after a preparatory measurement we have to assign to the system the family of all states Y(t)which satisfy (1.26). This family contains, in general, more than one element. However, in practice, instead of dealing with such a family of state vectors, one prefers to make some reasonable statistical assumption based on some additional knowledge or insight which one might have regarding the ensemble of systems used in the experiment. In order to simplify the discussion from the mathematical point of view, let us assume that A, ,..., A, represent a complete set of observables with pure point spectra, and that for every A = (A, ,...,A,) E SA1 x x SAm we can choose a normalized vector e, such that A,e, = A,e,, K = 1,..., n. Consequently, the family of all vectors e, , A E SA1 x -..x SA-, constitutes an orthonormal basis in X . Assume that the preparatory measurement which has been carried out on the system at to provides us with the information that the prepared simultaneous values of A, ,..., A, are at to within the Borel set B, where B C SA1 x x SAn. Let us further simplify the considerations by assuming that B is finite. Then the prepared family of states consists of all the states Y(t)satisfying

---

yA(tOn(to)

= eA

9

B,

and any linear combination of such states. Suppose that we submit a large ensemble of identical systems to the described preparatory procedure, and that we know that in this ensemble (1o&.uA)% of the systems have at time to the simultaneous values A, ,..., A, , (A, ,..., A,) E B, for A, ,..., A, . As we already know, the theory predicts that the mean value of the outcome of a determinative measure-

8. Completely Continuous Operators and Statistical Operators

ment at time t A, is

> to of an observable represented

(8.29)


39 1

by a bounded operator

AOYA(t))

if the measurement is carried out only on that part of the ensemble consisting of particles which at to had the simultaneous values A, ,..., A, for A, ,..., A, , respectively. Hence, the mean value of A, for the entire ensemble is (8.30)

Let us introduce for each instant t the operator (8.31)

dt) =

WAE'f'A(t)

I

EyA(t) =

9

A€B

yA(t))
which is called the density operator or the statistical operator of the considered ensemble. From the orthonormality property of the set of all vectors !PA(t), A E SA1 x .-.x SAn, at any fixed t , it follows immediately that (8.32)

em (4 =

c

.

AESA1X. * .xsAn

The above relation can be written in the form (8.33)


provided that A , p ( t ) is of trace class. Theorem 8.8 shows that this is indeed the case.

Theorem 8.8. Suppose {e, , e2 ,...} is an orthonormal basis in X . If (8.34)

A

=

cI

ek>

(ek:

1,

k

where Ak (8.35)

0 and Tr[A]

=

hk

< +03,

k

and if A, is a bounded operator, then A,A is of trace class. The above theorem can be easily derived from Theorem 8.7 by showing that the operators (A,A)*A,A and (AA,*AoA)lIaare of trace class. As a matter of fact, it is sufficient to show that (AA,*A,A)lI2 is of

IV. The Axiomatic Structure of Quantum Mechanics

392

trace class, since it is easy to see that in that case AAo*A,,A is also of trace class. T o establish that (AAo*AoA)l12is of trace class, note that (AAo*AoA)112= (AAO*AOA)l” (AA0*AJ)”4.

Then it follows from Lemma 8.3 that the sum

has the same value in any orthonormal basis {el , e2 ,...}. Let el , e2 ,... be the vectors appearing in (8.34). We have

and consequently

Thus, (AAo*A0A)lI2 is of trace class and Theorem 8.8 is established.

8.6. THEQUANTUM MECHANICAL STATEOF

AN

ENSEMBLE

Theorem 8.8 establishes that (8.32) can be written in the form (8.33) when A,, is bounded, since p ( t ) is a positive-definite operator with a pure point spectrum and of finite trace:

In the experiment described at the beginning of Section 8.5, in which we were concerned with a statistical description of an ensemble of identical independent systems, the relevant information about the outcome of any measurement can be obtained from the operator-valued function p ( t ) ; thus, it is justifiable to state that p ( t ) represents the state of the ensemble in the Schroedinger picture. In the light of the above discussion the following postulate is a natural generalization of (8.31).

8. Completely Continuous Operators and Statistical Operators

393

Axiom E. Consider a statistical ensemble of identical systems, and let &' be the Hilbert space associated with each system in the ensemble (Axiom 0, $2). Any state of the ensemble can be represented in the Schroedinger picture by an operator-valued function p ( t )

(8.37)

and {Yl(t),y/z(t),...) is an orthonormal system of state vectors whose time dependence is governed by the relation

where U(t, to) is the evolution operator of the systems in the ensemble. We note that the uniform limit in (8.36) exists on account of (8.37). This can be seen easily if we introduce

by means of the above strong limit, which obviously defines a linear operator. Since

c

k=n+l

+

c m

W

l P - i w k E Y k I < k=l

wkIIEYkII=

wk

k=n+l

converges to zero when n + co,p is the uniform limit of the partial sums in (8.36). An ensemble in the state (8.36) is viewed as consisting of systems in the states !Pl(t),!P2(t),... in the respective proportions w l , w2 ,... . If wk = 1 for k = k, and therefore wk = 0 for k # k, , the described state of the ensemble is one in which each of the systems is in the state y k , ( t ) . In that case p(t) is called a pure state of the ensemble. If a few of the W k are nonzero, p ( t ) is called a mixed state or a mixture of all the singlesystem states v k ( t ) for which wk > 0. It is easy to derive from (8.36) and (8.38)

394

IV. The Axiomatic Structure of Quantum Mechanics

which shows that

From the above relation we get

+

In the limit n + coy we obtain (8.39) by using the basic rules for dealing with uniform limits (see Exercise 8.7). In view of (8.33) and of Born’s correspondence rule in Definition 1.3 it is natural to assume that if a determinative measurement of a bounded observable A, is carried out at time t on each system in an ensemble in a state p(t), then the mean value of A, will be Tr[p(t) A,]. Thus, as a generalization of (1.27), we can state that the ratio of the number vAJ,...,A,(B) of systems having at t simultaneous values of A, ,..., A, within the set B, to the total number N of systems in the ensemble, should be equal to

If a preparatory procedure on the systems of an ensemble does not leave the ensemble in a pure state, i.e., if there are more than one linearly independent state vectors satisfying (1.26) in the preparatory measurement of a general type described in Definition 1.4, then usually the principle of equal a priori probabilities is invoked in order to assign to the ensemble a state. According to this principle, the proposed state (8.36) of the ensemble is the one in which equal nonzero weights are assigned to the single-system states Yk(t)satisfying (1.26), and zero weights to those states not satisfying (1.26). Thus, if Y,(t),..., YJt) are all the mutually orthogonal states satisfying (1.26), i.e., those states for which

8. Completely Continuous Operators and Statistical Operators

395

or equivalently

then obviously

Theorem 8.9 provides a simple algebraic criterion for distinguishing between pure states and mixed states of a statistical ensemble.

Theorem 8.9. only if p ( t )

=

The state p ( t ) of an ensemble is a pure state if and

pz(t).

Proof. If the ensemble is in a pure state in which the states of each system in the ensemble is Y(t),the statistical operator p ( t ) is at each instant t the projector onto Y(t),and therefore (8.41)

PV)

=

Conversely, assume that (8.41) is true. If we write p ( t ) in the form (8.36) and compare it with the following expression for p 2 ( t ) (see Chapter 111, Exercise 5.10)

we obtain wk2 = wk . Thus, wk can assume only the values 0 and 1. This, in conjuction with the condition wk = 1, shows that wk = 0 except for one value k, of k at which wk, = 1. Q.E.D.

x;=l

We have arrived at the concept of "state vector in the Schroedinger picture" by considering the possible values of all states Y(t)of the system at a given instant. In this same manner we arrive at the concept of statistical operators by considering the possible values of a state p ( t ) of an ensemble at instant t.

Definition 8.4. Let 2 be a separable Hilbert space associated with a quantum mechanical system. A linear operator p on 8 is a statistical

396

IV. The Axiomatic Structure of Quantum Mechanics

operator (density operator) if it can be written in the form

c wkE,, c w,E,,, n

(8.42)

p = u-lim n+m

m

=

k=l

k=l

where the real numbers wl, w2 ,... satisfy (8.37) and {Yl , Y2,...} is an orthonormal system of state vectors in &. Any statistical operator p is obviously a positive definite self-adjoint operator. Hence, according to Lemma 8.3, T r p exists if the sum Ci (ei I pei) is finite in some orthonormal basis {el , e2 ,...}. We can Y2,...} to an orthonormal basis obtain such a basis by completing {Y1, in 2.Thus, we have m

(8.43)

m

Trp=C(YkIpYk)=CWk=1. k=l

k=l

Conversely, Theorem 8.7 tells us that in the absence of superselection rules any positive self-adjoint operator of unit trace is a density operator. EQUATION IN LIOUVILLE SPACE 8.7. THEVON NEUMANN Let us concentrate now on the case of time-independent Hamiltonians

H ; i.e., in physical terms, on the case where each element of the ensemble, whose density operator is described (in accordance with Axiom E) by p(t), is isolated from its environment so that its total energy is conserved. In that case U ( t , to) in (8.39) is given by (3.2). Differentiating the right-hand side of (8.39) formally, we arrive at the zlon Neumann equation:

This equation bears a remarkable formal resemblance to the Heisenberg equation (3.25), but there are both mathematical and physical distinctions: in (3.25) A ( t ) is in general an unbounded self-adjoint operator representing an observable in the Heisenberg picture, whereas in (8.44) p(t) is a (self-adjoint) trace-class operator of unit trace and of bound /I dt)ll = sup wk < 1 , k= 1,2,

...

and this p(t) represents a (mixed or pure) state in the Schroedinger picture. Hence we have to study the precise meaning of the limit on the left-hand side and of the commutator on the right-hand side of (8.44), respectively, on their own rather than as corollaries to the results of $3.4.

8. Completely Continuous Operators and Statistical Operators

397

Such a study is greatly facilitated by the fact that the family Y ( 2 ) of all Hilbert-Schmidt operators on 2 (called in mathematical literature the Hilbert-Schmidt class* and in physical literature the Liouville space) is a Hilbert space, and that as a trace-class operator p ( t ) belongs to 9(&') by Theorem 8.7.

Theorem 8.10. The family 2 ( X )of all Hilbert-Schmidt operators on a given (separable) Hilbert space 2 is itself a (separable) Hilbert space (called the Liouville space over 2)with respect to the inner product (8.45)

(A, I

A2)2 =

Tr(A,*A,),

A, 9 A, E

=%m.

Proof. The argument at the end of $8.3, which establishes the fact that (8.17) is a norm, simultaneously establishes that 9 ( X )is a linear space. Hence, if A , , A, E 9(2) then A , + A , , A , iA, E Y ( X ) , and therefore

+

(8.46) Al*A,

=

+ A,)*(A, + A,) + (2i)-l(A1 + iA,)*(A, + iA2) - 8(1 - i)(Al*A, + A,*A2)

*(A,

is of trace-class by Theorem 8.5, so that (8.45) exists. The linearity features of the trace operation, (8.47)

Tr(a,A(l)

+ U ~ A (=~ a,) )T r A(') + a2T r A ( 2 )

that is evident fromt (8.8), immediately leads to the conclusion that (8.45) has the properties (2)-(4) in Definition 2.1 of Chapter I, whereas property ( I ) follows from (8.18). To establish that the Euclidean space 2 ( X )is complete and separable if % is a separable infinite dimensional Hilbert space,: we introduce *In Definition 8.2 we had restricted ourselves to separable Hilbert spaces, which are of exclusive interest in quantum mechanics, but that definition can be extended to the nonseparable case (see Schatten [1960]). + T h e trace-class is often denoted by 11(#), the Hilbert-Schmidt class by I , ( Z ) , the class of compact operators by I m ( Z the ) , Cx-algebra a(%)of Theorem 1.1 in We then have a,(%) C I,(&) C .*. C I,(*) C I(#) where Chapter I11 by I(&). al(.X), 12(%), and Im(%') are Banach spaces (and, moreover, so-called minimal norm ideals in I ( Z )= a(&)) with respect to the trace-norm 11 . 1 , , Hilbert-Schmidt norm 11 * /Iz , and operator bound 11 . 11, respectively). However,among all I i ( Z )i,= 1, 2,..., co (where appropriate classes and norms are defined for i = 3, 4, ...),only B , ( Z ) = Y ( Z ) is a Hilbert space (see Schatten [1960] for details).

* See Schatten [1960] for the case when Z is nonseparable.

IV. The Axiomatic Structure of Quantum Mechanics

398

an orthonormal basis { e l , e, ,...} in Z , and note that for any Y A, E 9(2)

4

m

(8.48)

< A 1 I4)2

=

m

m

I A I * A z ~ E=) 1 C (ei I Ale~)*<eiI A z ~ E ) .

C

E = l i=l

k=l

Consequently, the mapping assigning to A E 9 ( X ) the element Aek)}i,kin the separable Hilbert space Z 2 ( d ) = P(m) @ Z2(co) (see Theorem 4.7 in Chapter I and Theorems 6.9 and 6.10 in Chapter 11) of matrices (aik} with inner product {{(ei j

into Z2(co2). T o see that this is an isometric transformation of 9(&) transformation is actually unitary note that by Theorem 8.5 A E 9(&) if and only if m

(8.49)

l(ei

1 Aek)I2

= Tr(A*A)<

co.

i.k=l

Thus 9 ( X ) is isomorphic to Z2(co2), and therefore it is a separable Hilbert space. Q.E.D. I n physical literature operators acting on a Liouville space 9(&) are called superoperators (whereas in mathematical literature either the term operator is retained, since after all 9 ( Z )is a Hilbert space itself, or the term transformer might be applied). As examples of superconsider operators acting on A E 9(&)

u* = u-1,

(8.50)

UA

=

(8.51)

HA

= fi-l(HA - A H ) ,

UAU*,

H

= H*.

The superoperator U has domain 9” = A?(%) and is unitary. Indeed by (8.46) and (8.1 1) the infinite double series in (8.48) converges unconditionally, so that by setting A = A,*and B = A, we conclude that m

c1 m

=

m

m

m

i=l k = l

c m

( e , I Be,)(e, I Aei)

=

k=l

(ei I BAe,) = T r ( B A )

399

8. Completely Continuous Operators and Statistical Operators

for arbitrary A, B (UA I UA),

E 9(Z). Therefore,

= Tr[U(A*AU*)] =

Tr[(A*AU*)U]

=

( A I A), ,

and since obviously U-lA = U*AU, we also have that A?u = 9 ( Z ) , which establishes the unitarity of U. Similarly, assuming that in (8.51) H is some bouunded self-adjoint operator with gH= 2,(8.52) yields W

l

I HA,),

= Tr[A,*(HA, = Tr[(HA,)*A,

-

A2H)I

-

HA1*A2]

= h(HA,

I A,),,

so that H is a self-adjoint superoperator on 9 ( Z )On . the other hand, if H is unbounded we run into technical problems with domains of definition, so that we have to be more precise as to the meaning of H in (8.51). T h e following result illucidates, however, the relationship between the domains, spectra, and spectral measures of the operator H on A@ and the superoperator H on 9 ( Z ) .

Lemma 8.5. If H is any self-adjoint operator in 2,then the superoperator H acting in 9(Z)as follows (8.53)

HA

= h-l[H,

A]** = h-'(HA - A H ) --

and with domain gHconsisting of all A E 9 ( Z )for which H A , AH 2 ( A @ ) , is self-adjoint in the Liouville space 9 ( Z )furthermore, ; SH =

] A'

-

h

x"

: A', A" E

sq,

SpH =

] A' -fi A"

: At, A" E

E

s*q ,

and the action of the spectral function EAH of the superoperator H on any A E can ~ be expressed ~ in terms of the spectral function EAH of the operator H by means of the following spectral integral" (8.54)

If we denote by h-lim the limit in the Hilbert-Schmidt norm on 9 ( Z ) (so that A = h-lim A, means that 11 A - A, 1, + 0 as n + co), then we easily deduce from Theorems 3.1 and 8.4, Lemma 8.5, and the properties of (8.50) that the following is true (see PrugoveEki and T i p [1975]).

+

* See 93.7 in Chapter V for a definition of spectral integrals. The proof of (8.54) is a corollary of Theorem A.2 by PrugoveEki [1972b], whereas the proof of the remainder of Lemma 8.5 follows from the results in Section 3 of PrugoveEki and Tip [1975].

IV. The Axiomatic Structure of Quantum Mechanics

400

Theorem 8.11. If H is any self-adjoint operator in 3 and Utis the superoperator that acts on all A E 9(3) as follows, U t A = e(i/A)HtAe-(i/h)Ht, t E R1,

(8.55)

then Utis unitary on 9(3) and Ut,+t, = ut,ut,’

(8.56) (8.57)

U,A

=A =

iHA

(8.58)

t , ’ t , E R’,

h-lim UtA,

=

A E 9(#),

t+O

h-lim(U,A - A)/t. t+O

We recall how Theorem 3.1 was used in clarifying the meaning of the Schroedinger equation (3.1) and we observe that the same arguments lead to the conclusion that U(t, t o )

=

Ut-,,

=

UtUG1

leaves 9Hinvariant, and that (8.59)

A(t

d A ( t) = h-lim idt

+At)

-

A(t)

At

At+O

=

HA(t)

whenever A(to)E gH and A(t) evolves in accordance with (8.39): (8.60)

4 t ) = U(t, t o ) 4

0

)

=

W ’t o ) 4

0

)

U*(t, to).

When we now compare (8.44) with (8.53) and (8.59) we immediately uncover the mathematical meaning as well as the physical significance of von Neumann’s equation: (8.44) is in fact a Schroedinger equation on Liouville space, which consequently governs not the time evolution of state vectors (i.e., merely of pure states), but that of density operators (i.e., of either pure or mixed states)! ON SPECTRAL REPRESENTATION SPACES 8.8. DENSITYMATRICES

In quantum mechanics actual computations of quantities describing the behavior of a given system are performed in some spectral representation space L2(Rn,p ) of a complete set of observables {A, ,..., A,} for that system. Such spaces were introduced in the abstract in Definition 5.2, and concrete realizations of particular importance were encountered in 97 in the form of configuration, momentum, and angular momentum representation spaces.

8. Completely Continuous Operators and Statistical Operators

401

The same observation stays true in quantum statistical mechanics by virtue of the identification of the Liouville space 9 ( Z )over Af = L2(Rn, p) with the Hilbert space L2(Iw2",p x p).

Theorem 8.12. T o each Hilbert-Schmidt operator A in the Liouville space 9(Z)over Z = L 2 ( I w m , p) there corresponds a unique h, EL*(FP, p x p) such that (8.61)

(f I A g )

=

J

f*(4 h A ( X ' , 4g ( 4 dcL(x') dcL(x">

R2"

for all f, g €L2([Wn,p), and the mapping A t+ h, is a unitary transformation of 9 ( L z ( R n , p)) onto L2(Iw2",p x p). , are functions representing the Proof. If ek(x), ekr(x)E L ( , ) ( R ~ p) elements ek , ekr EL~(IW~, p) in the canonical representation (8.3) of A E 9(Z), then we observe that by (8.14) n

m

and if we set

then (8.62)

Consequently, by the Riesz-Fischer Chapter 11), (8.63)

hA(x', x")

=

theorem (Theorem 4.4 in

1.i.m. hA,(x', x") n+m

and since f ( x ' ) g * ( x " )E L ( ~ ) ( Rp~ "x, p), we deduce from (8.62) that (8.61) is satisfied.

IV. The Axiomatic Structure of Quantum Mechanics

402

The function hA(x’,x”) is not uniquely determined by A since, in general, the canonical decomposition (8.3) is not unique [e.g., that is the case if some A, is a degenerate eigenvalue of some A = A * E 9(2)]. However, if hA’(xt,x”) also satisfies (8.61), then we have [ f ( x ’ )g*(x”)]*[h,(x’, x”) - h A ’ ( X ’ , x”)] dp(x’) dp(x”) = 0

for all f,g E 2 = L2([w”,p), and since by Theorem 6.9 in Chapter I1 the linear manifold spanned by all functions f ( x ’ ) g*(x”) is dense in L*(R2,, p x p), we conclude that hA(x’,x”) and hA’(x’,x”) have to be almost everywhere equal with respect to p x p, and therefore they represent one and the same element of L2([w2,,p x p). Thus A eh, is defined for all A E 9(3), and it obviously represents a linear mapping. T o prove that this mapping is an isometry we note that by (8.14) n

Tr(A*A)

=

Xk2 = lirn Tr(An*An)

lirn

n-m

k=l

n-w

and consequently, by (8.17), (8.64)

II A 112

=

[l I

112

~ A ( x ’ ,x”)I2

4 4 ~ ‘C)( x ” ) ]

= I1 h~ l

l .~

R2n

The unitarity of this same mapping is finally established by noting that to any element h(x’,x”) of L(2)([w2”, p x p) there corresponds a unique operator A such that when h, is replaced by h in (8.61) the outcome is an equality valid for all f,g E L2(1Wn,p) (see Exercise 5.3 in Chapter V). Q.E.D. If p E 9(L2([w2,))is a density operator, then (8.3) assumes the form (8.42), and we shall write (8.65)

(x‘ I p I x”>

n

= h,(x’, x”) = 1.i.m. n+m

2 Yk(x’)wkYk*(x”). k=l

Following a long-established tradition in physics literature, we shall call the function (x’ I p 1 x”) a density matrix on the spectral representation space L2(1Wn,p)-albeit this function can be identified with a matrix only in the case where the spectra of A, ,..., A, in Definition 5.2 are all point spectra so that the support S of the measure p is a discrete set.

8. Completely Continuous Operators and Statistical Operators

403

This last case, however, almost never occurs in practice. For example, in the important case of the configuration to the representation (6.8) of the canonical commutation relations (concretely, if in the considerations of 97.2 we have N = n/3 spinless particles), the Hilbert space X equals L2([w"),and therefore (x' 1 p I x") is an element of L(2)(lw2n). To understand the physical significance of the density matrix, we note that the probability appearing in (8.40) can be expressed as follows: (8.66)

Indeed, choosing an orthonormal basis e l , e2 ,... in the subspace of L2(Rn,p) onto which E A I ~ . - . J ~ (projects, B) and then completing it to an orthonormal basis in L2(Rn,p) by adding to it el', e2', ... with supp e,'(x) C B' = 08" - B, we obtain

n

through the use of theorems on the unconditional convergence of double series with positive sums [already used in (8.1 I)]. The right-hand sides of (8.66) and (8.67), however, coincide. We now set by definition (8.68)

at all X E [w" where the above series converges, and by Lebesgue's monotone convergence theorem we obtain in accordance with (8.66) the basic result (8.69)

I

P,"'*.***An(B) = Tr[pEA1*...*An(B)] = (x I P I x> dp(x), B

which represents the most general integral-form expression for the probability that a simultaneous determinative measurement of the complete set {A, ,..., A,} of observables of a system in a (pure or mixed) quantum state p would yield values in the Bore1 set B €gn.For the special case B = [w" (8.69) assumes the form of a trace formula: (8.70)

404

IV. The Axiomatic Structure of Quantum Mechanics

I t should be emphasized that in general the defining formula (8.68) of the so-called diagonal elements (x I p I x) of the density matrix (x’ I p 1 x”) is not derivable merely by setting x f = x c = x in (8.65). Indeed, (8.65) involves a limit-in-the-mean, which in general implies pointwise convergence only almost everywhere on R2” for some suitable subsequence of the partial sums of the given infinite series (see Chapter 11, Theorem 4.5), whereas the set {(d,x”) I xf = x”} might be of p-measure zero in RZn(e.g., that is the case when p is a LebesgueStieltjes measure). On the other hand, all the terms of the series in (8.68) are nonnegative and

so that (8.69) is true by Theorem 3.10 in Chapter I1 (cf. also Fatou’s lemma). Furthermore, if the equivalence class of all functions hp(xf,x”) E 15(~)(IW2n)which satisfy (8.65) contains a continuous function, then we can set by definition m

<x’ I p

I %”>

=

yk(x’)

wkyJc*(x”)

k=l

for appropriate choices of ?Pk(x), k = 1, 2, ... . Indeed, in that case by Mercer’s theorem (see Riesz and Sz. Nagy [1955], Section 98), the func, 1, 2,..., appearing in (8.65) can be chosen to tions ?Pk(x)E L ( ~ ) ( R k~ )= be continuous, and then the above series converges pointwise (and, in fact, also uniformly on compact sets). Hence, in the practically important case of continuous density matrices (x’ I p I x”) E L ( ~ ) ( R ~ ~ ) , we can obtain the “diagonal” values defined in (8.68) simply by setting xf = x” = x in the chosen continuous density matrix ( x f I p 1 x“). T h e most frequently used representations of density matrices are those in configuration and in momentum space. For example, for a system of N spinless particles possessing a total of n = 3N degrees of freedom, the frequent choice of spectral representation space is L2(Rn), and therefore the Liouville space can be identified with L2(RZn),where the measure p is in this case the Lebesgue measure on the measurable space (Rn, an).

* *8.9.

APPENDIX:CLASSICAL AND QUANTUM STATISTICAL MECHANICS IN MASTER LIOUVILLE SPACE

Classical and quantum statistical mechanics are both statistical theories describing the same body of phenomena, and it is commonly held that when both classical and quantum theory are applied to one and the

8. Completely Continuous Operators and Statistical Operators

405

same system (e.g., a Brownian particle moving in some gas, such as air) then the classical description should provide in some sense an approximation to the quantum one. Yet, the conventional mathematical formalisms for these two cases are so very dissimilar that it is not at all evident that this observation is of universal validity. Thus, whereas, as we have seen in 998.6-8.8, the states of an ensemble are represented in the quantum case by density operators p in a Liouville space, their evolution in time (in the Schroedinger picture) being governed by the von Neumann equation (8.44),in the classical case the states of the same ensemble are represented by distribution functions f ( q , p ; t ) on phase space r (where r = R2n in case of n degrees of freedom) which obey the Liouville equation (8.71)

at almost all (q, p) E I', where Hcl is the classical Hamiltonian; e.g., for time-independent potentials V(q) and N particles moving in three dimensions (8.72)

where n = 3N, q = (4, ,..., qn),p = ( p , ,..., pn), and mSffl = m3i+2 = m,i+3 , j = 0,..., N - 1. In this appendix we shall indicate how one can take advantage of the very general approach to quantum mechanics developed in this chapter to treat classical and quantum statistical mechanics in a unified manner by working in both instances on the same Hilbert space. I n the aforementioned case of n degrees of freedom this Hilbert space will be L 2 ( r ) . We shall refer to the Liouville space .5$ over L2(I')as the master Liouville space for both theories. * We represent in 9 r the classical state of an ensemble by the equivalence class of all density operators pcl E 9 r for which n

(8.73)

f (4, P) = (4, P I Pc' I q, P> = bz k=l wk

Iyk(q,

p)12,

* It is also possible to recast quantum statistical mechanics in terms of a formalism based on phase space distribution functions. For these distribution functions one can adopt the Wigner transforms of density matrices (see, e.g., Balescu [1975]). However, this transform is not positive definite, so it cannot be interpreted as a probability density on phase space (see, e.g., Srinivas and Wolf [1975]). The alternative is to use the stochastic phase space probability densities briefly described later in this section (see the review articles by PrugoveEki [1979, 1981a, b] and Ali and PrugoveEki [1980] for details and further references).

IV. The Axiomatic Structure of Quantum Mechanics

406

almost everywhere in L2(F),wheref(q, p ) is the r distribution function the superoperator describing that state. Introducing in (8.74)

it is easily verified" that if we restrict ourselves to diagonal elements, by setting q' = q", p' = p" in the (classical) von Neumann equation (8.75) for any density matrix representing an evolving state in accordance with (8.73), then (8.75) coincides with the Liouville equation (8.71). T o formulate quantum statistical mechanics on 64,, we adopt in L 2 ( r )the following representation of the canonical commutation relations (8.76)

a > Qk2 ifi apk

fS,>pk-

a

ifip %k

'

k

=

I , ..., n.

Of course, this representation is highly reducible and according to Theorem 6.4 it can be decomposed into a (countable) direct sum of irreducible representations. Theorem 6.4, however, does not claim that the decomposition is unique, and indeed an uncountable infinity of such decompositions exists. T o uncover those closed subspaces of L 2 ( r )carrying irreducible representations of physical interest, we consider positive operator-valued measures of the form

*: Further details and proofs of all the statements in this section can be found in Ali and PrugoveEki [1977a, b] and PrugoveEki [1978a-d, 1981bl.

8. Completely Continuous Operators and Statistical Operators

407

represents an element of L 2 ( r ) if y €L2(1Wn). It turns out that P,,(r) is the projector onto a closed subspace of L 2 ( r ) when (27rfi)"/2y(x) is normalized, and that this subspace L2(r,) is left invariant by the operators (8.76) that induce in L2(ry)an irreducible representation of the canonical commutation relations. T h e last fact is established by proving that the mapping (8.79)

9w * $(%PI J=

P . XI Y * ( X

exP[-(i/4

-

4)#(4d " X

Rn

supplies a unitary transformation U,, of L2(R") onto L2(ry),and that the unitary transforms UyQkU;l and UyPkU;' of the Schroedinger representation (6.8) of the canonical commutation relation for n degrees of freedom coincide with the restrictions of the operators Qk and pk in (8.76) to the closed subspace L2(r,)of L2(T)[note that L2(r,)is itself a Hilbert space!]. In general, distinct choices of y €L2(Rn)will lead to distinct spaces L2(r,,)(that have only the zero vector in common) as illustrated by the choice (8.80)

Y ( ~ ) ( X ) = ( 7 ~ P s ~ ) - ~ exp(--x2/2h2), /4

s

> 0,

providing a continuum of different subspaces L2(&8J)as the parameter s varies over (0, 00). The fact that the mapping U, defined in (8.79) is unitary for each y E L ~ ( Windicates ) that one and the same quantum state of an ensemble will be represented in Zr by an equivalence class of density operators, albeit the equivalence relation has to be defined in a way that is different from the method used in the classical case, namely to each p E ,EP(L2((r)) = Yr,given in the configuration representation by the density matrix (x' I p 1 x"), we assign a unique p y = U,,PU;~ in each of the Liouville spaces pryover L2(r,).The essential point, however, is that a (quantum) von Neumann equation (8.81)

w, P' I

%%/atI 4",P") =

(4's P' I f i P t I q", P"),

representing a counterpart of (8.75) can be introduced globally on L 2 ( r ) by setting (8.82)

fi = ti-l[fi0 + Z?, , .I,

n

fi0

=

1 p i 2 / 2 m i , fiI = V(&), i=l

and that due to the unitarity of U,, the restriction of this global equation to each pry supplies a time evolution for each (8.83)

ptY

= exp[i(A,

+ fi*)t]pY,

tE

EP,

IV. The Axiomatic Structure of Quantum Mechanics

408

that is unitarily equivalent to that supplied by the conventional von Neumann equation in the configuration representation (naturally, the self-adjointness of &k and p k guarantees that of A,, and I?, defined as operator-valued functions in accordance with Theorem 2.5, and the question of the self-adjointness of I?,, I?, can be handled with the methods of 97); furthermore, when the restriction to diagonal elements is executed in (8.75) and (8.81), a detailed comparison shows that in the zeroth order in A the two equations coincide (cf. PrugoveEki [1978b]). This last observation supplies the first indication as to the exact sense in which the classical theory approximates the quantum theory when in the later case terms of first and higher orders in A are neglected. However, the physical validity of this conclusion depends on the existence of a physical interpretation for the diagonal elements of density matrices in g r .In the classical case such an interpretation has been implicitly imposed from the very beginning by the relation (8.73) between I' distribution functions and density matrices on 9 r . T o discover the physical meaning of (q, p I py I q, p) in the quantum case, we use the mapping

+

(8.84)

Pi-+ py =

u,pu;l,

p E 9(L2(1W")), py E dp,, ,

to establish that as a consequence of the definition (8.79) of U, ,

where (k I $ 1 k) denotes the density matrix in the momentum representation, i.e. (cf. 98.8), m

W

T h e physical meaning of the integrals in (8.85) and (8.86) emerges as

8. Completely Continuous Operators and Statistical Operators

409

soon as the resulting measures (8.87)

-

pol-* * on (B) OY = Tr[pYPy(Bx

Wn)] =

1 1 dnq

(8.88)

(4

xoy(x)(x

1 p I x) dnx,

W"

B

ppl****.pn OY

= Tr[pYIPy(Wnx

B)] =

1 / p(K)(KI dnp

B

$ I K) dnK,

Rn

on (W, 9P)are compared with their conventional counterparts obtained by specializing (8.69) to the configuration and momentum representations, respectively:

(8.90)

/

PP1*****'n(B) = Tr[pEpls***.pn(B)] = (K 1 p" I K) dnK. B

Indeed, if we consider (8.78) and (8.88) for the case of y given in (8.80), we easily establish that (8.91)

For s E (0, +a),or for the case of a general y , (8.87) and (8.88) also have an obvious interpretation: whereas (8.89) and (8.90) are the probabilities that would be measured if in each random sample of values of position and momentum, respectively, each value x E Wn and K E Wn would be determined with absolute precision, (8.87) and (8.88) are probabilities obtainable with realistic measurements performed with actual instruments whose readings necessarily involve a margin of uncertainty (cf. Dietrich [19731). I n other words, by executing the calibration of such instruments we always find out that, say, a reading q in reality means that for any interval I containing q the confidence probability of the actual value x being within I equals py(I), and that the measure p ( B ) , B is never a &measure centered at x. Thus, in our case the respective confidence measures would be (8.93) and we can regard (8.78) and (8.88) as the probabilities of obtaining

IV. The Axiomatic Structure of Quantum Mechanics

410

stochastic values (4, pgy) and ( p , & y ) , respectively, with q E B and p E B. Naturally, under normal circumstances the difference between (8.87) and (8.89), or (8.88) and (8.90), might be negligible by comparison with the statistical fluctuation displayed by different samples. Yet, there is one obvious advantage of dealing with stochastic values in quantum mechanics: if we do that we can then consider the possibility of simultaneous stochastic values ((4, p q Y ) , ( p, & y ) ) for position and momentum without violating the uncertainty principle, since in view of the definitions in (8.85) and (8.86) of the con.dence functions xgy(x) and g p y ( K ) , the spreads of these functions automatically obey the resulting space" of all such stochastic that principle. Denoting by values obtained as (q, p ) varies over r with y fixed, we can consistently extend the standard interpretation of quantum mechanics based on Definition 1.3 by interpreting

ry

as the probability that a determinative measurement of stochastic values in ryon an ensemble in state p would yield outcomes ((q, pgy), (p, & y ) ) with (4,p) E B. Indeed, by (8.87) and (8.88), (8.95)

p Ql..... Qn,P,,....P

(8.96)

pol.....o , . p , .....P"([w"

n ( ~

[wn) =

p 4 ....

I

PY

OY

*"(a

B ) = p P 1 .....P,

DV

PY

(B),

so that this interpretation is simply an extrapolation of the conventional one. One of the many attractive features of such an interpretation is that it naturally leads to a concept of probability current in quantum mechanics without any need for extraneous new postulates. Consider for the sake of simplicity the case of a single particle, i.e., of three degrees of freedom and of r = Rs. Setting in the classical case (8.97) (8.98)

jY(r) =

j

Vjt(r,PIdp,

v

= p/m,

w3

we see that as the average value of velocity, j P'(r) represents a probability current at r E R3. Specializing (8.71) and (8.72) to n = 3 and integrating

* This is

called stochastic phase space; cf. PrugoveEki [1981a].

8. Completely Continuous Operators and Statistical Operators

41 1

both sides of the Liouville equation (8.71) in p over R3,we obtain the well-known continuity equation (8.99)

atpzl(r)

+v

*

jl(r)

=

o

expressing probability conservation. Let us similarly define in the quantum case (8.100)

We can then easily establish that (cf. Prugovezki [1978a]) (8.102)

atpis)(r)

+ v - jis)(r) = 0,

so that jf?)(r)possesses all the natural features of a probability current. Moreover, -

m

(8.103)

lim ,dS)(r) =

s++o

C wk 1 1CIk(r)l2= (r I P I r> k=l

whenever Vi+bk(r),k = 1, 2,... are continuous in a neighborhood of r and the two series can be suitably majorized. T h e expression on the right-hand side of (8.104) coincides with the probability current found in all text-books on quantum mechanics (see, e.g., Messiah [1962]).

EXERCISES 8.1. Show that the projector EM onto an infinite-dimensional closed subspace M of 2 is not completely continuous. 8.2, Prove that if the continuous spectrum SA, of a self-adjoint operator A is not empty and contains points which are not accumulation points of the point spectrum, then :S contains infinitely many points.

8.3. Show that a bounded self-adjoint operator with a pure point spectrum, containing only eigenvalues of finite degeneracy which have no accumulation points with the possible exception of zero, is a completely continuous operator.

412

IV. The Axiomatic Structure of Quantum Mechanics

8.4. Show that the trace of a projector is equal to the dimension of the space onto which it projects.

8.5.

Show that if a self-adjoint operator A is positive definite, i.e.,

( f I Af) 3 0 for all f E g A, then its spectrum does not contain negative values. 8.6. Prove that if Tr A is finite and A = A*, then the traces of the operators A‘+’ and A ( - )defined in (8.24) are also finite. 8.7. Show that if the linear operators A, B, C are the uniform limits of the sequences {An},{B,}, {C,}, then ABC is the uniform limit of the sequence {A, B, C,}.

References for Further Study The theory of quantum measurement has many ramifications that traditionally are surveyed primarily in monographs on the philosophy of quantum mechanics, such as d’Espagnat [I9761 or Jammer [1974], or in specialized collections of review articles, such as those by Price and Chissick [1977] or Marlow [1978]; for basic aspects see, however, Messiah [19621, Gottfried [1966], and Ballentine [ 19701. A great deal of functional analysis pertinent to the mathematical background of $92-7 can be found in Riesz and Sz. Nagy [1955], Akhiezer and Glazman [1961], Stone [1964], Schechter [1971], Reed and Simon [1972, 19751, and Yosida [1974]. The operator-theoretical treatment of complete sets of observables of 95 is taken from PrugoveCki [1969a], whereas other rigorous treatments are based on a W*-algebra approach (Jauch and Misra [1975]). The case of infinitely many observables has been given a careful mathematical treatment by de Dormal and Gautrin [1975], and relevant results are also reported by Berezanskii [1978] and Ascoli et al. [1978]. The subject of canonical commutation relations for a finite number of degrees of freedom treated in 96 is thoroughly surveyed by Putnam [1967], who also provides an extensive bibliography on the subject. An introduction to the case of infinitely many degrees of freedom that contains further references can be found in Bogolubov et al.

[1975]. Many of the results on essential self-adjointness of Schroedinger operators that were only briefly dealt with in 97 are reviewed by Kato [1967] and by Reed and Simon [1975]. The subject of group theory in quantum mechanics (barely touched upon in 97.8 since it lies beyond the scope of the present book) has been given careful mathematical treatment in many monographs, such as those by Miller [1972] and by Barut and Rqczka [1977]. The concept of density operator was first introduced by von Neumann [1955] in the original German edition of his classic book, and by now it forms the basis of all standard formulations of the concept of state in quantum statistical mechanics (see, e.g., Balescu [1975]). However, other formulations exist, such as those based on the C*-algebra approach (see Emch [1972]). Many questions pertinent to the related subject of completely continuous, Hilbert-Schmidt and trace-class operators are systematically treated by Schatten [1960].

References for Further Study

413

It should be mentioned that there have been many attempts to generalize the formalism of quantum mechanics by employing mathematical tools other than Hilbert space theory. Perhaps the best known is that employing lattice theory, and a very readable account can be found in Jauch [I 9681, whereas Varadarajan [1968] covers exhaustively the more technical aspects of this subject. In addition, other mathematical disciplines, ranging from W*- and C*-algebras (von Neumann [1936], Segal [1947]) to measure theory (PrugoveEki [1966]) have been used. Yet, Hilbert space theory has remained the one and only practically viable alternative that enjoys universal acceptance in its role of providing the mathematical background for nonrelativistic quantum mechanics.

CHAPTER

v

Quantum Mechanical Scattering Theory

I n the preceding four chapters we have given the basic mathematical tools needed for a mathematically rigorous study of conventional nonrelativistic quantum mechanics. The actual applications have been limited either to obtaining very general, albeit fundamental, results of quantum mechanics or to computing the bound states of specific systems (e.g., the hydrogen atom). Within the context of nonrelativistic quantum mechanics, scattering theory deals with collision phenomena of particles moving at velocities small by comparison with the speed of light. Since scattering theory is concerned with completely or partially free particles, it is of utmost interest from the experimental point of view. From the theoretical point of view, scattering theory is still the object of intensive research for the physicist as well as the mathematician. We shall give in this chapter the most basic ideas of nonrelativistic quantum scattering theory. In illustrating these ideas, we shall limit ourselves to two-particle scattering, which is the only case which has received a thorough treatment until now. The scattering of n particles, usually referred to as n-body or multichannel scattering theory, is a field of active current research, which deserves at least one entire volume for a minimally adequate treatment. A very short introduction to this general case will be given in $8 of the present chapter. 1.

Basic Concepts in Scattering Theory of Two Particles

1.1. SCATTERING THEORY AND

THE

INITIAL-VALUE PROBLEM

I n studying in Chapter I1 n-particle systems in quantum mechanics, we were concerned exclusively with problems involving bound states. 414

1. Basic Concepts in Scattering Theory of Two Particles

415

Such states are related to the point spectrum of the internal energy operator of the system, as described in 95, Chapter 11. We shall now turn our attention to two-particle states which are not bound states. The solutjon of the general initial-value problem for an n-particle system is in essence contained in Axiom S3 of Chapter IV, $3-namely, if we could prepare by some means a state Yoof the system at the instant t o , and if the Hamiltonian H were given, then, in principle, we could compute the state* Y(t)= exp[-iH(t - to)] Yo

at all later instants t. However, such a computational job, besides being in most practical cases unmanageable due to its extremely great computational complexity, is not warranted by the needs of the experimentalist who carries out experiments at the present level of technical sophistication. The experiments carried out at present in molecular, atomic, nuclear, and elementary particle physics which involve, from the theoretical point of view, an initial value problem are exclusively scattering experiments. Such experiments are characterized by the fact that the system undergoes a preparatory measurement at some instant t o , a determinative measurement at a later instant t , , and that for most of the time span 1, - to of the experiment there is no interaction (or practically negligeable interaction) between the different constituent parts of the system. As a matter of fact, these parts interact with one another only for a very small part of the duration t , - to of the experiment. For the rest of the time they are spatially too far apart to have the very shortrange forces make themselves felt. This and some other features of scattering experiments, which will be mentioned later, make it unnecessary to solve completely the initial-value problem in order to relate the theoretical predictions to the available experimental data. It is clear from the above remarks that for the greater part of a scattering experiment, the different constituent parts of the physical system under observation move independently from one another, i.e., they are “almost free.” I n order to elucidate this intuitive concept of “almost free’’ motion in a mathematically clear and concise form, it is desirable to consider this concept first in the framework of classical mechanics, which readily appeals to intuition.

* In this chapter we adopt a system of units in which ti

=

1.

V. Quantum Mechanical Scattering Theory

416

1.2. ASYMPTOTIC STATESIN CLASSICAL MECHANICS Let us consider a two-particle classical system. I n the Newtonian formalism of classical mechanics, the state of such a system is given by a vector-valued function

The trajectory of the Kth particle, k = 1,2, can be described by a threedimensional vector-valued function rk(t),which geometrically is a curve in three dimensions. If the two particles of the system undergo the kind of motion which was depicted earlier as typical in a scattering experiment, then the trajectories rk(t),K = 1, 2, should have as characteristic the main qualitative features of the curves in Fig. 4: at the beginning t M to

FIG.4. Asymptotic states in classical mechanics.

of the experiment and for a relatively long time afterwards the motion of each particle should be “almost free,” i.e., “almost” along a straight line rF(t) at a uniform speed vp, k = 1,2; then, during a relatively short period, while the particles are interacting with one another, the motion is, in general, very complex; finally, at the end, t M t , , of the experiment and for some time before that, the motion of the particles should be again along straight lines riut(t)and at “uniform speed” viUt. Now, in classical mechanics a particle is said to be in free motion if its state is of the form r(t) = ro vt,

+

so that r ( t )

=

0. Hence, at times t rk(t)M rf

M

+ vft,

t o , we expect that k

= 1,2,

1. Basic Concepts in Scattering Theory of Two Particles

417

and similarly for t m t ,

The vectors rf and

vin characterize

a free state of the kth particle,

k = 1, 2, which is called the incoming asymptotic state. Similarly, the free state characterized by riut and vzut is called the outgoing asymptotic state of the kth particle. It is not a priori clear in what precise sense the approximation signs in (1.2) and (1.3) should be understood. One obvious choice is to take them to mean that (1.4)

1 rk(t)- (rf I i.&)

+ vft)~ - vf

1

s",

0,

t

w to,

s",

0,

t

M

to,

where I I above denotes the length of three-dimensional vectors. Since one can reasonably expect that the situation does not intrinsically change if we increase t , - t o , and, moreover, the above approximations might hold with increasing accuracy when to + -a,it seems natural to give to (1.4) the following precise meaning: lim I r,(t) - (rf

t+-m

(1.5)

lim

t+-m

+ vft)~

= 0,

1 r k ( t )- v In k I = 0.

Similar considerations lead to the following corresponding requirements:

The conditions (15 ) and (I .6) define in a precise way the asymptotic states of a single particle. Naturally, the asymptotic states xin(t) and xout(t)of the entire system is the aggregate of the asymptotic states of its two constituent particles. Incoming asymptotic states defined by (1.5) can be thought of as being states in which the particles of the system would be at all times if there were no interaction between the two particles. Similarly, the outgoing states would correspond to the fictitious case of particles which have interacted in the "infinite past," and afterwards move independently.

418

V. Quantum Mechanical Scattering Theory

We note that according to the definitions in (1.5) and (1.6), if asymptotic states exist at all for a given classical physical process, then they are unique for each state, since vy and viUtare uniquely determined by the second of the respective relations in (1.5) and in (1 .6), and then rp and riUeare uniquely determined by the first relations in (1.5) and (1.6), respectively. It is, however, natural to ask whether asymptotic states in the above sense exist for every classical physical process of importance. T h e answer to the above question is negative. I n the case of long-’ range forces, i.e., forces which are of significant intensity even at arbitrarily large separations between particles, asymptotic states in the above sense do not exist. A notable example of such a force is the Coulomb force (see Exercises 1.1 and 1.2). T h e intuitive physical explanation of this phenomenon is that long-range forces never sufficiently loosen their grip on the particles to allow them to travel in a free manner when they are at large distances from one another. A generalization of the above concept of asymptotic state is obviously necessary to deal with this situation. However, the already introduced concept is of sufficient generality to cope with most of the practically important cases, Even the Coulomb potential is in practice most often “screened,” i.e., instead of having to deal with the Coulomb potential V ( r ) l / r , we are faced with a potential which at large distances decreases much faster than l/r, and for which asymptotic states, in the above sense, exist.

-

STATESAND SCATTERING STATES 1.3. ASYMPTOTIC IN THE SCHROEDINGER PICTURE It is straightforward to adapt the concept of asymptotic states defined in the classical two-particle case by (1.5) and (1.6) to the case of quantum mechanics. We can achieve this by stating that in the Schroedinger picture, the “free states” Y i n ( t ) and Yout(t) are, respectively, the strong (or Moller) asymptotic incoming and outgoing states of a twoparticle system in state Y ( t ) if and only if*

I n order to define precisely the meaning of the concept of a “free state” we must assume that in addition to the “total” Hamiltonian H ( 2 ) of the two-particle quantum mechanical system we have at our disposal a “free” Hamiltonian Hi2),which would describe a system consisting of two particles with the same physical characteristics (mass, intrinsic

* See

82.7 for a more general concept of quantum asymptotic state.

1. Basic Concepts in Scattering Theory of Two Particles

419

spin, etc.), but which do not interact among themselves. For instance, in the case where is given essentially’ by r

=

r2- r l ,

then Hh2’ is taken to be essentially 1 A1 -

2m1

where the domains of definition of the above differential operators are adequately defined,+ as was done in 97 of Chapter IV. However, if both particles move in an external force field, given by a potential Vext(rl , r2),then instead of (1.8) we have

and instead of (1.9) we have --1

2m1

A1

-

This will be the case when, for example, the system under consideration consists of two spinless charged particles moving in the force field generated by a much heavier particle (such as the heavy nucleus of some atom). Since the heavier particle is practically unaffected by the motion of the lighter particles, such a problem can be treated fairly accurately as a two-body problem in an external field rather than a three-body problem. Consider now the general case when the “free” Hamiltonian Hi2’ and the total Hamiltonian W2) are given by the self-adjoint operators acting in the Hilbert space A?(2)associated with the system. We say that a nonzero vector-valued function ! W ( t ) ,t E R1, is a free state in the Schroedinger picture if and only if it satisfies the relation Y(’)(t)= exp( - i H ( y t ) Y(f)(O), t E R1,

and if Y(t)(O) is orthogonal to the closed linear subspace Zit)of bound states in .#(2) of the system with Hamiltonian HA2). I n the absence of See the discussion in 97 of Chapter IV on the essential self-adjointness of the Schroedinger operator. t We note that in the preceding two operators, as well as throughout this chapter, we have adopted a system of units in which fi = 1.

V. Quantum Mechanical Scattering Theory

420

external forces both HA’) and its internal energy part Hi:), obtained after subtraction from Hi2) of the center of mass motion, are operators with a pure continuous spectrum so that = (0). For example, this is the case when Hi2)is given by (1.9), and therefore HA:) is essentially -(1/2m,)A acting on wave functions #(R,r), where A is the Laplacian containing derivatives with respect to the relative position coordinates m2)-l. of r and m, = m,m,(m, In reality, the particles interact, and any Schroedinger picture state Y(t) in which they can be found, which will be called occasionally an interacting state, has to satisfy the relation

):A&

-+

Y(t)= exp(-iH(%)t)Y(O),

t E R1.

Due to the unitarity of the operator exp(-W2)t) we have (1.10) 11 Y ( S) ~ ‘ “ ( t )= l ~ 11 exp(-i~(’)t) Y(O)- exp(-iHt)t) Y”(O)~I = 11

Y(O) - exp(iH(%)exp(-iH>)t) ~““(0)ll

= 11 exp(iHf)t) exp(-iH(2)t)

Y(O)- ~‘~(o)ll,

where “ex” is a symbol which will be frequently used in the future, and which stands for “in” and “out.” In view of the preceding notation, we see that (1.7) implies that

or, equivalently,

where the interpretation of the above notation is that one should let t -+ - co when “ex” stands for “in,” and t -+ 00 when “ex” stands for “out.” Since, conversely, (1.11) or (1.12) implies that (1.7) is true, we can state the following theorem.

+

Theorem 1.1. The state Y(t) = exp(-iH(2)t) Y(0)possesses strong incoming and outgoing asymptotic states Yex(t): (a) if and only if the strong limits (1.13)

P””= s-lim exp(i@)t) exp(-iH%) ~ ( 0 1 , t+Fm

421

1. Basic Concepts in Scattering Theory of Two Particles E3@ (b) if and only if there are vectors Yex(0) such that

#At), ex = in, out,

Y(O)= s l i m exp(id2)t) exp( - i ~ t ’ t )~“”(0).

(1.14)

t+Tm

An interacting state Y(t)which has an incoming as well as an outgoing asymptotic state is called a scattering state. We recall from relation (3.43) of Chapter I V that

P(t)= exp(i~~)t)exp(--iH(2’t) Y(O) represents in the interaction picture the same state which is represented in the Schroedinger picture by Y(t)= exp(-W2)t) Y(0).Thus, pex, ex = in, out, can be thought of as being the asymptotic states of p(t) in the interaction picture.

1.4.

WAVE OPERATORS

M0LLER

Denote by RY) and R!? the set of all vectors which are the limits (1.14) for some Y e x ( 0 ) E i W 2 ) QX‘h2,’ (where Xht) is spanned by the bound states of HA2))when t -+- m and t --+ +m, respectively, i.e.,

(1.15)

RY’

= {f+:f+ = s-lim

R?’

= {j-:f- =

t+-m

d 2 ’ ( t ) f , f &‘ ~ @ Sit)},

s-lim d 2 ’ ( t > f , fi~ ~ t++m

0

d 2 ) ( t )= exp(iH(”t) exp( - i ~ t ’ t ) .

We see from (1.14) that if a nonzero vector g belongs to either RY) or R!?, then exp(-iH(2)t)g does not represent a bound state, since it is intuitively obvious that a bound state cannot be asymptotically free. Hence, it seems natural to assume that in any realistic quantum mechanical two-particle theory

RY’ = R P ’ = S b (2’1 where HL2)is the space of bound states of H ( 2 ) .We shall see later that the conclusion of this heuristic argument is confirmed when dealing with potentials which usually occur in practice.

Lemma 1.I.

If A(t) is a uniformly bounded family of operators,

II 4 ) l l

< c,

--oo


V. Quantum Mechanical Scattering Theory

422

then the set M of all vectors f E X for which s-lim A(t)f,

-a

t-f

< to < +a,

exists is a closed linear subspace of X .

Proof. Since the existence of the strong limits of A(t)f and A ( t ) g for t --+ to implies the existence of the strong limits of A(t)(uf bg) = u A ( t ) f b A ( t ) g ,it follows that M is a linear subspace of X . Letfbe a vector in X which is the strong limit of a sequence fl ,fz,... of vectors from M. If

+

+

g,

=

s-lim A ( t ) f n , t+t,

then we can write (1.16)

lG" =

- 4tZ)lfll

II[A(tl) - A(tz)l(f

-fn)

+

[ 4 1 ) -4

< II[A(tJ - A(tJl(f -fn)ll + II A(t1)fn < II 4 t d f -fn)ll + II 4)(f -fn)ll

Z ) l f T i

-

II

gn - [A(t~)fvz - gnlll

+ II A(t1)fn -En II + I1 4 A f n - gn II

< 2c Ilf - f n

II

+ II A(tl)f,

- g,

By choosing some sufficiently large n 2c Ilf

-fn,

=

II

+ II &)f,

-

g,

II.

n o , we can obtain

II < 4 3 .

Since, for this fixed n o , we have

II A(t)fno- gn, II < 6/3 for all t sufficiently close" to to , we see by glancing at the right-hand side of (1.16) that

II 4,)f - A(t2)fll < 6 for all such t , , t , which are sufficiently close to t o . Hence, the strong limit of A(t)f for t -+ to exists, and thereforefE M. Q.E.D. The operator function S2(2)(t)defined in (1.15) assumes as value a unitary operator for every value o f t E OX1, and consequently it is uniformly bounded: --co < t < +a. 11 Q(yt)ll = 1,

* If to =

&a,sufficiently "close" to tomeans sufficiently large.

1. Basic Concepts in Scattering Theory of Two Particles

423

Thus, we can apply Lemma 1.1 to conclude that the families Mf’ of all vectors f E Xc2’ 0S‘bz for which the strong limits 52c2)(t)f for t --t - 00 and t -+ co, respectively, exist are closed linear subspaces of 02;:). Hence, the family

+

MF) = M(2) n MP)

+

of all vectors f for which both limits, s-lim LY2)(t)f = f* , t+F‘m

exist is a closed linear subspace of 2(2). Let EMt)be the projectors onto MT), EMp%?(2’ = M(2) f.

(1.17)

Then, according to the above definition of MY’, the limits s-lim Q 2 ) ( t ) EM2’f t+Fm

exist for all vectors f E X , because E $ ~ EMY). Then the operators (1.18)

exist and are defined on the entire Hilbert space iXF2).The operator 52y’ is called the incoming M ~ l l e wave r operator and Q!? is called the outgoing Meller wave operator. We shall refer to these two operators as the “in” and “out” wave operators.

1.5. THESCATTERING OPERATOR The scattering operator (or S operator) S2)for the problem at hand is defined in terms of the wave operators Q‘$,

Since the wave operators 52:) are strong limits of uniformly bounded operators 52c2)(t),)I 52(2)(t)l) 1, they are bounded. Furthermore, we have seen that are defined everywhere in iW2).Hence, the adjoints Qf)* exist and are bounded (see Chapter 111, Theorem 2.7). Thus, S2) is a bounded linear operator, defined on the entire Hilbert space

522)

<

V. Quantum Mechanical Scattering Theory

424

The importance of the scattering operator lies in the fact that many quantities which are readily measurable by scattering experiments are easy to compute if we know S2). T o provide an illustration, assume that a scattering state Y+(t)of the system has been prepared at the beginning of the scattering experiment, and that at the instant to of the preparatory measurement Y+(t)had been essentially a free state Yp(t), lim 11 ~ + ( t-) ~ p ( t ) l= l 0,

1--m

~ p ( t= ) exp(--i@)t)

Y~(o).

Furthermore, assume that a determinative measurement at an instant , long after the two particles of the system have interacted, can determine the value of the observable Ell of the form

t,

which is represented by the projector on the scattering state Y-(t) at t = t , . Then, if Y+(t)is normalized, the mean value of E,, is given by the expression (1.20)

PY++F-

= (‘u+(t1)

I 4, Y+(t1)>= IW+(tJI Y-(tA)I2*

We recall from the discussion at the end of $1 in Chapter IV that the above expression is usually referred to as the transition probability from Y+(t)to Y-(t),due to the usual assumption that if the outcome of the determinative measurement of E,, is X = 1, the system is left afterwards in the state Y-(t). We note that the expression in (1.20) is actually independent of t , ,

due to the fact that the operator exp(-zW2)t) is unitary. Let Yozt(t) be the strong outgoing asymptotic state of Y-(t): lim 11 ~ - ( t ) ~?(t)ll

t++m

Since according to (1.14)

= 0,

P U t ( t ) = exp(--i~F)t)

PUt(o>.

425

1. Basic Concepts in Scattering Theory of Two Particles

we arrive at the following expression for the transition probability:

This expression is very convenient and helpful in practice, since it involves free states which in the typical scattering experiment are computationally easy to derive from the available experimental data.

SCATTERING CROSSSECTION 1.6. THEDIFFERENTIAL In practice, the typical two-particle scattering experiment is of the kind depicted in Figure 5: a beam of particles 6, impinges on a target

INCIDENT BEAM

I ,A’--L Yo

FIG. 5. Scattering of a beam of particles off a finite planar target.

consisting of particles 6,. Assuming that the particles in the beam do not interact with one another, and that each particle 6,in the beam interacts with only one particle 6, in the target, the experiment can be viewed as consisting of a large ensemble of independent scattering experiments of two-particle systems” 6 = (6, , G2}. The two mentioned conditions can be satisfied by taking a “weak” beam of particles (i.e., a beam with few particles per unit volume), and taking for the target a slab of material which is sufficiently thin to eliminate the possibility of a particle G, scattering from more than one particle in the target.

V. Quantum Mechanical Scattering Theory

426

The above experimental setup is ideally suited to determine the percentage of the particles in the beam which can be found in a given volume of space, occupied by a detector, after they have been scattered by the target. I n any scattering experiment the detector is placed as depicted ip Fig. 5, i.e., sufficiently far from the target so that it never interferes with the scattering process. Hence, the experiment obviously provides information about the probability that a particle of the beam, coming towards the target from the direction wo = ( 4 0

9

OO),

0 <4 0

< 2.rr,

0 < 00

< .rr,

will be scattered within the solid angle dw around the direction (1.23)

=

(4, e),

o G 4 G 2T,

oGeG

TTT.

If N denotes the number of two-particle scatterings per unit time at the energy of relative motion E, then the number of particles scattered within the solid angle dw can be written in the form Nr(E, wo , w ) dw, since it is obviously proportional to N as long as the two earlier mentioned conditions on beam and target are satisfied. It can be expected that r ( E , wo ,w ) is dependent on the energies E and E' for the relative motion of the particles of the system 6 = {GI,G2} before and after collision. However, since energy is conserved, we must have E = E', so that the dependence on E' does not have to be displayed. It has become customary to orient the frame of reference so that c $ ~ = Oo = 0, and to call the quantities

(where Jo is the incident jlux, i.e., the number of incident particles per unit time and unit beam cross section) the dzrerential cross section and the total cross section, respectively. The notation daldw instead of a(E, w) is often encountered in literature.

1.7. THETRANSITION OPERATOR and Hi2) are given by (1.8) and (1.9), and In potential scattering one can easily relate a(E, a)to the S operator. T o establish this relation, it is computationally advantageous to express S2)in terms of a new operator, (1.25)

s(2)=

1 - 2TiT(2).

We shall refer to T ( 2 )as a transition operator [not to be confused, however, with t h e y ( < ) operators introduced in W.8J since in the case when

1. Basic Concepts in Scattering Theory of Two Particles

427

there is no interaction present, the system stays in the same free state (i.e., there are no “transitions”), and the operator F 2 ) is zero. We easily establish that F 2 )= 0 when there is no interaction by noting that the “absence of interaction” means that W 2 )= HA2) and therefore S2(2)(t)= 1, i.e., Q,(2) - Q(2) - = s‘2’= 1.

We have seen already in 97 of Chapter I1 that in the absence of external forces the Schroedinger operator HA2’ of a two-particle system can be written in the form (2)

- H(Z) + H(2)

Hs -

c

t

where Hi2’ is the center-of-mass energy operator and Hi2) is the internal energy operator. The Hilbert space 8(2) = L2(Rs)of wave functions #(R,r) is obviously the tensor product x(2) = #‘1’@

(1 -26)

S(1)

of the Hilbert spaces X‘,l) = L2(R3)and 8 ( l )= L2(Rs)of wavefunctions #,(R) and t,h(r), respectively, where R and r are the center of mass variables given in (7.2) of Chapter 11. It becomes evident from looking at the differential operators

and (7.3) in Chapter I1 that Hi2) = Ha’ 0 1 and Hi2) = 1 @Hi1), where Hi1) and H{l’ act in XL1)and &(l), respectively. It is natural to require that, in general and when no external forces are present, the free and total Hamiltonians can be written in the form H p (1.28)

=

H.p’

+ Hi;’,

H ( 2 )= Hi2)+ H/2),

H$) = 1 @ H$, Hf2)= 1 @ HP).

HC(2’ = H,’“ @ 1

Since the operators HA2) and HAT), as well as Hi2) and Hi2),obviously commute, we can use Theorem 2.4 of Chapter IV to infer that

V. Quantum Mechanical Scattering Theory

428

The above two relations are derivable directly from (2.13) in Chapter IV by noting that in the above case all operators are bounded and defined on the entire Hilbert space 3Y2). From (1.28) we get (1.29)

d 2 ’ ( t )= exp(iH%)

exp( --iHf’t)

= exp(iHp’t) exp(iH,j%)exp( - - i ~ e ’ texp( ) -i~$’t)

= exp(iH/%) exp(-iH$’t) = 1

0exp(iHl(’)t) exp(--iH$’t).

Inserting the above expression for D 2 ) ( t )in (1.18) and (1.19), we easily arrive at the following result.

Theorem 1.2. If the free and total Hamiltonians HA2) and H @ )of two-body system can be decomposed into sums (1.28) of commuting Hamiltonians for center of mass and internal motion, then

a

where Qy) and S“) are operators in the internal-motion Hilbert space &“‘) of the tensor product 2?(2) = 2;’) 0if(’). T h e above theorem shows that in case of two-body potential scattering in which no external forces are involved, we can work in the Hilbert space which is related to the internal motion of the system. If we work in momentum space, we can introduce the relative momentum variables p, related to the relative position vector r. We might hope that which, the S(1)operator could be written as an integral operator in M1), in case of spinless particles, would be of the form

However, we shall see in the 92 that S“) commutes with the free Hamiltonian HA:). Lemma 1.2 shows that, on account of this fact, the representation (1.30) is not feasible.

Lemma 1.2.

Let K be a bounded integral operator on L2(R3)

429

1. Basic Concepts in Scattering Theory of Two Particles

with a kernel K(p, p') which is Borel measurable in R6. If K commutes with the self-adjoint operator Hi:),

then K is the zero operator.

If f(p) is a function of compact support A , then EP(d)f = ED(d)is the spectral measure obtained as the product (1.13) in Chapter IV of the spectral measures of the three momentum component observables: Proof.

f~ 9&), where

(D(4g)" (PI

= XAP) g"(P)

The operator Hi;'E'(d) is obviously bounded, and commutes with K,

=0

(almost everywhere).

Since the above expression has to vanish almost everywhere for all f(p) of compact support, we have K(P, P') (K "' 0

-)2%

p2

=0

(almost everywhere in Rs x R3 = Re).

Now, the set of points in R6 for which I p I = I p' I (called the energy shell) is of Lebesgue measure zero. Hence, we deduce that K(p,p') vanishes almost everywhere in R6. If f(p) and g'(p), f, g €L2(R3), are bounded functions of compact support, then t * ( p ) K(p, p')f(p') is certainly integrable on R6 since K(p, p') is Borel measurable and vanishes almost everywhere. From the relation

s..

g"*(P)K(P, P')f(P') dP dP'

= 0,

we get by applying Fubini's theorem and Theorem 4.6 in Chapter 111,
1 dP'K(P9 P')f(P')

dPg"*(P) P3

= 0,

w3

for all 3, g' E e(R3). Since e ( R 3 ) is dense in L2(R3), we get Kf = 0 for all3 E Pb(R3). Thus, K is a bounded linear operator which is zero.on a dense subset e ( R 3 ) of L2(R3), and therefore K = 0. Q.E.D.

430

V. Quantum Mechanical Scattering Theory

Let us introduce spherical coordinates ( p, W ) = (p, 0, +), 0 < p < m, 0 < 0 < n, 0 < C$ < 2n, in the space R3 of the variables p. We shall see later in this chapter [cf. (4.96) and (4.9711 that the 2"l)

+

operator can be written as an integral operator with respect to the variable w ,

over the set SZ,, defined in $7.2 of Chapter 11, which is essentially isomorphic to the unit sphere in three dimensions. The function T(])(p;w , w ' ) is uniquely determined by the operator T(l) (see Exercise 1.6) if we restrict ourselves to continuous kernels for the integral operator in (1.31). *1.8.

THET-MATRIXFORMULA FOR THE

DIFFERENTIAL CROSSSECTION

The differential cross section o(E, w ) can be computed from the function T")(p; w , w ' ) which occurs in the integral representation (1.31) of the T operator. In order to arrive at an expression relating o(E, W ) and T(l)(p;w , w'), we have to express in theoretical terms those essential features of a scattering experiment which enter in the definition of the differential cross section. I n any such scattering experiment, a beam of particles scatters from the particles of a target. The experimental setup is such that the relative momentum p of the incoming particles lies within a very narrow range of values. Let us introduce spherical coordinates (p, 0, C$) in the p space, and suppose that the incoming particles are prepared in a scattering state Y((t),which is such that p lies within the set (1.32)

L = [ p 0 ,P, + 4,1,

L,= reo,eo + deOix M ~ ,4,

+~ 4 ~ 1 .

Denote by Yin(t) the incoming asymptotic state of Y((t).Suppose that P ( 0 ) is represented by the function (see Exercise 1.5) (1.33)

$in(p,P) =

$m$o(P),

of the center of mass momentum P and relative momentum p. Let us denote by F"(B) the spectral measure (1.34)

(E'D'(B)#)-(P,P) = XdP) $(R

P).

43 1

I . Basic Concepts in Scattering Theory of Two Particles

It can be shown (see Exercise 1.8) that for any Bore1 set B E g3

Since the support of the function $(P,p; t ) representing Y(t)is within c ( ( O ) , we deduce from the above relation that

R3 x

Hence, ~?~(p) vanishes almost everywhere outside C(O). Denote by Yo"t(t) the outgoing asymptotic state of Y(t).It is easy to show (see Exercise 1.4) that (1.36)

Yo"t(o) = S'2'Yy'n(o) = ( 1 x S'") Y'"(0).

Hence, by applying the same type of argument which led to (1.34) to 'Pout(t), we obtain

=

I

B

I(S(l)#O)"(P)I2dP.

Consequently, the probability P(luo+ Iuo,) of finding that the relative momentum p of the system, a long time t after the scattering had taken place, is within the cone C(+) = {P:(4 4) E Lo,},

Lo,= Pi,47' + 4 7

x

[do', 4;

+

4 0 1

is given by (see also Exercise 1.12) (1.37)

wuo -Lo,>= I=,+,

I(S(1VO)"

(P)I2 dP.

We shall see in the next section that S(l)commutes with EHA:'(B) (see Theorem 2.3), where (EH"'(B)#)"(P)

= X { D * , / ~ ~ ~ G B$(PI' )(P)

V. Quantum Mechanical Scattering Theory

432

It follows that for the complement B, E g1of the set {p*/2m:p E ZpJ, we have EH?(B1) S(1)#, = S(l)EHd.:)(B,) #o = 0

since E#'(B,) y50 = 0 because $o(p) vanishes outside &). This means that ( S ~ l ) # o ~vanishes (p) when p8/2m E B, , i.e., when 1 p I 4 Zpo . Therefore, if the integral in (1.37) is expressed in spherical coordinates, the integration in p extends in effect only over the set Zpo. Thus, we can write

Let us consider the case when the direction of relative motion of the two particles in the system has changed as a result of interaction, i.e., when wo # w,,' and Iwon = ~zr. If we use the definition (1.25) of the transition operator to write S")& = $0 - 2m'T"'#o

and recall that $,,'(p, we arrive at the result (1.39)

P(Za0+I,.)

w) =

J0(p) vanishes when

w

is outside Iao,

= 4d

Hence, we get for the probability density P(wo')(per unit solid angle) of scattering in the direction w,,'

provided that (T(14&)'( p, 0') is continuous in w' in some neighborhood of wg). Substituting in the above relation for the operator T(')its integral representation (1.31), we obtain

1. Basic Concepts in Scattering Theory of Two Particles

433

where the integrations in a, and w2 extend over the unit sphere in three dimensions. The request that (T(l)+,’)“(p,w ‘ ) be continuous can be met by demanding that T(l)(p; o’, w ) be continuous in o‘. At this stage one must remember that the expression (1.41) is the probability density of scattering of two particles in the relative direction wo’ with respect to one another. We have already indicated, however, that in a scattering experiment a beam of particles is being scattered on a target consisting of a large number of particles. Thus, in a scattering experiment, a great many two-body scatterings are observed simultaneously. These scatterings between particles in the beam and particles in the target are taking place at different locations with respect to the laboratory frame of reference. Hence, in order to arrive at an expression for a planar differential cross section, an averaging of P(w:) over all these two-body scatterings at different locations has to be carried out. If a particle 6,” in the target is at a location roin relation to another particle G,’, while 6,‘ is in position r with respect to a particle 6, in the beam, then 6,” is at the location r ro in relation to the incident particle 6, . Since, with the exception of location, all the other variables of 6,’ and 6,” are the same, we conclude that if

+

represents Yin(0)for (6, , G,’}, the wave function

represents Yin(0) for {G, , 6,”). By taking Fourier transforms, for (6, , G2”}we easily get

$ro(P)

= ~ X P (i ~ r oJOO(P), )

where &(p) refers to (6, , Gz’}, and Jr0(p)refers to (6, , 6,”). Hence, the probability Pro(oo‘) for {G, , 6,”) is (1.42) Pr0(wgl)= 479

I

I

d p p 2 dw, dw, T*(p;w,,’,

w,)

1%

x

w;

Wo’, W 2 )

exP[iro(Pz - PA1 JO*(PA$O (P2)IPI-llp

instead of (1.41). We shall now make the assumption that the scatterers in the target are uniformly distributed, so that No scatterers interact with the incident particles per unit target volume and unit time. Then the total number

434

V. Quantum Mechanical Scattering Theory

of particles scattered* per unit time and solid angle in the direction is on the average equal to (1.43) ~ ( W O ' )=

~o

1

w,,'

pro(wi) dro

= 4T2No

s s s dro dp p 2

dw, dw, exp[iro(p2- p,)] P ) * (p ; wf0 ,w,)

assuming that the scatterers in the target are sufficiently dense that we may, with sufficient accuracy, replace the sum over the individual scatterers with an integral over the volume I'of the target. We shall evaluate the integral (1.43)under the assumption that the target is a slab of thickness 6, which extends to infinity in the directions x, and yo (see Fig. 5). Setting k = p2 - p1 and reversing the orders of integration in (1.43), we obtain (1.44)

~ ( w i= ) 4?'r2NO dw,

dro j d p dwzp2eikrG F(wo';p , w1 , w 2 ) ,

where we have introduced the function (1.45)

F(wo';P, wi =

9

wz)

T'l'*(p;wo', w1) T Y p ; W"',

%a)

&*(A W l ) $JP,

wz).

Since 1 p1 I = I p2 I = p, only two of the components of k are independent. We can substitute for w2 the variable w which represents the spherical coordinates 0 and r$ of p2 in a coordinate system with its z axis in the direction of p1 . Hence, plz = p , = 0 and

By substituting for w p2

dp dw

= p 2 sin

k,

= p,, = p s i n

k,

= pZv =

=

0 cos+,

p sin 0 sin+.

(8,r$) the variables k, and k, , we obtain

0 dp d0 d+

= p 2 sin 0

a@,

' )' 3

k,)

dp dk, dk,

=

1 dp dk, dk, .

cos e

i+In practice, the scattsrers In the target are approximately at rest with respect to the laboratory frame, so that the relative scattering direction wo' computed in the center-ofmass frame is very often almost equal to the scattering angle measured in the laboratory reference frame.

435

1. Basic Concepts in Scattering Theory of Two Particles

Using the properties of Fourier transforms applied to the variables

k, ,kv and their conjugates x, and y o , we obtain

Since obviously dx, dy, dz, = dr, , the above relation can be used in computing the integral in (1.44). When k, = kv = 0, we have +1 = and 8, = 8, or = +2 and 8, = T - 8,. Since F(wo';p , wl, w,) vanishes when w1 or w, are outside I", , it follows that when I",is sufficiently small only the first alternative dl = #, and 8, = 8, yields a nonzero result. This means that A, = 0, 8 = 0, and the integration in a, in (1.44) can be immediately performed:

+,

We note that N,6 = Nl is the total number of scatterings per unit time and target area, i.e., the incident flux J0 . Let us choose Ap, , AB,, and A#, very small, and the direction of the incident beam along the z axis, so that do = = 0. Then, in the limit

+,

dP0

9

A00

7

4

0

-

+o,

I $O(P)l2

-

6YP - Po),

we get by the mean-value theorem of the integral calculus lim

Jo I T'l)(p;w,,', wO)l2,

v(w,,') = (2X)4

140 1 8 4 9

P2

W,

= (0,O)

We note that the application of the mean-value theorem is warranted on account of the implict assumption that T t l ) ( p ;w', w) is continuous in some neighborhood of w' = (0,O). Since ~ ( w , , ' )is the number of scatterings off an infinite planar" target (cf. Fig. 5 ) in the direction w,,' per unit time, we have by (1.24) o(E, w0')

1 lim Jo 1 6I*+ss ~

= --

v(w0').

* By qualifying the above differential cross section as "planar" we depart from conventional terminology, which in the quantum context recognizes only the cross section given by Eq. (1.47) or, equivalently, by its more standard counterparts (4.98) and (4.99) (see 14.10 for a discussion of this point).

V. Quantum Mechanical Scattering Theory

436

Thus, we have obtained the expression

for the planar differential scattering cross section under the assumption that P ) ( p ;wl', wl) is continuous in wl' and w1 when wl' is in a neighborhood of w' and w1 in a neighborhood of wo = (0,O)for a given P E [O, a).

EXERCISES

+

1.1. Let rEx vixt,K = 1, 2, be the asymptotic states for the state rk(t),k = 1, 2, of a classical two-particle system. In the center of mass coordinates (7.2) of Chapter I1 the state of the system is given by R(t) and r(t), where R(t) = R, V,t describes the uniform motion of the center of mass of the system. Show that if rex = rlx - rleX, v e x = vgx - vgx, then I r(t) - (rex vex t)l ,0, (l/t) r(t) + vexand I r(t)l - t I vex I -+ rex* vex/)vex I when t + T co, respectively.

+

+

1.2. Prove that a system of two particles interacting via a potential V(Y)= 1 / ~(gravitational or Coulomb potential) does not possess any asymptotic states satisfying (1.5) and (1.6). 1.3.

Show that the adjoint K* of a bounded integral operator K ,

defined on L2(1Wn), is an integral operator of the form (K*g)(4 =

1,

(K(x',

XI)*

g(4

1.4. Prove that if Yex(t) E Mo(ex = in, out) are the asymptotic states of Y(t),and S is the scattering operator, then YoUt(0)= SYin(0).

1.5. Show that the formula (1.47) for the scattering cross section stays valid in the general case when P ( 0 ) is represented by an arbitrary wave function t,@(P,p), which is not necessarily in the form of the product occurring in (1.33).

; , w ' ) and Ti1)@;w , w ' ) are two kernels 1.6. Prove that if T ( l ) ( p w satisfying (1.31) for a given operator T ( l ) , then Tkl)(p;w , w ' ) = T")(p;w , 0)') almost everywhere with respect to the measure p:') x pp x pp on [0, 00) x Qp x Q p .

+

1. Basic Concepts in Scattering Theory of Two Particles

437

1.7. Suppose Y(t) and Yo(t),t E R1, are vector-valued functions assuming values in the Hilbert space H, and that there is a constant C such that 11 Y(t)ll C and 11 Yo(t)ll C for all t E R1. Let A be a bounded operator on H.Show that limf-rTco 11 Y(t)- Y,(t)ll = 0 implies that

<

<

Use the result of Exercise 1.7 to prove that

1.8.

lim (!P(t) I E(p)(B)Y(t))= ( Y e x ( 0 ) I E(D)(B)??‘ex(O)),

t-l yco

+

where (HA’) #)-(P,p) = (P2/2M0 p2/2m0) $(P,p), with M , and m, denoting, respectively, the total and the reduced mass of the system. 1.9.

Consider the case of one-particle wave mechanics, when

H, 2 (1 /2m)A. Using the well-known relation (exp( -iH,t)+)“(k)

= exp[ -i(k2/2m)t]

$(k)

derive that, for functions #(r) which are Lebesgue integrable as well as square integrable, (exp(-iH,t)+)(r)

=

(~~/2&)3/2 exp[ i(m/2t)(r - r’)2]+(r’) dr’ L a

almost everywhere in R3. 1.10.

Verify that the operators Ui’) and Ui2),defined for real t # 0

by ( Ut(l)+)(r)= (m/it)3//”exp[im(r2/2t)l $(mr/t),

(UP)+)(r) = exp[ im(r2/2t)l+(TI,

are unitary. Use the result of the preceding exercise to show that for H, 2 -(I /2m)A e-fHot

=

u?)~?).

1.11. Use the result of Exercise 1.10 to prove that the operator families e-iHot and Uil) defined there satisfy

for all.4,1 E L2(R3).

438

V. Quantum Mechanical Scattering Theory

1.12.

Consider the cones

c(+) = {( e,4): I,

2 0, 0, G e G 0,

+ do, 4, G 4 G 4, +

and C(-)= {r: -r E C+}in R3. Use the results of Exercises 1.7 and 1.11 to prove that for H , 2 -(1 /2m)A

Remark. This result indicates that asymptotically in the future the probability of finding the particle spatially located in the cone C(+) is equal to the probability of finding its momentum within C(+).

2. General Time-Dependent Two-Body Scattering Theory

2.1. THEINTERTWINING PROPERTY OF WAVEOPERATORS I n $1 we defined the M0IIer wave operators Q* by means of strong limits of Q(t), Q+ = s-lim Q(t) EM+, t+Tm

(2.1) = eiHt e-iW ,

H

= H*,

H,

= Ho*,

in the time parameter t. In this definition, the closed linear subspaces M, of A? are spanned by all vectorsf which are such that c i H o f f is a free state and the respective strong limits s-limf++mQ(t)f exist." The version of scattering theory based on this procedure of computing the wave operators, and eventually the scattering operator (2.2)

s = Q-*Q+,

is called time-dependent scattering theory. In this section we shall derive some of the basic properties of wave and scattering operators, which follow from the general definitions (2.1) and (2.2), and which do not require the specification of H , and H within

* A mathematically common (but physically unfounded) way of defining Sa* features EH(S2) instead of EM+in (2.1), where Sto is the absolutely continuous spectrum of H

(in practice, S," = S$ , though in general SE C S,"). Since, in general, the strong limit co might not exist, this alternative definition gives rise to the of Sa(t)Eu(S&) for t -P question of existence of wave operators. In our version, the wave operators always exist, but one must check in each particular case on the size of M* and R* (see $97.1-7.2).

439

2. General Time-Dependent Two-Body Scattering Theory

the confines of some particular model. It can be easily seen by comparing (2.1), (2.2) with (1.18), (1.28), and (1.29) that the derived theorems apply and ,S2)as well as to their counterparts Qy)and S1) to the operators in the "internal motion" Hilbert space &?(l).Naturally, in the first case one should take H = H(2)and H , = Hi2),while in the second case one should take H, = Hi1) and H = H ( l ) .

Qz)

Theorem 2.1.

T h e wave operators 52,

have the intertwining

properties (2.3)

'QkeitHO

,

= e'tHQ,

tE

B E a'.

Q*EHO(B)= EH(B)Q+ ,

(2.4)

R',

Proof. If fl belongs to the respective subspaces M, , then according to the definition of M,, the respective strong limits of Q(tl)fl exist when t, ---f co. Thus, we can write

I\exp(iHt)Q*fl = lim

tpTm

=

- s-lim[exp(iHtl) exp( -iHotl)] tl+%v

exp(iHot)fl II

11 exp(iHt) Qif,- (exp(iHt,) exp(--iHOtl))exp(iHot)fl II

lim 11 exp(iHt) Q*fl - exp(iHt) exp[iH(t, - t)] exp[-iHo(tl - 91f1 II

tl+Fm

= lim

ta+Fw

11 Q+fl

- exp[iH(t, - t ) ] exp[-iH,(t,

Hence, exp(iHot)fl also belongs to

- t)]fl

11

= 0.

M, , and since

s-lim exp(iHt,) exp(-iH,t,) exp(iH,t)f, = Q+ exp(iH,t)f, , tl+Tm

we have proved that Q+ exp(iHot)fl = exp(iHt) Qf,

for all t E W. Now, if f 2 E Mi fact, for allfl E M,

=

&? 0M,

<exp(iH,t)f2 I fl)

,

flE M, ,

, exp(iHot)f2 also belongs to Mi . I n

= (fa

I exp(--iHot)f,)

= 0,

440

V. Quantum Mechanical Scattering Theory

since exp(iHot)fiE M, according to the preceding discussion. Thus, due to definition (2.1) of Q, , we have SZ, f s = 0 and Q* exp(iH,t) f s = 0. Consequently (2.6)

QA

exp(iHot)f 2 = exp(iHt) Qfa = 0,

f a E MI,

.

Since everyf E if can be decomposed in the form

we immediately obtain (2.3) from (2.5) and (2.6). The relation (2.4) can be easily derived from (2.3) by methods frequently employed in Chapter IV (see Exercise 2.1). Finally, by making use of the result (3.6) in Theorem 3.1, Chapter IV, and of the fact that Q, are bounded and therefore continuous, we easily derive that for f E gH0 QJY, f

= Q*

1

s-lim - (exp(iH,t) - 1)f t+O it

1

= s-lim - (Q, 1-0

= s-lim 1-0

= s-lip

it

1 zt

1

exp(iHot)- Q,) f

(exp(iHt)Q* - Q*)f

it(exp(iHt) - 1) Q*f

= H&f.

Q.E.D.

2.2. THEPARTIAL ISOMETRY OF WAVEOPERATORS We recall that in the last section M, were defined to be the sets of all for which the respective limits vectors f E if 0ifob

exist, while R, are the sets of all vectors f, . According to Lemma 1.1, the sets M, are a closed linear subspace of if.We shall prove the same statement for R, , by showing that Q, are partially isometric operators.

Definition 2.1. A bounded linear operator U defined on the Hilbert space if is said to be partially isometric with initial domain M and final domain N if U maps M isometrically onto N and maps M* = if 0M into the zero vector, i.e., if I( Uf(1 = 11f 11 for f E M and )I Ufll = 0 f o r f ~ X ' G M .

441

2. General Time-Dependent Two-Body Scattering Theory

Theorem 2.2. The ranges' of' Q+ are closed linear subspaces of .%; the wave operators 52, are partially isometric with initial domains M, and final domains R, , respectively. Proof. According to the definition of R, , any vector f, strong limit f+ = 8 ( t )f = 8,j

E

R, is the

s;!?

for some vector f E M,

. Moreover, since Q ( t ) is unitary, II Q+fIt =

(2.7)

II QWf II = I l f II

i f f € M, . Hence, the restriction 0, of 52, to M, has an inverse , which is bounded on account of (2.7). To prove that R, is closed, let h E .% be the strong limit of a sequence h , , h, ,... E R,. By (2.7), fl = &lhl, f, = &%, ,... is a Cauchy sequence of elements from M, . Since M, is closed; fl ,f, ,... has a limit f E M, Therefore, h = fl, f E R, , which proves that R, is a closed set. The linearity of R, is an obvious consequence of the linearity of Thus, R, is a closed linear subspace of %,and 52+ maps M, isometrically onto R, . Moreover, if g 1M, , then

.

a+.

sZ+g

= s-lim 8 ( t ) EM+g= 0. t-t-m

Thus, 52, is partially isometric with initial domain M, and final domain R, . The corresponding result for R- is established in a similar manner. Q.E.D.

Lemma 2.1. If U is a partially isometric operator on .% with initial domain M and final domain N, then U* is a partial isometric operator with initial domain N and final domain M, and the following relations are true: (2.8)

U*U

= EM,

(2.9)

UU*

= EN.

Proof. If g 1M, then Ug = 0 and consequently U*Ug = 0. Hence, iff E M, then ( g I U*Uf> = ( U * Q

If>= 0

for all g E MI, and therefore U*Uf E M. Since M is linear, we also have E M for any f E M. On the other hand,

f - U*Uf

( h If - U * U f )

= (h

I f ) - (Uh I Uf> = 0

V. Quantum Mechanical Scattering Theory

442

for all h, f E M (see Exercise 2.2), which implies that f = U*Uf. Thus, U * U f = f f o r a l l f E M a n d U * U g = O f o r a l l g L M , i . e . , U * U = EM and (2.8) is established. We shall prove now that U* is partially isometric. If h E N, there is a unique f E M such that h = Uf, (1 f (1 = 1) h 11. Consequently, U*h = U*Uf = f by (2.8). On the other hand, if h, 1N, then (U*h, If,) = (h, I Ufl) = 0 for all fl E S, since Ufl E N always. Consequently, U*h, = 0 whenever h, 1N. Thus, U* is partially isometric with initial domain N and final domain M. By reversing the roles of U and U* in (2.8) we show that (2.9) is true. Q.E.D.

If we apply this lemma to the operators Q, relations:

,we arrive at the following

I n particular, if M, = M- = M, = 8,we would have Q,*rR, = 1. We shall see in later sections that in problems of physical interest M a ) coincides with the subspace 3Y)@ Zit) of all vectors which are = )8(’) @ where [see (1.2611 orthogonal to the subspace 8:; C is the space spanned by the eigenvectors of the “internal” free Hamiltonian HA;) given in (1.28). Ordinarily, Hit’ has a pure continuous spectrum, in which case 8 i : = (0} and Mi2’ = W 2 ) .

862,

s

OPERATOR 2.3. PROPERTIES OF THE We turn our attention now to the S operator, giving special attention to the practically very important property of its “unitarity.”

Theorem 2.3. The S operator commutes with the free Hamiltonian H , , and Y o u t ( t ) = SYin(t)if Yex(t), ex = in, out, are the strong asymptotic states of some Y(t)= eiH6Y(0). Proof. Let EHo(B)be the spectral measure of H, . Using the intertwining property (2.4) and its equivalent (2.12)

Q**EH(B)= EHO((B)Q**,

derived from (2.4) by taking adjoints, we obtain

B E a’,

443

2. General Time-Dependent Two-Body Scattering Theory

.

Thus, S commutes with H , Since by (2.1) [cf. (1.14)], Y(0)= Q,YeX(O) so that Y(0) = R, n R- , we get Y O u t ( t ) = SPin(t)at t = 0 by using (2.1 1). By virtue of [S, H,] = 0, this stays true for all t E UP. Q.E.D. We recall that Theorem 2.3 has been used in proving that (1.30) is not feasible. In Theorems 2.4 and 2.5 we show that S*S = En, if M, = M, .Thus, the condition that S # 0 coincides with the condition that M, # {0}, i.e., that there are asymptotic states and that the wave operators do not vanish.

Theorem 2.4. The S operator maps the subspace M, into M-, and maps its orthogonal complement &' @ M, into zero; moreover it maps M, isometrically into M- (i.e., 11 Sfll = l l f l l for all f E M,), if and only if R, C R- . Proof. According to Lemma 2.1, the final domain of sZ-* is identical to the initial domain M- of sZ- . Therefore, Sf = sZ-*(sZ,f) E M- for all f E [email protected], iff 1M, , then Q, f = 0 and consequently Sf = 0, which establishes the first statement in the theorem. According to Theorem 2.2, Q, maps M, isometrically onto R,, i.e., 11 sZ,gI = Ilgll for g E M,. By Lemma 2.1, sZ-* maps R- isometrically onto M- . Thus, we have 11 sZ-*sZ+g 11 = 11 sZ+g 11 = I(g11 for all g E M, if R, C R- . However, if R, 0 (R- n R,) # {0}, then the elements of R, 0(R- n R,) are mapped by Q-* into zero, and S = sZ-*sZ+ is not isometric on M, , Q.E.D. The above theorem indicates that the S operator might be unitary if M, = M- = M, and S is restricted to M,. Theorem 2.5 gives an important necessary and sufficient condition for this case to occur.

Theorem 2.5. The S operator is partially isometric with initial domain M, and final domain M- and satisfies the relations (2.13) if and only if R,

S*S = Ell+,

= R-

SS* = En-

.

Proof. Assume that R,

= R- . Using &*Q=t

= Q=t

,

and the identity (2.11), we obtain that (2.13) holds: S*S = Q,*Q-Q-*Q+= Q,*ER-Q+

s s* - Q-*Q Q +

+

= Q+*ER+Q+ = Q+*Q+ = E n + ,

*Q- = Q-*E R+ Q- = Q-*ER-Q-

= Q-*Q-

= En-.

V. Quantum Mechanical Scattering Theory

444

Now, by Theorem 2.4, when R- = R, , the S operator is partially isometric, with initial domain M, and a final domain equal to its range 9ts. Since, according to the same theorem, 9ts C M- , we obtain by using (2.13) and Lemma 2.1 that 9ts = M- . Conversely, assume that the relations (2.13) are true. We get by expressing in (2.13) S*, S, and EM, in terms of the wave operators

sz* :

Q+*Q-Q_*Q+ = Q+*Q+ ,

Q-*Q+Q+*Q-. = Q-*Q-

.

Multiplying the above two relations from the left by Q* and from the right by Q**, respectively, we obtain, after using (2.10) and (2.11), (2.14)

ER+ER-ER+= E i +

= ER+,

Thus, we have for any f E 2'

11 ER+f112 = (f 1 ER+f) = (f 1 E R + E R - E R + f ) = (ER+f

I ER-ERf)

= 11 E R - E R + f l l Z -

The above equality implies (see Exercise 2.3) that ER+f all f E Z , i.e., ER-ER+ = ER+ = Ei+ = ER+ER-.

=

ER-ER+ f for

Inserting the above relation in (2.15), we get ER-

= ER-ER+ER- = Ei-ER+ = ER-ER+ = E R + .

Q.E.D.

In the case of two-body potential scattering, i.e., scattering theory in which H and Ho are essentially of the form (1.8) and (1.9), we will be able to prove that indeed M, = M- = M, and R - R- when the +.potential V(r) fulfills certain conditions. However, it is still interesting to investigate the physical consequences of a theory in which R, # R- , i.e., when the S operator is not a partial isometry from M, to M- . Let us assume that go is a vector which belongs to R, but does not belong to R- . According to the definition of R+ , the fact that go E R+ implies that there is a vector g,, E M, such that go = sZ,gi, , i.e.,

Hence, the interacting state

e-iHt

go has an incoming asymptotic free

445

2. General Time-Dependent Two-Body Scattering Theory

state. This means that e-iHrg, is not a bound state. However, since go$R-,

does not exist. Hence, by Theorem 1.1 e-*H1go has no outgoing asymptotic state. Thus, e-iH1gois a “quasi-bound” state, which is approximately a free state in the distant past, but which does not become a free state again at any time in the future (see Pearson [1975] for examples). On the other hand, if there is a vector g, E R- which does not belong to R,, then the theory possesses an interacting state e-iH1g, which becomes a free state in the distant future, but wich has never been a free state in the past. Thus, the condition of “unitarity” of the S operator, which is usually imposed on scattering theories, is equivalent to the requirement of not having any quasi-bound states in the theory.

2.4. INITIAL AND FINAL DOMAINS OF WAVEOPERATORS Thus far in this section we have never used the basic assumption that a state e-iHolg is called a free state only if it is not a bound state of the Hamiltonian H , . As a matter of fact, all the theorems of this section stay valid if we define M, to be the sets of all vectors f E # for which the respective strong limits of Q ( t ) f , t + co, exist. If M, are defined this way and we choose H identical to H , , we would have Q(t) = 1 and M, = M, = 2,always. However, if H, has bound states, then not all e-%‘f, f E #, are free states, and M, defined in $1 is a nontrivial subspace of A?. Let M, = En( Sf)# denote the closed subspace of A? onto which the spectral measure EH( St) belonging to the continuous spectrum of H projects.

Theorem 2.6. The closed subspaces R, of # are also subspaces and are left invariant by EH(B), i.e., EH(B)f E R, of M, = En(?$)#, for all B E whenever f E R, . Proof. Since R, are the ranges of A?, , we can find for any g , , such that g , = Q, f* . Hence, using (2.4), we get

E

R,

vectors f*

EH(B)gh = QhEHo(B)f* E R, ,

B E L%l,

which establishes that R, are left invariant by H. We shall prove that R, C M, by showing that SZJ, all the eigenvectors of H.

is orthogonal to

V. Quantum Mechanical Scattering Theory

446

Let g be an eigenvector of H with the eigenvalue A , , EH({A,})g = g. Given any f, E M , , we can write (g

so that

I Q*fl>= (EH(P0))g I Qifl) = ( Q * * E H ( { U g I f l > = <EH"{~o)) Q**g

I fl>

= (g

I Q*E"~({U)fl> = 0,

where we have made use of (2.12) to derive the third equality; the last inner product vanishes because EHo({A,})f l = 0, since no f, E M, is an eigenvector of Ho . Since SZ, f 2 = 0 whenever f2 1M, ,we derive, by setting f =f l f 2 , f l E M * , f 2 IM , ,

+

(g

I Qif) = (g I Q*fl>

for any f €8. Hence, R, g o f H . Q.E.D.

+( g I

Q*f2>

=0

are orthogonal to any eigenvectors

We will be able to prove in later sections that in potential scattering with certain classes of potentials, we have R, = R- = M , . I n order to arrive at results of this kind, one usually starts by proving first that R, = R- . This establishes immediately that the S operator is unitary. Now we shall prove a theorem which states a sufficient condition for the identity of R, and R- . This theorem applies to operators with a simple continuous spectrum. The concept of one operator A with a simple spectrum was introduced in 95 of Chapter IV: the self-adjoint operator A has a simple spectrum if the one-element set {A) is a complete set of operators. According to the theorems of 95 in Chapter IV it is equivalent to say that A has a simple spectrum if there is a vector fo which is cyclic with respect to A. If SA, and S< denote the continuous and point spectrum of A, then the closed subspaces XA, = EA(SA,)X and X< = E A ( S < ) X onto which E A () :S and E A ( S<) project, respectively, are left invariant by A. T o see that, note that X = &'A, @ A?< and that

iff E 2 2 a n d g E X< n g A or , iff E X < and g E 2: n g A . The restrictions A, and A, of A to X, and Xp.,respectively, are obviously self-adjoint operators with spectral resolutions (2.16) (2.17)

4 I

A -

XdE?,

EAc(B) = EA(Bn Sf),

A,

X dEp,

EAp(B) = EA(B n S;),

=

2. General Time-Dependent Two-Body Scattering Theory

447

respectively. When A, or A, are operators with a simple spectrum, we say that A has a simple continuous spectrum or a simple point spectrum, respectively. I t is easy to establish that A has a simple spectrum if and only if both its continuous spectrum and its point spectrum are simple (see Exercise 2.4).

*Lemma 2.2. Let A be a self-adjoint operator with a simple spectrum, and let M,, K = 1,2, be invariant subspaces of A, i.e., for any f E M, n gAwe have A f E M, , K = 1,2. If the restrictions A, , K = 1,2, of A to M, have identical spectra, then M, = M, . Proof. Since M, is left invariant by A, the operators A and EM, commute (see Exercise 2.5). Hence, by Theorem 5.6 of Chapter IV, the projector EM, is a function of A, i.e., EM,= F,(A). From the properties EM, = E,*, and E;, = E M , , by using (2.26) of Chapter IV we prove that F,(A) is a real function satisfying

and therefore F,(A) can assume only the values 0 and 1 for A E SA. Thus, if we set Fk(h) = 0 for A 4 SA, we can state that F,(h) is the characteristic function of a Bore1 subset Bk of SA. We obviously have A,

= AEM, = AF,(A).

Hence, the spectrum S A k of A, coincides with B , . Consequently, Q.E.D. S A 1 = S A e implies that F,(A) = F2(A),i.e., EM, = EM,. The following theorem can be very useful in establishing that S is unitary.

Theorem 2.7. If the continuous spectrum of H is simple and M, = M- , then R, = R- . Proof. Let H , be the restriction of H to the closed subspace M, = E H ( S f ) X , T h e operator H , is obviously a self-adjoint operator with a pure continuous spectrum which is identical to the continuous spectrum of H . Hence, according to the assumption, the spectrum S H 1 of H , is simple. According to Theorem 2.6, R, are closed subspaces of M, which H , leaves invariant. Since the spectrum of H I is simple, by (2.4) and (2.1 1) we can apply Lemma 2.2 and infer that R, = R- . Q.E.D.

448

2.5.

V. Quantum Mechanical Scattering Theory

DYSON'S PERTURBATION EXPANSION

We saw in $1 that the knowledge of the S operator is sufficient for the calculation of the differential cross section and of transition probabilities. This shows that a complete solution of the dynamical problem, which would consist in computing the evolution operator U(t, to) introduced in $3 of Chapter IV, is not necessary for the theoretical description of a scattering experiment. Since the computation of Y(t)= U(t, to) Yo for given Yo can be a difficult and intricate computational task, it is desirable to develop methods for computing the S operator in a direct manner. Dyson's perturbation theory provides one such method. It follows from (2.1) that for any f,g E X (2.18) ( f I Q-*g)

= (Q-f

I g)

= !$(Q(t)

EM-f

I g> = 2$,
Since Q*(t) is unitary, we have 11 Q*(t)g 11 = 11 g 11. On the other hand, if g E R- , it follows from the fact that Q-* is a partially isometric operator with initial domain R- (see Theorem 2.2 and Lemma 2.1) that 11 Q-*g 11 = 11 g 11, so that 11 Q * ( t ) g 11 = 11 Q-*g 11. According to Lemma 6.2 of Chapter IV, this result in conjunction with (2.18) implies that 52-*g = s-lim EM-Q*(t)g, g E R- . t++m

= R- , we have

Hence, if S is partially isometric, so that R, S

(2.19)

= EM-

s-lim Q*(t)(s-lim Q(to)EM+)

= EM-

s-lim(s-lim Q*(t) Q(to))EM+,

t0+-m

t++m

t++m

to+-m

where the operator (2.20)

V(t,to) = Q*(q Q(t,)

= eiHot

e-iH(t-to)

e-"ntO

is obviously unitary for any t, to E W. Comparison of ~ * ( t=) eiHot e--fHt

with (3.43) of Chapter IV shows that Q*(t) is the operator which determines the time dependence of the state p(t)in the interaction picture. Hence, we have (see Chapter IV, Exercise 3.4)

(2.22)

H I ( t ) = ei H n t ( H - H,)

for any f E g Hn g x 0 .

449

2. General Time-Dependent Two-Body Scattering Theory

We have seen in 47 of Chapter I V that in potential scattering, under very reasonable assumptions on the potential, g Hcoincides with gH0 . When this is the case, and the vector f belongs to g H3 g H 0we , have exp(-iHoto)fE gH0and exp(iHto) exp(--iHoto) f E g H (see Exercise 2.6), so that Q(to)f E g H= g Hn gH0 and

Thus, for any vector h E &? (a/at)
(2.24)

--i(h

=

IHdt)

v

7

to)f>.

If the function (h 1 H,(t) V ( t ,t o ) f ) is integrable in t, we obtain from (2.24), after having observed that V ( t o ,to) = 1, the following relation: (2.25)

Since (h

v(t7b)f> = ( h I f > I V(t,t J f ) and (h If)

(h I

t

--I

It, ( h I

-i

51

( h I Hdh) v(t1 t ~ ) f > .

t0

are bounded functionals, HI(t1) Vl 9 t o m

4

is defined for any h E &?, it determines for each fixed f E BHo a continuous antilinear functional in h, and therefore by Riesz's Theorem (Theorem 2.3, Chapter 111),it defines a mappingf t+ Wl(t,to)$ It is easy to see that Wl(t,to) is a linear operator defined on gH0 . We write, symbolically, t

(2.26)

=

w1(t7

v(tl

dtl

7

*

t0

In this notation (2.25) is equivalent to the statement that (2-27)

v(t,to)f

I H~(ti) t

=

(1 - i

v(t1 to) dt1)f3

f

9H,,

to

If we treat the above integral equation recursively in V(t,to),we obtain after n steps

v(t,t O ) f

=

[I

vl(t,to)

+ ". f

vn-1(f7

to)

+

wn(t7 t O ) ] f ,

'f

gHo

9

where Vk(t,to) and W J t , to) are defined by the relations

(2.29) W,(t, to) = (-i)n

j

dt, to

I

tl

t

dt, to

r-'

dt, HI(tl)

$0

HI(t,) V(t, ,to),

V. Quantum Mechanical Scattering Theory

450

whose precise meanings are established in the same manner as in the case of W l ( t , t o ) . It has become customary to introduce a chronological ordering operation T by setting (2.30)

T[HI(tl)HI(&)* * ’

HI(tk)l

”’

=

til >, ti, 2

>, ti, ,

**.

This operation orders the time-dependent operators in the product H,(t,) .*.H I ( t k )in a chronological sequence of increasing values of the time variable from right to left, so that the integrand in (2.28) becomes symmetric under permutations of t , ,..., t, when T is set in front of it. Since there are n! such permutations, it is easy to check that

Suppose that

exists for every k (2.33)

=

1, 2, ..., and that Wnf = w-lim(w-lim t++m tp-m Wn(t,t o ) f )

also exists and converges (at least in a weak sense) to the zero vector when n + +m. Under these assumptions we obtain from (2.19), (2.20), (2.30), and (2.31) the following expansion:

sf = EM-(1 + vl

+

v2

f

“‘)EM+f*

The above expansion is known as a perturbation expansion of the S operator. I n the zero order for the case M, = M,, it becomes S = E;, = EM , i.e., it describes the case when there is no scattering. The compdtation to higher orders is facilitated in practice by an explicit knowledge of the commutator [H,(t,), H1(t,)] = D(tl , f,). This makes possible the computation of the integrand of V , by successive use of the relation

T[Hl(tl)H&)l = HI(t1) W

2

)

- O(t2 - t l ) D(tl Y ta).

45 1

2. General Time-Dependent Two-Body Scattering Theory

Computations of this nature occur very frequently in quantum field theory, and techniques have been developed for coping with them in a systematic manner.

2.6. CRITERIA FOR EXISTENCE OF STRONG ASYMPTOTIC STATES There are cases even in potential scattering in which M, = {0}, i.e., when there are no asymptotic states in the sense of (1.7). One notable example is the case of Coulomb scattering, when we are dealing with the Schroedinger operator H with a Coulomb potential. Namely, it can be shown that quantum Coulomb scattering shares with the classical Coulomb scattering the feature of not having asymptotic states in the conventional sense (1.7), though asymptotic states can still be introduced in a somewhat different sense of the word." This particular example indicates that it is desirable to produce some criteria for the existence of strong asymptotic states. The rest of this section deals with the derivation of Theorem 2.8, which will be found very useful in establishing the existence of strong asymptotic states in potential scattering (see Exercise 2.8 and $7.1).

Theorem 2.8. Suppose g , , $9, C BHn g H 0is, a linear subwhich is such that r i H o f f E g Hn gH0 for all f E 9, . manifold of Then the closure a, of $3, belongs to the initial domains M of the wave f operators A2* (Mdler, i.e. strong, operators) if and only if when t , + -c o and t, + co the respective strong limits of

*

+

1

ta

(2.34)

Il(tl, t z ) f =

eiHtH, e-iHoYOtfdt,

H,

=H

-

Ho ,

tl

exist for every f E 9, . A sufficient condition for the existence of these limits is that for some t, , t, E W1

5

ta

-m

+m

II Hl exp(--iH,t)fII dt < +m,

II H1exp(--iHot)fII & < +a,

tl

respectively. To understand the above theorem, we must comprehend the meaning of the integral appearing in (2.34) (see also $3.7). This integral is defined by the requirement that for fixed f E 9,

* For details, see

94.9.

452

V. Quantum Mechanical Scattering Theory

determines a continuous linear functional in g. I n fact, if the integral in (2.35) exists, then, by Riesz’ theorem (Chapter 111, Theorem 2.3), there is a unique vector f, = Il(tl , t2)f such that (fl I g ) = $r(g;t, ,t2). Thus, in order to know that Il(tl , t2) is well defined we only have to establish that the Riemann integral in (2.35) really exists. For this task we need Lemma 2.3. For any two vectors f E gH0 and g E g Hthe function

Lemma 2.3.

(2.36) F(t) = (exp( --iH,t)f I H exp(--iHt) g) - ( H , exp(--iH,t)f I exp( --iHt) g)

is continuous in t E R1, and (2.37)

f;((exp( -iHot)f =

I H exp(- - i ~ t g> ) - ( H , exp(- - i ~ , t ) f I exp(- - i ~ t g)) ) dt

-i ( [ W 2 ) - W1)l.fI g>.

Proof. T o establish the continuity of the first term in (2.36), we note that for any r E R1 (2.38) I(exp[--iH,(t .)If I H exp[--iH(t .)I g> - <exp(--iH,t)f I H exp( --iHt)g)l < I(exp[--iHo(t .)If1 H(exp[--iH(t .)I - exp(--iHt))g)l I((exp[--iH,(t T)] - exp(--iH,t))fI H exp(--iHt)g)l

+

+

+

+

+ + d llfll II H(exp[--iH(t + 41 - exp(--iWg II + Il(exp[--iHo(t + 41 - exp(--iH,t))fII II H exp(--iHt)g

II.

We recall now that by Theorem 2.7 of Chapter IV, the operator H(exp[-iH(t T)] - exp(-iHt) is a function of the self-adjoint operator H, and consequently

+

H(exp[-iH(t

+

T)]

- exp(--iHt)) g

= (exp[-iH(t

+

T)]

- exp(--iHt))

Hg.

Since exp(-iHt) and exp(-iH,,t) are strongly continuous operatorvalued functions in t (see Chapter IV, §6.3), the expression on the righthand side of (2.38) converges to zero when r --+ 0. T h e continuity in t of the second term in (2.36) is even easier to prove by following the same line of reasoning, since we have

+

+

( H , exp[--iH,(t .)If I exp[--iH(t 713 g> - ( H , exp(--iH,t)fI exp(--iHt)g)l II Ho(exp[--iH,(t 41 - exp(--iH,t))fII Ilg II II H , exp(--iH,t)f!I Il(exp[--iH(t .)I - exp(--iHt))g (I.

<

+

+

+

2. General Time-Dependent Two-Body Scattering Theory

453

The continuity of F ( t ) in the variable t implies the existence of the integral in (2.37) for any given t , , t, E W. Thus, the first part of the lemma is established. To establish the second part, we recall from Theorem 3.1 of Chapter IV that whenever g E gA 2

s-llm 7 (exp[-iA(t

+

T)]

- exp( -iAt)) g = A

exp(-iAt) g

for any A = A* in general, and consequently for A = H and A = H , in particular. In view of this fact, a glimpse at (2.36) suffices to show that

I

~ ( t= ) i ( d / d t ) ( e P o y e-"ntg).

Hence, (2.37) immediately follows. Q.E.D. Let us return now to the proof of Theorem 2.8. When

then exp(--iH,t)f (exp( -iHot)f

f €91 9€f BHe E .%,ngH, so that I H exp(-iHt) g) = (exp(iHt) H exp( -iH,t)f I g),

and F(t) can be written in the form F(t) = (exp(iHt) H , exp( -iHot)f 1 g).

Thus, according to (2.37), we have (2.39) ([Q(t,) - Q(tl)]f I g)

=i

It'

(exp(iHt) Hl exp( -iH,t)f I g) dt

tl

for any f E g1and g E g H .Since gHis dense in A?, the integral on the right-hand side of the above relation can be said to determine a unique linear functional in g E 2.Thus, we can write symbolically

It follows from the above relation that in the case s-lim Q ( t ) f exists for t --+ -00 or t + 00, the respective strong limits of the integral in (2.34) for tl + - 00 or t, + 00 also exist. Conversely, let us assume that

+

tp-m

+

exp(iHt) HI exp(-iHot)f dt

V. Quantum Mechanical Scattering Theory

454

exist for all f E g1, Then, it follows from (2.40) that s-lim Q ( t l ) f also exists when t , --t - co andfrz g1. However, by Lemma 1.1, the set of all vectorsf for which this last limit exists is closed, and consequently the closure 6, of gl must be contained in it. Thus, the theorem is established. For all practical purposes, the investigation of the existence of the strong limits in t , and t , of the integral in (2.34) reduces to the investigation of the existence of integrals

for some t , E W,and of

for some t ,

E

W.I n fact, by using the Schwarz-Cauchy

inequality we get

(2.43)

Hence, the existence of (2.42) implies, in conjunction with (2.39), that f+ = w-lim Q ( t l ) f exists when t l + -a.Since 11 Q(tl)fll < l l f l l = ~ ~ fwe + infer ~ ~ by , using Lemma 6.2 of Chapter IV that s-lim Q(tl)f also exists for t , + - 00. The argument for t , + + co is completely analogous.

2.7. THEPHYSICAL ASYMPTOTIC CONDITION The condition which the Schroedinger-picture strong asymptotic states Y e x ( t ) (ex = in, out) of the interacting state Y(t)have to satisfy,

has no direct physical meaning, since the above vector norm is not a measurable quantity. We recall from §§l and 2 of Chapter IV that the directly measurable quantities are the probabilities

455

2. General Time-Dependent Two-Body Scattering Theory

for any set of compatible fundamental” observables. Hence, we expect asymptotic states to satisfy some kind of physical asymptotic condition which would specify that the probabilities (2.45) for the interacting state Y ( t ) approximate arbitrarily closely the corresponding probabilities for the respective asymptotic states Yin,out(t),when t + fa.That this is actually the case is stated in Theorem 2.9.

*Theorem 2.9. Let the Hamiltonian H and a set 0, of additional fundamental observables be such that any other observables of the interacting system or the free system is a function of compatible observables from 0,= {H}u 0, or 0teX= {H,} u 0, , respectively. A,’,..., A,‘ E 0,, Furthermore, assume that whenever {H,Al’,..., Am’}, is a set of compatible observables, then H , , Al’,..., A,’ are also compatible. I n that case the strong asymptotic states Y e x ( t ) (ex = in, out) of the scattering state Y ( t ) satisfy the physical asymptotic conditions

for any sets {A, observables.

,..., A,}

C 0,, { H , A,‘,..., A,’} C

Of of compatible

Proof. It is straightforward (see Exercise 1.7) to prove that (2.46) holds, i.e., lim I(Y(t) I

t+fm

EA1*-.*Am(B)

Y(t))- (Yex(t)

I EA1****+”’ ( B )Yex(t)>I = 0

when (2.44) is satisfied. Since Al’ ,..., A,‘ commute with H as well as with H , , we infer that for arbitrary B2€93, eiHL

e-iHotEAl‘.

...,A,’ (B,) = EAI’.....A,’(B,)

e i H t e-%L.

By applying the operators on both sides of the above equation to vectors + fa, we obtain immediately E A 1 ‘ 9 . . . * “ - ’ ( B Z ) f ~ M, and

j~ M, and letting t (2.48)

Q&Al’.---.~‘(B,)

f = EAI’*.--*An

(B,) Qif.

* See the discussion in the beginning of $2 in Chapter IV concerning fundamental observables.

456

V. Quantum Mechanical Scattering Theory

Furthermore, if g

1M, ,then

(EA”.””A”’(B,)gIf)

=(

g I EA1‘”’*.””‘(B2)f)

=0

for all f~ M, , and therefore E A 1 ’ * * * * * A n ’(B,)g 1M, . Consequently, (2.48) holds for arbitrary f E S. Combining this relation with (2.4), we conclude that (2.49)

Q,EHO(B,)

E”1’.***.AqB2)=

EH(B,) E A ” . . * . . A ” ’ ( B , )

Q,

for any B, E B1and B, E an. As a, consequence of the basic properties of spectral measures (see Chapter 111, the proof of Theorem 5 4 , we can write for any B E B n + l

where the projector infimum is taken only over finite unions of sets Bik) x Bkk’, with Bik)E B1,Bkk)E Bn. Hence, E H o * A 1 * . * * , A n ( B ) is the strong limit of some sequence of finite sums

c

E~l’.....~n’(~f))

k

corresponding to a sequence of sets uk(Bik)x Bkk’),with B as an intersection. This result in conjunction with (2.49) implies that

(2.50)

.....A.,‘(B)

Q*EH~.A~‘

= EH.A~‘,....A,,’(B)

Q&

2. General Time-Dependent Two-Body Scattering Theory

457

where the last step follows from the fact that 52,*Q,YeX(O) = Yex(0), since Yex(0) E M, . Hence, (2.47) is established. Q.E.D. All nonrelativistic quantum mechanical theories satisfy the conditions of the above theorem. For example, in wave mechanics for a single spinless particle interacting with a potential V(r), one can take H 2 -(1/2m)A

+ V(r)

and 0,consisting of L2, L,(n E R3, I n I = l), Qkand Pk(k = 1,2,3); here, L2denotes the total angular momentum operator (encountered in $7 of

-

Chapter IV), L, = n L is the angular momentum projection in the direction n, and Qk , Pk are the position and momentum components with respect to the Kth axis of a Cartesian inertial reference frame. From the physical point of view, it is sufficient if asymptotic states satisfy the physical asymptotic conditions (2.46) and (2.47), rather than (2.44). As a matter of fact, albeit that for long-range potentials (such as the Coulomb potential or V(r)-rO,0 < 01 < 1) there are no free states satisfying (2.44), there are free states satisfying (2.46) and (2.47). One could call such states physical asymptotic states, and develop" a scattering theory based on the existence of these states. As a matter of fact, in this case one could define the wave operator Q+ to be the operator which maps an incoming physical asymptotic state into the corresponding interacting state, i.e., as the operator satisfying (2.51); Q- could be defined in a similar manner. As soon as the wave operators are defined, one can proceed in the usual manner and introduce the scattering operator by means of (2.2) and the transition operator by the old formula (2.53)

T

= (1 - S)/2ni.

For the rest of the quantities of importance in scattering theory (transition probabilities, differential scattering cross section, etc.), one can also retain the old definitions (see $4.9).

EXERCISES 2.1. Assume that C is a bounded operator in &', and A, , A, are self-adjoint operators in 2 with spectral measures EA1(B)and E"e(B), respectively. Prove that the relation eiAltC = CeiAlt is satisfied for all t E R1 if and only if E"i(B)C = CE"a(B) for all B E @.

* For details see the articles by PrugoveEki [1971a, 1978b], which also contain further references.

458

V. Quantum Mechanical Scattering Theory

2.2. Show that if U is a partially isometric operator with initial domain M, then (Uf I Ug) = (f I g) for all f,g E M. 2.3. Show that if E is a projector and then f = Ef.

I( Ef (1

=

llfll

for some f E X ,

2.4. Show that fo is a cyclic vector with respect to A if and only if EA(St)fo is a cyclic vector of A, and EA(Sg)fo is a cyclic vector of A, , 2.5. Prove that if a self-adjoint operator A leaves the closed linear subspace M invariant, then [EA(B),EM]= 0 for all B E B1. 2.6. Show that if A is self-adjoint and f E g A then , eiA6f E g Afor all t E W. 2.7. Suppose that the potential V(r) is Lebesgue square integrable on R3 so that 1) V 11 < m. Apply the result of Exercise 1.9 to prove that if #(r) is Lebesgue integrable as well as square integrable on R3,then

+

almost everywhere in OX3, and consequently

where ( Ve-iHot#)(r)

=

V(r)(e-iHol +)(r).

2.8. If the potential V(r) is Lebesgue square integrable on W3, then M, = M- = L2(R3)in potential scattering for V(r). Explain how this statement follows from Theorem 2.8 and Exercise 2.7. 3. General Time-Independent Two-Body Scattering Theory 3.1.

THERELATIONOF TO THE

THE

TIME-INDEPENDENT

TIME-DEPENDENT APPROACH

It is possible to construct the Mdler wave operators Q, by taking limits with respect to parameters other than the time parameter t. Scattering theory based on such time-independent procedures for computing SZ,, S, and T is called time independent or stationary. The relation of the time-dependent to the time-independent approach is best exhibited by showing that SZ, can be obtained as strong limits of operator functions Q, for E -+ +0, where the parameter E ,

3. General Time-Independent Two-Body Scattering Theory

459

0 < E < + 00, is not related to the time parameter t and has no physical meaning, and the operators SZ,, satisfy the equations (g

(3.1)

(g

for allf, g E X and 0 < E < +a. Let-us-show that the relations (3.1) can play the role of definitions of linear operators SZ,, . Since eiHLand e-iHot are strongly continuous functions of t (see 96 of Chapter IV), Q ( t ) = eiHt r i H o f is also strongly continuous (see the function Chapter 111, Exercise 5.6). Hence, for any f, g E S, (g I Q(t)f>

=

(g I eiHte-"O"f>,

tE

UP,

is continuous, as well as bounded in t:

l(g I ea'Ht e-*Hot f >I

< IIg II II eiHte-iHo"fl

= llfll

IIg II.

Consequently, the first improper Riemann integral in (3.l),

1:"

(3.2)

e-ct(g

I eiHt

-iHot

e

f>dt =

(glf),

converges absolutely for any f, g E S:

Thus, the functional ( g I f ) is defined on X x S. It is straightforward to verify that ( g I f ), f,g E S, is a bilinear form. Furthermore, we see from (3.3) that 1 is bounded. Hence, there is a unique bounded linear operator (see Chapter 111, Exercise 2.5), which we denote by SZ-, , such that ( a

(3.4)

(g 1 Q - f >

=

(g I f ) = c

J

a)

m 0

e-St(g 1 eiHte

f>dt

for allf, g E X . We write the above relation in the symbolic form

I

m

(3.5)

SZ,

=E

0

eCt eiHte-iHotdt,

E

> 0,

V. Quantum Mechanical Scattering Theory

460

under the agreement that its real meaning is given by (3.4). I n fact, (3.5) is an example of a Bochner integral (see Theorem 3.4 in Appendix 3.7 to this section). It can be shown in a completely analogous manner that for any E > 0 there is a unique bounded linear operator Q,, which satisfies

1 0

( g IQ f f )

(3.6)

=E

eft(g I eiHt e-iHoytf>dt

-m

for all f,g E 2.The above relation is written in the symbolic form Q ,

=

1'

ert e i H t

e-iH t dt,

E

> 0.

-m

We are interested in the existence of the strong limits of Q*,f

+o.

€+

as

From (3.3) we immediately infer that

By applying Lemma 1.1 we deduce that the sets N, and N- of all vectors f E S for which the respective limits of Q, f exist, N+ = {f: s-lim S+JE X ] €++O

are closed linear subspaces of 2. We shall prove now that and M- C N- (the converse not being generally true!).

M+ C N+

Theorem 3.1. If for some f E X , s-limt++mQ ( t ) f exists, then s-lim,++oQ-, f also exists, and (3.9)

s-lim S ( t ) f t++m

= s-lim f2-J €++O

Similarly, if for some g E S, s-limt+-m Q(t) g exists, then s-lim,++,, Q+,g also exists, and (3.10)

s-lim Q(t)g t-r-m

=

s-lim Q,g. €++O

Proof. Let us write f-

= s-limQ(t)f t++m

461

3. General Time-Independent Two-Body Scattering Theory

when the above limit exists. Then, for any g E A? l(g I Q-,f

I

= c

-f->I

J

*

co

e-€t(g I ~ ( t ) fdt > -c

e-ct
0

if->

dtl

We set out to prove that the above integral is smaller than any given > 0 for all sufficiently small c > 0. Choose some a > 0 and split the interval of integration:

q

f; 1: =

+

1:

*

For the first integral we obtain the following upper bound:

Now, take some a satisfying 1

o<.
E

2 llfll

+ Ilf-

II '

> 0 so small that II Q(4f -f- II < rl/2

for all t >, a/€.For any such

E,

Consequently, for any given 7

we compute that

> 0 we have -f->I < rl Ilg II

l
for all E > 0 which are such that (3.11) is true whenever Hence, we have established that (3.12)

f- = w-lim Q-, f. €-MI

t

>, a/€.

V. Quantum Mechanical Scattering Theory

462

Since 11 f - 11 = lim,,,, 11 Q(t)f 11 = 11 f 11 >, 11 Q-,f 11, we have shown that the weak limit (3.12) is also a strong limit (see Chapter IV, Lemma 6.2), f-

= s-lim Q-,f, r++O

i.e., (3.9) is true. The proof of (3.10) proceeds along identical lines.

Q.E.D.

As an immediate consequence of the above theorem we obtain the important result (3.13)

Q, =

s;>+y Q+EE M &

3

which can be considered as a time-independent alternative definition of the wave operators. It should be noted, however, that Theorem 3.1 establishes only that M, C N, , and not that M, = N, . Since the conconverse of Theorem 3.1 [which would state that the existence of s-lim,+&J implies the existence of s-lim,+,,L?(t)f] is not generally true (see Howland [1967]), it has been suggested (see Jordan [1962a, Section 61) that 42, could be defined in a time-independent manner, by taking limits of Q,,f, even in cases when the time-dependent definition (2.1) is not adequate because M, = (0). In potential scattering, for classes of potentials which have been considered in the last section, the above problem does not arise, since in those cases M, = X , and consequently Q* = s-limQ(t) = s-limQb t+Tm

€++O

From (3.13) we get for any f,g E X

Since Q,* are partially isometric with initial domains R,, verify that


I g> = !$$,<EM&QzERlf

we easily

I g>

by noting that for f E R, the above relation is equivalent to the relation preceding it, while for f 1R,, the expressions on both sides of the above relation vanish. In view of the fact that

463

3. General Time-IndependentTwo-Body Scattering Theory

we infer from this result (by using Lemma 6.2 of Chapter IV) that Q** = s-lim En,Qf,ER*

(3.14)

3.2.

r++0

.

LIPPMANN-SCHWINGER EQUATIONS IN HILBERT SPACE

One of the advantages of the time-independent limits (3.13) for determining Q, is that they lead to integral equations which can be used to compute Q, without having to compute Q(t). Using the spectral decomposition H = J h d E f and recalling that eiH' is a function of H, by employing Theorem 2.1 of Chapter IV we obtain (3.15)

(g 1 Q - f )

I, m

=E

e-'t(g

I eiwt e

f>dt

Now, we would like to interchange in the above relation the order of the integrations in A and t, and write (3.16) ( g I Q , f )

=a

1 dA1 exp[-(a - ih)t](g I E f e x p ( - i H o t ) f ) dt. +m

m

--oo

o

However, before we ask whether such an interchange of the order of integration is correct, we must make clear the meaning of the integration in h occurring in (3.16). We will show that the interchange of the order of integration can be easily justified if the integral in A is taken to be a Riemann-Stieltjes integral?

Definition 3.1.

T h e Riemann-Stieltjes integral

of the complex function F(h) with respect to the complex function .(A), a h b, is said to exist if for any sequence of subdivisions a = h'0) < h'l' < < A(") = b of [a,b] which is such that the norm 6 = maxk,l,...,n(X(k) - h'k-l)) of the subdivisions in the sequence

< <

...

"Measure theory (i.e., Fubini's theorem) cannot be applied since in the resulting

formula (3.30) the Riemann-Stieltjes integral cannot be interpreted as an integral with

Indeed, if such a measure existed, then, by the Riemannrespect to a measure on (Rz,a'"). are absolutely Lebesgue lemma we would obtain that = 0 if :S and continuous.

Sp

V. Quantum Mechanical Scattering Theory

464

tends to zero, and for any choice of A'(k) E [A(k-l), A ( k ) ] , k = 1,..., n, the limit of Riemann-Stieltjes sums n

(3.17)

l i i

1 F(hl(qu(X(k))- u(P-1))) K-1

exists, and is independent of the selected sequence of subdivisions; if the integral (3.17) exists, then its value is taken to be (3.18)

If F(A) and .(A) are defined on R1, then the improper Riemann-Stieltjes integral on W1 is defined as (3.19)

provided that the two limits in a and b exist. For functions F(A, &) and &) of two variables A, A. E R1, we define the cross-iterated RimannStieltjes integral on [a, b] x I , where I is a nondegenerate interval in W1, by requiring that for any of the above sequences of subdivisions of [a, b], Q(A,

where the above limit should be finite and of a value which is independent of the chosen sequence of subdivisions. It is easy to relate Riemann-Stieltjes integrals to integrals with respect to measures when F(A) and .(A) are Bore1 measurable functions (see Exercise 3.3). It is equally easy to prove in a direct manner that most of the usual properties of Riemann integrals (which are obviously special cases of Riemann-Stieltjes integrals in which .(A) = A) are also properties of Riemann-Stieltjes integrals (see Exercises 3.4-3.8). We find the concept of Riemann-Stieltjes integration of importance because it enables us to give a meaning to the integral in (3.16)) namely, the integral in (3.16) can be written as a cross-iterated Riemann-Stieltjes integral with respect to the function (see also Definition 3.4) (3.21)

465

3. General Time-IndependentTwo-Body Scattering Theory

Let us introduce the following convenient notation: (3.22) o((kA'] x (A,

9

4'1) =

+', A,')

-

dh',43) - 4, A,')

+

o(A,&).

We can establish that (3.16) is true by means of Lemma 3.1.

*Lemma 3.1. Suppose the real function o(A, A,) is of bounded oariation on [a, b] x [c, d ] , i.e., that for any A E [a, b] and A, E [c, d] we have

2 2 I U((A(~-'), A'"] m

(3.23)

w(A, A,)

= sup

n

x A(1-:)

i=1 k=l

where the supremum is taken over all subdivisions (3.24) a


=

< ...
= A,

c =$ ,)'

<*)

< ...

=

,),,

of [a, A] x [c, A,]. If the complex function F(h, A,) is continuous on [a, b] X [c, d ] , then both its cross-iterated Riemann-Stieltjes integrals with respect to o(A, A,) exist and

Proof. It is easy to see that the functions o&,U

=

! M A , A,)

+ 4%4 f 4, &)I

are nondecreasing in [a, b] x [c, d ] in the sense that u*( (A, A'lx (A,, A,,'I)>O whenever A A' and A, A,'. Since

<

<

4 4 4)) = o+@, A,)

- .-(A,

A,),

we can exploit the linearity properties (see Exercises 6.4 and 6.7) of cross-iterated integrals to reduce the integrals in (3.25) to a sum of integrals of real functions with respect to a+(A,A,) and o-(A, A,,). Hence, it is sufficient to prove the lemma for the case of a nondecreasing .(A, A,,) of bounded variation and for real F(A, A,). By using the mean-value theorem for Riemann-Stieltjes integrals (see Exercise 3.8), we obtain

V. Quantum Mechanical Scattering Theory

466

<

for some AAk-l) Ah(k) relation (3.20), we get

m

=

lim

<

Combining this result with the defining

a

C C ~ ( x ( ’A?)) ) , u ( ( ~ ( i - 1 ,) A‘”’]

x

(@-I),

At’])

i=l k = l

in the limit of finer and finer subdivisions (3.24) of [a, b] x [c, d]. The existence of this limit and its independence of the chosen sequence of subdivisions {(A(i), Ahk))} is obviously a necessary and sufficient condition for the existence of the integral. T o establish the existence of this limit and its independence on the sequence, it is sufficient to note that if {(A(c*j), Ahk*’))) is a subdivision of [u, b] x [c, d], which contains and is finer than the subdivision {(Ace), gk))}, and if (A’(ivj), A ~ c k ~ zare ) ) points within the meshes determined by {(A($lj), gk9 ‘I)},then due to the uniform continuity of F(A, A,) we can achieve for any given E > 0

I ~ ( x ‘ i . .A;(k,Z)) ”,

-

~ ( x ( ix$k))l ); <

for all sufficiently fine subdivisions {(A(i), ASk))}. Consequently, if i(iJ) and iAk*’) denote the points preceding A(iJ) and hik,’), respectively, in the subdivision, then

= EU((U,

b] x (c, d ] ) .

This establishes the existence of the considered limit, as well as its independence of the sequence of subdivisions. By the same token

467

3. General Time-Independent Two-Body Scattering Theory

also exists. By exploiting again the uniform continuity of F(A, A,,) [a, b] x [c, 4, we conclude that for any > 0

I F(A"(f), y(k))

- F(X(<),&(k))(

in

<

for all sufficiently fine subdivisions (3.24) of [a, b] x [c, 4. Hence, the absolute value of the difference of the two integrals in (3.25) is smaller than

cc m

<€

n

o((A(i-1)

,$"I

tgk-", gk)])

1-1 k-1

= CU((U,

b] x (c, d ] ) .

Since this is true for arbitrarily small values of that (3.25) holds. Q.E.D.

E

> 0,

we conclude

Let us compute the total ~ ~ ~ i a tfunction ion o(A, A,), defined by (3.23), for the function .(A, A,) given in (3.21). By using in the process a decomposition of the type (5.13) in Chapter 111, we easily derive

Since the integrand in the above integral is continuous and therefore integrable, it follows that o(h, A,) is of bounded variation. Consequently, Lemma 3.1 can be applied to infer that

V. Quantum Mechanical Scattering Theory

468

To establish the validity of (3.16), we have to prove that the above relation remains true in the limit when a + - co and b, T + CO. In view of the relations (see Exercise 3.5)

+

1:" (3.27)

-

sI1s:"

f o

--m

-

=

,:s:

1:"

J+m -m

Q

=

:s 1:"

+

s: /Im s: s":. +

+m

+ JYm

J: 1, +

7

10

'

and (3.26), we see that this will be the case if and only if the integrals on the right-hand side of the equalities (3.27) exist and vanish when a + -CO and b, T --t +a. Employing the procedure used in deriving (3.26) we arrive at the following estimate:

+ (e-jKOt + ig I P ( ( a , ,b , ~ ) ( e - ~+~ig)) o~

+ G<e-iHotf I E%,,

,boll e-"otf)

+ G ( g I ~ " ( ( a,, b,l)g)} dt.

By Lemma 3.1 in Chapter IV, the integral on the right-hand side of the above inequality can be made arbitrarily small, e.g., for sufficiently large values of either c, (for arbitrary values of do > c,), or of a, (for arbitrary values of b, > a,). Since, according to Lemma 3.1, the order of integration in h and t can be reversed without changing the value of this integral, we conclude that all integrals on the right-hand sides of the relations (3.27) vanish in the limit when a + - 00 or b, T + m; for example, for the representative case of the first of these integrals, we can choose for given T > 0 a r ( ~and ) a sequence T(T) = T, < r1 < r2 < ... diverging to infinity which is such that

+

and consequently

3. General Time-IndependentTwo-Body Scattering Theory

469

Thus, we conclude that (3.26) stays valid in the limit a + -a and b, 7 + +a,and therefore (3.16) is true. Using the fact that e-iHot is a function of Ho , and applying Lemma 3.1 in the same manner as before to interchange orders of integration, we arrive at the result

I exp[-(e I m

- ih’)t](g

1 EFexp[-iHot] f ) dt

0

m

=

dt exp[-(e - ix)t]

0

=

Smexp[-(4

dAo

0

I

+m -m

- ih’

exp[-d,,t]

dAo(EFgI E2f)

+ i&)t](E\”g I E2f) dt

Inserting this result in (3.16), we obtain

I n a completely analogous manner we can derive a corresponding relation for Q+,:

The two relations (3.28) and (3.29) are equivalent to the single relation

if the parameter E > 0 is replaced by the new parameter 7 # 0, which can assume also negative values. Using the defining formula (3.20) for cross-iterated integrals, we obtain from (3.30) (Qn*g

If) = (g I Q n f )

where the last integral is deJined by the limit preceding it.”

* See also Definition 3.4 for integrals of this type in Appendix 3.7 to this section.

V. Quantum Mechanical Scattering Theory

470

According to Theorem 2.2 of Chapter IV,

and consequently we have (3.31)

+

+

Since iq(A - H, iq)-l and ( A - H,)(A - Ho iq)-l are bounded functions of H , , they are defined everywhere in &. According to Theorem 2.4 of Chapter IV, we can add and multiply these operator functions as we would ordinary functions, and therefore we have the identity h - Ho i.l (3.32) h - Ho i7 h - Ho iq

+

+

Substituting the right-hand side of the above equation in (3.31), and using symbolic notation (see also Appendix 3.7 to this section), we arrive at the following expression for Qn*: (3.33) Consider a free state e-iHotY, Y E M, , which is the incoming asymptotic state of e - W P or the outgoing asymptotic state of eciHtY-. Since obviously y* = s-~imeiHt e--iNOtY =&Y, t4Fm

we deduce from (3.14) and (2.10) s-lim EM*Qn*Y* = Q+*Q*Y= Y. $+fO

By Theorem 3.5 in Appendix 3.7 this result, combined for M, = 2' with (3.33) leads, after a rearrangement of terms, to the equations (3.34) Y* = Y

+ d i m S,

+m

n+O

- Ho

+ iq HI d,EfY*,

HI

=H

-Ho.

The above two relations can be considered to be equations for determining Y- and Y+,respectively. We shall refer to them as LtppmannSchwinger equations in Hilbert space. Indeed, in the next sections we shall see that in the special case of potential scattering these equations

3. General Time-Independent Two-Body Scattering Theory

47 1

lead to a commonly encountered form of integral equations for distorted plane waves, usually called Lippmann-Schwinger equations. 3.3.

SPECTRAL INTEGRAL REPRESENTATIONS OF WAVEOPERATORS

The main drawback of (3.34), viewed as equations for finding Y* when Y is given, is that they require a knowledge of the spectral function EF of the total Hamiltonian H. Now, in practice, E f is an unknown quantity which is very difficult to compute. In fact, if E f were known, then one could easily compute the time evolution operator U(t, to) = exp[-iH(t - to)], and thus solve the dynamics of the problem generally, without even having to resort to scattering theory techniques. A practically more suitable set of equations relating Y and Y*,which yield themselves to perturbation techniques, can be derived from (3.1) by applying the spectral theorem first to Ho and then to H, and otherwise proceeding in the same manner as in the derivation of (3.16): +W

(3.35)
=E

dte-'t 0

I

t

+w e-iA O

dAo<e-iHtg I

-W

E2f)

Hence, we end up with the relation

An analogous relation can be derived for Q, . After reintroducing the parameter q # 0, we can write (3.36) and the corresponding relation for Q+, jointly in the symbolic form (3.37)

Naturally, if we exploit the relation (3.38)

-iq

H-h,-iv

= I -

H - b H--h,-iq'

V. Quantum Mechanical Scattering Theory

472

(3.37) can be recast as (3.39)

Hence, if we again write (3.40)

Y* = Q*Y = s-lim QnY, V+*O

Y E M*

,

then in view of (3.39) and of Theorem 3.5, HI = H

(3.41)

- Ho.

The above two equations are the solution-type Lippmann-Schwinger equations in Hilbert space, since they can be viewed as solutions of (3.34). However, they actually are not explicit solutions of the equations (3.34) since they contain the operator function (A, - H iq)-l (H - Ho),.which is unknown in practice. On the other hand, we shall see later in this section [see (3.67)] that (3.41) can be solved by iterative procedures. Let us write (3.39) in the form

+

Taking adjoints on both sides, we obtain

In symbolic notation the above relation assumes the form (3.43)

which will be very useful further on. 3.4. THETRANSITION AMPLITUDE It was mentioned in $1 of Chapter IV that if Y+(t)and Y J t ) are states in the Schroedinger picture, the I (Y+(t)I Y-(t)) l2 is called the transition probability from Y J t ) to Y+(t); furthermore, the complex number (Y+(t)I Y - ( t ) ) is called the transition (probability) amplitude from Y J t )to Y + ( t ) .

3. General Time-Independent Two-Body Scattering Theory

473

Let us assume that Y+(t)and YJt) are scattering states, with asymptotic states Yy(t)and !PEx(t), respectively. Since Y*(O)= Q+Y?(O)= Q_YFt(0),

we easily obtain (Y+(t)I Y-(t))= (Y+(O)I Y-(O))= (Yy“0)I SYF(0)).

Let us simplify the notation by writing Y!n(O)

=

Yi ,

YYt(0)= Yf ,

where the indices i and f stand for “initial” and “final,” respectively. Using the properties of strong limits, we obtain

Substituting the expression (3.39) for 52-, in (Q-, Y’f

I Q+?Pi>, we get

On the other hand, after using (3.43) to express the identity operator in terms of the other two operators occuring in that equation and then applying it to the vector 52+Yi, we arrive at the relation

After substituting the above expression for Q+Yiin the first term on the right-hand side of (3.45) and carrying out an obvious summation, we obtain

In view of (3.44) and the fact that if M+

=

M- ,

V. Quantum Mechanical Scattering Theory

474

we deduce from (3.47) that

I n terms of the transition operator T = - 1-s (3.50) 2rri ’ and provided that M, = M- , we can write (3.49) in the form (3.51)

(lui I TYi)

by taking advantage of the intertwining property (2.4). Using Lemma 3.3, we immediately obtain from (3.51) (3.52)

1

1

{Yf1 TYi) = rr- r++O lim (Yf j+m dAoE2HlQ+ -m

€

(H, -

+

e2

It is interesting to note that on account of the relations (3.53) (3.54)

in Yi = Y(0) = s-lim eiHot e-%P-(O), t+-m

=

~ +‘ ~ ~=( s-llme 0 . ) iHOt e- i H t y +(o), t++m

the vectors Yf and !Pi can be considered to be the asymptotic states of the interaction-picture states [see (3.43) of Chapter IV] (3.55)

P+(t)= eiHot e-”tYi(0),

respectively. Thus, it can be said that (3.52) provides an expression for the transition amplitude (!Pf I TY,) from the incoming interactionpicture asymptotic state Yi to the outgoing interaction-picture asymptotic state Yf .

3.5. THERESOLVENT OF

AN

OPERATOR

In the Lippmann-Schwinger equations (3.34) and (3.41) we encountered operators of the form (A - h - i~)-l, where in the first case A stands for H , and in the second case A stands for H . Operators of the form (A - [)-l, where [ is in general a complex number and A is

3. General Time-Independent Two-Body Scattering Theory

475

a linear operator in S,play a very important role in functional analysis. Since they have been systematically studied, it is advisable to acquaint oneself with the results of such studies, and then apply these results to the special problems at hand.

Definition 3.2. If A is any linear operator in A?, the resolvent It,,(() of A is the operator-valued function

&’, 5 E @,’ defined at all complex values of 5 at which (A - {)-’ exists.

(3.56)

RA(5)

= ( A-

We note that, according to Theorem 4.2 of Chapter 111, (A - {)-l exists if and only if the equation (A - 5)f = 0 has the unique solution f = 0, i.e., if and only if 5 is not an eigenvalue of A. We can classify the points 5 in the complex plane C1in relation to a densely defined linear operator A as follows.

Definition 3.3. The set of complex numbers 5 consisting of all points 6 E C1at which the resolvent RA(<)is a bounded operator defined densely in A? is called the resolwent set of A . The complement of the resolvent set is called the spectrum S A of A . The spectrum S A is the disjoint union of the point spectrum S<,continuous spectrum :S , and residual spectrum S :, which consist of all points which have the following respective properties: Sg of the points 5 E S A at which RA(5)does not exist; St of all 5 E S A for which RA(5)exists, is unbounded and defined on a dense subset of A?; Sf of all 5 E S A at which ItA(() exists but it is not defined in a dense subset of A?. In the case where A is a self-adjoint operator, the above definition of the spectrum coincides with Definition 6.2 of Chapter 111. T o establish this equivalence, we need to show that (3.57)

Iff belongs to the domain of (A - [)-I, there is a vector g E 9”such thatf = (A - 5)g. Consequently, we have

Conversely, if

V. Quantum Mechanical Scattering Theory

476

then, according to Theorem 2.5 of Chapter IV 1 1 -dEff A-[f=I,ih-[

(3.58)

is defined. Moreover, g = [1/(A - 5)] f belongs to gAbecause 1

JRl I E f nf I 2 I, 1 h - -5 A2

=

12

dII Eff (I2

< +a.

Hence, by (2.25) of Chapter IV, (A--5)g=(A--5)-f=j 1 A--5

, ( X - O m d1 E f f = f , R

i.e., f is in the domain of (A - <)-'. Thus, (3.57) is established. By comparing (3.58) here and (2.19) of Chapter IV, we see that the above argument also establishes that when A is self-adjoint, 1

(3.59)

RA(-5) =

jRl m

dEf = 1 A--5'

Hence, for self-adjoint operators, Definition 6.3 of Chapter I11 and Definition 3.3 of the point spectrum Sg of A coincide, since E A ( ( 5 ) ) # 0, 5 E R', if and only if 5 is an eigenvalue of A, in which case (A - r)-l does not exist. We shall prove now the equivalence of the two definitions (6.3 of Chapter I11 and 3.3) for the continuous spectrum S$ of a self-adjoint operator. €1) # 0 for all E > 0 When, for a fixed real 5, EA([C- E , 5 and E A ( ( 5 ) ) = 0, we must have either EA([{ - E, 5)) # 0 or EA((5,5 €1) # 0 for all E > 0. Let us say that the first alternative is the case. Then the equations

+

+

w-5can be satisfied for all E

€7

-5))f.

,

=f€

llf€It = 1,

> 0. Inasmuch as

we conclude that RA(5)is an unbounded operator. On the other hand, RA(5) is defined on the dense set of all vectors g satisfying E A ( [ ( - E, 5 ~ ] ) g= 0 for some E > 0. Thus, if 5 is an element of the continuous spectrum in the sense of Definition 6.3 in Chapter 111, then

+

477

3. General Time-Independent Two-Body Scattering Theory

it belongs to the continuous spectrum in the sense of Definition 3.3. The complete equivalence of these two definitions of St for a self-adjoint operator A follows from Lemma 3.2.

Lemma 3.2. The resolvent RA(<)of a self-adjoint operator A is a bounded linear operator if and only if either Im 5 # 0 or 4 is real but I?"([( - E, 5 €1) = 0 for some E > 0; in either case 9RA(t) = 3?.

+

<

+

Proof. We have seen earlier that if is real and I?"([[ - E, 5 E ) ] # 0 for all E > 0, then RA(5)either does not exist or, if it exists, it is an unbounded operator. E ~ ) )= 0 for some c0 > 0. Assume now that 5 is real and EA((5 - e0, 5 Since the function 1/(5 - A) is bounded on

+

(-a,{ - €01 u

r5 + € 0 , +a),

we conclude that

Hence, it follows from (3.57) that RA(<)is bounded and defined for all f E 2. In the case where I m 5 # 0, we have

The above inequality establishes that when Im

5

# 0,

(3.60) and R,({) is defined for all f E A?. Q.E.D. It follows from the above lemma that a self-adjoint operator has no residual spectrum.

3.6. THERESOLVENTMETHODIN SCATTERING THEORY The two resolvents we encounter in time-independent scattering theory are the resolvent Ro(5) of the free Hamiltonian Ho and the resolvent R(5) of the total Hamiltonian H. However, while Ro(5) is easy to compute in practice, R(5) is an unknown quantity. Thus, Theorem 3.2, which relates R(5) and Ro(5),proves to be very useful.

478

V. Quantum Mechanical Scattering Theory

Theorem 3.2. Let H and H, be two closed (not necessarily selfadjoint) operators in the Hilbert space &', having dense domains of definition gH= g H 0The . following second resolvent equations are satisfied by the resolvents R(5) and I?,(() of H and H,, respectively, (3.61) R(5) - Ro(5) = -Ro(O(H

-

ffo)R(t) = --R(t)(H - Ho) R o ( ~ )

<

for any which belongs to the resolvent sets of both H and H , , i.e., for 5 $ SH v S H 0 .

Proof. Since H and H,, are closed, R(<)and R,(5) are defined on the entire Hilbert space &' whenever 5 does not belong to the spectra of H and H , . Hence, for every f €A? there are vectors g € g H and h e g H 0 such that f = (H - 5 ) g = ( H , - <)h.Since gH= gHo ,g also belongs to gH0, and therefore Ro(t)(H - H0)g

=

Ro(t)(H - Ho) R(0.f

is defined. Setting above (H - H,)g the relation Ro(t)(H - Ho)g

=

[H

-

5 - ( H , - <)]g, we

get

= RO(C)(H - 1)g - g = RO(0f - R(t)f,

which is equivalent to the first of the identities (3.61). The second identity in (3.61) can be established by a similar procedure. Q.E.D. Before proceeding with the applications of the second resolvent equations, we shall derive, for the sake of completeness, the$rst resolvent equations.

c1

Theorem 3.3. If and 5, belong to the resolvent set of the closed and densely defined linear operator A, then the first resolvent equations (3.62)

RA(51)

- R 4 ( 1 2 ) = (1,

- 1 2 ) RA(51) R A ( 5 2 )

are satisfied.

Proof. The range of A - 5 is identical to the domain of definition of R,([) = (A - [)-I, which coincides with &' when 5 is in the resolvent set of A. Hence, for any f E 2,a vector g can be found so that f = (A - <,)(A - &)g, and therefore (3.63)

(5, - 52)

R A ( 1 1 ) RA(12)f =

(tl - 5 2 k .

On the other hand, we have (3.64)

[RA(~I) - R~(t2)l.f= ( A - 1dg - ( A - &)g = (61 Since (3.63) and (3.64) hold for any f E &', (3.62) follows. Q.E.D.

479

3. General Time-Independent Two-Body Scattering Theory

Let us return now to our main task of giving a perturbational method for computing transition probabilities. Using Theorem 3.2 we infer that, if H , = Ho*, H = H*, and

,

9 I i = %Io

(3.65)

R(5) = %(5) - %(5)

whenever I m we obtain (3.66)

5

WqI),

4=H

- Ho

3

# 0. After n successive iterations of the above relation, R(5) = %(5)

f

(4%(5))k

k=O

+ R,

9

where the remainder R, is (3.67)

R,

R(5).

= (-1)"+'(Ro([)

+

If the above remainder converges to zero, in some sense, * when n -+ co, then (3.66) provides us with a means of computing R(5) when Ro(5) is known: (3.68) R(5) = &(5) - Ro(5) HiRo(5)

+

HiRo(5) HiRo(O f . * *

-

This result combined with (3.67) leads eventually to a "perturbation" solution of the Lippmann-Schwinger equations. This solution is obtained by substituting for R(h, iq) = ( H - A, - i ~ ) - lin (3.41) the series on the right-hand side of (3.68). If the series so obtained for the integral in (3.41) converges, and the order of summation and integration can be interchanged, then Y*can be computed to any order.

+

3.7. APPENDIX: INTEGRATION OF VECTORAND OPERATOR-VALUED FUNCTIONS In the last two sections we encountered integrals, such as ( 3 . 9 , which are special cases of Bochner integrals. The theory of Bochner integration is based on the following theorem.

Theorem 3.4. Suppose p is a measure in the measurable space AY and that f( t),f E R, is a vector-valued function on the measurable set R, assuming values in a Hilbert space 2.If (g I f(g)) is measurable on R for anyg E %' and if (3.69)

* See also Lemma 5.2 and Theorem 5.9 of this chapter.

V. Quantum Mechanical Scattering Theory

480

then: (a) there is a unique vector h E 8,called the Bochner integral of f (4) on R and denoted by

which is such that for all g E 8

(b) if A is a bounded operator on 8 and then (3.71)

A

J R f ( l448 ,

=

J

R

11 Af (()I1 is measurable,

Af(5)dP(E).

Proof. (a) The linear functional

is bounded on account of (3.69): I+(g)l

< JR l(f(5) I g>l 4 4 5 ) < II g II JR llf(l)IldP(l).

Hence, by Riesz’ theorem there is a unique vector h which is such that +(g) = (h I g ) for all g E 8. (b) The Bochner integral of Af(6) exists since ( g I Af(6)) = (A*g I f(4)) is measurable for any g E &, and

In order to prove (3.71) it is necessary and sufficient to show that

for all g E &. Using the definition of the Bochner integral, we get (g I Ah) =
thus establishing that (3.72) is true.

J

R


Q.E.D.

48 I

3. General Time-Independent Two-Body Scattering Theory

The Bochner integral for vector-valued functions can be used to define Bochner integrals for operator-valued functions A( 5). As a matter of fact, if the Bochner integral AR[fl

=

J R [A(5>fIdP(5)

exists for all f E &‘, then the mapping f -+ AR[flis linear since

(3.73)

can be defined to be the Bochner integral of A( f ) on R. It is useful to note that the integral in (3.73) has the property (see Exercise 3.9) (3.74)

AR* =

/

R

A *(5) 4 4 5 ) -

Naturally, all of the above results can be immediately generalized to complex measures p by decomposing such measures into a sum of real measures. A certain kind of generalization of Bochner integrals has been encountered in (3.33), (3.34), (3.41), and (3.43). We define such integrals in a more general context by means of the following concept of integration.

Definition 3.4. Let f, and A(h) be functions in h which assign to every X E [a, b] a vector f, in &‘ and a bounded operator A(X) on S,respectively. Suppose that for any sequence of subdivisions a = X(0) < h ( l ) < .-* < X(n) = b of [a, b] with norm 6 = max{l

- X(K-1)

I :k = 1, ..., n}

converging to zero, and that for any A’(k) E (h(k-l), (3.75)

1

482

V. Quantum Mechanical Scattering Theory

exists for some fixed g E #. If this limit is the same for any choice of sequences of subdivisions with shrinking norm 6 -+ 0 and for any choice of A‘(k) E (A(k-l), A(k)], then it is denoted by the integral symbol (3.76)

A corresponding improper integral is defined as

with a similar definition holding when a -+ -a. We encountered integrals of the type (3.76) in (3.31) and (3.39). In those cases, f A is of the form EAf,where EA is a spectral function and f is a fixed vector in #. For any given interval I C R1, the functional

is obviously linear. If $ I ( g ) is defined for all g E X and if it is bounded, then by Riesz’ theorem there is a unique vector hI which satisfies the equality (hI [ g ) = y$(g) for all g E X . We denote this vector by the integration symbol

Suppose K Ais a operator-valued function. It is easy to see that

and that the existence of the integrals on the right-hand side of the above relation implies the existence of the integral on the left-hand side. Hence, the mapping which takes f into

is linear. We denote the linear operator A, by the integration symbol

3. General Time-Independent Two-Body Scattering Theory

483

and refer to it as the weak# Riemann-Stieltjes integral over I of A(A) with respect to K A .If K Ais a spectral function then (3.79) is called a (weak”) spectral integral.

Lemma 3.3. Suppose that E,” is the spectral function of a selfadjoint operator A and that F(A), A E R1, is a family of bounded operators such+that F * ( h ) f ~gAfor all f~ &. If 11 F(A) 11 C for all A E R1 then

<

and the existence of one of these weak spectral integrals implies the existence of the other.

Proof. We shall prove first that for any a

< b,

by establishing that the corresponding Riemann-Stieltjes sum converges weakly to zero in limit 6 --t +O described in Definition 3.4. Indeed,

since E$, - E $ k - l , is a projector and therefore self-adjoint and equal ~ . The absolute value of (3.82) to its own square and F * ( A ‘ ( k ) ) fgA is obviously smaller than

*In defining (3.79) we used weak limits of Riemann-Stieltjes sums such as (3.75), but the limit might exist and yield the same operator also in other topologies, such as the strong topology (see PrugoveEki [1972a] for a systematic treatment). + See Lemma 2.1 by PrugoveEki [1973a] for less restrictive conditions under which (3.80) is still true.

V. Quantum Mechanical Scattering Theory

484

< (6 k=l

(A‘k’

- A‘k-1’)

c 2

Ilfll2)lil()’/2

d s w ! llfll (b - .)l12 ll(Ef - E,A)gI1 d C(b - w2llfll II g I1 all2, where the second inequality follows by the Schwarz-Cauchy inequality on P(n) and the third inequality follows from the following relations: ll(At(k) - A)(E,Atk,- E , q k - l ) ) h

< (A‘k’ (g(k)

- A(’-’))

- Ef(k-l))(Ef
ll(Ef(k)

II

- Ef(k-l))h

A

A

- E ~ ( t - l ) ) = skz(EA(k)

11, A

- EA(k-1)).

Thus, we see that in the limit 6 - t +0, (3.82) approaches zero, and therefore (3.81) is true. I n view of the obvious fact that

c

+N

+m

F(A)(A- A ) dAE,Ag= w-lim N+*

n=-N

n+l n

F(A)(h - A ) d,,E,Ag,

we can conclude not only that (3.80) is true, but also that the existence of one of the integrals in (3.80) indeed implies the existence of the other. Q.E.D.

By using the Lemma 3.3 we can deduce the following general result, which in view of Theorem 3.1 immediately leads to the Hilbert space Lippmann-Schwinger equations (3.34) and (3.41).

Theorem 3.5. If H and H , are any two self-adjoint operators having a common domain of definition in a Hilbert space then for

485

3. General Time-Independent Two-Body Scattering Theory

any

E

(3.83)

>0 Q *, ==F E mL ' =

1-

J

efd ei H t e-iH ot d t

+m

( H - A, 7 k)-l(H - H,) dAoE2

-m

+m

=1-

J-

d,E,HH - Ho)(A - H ,

k)-'.

m

Proof. By reviewing the derivation of (3.33) and (3.39) we easily establish that the only prerequisite has been the self-adjointness of H and H, . If in addition g,, C gH0 then (H - A, f k)-l in the role of F(A,) satisfies both conditions of Lemma 3.3 since by (3.60) ll(H - A, f k)-lII

< 4,

A,

E

R1,

and obviously F(A,)f E g H. Hence the equality of the expressions in the first and second line of (3.83) follows by applying this lemma to (3.39). The last of the equalities in (3.83) can be derived in a similar manner by taking adjoints in both sides of (3.33). Q.E.D.

The most remarkable feature of the Hilbert space versions of the Lippmann-Schwinger equations (3.34) and (3.41) is their great generality due to very simple and modest assumptions required in Theorem 3.5from which they immediately follow. Consequently, not only do these equations lead to Lippmann-Schwinger equations for distorted waves" in two-body scattering (see $8.7), but their applicability extends beyond the realm of quantum mechanics proper (see $4). It should be noted, however, that these equations become trivial identities if M, consist of only the zero vector. This is exactly what happens even in two-body potential scattering if the potential is of long range, and in particular for the practically important case of Coulomb-like potentials. One can then impose, however, space cutoffs on the potential, thus recovering nontrivial Lippmann-Schwinger equations, and then study the behavior of these equations as the cutoff is removed (see $4.9 as well as Prugovetki and Zorbas [1973] for details and further references). Alternatively,

* These were the equations actually derived by Lippmann and Schwinger [1950]. Hilbert space equations equivalent to (3.34) and (3.41) made their first appearance in a paper by Tixaire [1959] (and were independently derived and generalized by PrugoveEki [1969, 1971b, 1973a]), whereas the Riemann-Stieltjes spectral integrals, which form their basis, were first introduced by Daletskii and Krein [1956]. In studying and applying these Hilbert space equations some authors (e.g., Amrein, Georgescu, and Jauch [1971], Chandler and Gibson [1973], Thomas [1975]) prefer using strong instead of weak spectral integrals, but then additional restrictions have to be imposed on H and Ho

.

V. Quantum Mechanical Scattering Theory

486

one can develop modified Lippmann-Schwinger equations (PrugoveEki [1973a]) that can be reduced to a form that contains a new effective potential and can be then solved by iterative techniques (see Masson and PrugoveEki [1976]).

* *3.8.

APPENDIX:SCATTERING THEORY I N LIOUVILLE SPACE

Scattering theory for density operators plays an important role in nonequilibrium statistical mechanics. * Through the use of the Liouville space formalism of $8 in Chapter IV we can immediately adapt the results of $92-3 to quantum statistical mechanics. Indeed, a Liouville space 9(#) is itself a Hilbert space. Therefore, all the general theorems that we have derived in 392-3 are as applicable to 9 ( Z )as they were to Z itself once it is established that their presuppositions are satisfied. Thus, the basic difference between quantum scattering theory in 8 and statistical-mechanics scattering theory on Y ( Z )stems from somewhat different physical asymptotic conditions in the two cases. The difference in these conditions reflects the difference between equations (1.14) and (8.66) in Chapter IV for state vectors and density operators, respectively. This leads to appropriate changes in the detailed form of (2.46) and (2.47), which in the case of density operators, requires trace operations rather than inner products. For example, if we work with the density matrices of a complete set of operators, the density operator counterpart of the asymptotic condition (2.46) states that, by (8.69) in Chapter IV, p A ,

.....A,(B)

- poAl , . . . A m

dt)

ex(t)

(B)

(3.84) =

[(x

I &)I

x> - (x

I Pex(t)l %>I 444

should converge to zero as t + f00. It is quite easy to establish that a condition such as (3.84) converging to zero for t + f00 is satisfied if

(3.85)

h-lim[p(t) - pex(t)] t-rTrn

= 0,

ex

=

in, out,

# See Prigogine [1962], Kirczenov and Marro [1974], and Balescu [1975]. A sample of specific applications can be found in the articles by Zwanzig [1963], Fano [1963], Snider and Sanctuary [1971], Tip [1971], Eu [1975], and Coombe et al. [1975]. The proofs for statements made in this section can be found in Jauch et al. [1968] and PrugoveEki [1972, 1978b].

487

3. General Time-Independent Two-Body Scattering Theory

although the converse is by no means true. On the other hand, since the limit in Hilbert-Schmidt norm appearing in (3.85) is in fact a strong limit in 9(2), and 9(2) is itself a Hilbert space with an inner product that gives rise to that norm, the general part of Theorem 1.1 is applicable in the present case so that we can state that (3.85) is satisfied if and only if (3.86)

p(0)

a(t)= eiHte

= h-lim a(t)pex(0), t+Tm

.

If we take into consideration Theorem 8.1 1 in Chapter IV and recall (2.1), we see that for any A E 9(S), (3.87) (3.88)

Q(t)A = Q ( t )AQ*(t)

Q+AQ** = h-lim t+Fw

where the superoperators E, (3.89)

Q ( t )EiA,

,

Ed

=

EM*AEPI*

are obviously idempotent, i.e., E,

= E,2,

and also self-adjoint in

P(-@), ( A , I Ed2)2

= T~[AA,*EM,A~EM,] = Tr[EM,A~*EM,A21 = @&I

I4

) 2

-

Therefore E, are superprojectors in 9(2). Consequently, (3.86) suggests the introduction of the wave superoperators (3.90)

= h-lim t-r?xn

a(t)E* ,

which by (3.87) we related to wave operators as follows: (3.91)

a*A = Q*AQ**,

A E Y(.#).

Starting with the definition (3.89) we can restate in a most striaghtforward manner every single theorem in $52-3 in terms of a, and the scattering superoperator S , where (3.92)

sp = aia+p= sps*.

In particular, we shall have by Theorem 3.1 (3.93) (3.94)

= h-lim E-+O

!&E+,

488

V. Quantum Mechanical Scattering Theory

and by the results of 563.2-3.4, for HI = H - H, and

+I

+m

(3.95)

84 = 1

E,

=

1,

(A - H f ie)-lH1 dAEy

-m

thus obtaining a spectral integral formula for the transition superoperator (3.96)

T = (2mi)-l(l - S) = h-lim f++O

=(

2 ~ h-lim(Qzc 4 ~ ~ - P* -,)a+ f++O

I-, dAEyH,{~/[(H- A)2 + +m

E~]}P+.

One of the advantages of the Liouville space approach to quantum scattering theory is greater generality. Indeed, if we define E, as the superprojector onto the closed subspace of 3(X) spanned by all pcex)(0) for which (3.86) exists [rather than in terms of M, by means of (3.89)], then we might have E, = 1 even when EM* = 0. For example, assume that M, = X for some choice of Ho and H in (2.1). If we then shift the energy scale by adding a constant c # 0 to H, thus c, we shall have that arriving at H' = H

+

s-lim t+?m

eiH't

-iHot

e

f =

eict

s-lim

eiHt

t+Tm

-iHot

e

f

obviously does not exist except iff = 0. Thus, although nonrelativistic theories should not be sensitive to shifts in the energy scale, since (nonrelativistically) only relative and not absolute energy values are measurable, the standard scattering theory in 2 espoused in St~2-3 displays a definite (and totally unphysical) dependence on the choice of origin of the energy scale of H in relation to Ho . This weakness is, however, removed in the Liouville space approach, since then obviously eiH't

e-iHot p = e

i H t -iHot

e

p

for any density operator p, so that E, are unaffected by such shifts in energy scale. An even more significant advantage of the Liouville space approach to scattering theory is its immediate applicability to classical statistical mechanics. This enables not only the application of techniques developed in the quantum context to classical theory (see, e.g., Resibois 119591, Miles and Dahler [1970], St. Pierre [1973], Leaf [1975]), but

489

3. General Time-Independent Two-Body Scattering Theory

through the use of the master Liouville space dpr introduced in $8.9 of Chapter IV, it makes possible the step by step comparison of classical and quantum effects when classical and quantum models for the same system are formulated in a common dpr (see Ali and PrugoveEki [1977b], PrugoveEki [1978b, c], as well as $4.10).

EXERCISES 3.1.

0

<E

Show that there are unique operators sZ-,(s)

< s, which satisfy the relations (g

1 Q-,(sl.f>

=6

J' e - Y g 1 JZPlf) at, 0

< 1.

for all f,g E A?, and that I( sZ,,(s)ll 3.2.

and Q+,(s),

Prove that f2dx d i m 8&). - s++w

3.3. Suppose that the Riemann-Stieltjes integral of F(X) with respect to o(h) exists in [a, b], that F(h) is Bore1 measurable, and that a@) is nondecreasing. Show that F(h) is integrable with respect to a measure p ( B ) which satisfies the relation p(( - CO, A]) = o(h), and that

3.4. Derive from the basic Definition 3.1 that the integrals (3.18)(3.20) represent linear operations; for example, that

1: 4

Jz ra,Fl(&4)

= a,

+

a2F2(A

MI 4 0 4 43) ,

J: 4 J FI(4 4) 4,+,4) + Z

a2

Jl 4 s

Z

F2(&

4J 4,+,4).

3.5. Prove that if the Riemann-Stieltjes integral on [a, b], [a, c] and [c, b] exists (a < c < b), then b

JC

a

F(h) do(/\)=

a

F(h) &(A)

+ J F(h) du(A), 0

a

< c < b.

V. Quantum Mechanical Scattering Theory

490

Extend the proof to show that for cross-iterated integrals

I: I, /:Il ,:Il, I,: I,!:. =

+

Remark. If u(A) as well as F(A) are discontinuous at c, the integral on [a, b] might not exist, although the integrals on [a, c] and [c, d ] exist.

3.6, Show that if .(A) is nondecreasing and F,(A) < F2(A),then

<

Using this result, show that if u(A, A,) is nondecreasing [i.e., a(A, A,) A,,') whenever A A' and A, A;] and Fl(A,A,)
<

u(A',

<

3.7. Assuming that the respective Riemann-Stieltjes integrals for u1 and u2 exist, prove that the corresponding integrals for the linear a2u2, a , ,a2 E C1,exist and combinations a,u,

+

where I and I, are arbitrary nondegenerate intervals in R1.

3.8. Using the result of Exercise 3.6, prove the mean-value theorem for Riemann-Stieltjes integrals: If F(A) is real and continuous, and if u(A) is nondecreasing and bounded on [a, b], then there is a point A' E [a, b] for which

s" a

b

F(X)dAu(h) = F(X')

a

dAu(h) = F(h') O((ay b]).

Show that the above integral exists if F(A) and u(A) satisfy the assumptions made earlier.

4. Eigenfunction Expansions in Potential Scattering

3.9.

49 1

Prove that if (3.73) exists, then the relation (3.74) holds.

3.10. Prove that the resolvent RA([)of a self-adjoint operator can be represented for Im 5 < 0 by a Bochner integral: +m

R A ( [ )= - i

ei(A-c)t

A

dt.

0

4. Eigenfunction Expansions in Two-Body Potential Scattering Theory 4.1.

FREEPLANEWAVESIN THREE DIMENSIONS

In 93 we presented a very general framework for time-independent scattering theory. However, in the practically important case of potential scattering one encounters much more specialized versions of that framework, which are, on the other hand, more convenient to deal with from a computational point of view. To derive these formulations rigorously from the general framework of 43, we need the existence of eigenfunction expansions for the Hamiltonians H, and H. Before formulating the concept of eigenfunction expansions on a general level, we shall elucidate this concept by studying it in the special case of the kinetic energy (or “free”) Schroedinger operator H , on L2(R3), defined by means of the differential operator form

If we consider H,’ to be an operator acting on the space of all everywhere twice-differentiable complex functions f(r), then H,’ has eigenfunctions Qk(r)

associated with the eigennumbers (4.3)

A(k)

= k2/2m.

In physical literature the functions Qk(r)are called free plane waves. However, these eigenfunctions are not square integrable on R3 in the Lebesgue measure, and therefore they do not represent eigenvectors of H , , considered as a self-adjoint operator in L2(R3).

492

V. Quantum Mechanical Scattering Theory

The family {Ok(r): k E R3}of eigenfunctions of H: has some remarkable properties, which follow from Theorems 4.5 and 4.6 of Chapter 111. According to Theorem 4.5 of Chapter 111, if EL~(IW~), and if B, ,B, ,... are any bounded measurable sets, the functions (XB,$)(r) have Fourier transforms UF(xB,$), since for every n = 1,2,... the function.xB,(r) +(r) is integrable as well as square integrable with respect to the Lebesgue measure. On the other hand, if B, = R3, then \I xB,+ - (1 -+ 0 when n -+ m, and therefore, in view of Theorem 4.6 in Chapter 111,

+

+

u:=,

+

UF+

(4.4)

= s-lim n+tm

U,(xB-+).

Let us introduce some convenient notation in Definition 4.1.

Definition 4.1. Suppose F(x, y ) is the kernel of an integral operator F from L2(9V, v) into L2(%, p) (4.5)

and that S is the family of all functions g(x) for which the integral in (4.5) exists. Then we shall write for some h €L2(9V,v) (Fh)(x) = 1.i.m.

L

F(x, y ) h(y) dv(y)

if the linear operator defined by (4.5) has an unique extension f = F h to h, i.e., if for any sequence g, ,g, ,... E 9converging strongly to h there is an unique f EL,(%, p) such that

We note that if F happens to be a bounded operator on S,then by the extension principle of bounded operators (Chapter 111, Theorem 2.6), 1.i.m. SF(x, y) h(y) dv(y) is defined for all h eL2(9V,v) and represents the extension of the operator F to h. I n particular, if F is the Fourier-Plancherel transform U , , then (4.8)

$(k)

= (uF#)(k) =

( 2 ~ ) -1.i.m. ~ / ~JR, cgkr$(r)dr

is well defined. Moreover, by Theorem 4.6 in Chapter I11 we also have (4.9)

L

(EP(B,)#)(r) = ( 2 ~ ) - ~ eirk$(k) /~ dk,

493

4. Eigenfunction Expansions in Potential Scattering

and consequently (4.10)

+(r)

=(

2 ~ ) - 1.i.m. ~/~

II

eirk $(k) dk.

I

The formal analogy of (4.10) and (4.8) with the expansion formula of a vector 4 E 2,

in an orthonormal basis ek becomes obvious if we rewrite these two formulas in the form (4.11) +(r) = 1.i.m.

I *$(k) Qk(r)dk, 68

$(k)

= 1.i.m.

I Ok*(r)$(r) dr

and introduce the convenient notation (4.12)

(@k

I +> = 1.i.m. J ak*(r)$(r) dr = $(k).

The fact that (akIf) is not related to the inner product in L2(R*) is emphasized in this notation by the round bracket in (. I .). Due to such analogies, the relations (4.11) are referred to as eigenfunction expansions of +(r) by means of free plane waves ak(r). It is important to realize that we can easily express the spectral measure EHo(B)of Ho with the help of (PL(r).Indeed, we recall that the momentum operators are essentially multiplication operators when they act on $(k), (4.13) (I""'+)"(k)

= k,$(k),

(P'+)" (k) = k&(k), (I""'+)"(k) = k&k),

so that the k variables are in this case identical to the momentum variables p. Since Ho is a function of P, (4.14)

P2 H --=A(P), O - 2m

we have, in the momentum space,

or equivalently, in the configuration space, (4.15)

(EHo(B)+)(r) = 1.i.m.

@k(r)$(jk) dk. I,,,,,

V. Quantum Mechanical Scattering Theory

494

4.2. DISTORTED PLANEWAVES Let us now consider a total Schroedinger operator H defined by means of the differential operator form Hr

(4.16)

=

-(1/2m)A

+ V(r).

In 556-7 we shall extend to H’ the above results on HG by proving that, under certain assumptions on the potential V(r), the following statements are true. For each vector k E R3 - 9; , there is a unique solution” &)(r)

= (24-3’2 (eikr

+ vk(r))

of the differential equation 1 2m

- -A&’(r)

(4.17)

+ V(r) &)(r)

for which, in spherical coordinates I, 8, and tically as follows:

4) -7

wk(r)

(4.18)

fk(e,

eikr,

Y

=

k*

= -@i+)(r)

2m

4,

vk(r) behaves asympto-

I r I + +GO,

K

=

I k I,

where fk(8, 4) is a function uniquely determined by vk(r). The family {@L+)(r): k E R3 - 9%) provides an eigenfunction expansion for any element E L$c(R3)in the sense that

++

(4.19)

++(r)= 1.i.m. JR.

@F)(r)$(k) dk,

where Llc(R3)= EH(S3L2(R3)denotes the closed linear subspace of Lz(RB)corresponding to the continuous spectrum Sf of H. The function $(k) appearing in the eigenfunction expansion (4.19) is not the Fourier-Plancherel transform of ++(r),but rather the FourierPlancherel transform of another function related to by the equation = a+#.In fact, one of the most important results (which will be derived in 57) of time-independent potential scattering is that Q+ is in the present case a partial isometry with initial domain M, = L2(R3)and final domain R+ = Lic(R3). Hence, to every eLlc(R3)corresponds a unique E Lg(R3) such that = Q++.

++

++

+

++

++

* 9: is an exceptional set of Lebesgue measure zero on which Fredholm techniques do not yield a unique @t*’(r)(see Theorem 6.1).

4. Eigenfunction Expansions in Potential Scattering

495

In physical literature the functions @+)(r)are referred to as the outgoing (or retarded) distorted plane waves, while fk(e,f$) is called the scattering amplitude since, as we shall see in 96, it is intimately related to the function T(l)(p;w , w ’ ) introduced in (1.31):

The justification of the term “distorted plane wave” is that in the special case when V ( r ) = 0 we have vk(r)= 0, and @ + ) ( rbecomes ) the free plane wave Gk(r).I n addition, the function

@p)(r; t ) = @(r)

(4.21)

exp[--i(ke/2m)t]

provides a solution of the time-dependent Schroedinger equation (4.22)

a

i-&)(r; at

t ) = [-(1/2m)A

+ V(r)] @i+)(r;t).

The “distorted part,” vk(r)exp[ -i(ka/2m)t] of this wave describes, heuristically speaking, a process which recedes away from the scattering center r = 0. The functions

are called incoming (or advanced) distorted plane waves for analogous reasons. They also satisfy (4.17), and for any i,h- EL,!~JR~), we have the expansion (4.24)

yL(r) = 1.i.m.

I,. @i-)(r)&k) dk,

where #- = Q-#, with i,h uniquely determined by 4- due to the fact that M, ILB(W3)and R- = L 3 R 3 ) . A further significant result of 96, which in fact represents a generalization of (4.19) and (4.24), is that the spectral measure EH(B)can be computed onLic(R8)by means of @L*)(r): (4.25)

(EH(B)+i)(r) = 1.i.m.

L-U,

@f’(r)$(k) dk, +* = Q i +

In particular, for B = R1, the relations (4.25) assume the form (4.19) or (4.24), respectively. It is important to realize that (4.15) is also a

V. Quantum Mechanical Scattering Theory

496

special case of (4.25), since when V(r) and Lio(Rs)= La(R8).

4.3.

3

0, we have Sa,

= Sa- =

1

FREEAND DISTORTED SPHERICAL WAVES

The free plane waves Gr(r) are not the only eigenfunctions of the differential operator H i . It is easy to verify that the functions (4.26)

@Ic2m(r,

&+I = (2/77)1/2~?1(kr) Ylm(64)

are also eigenfunctions of H i :

a a sine-] sin 0 a0 a0

i -.La = --

(4.28)

[-

I +--sin20 ap' a2

for any values (4.29)

0
< +a, I

,...,

= 0,1,2

m

= -1,

-1

+ 1,..., +Z.

The function @klm(r, 8 , + ) is called an outgoing free spherical or partial wave. It follows from (4.10) and the relation (7.70) in 97 that the family of all functions (4.26) provides us with an eigenfunction expansion:

where, using the notation Sap introduced in (7.10) of Chapter 11, we have Wzm(r)+(r, O,+) r 2 sin 0 dr d0 d+.

(4.31)

In the case that V(r) is spherically symmetric, i.e., V(r) = Vo(r),the differential operator H' has the eigenfunctions (4.32)

1/2 2

(*)

@2a(r,44) = ( 2 / 4 iRk2 ( r ) Yt"(4+).

These eigenfunctions are called, respectively, outgoing and incoming distorted spherical waves and are generalizations of @klm ; i.e., (4.33) f J r Q a r , 094)

=

1 k2 (- 2 m A + V"(4)@Eah 094) = g @E#,0, 41,

497

4. Eigenfunction Expansions in Potential Scattering

and for any

I#* ~Lic(lW~) m

+*(r) = C

(4.34)

Z=O

where

I#*

= Q*I#,

+z

C m=-2

1 @L:(r) m

1.i.m.

&,(k) k2 dk,

0

and $,,(k) is given by (4.31).

It is well known (see Butkov [1968, Section 9.101) that the asymptotic behavior for large r of the spherical Bessel functionjl(R~)is (4.35)

j,(kr)

-

(l/kr) sin(kr - l?r/2),

r 3 fa.

We shall see in 97 that the functions R&;)(r)have a similar asymptotic behavior

In physical literature a I ( K ) are called phase shifts, since they represent the change of phase from the free spherical waves corresponding to the noninteraction case V(r) = 0, to the distorted spherical waves. We shall see in 57 that they play a crucial role in the computation of the S operator, which can be written in the form (4.37)

(&%rn(k) = exp[2i~2(41 &n(k).

The function S,(K) = exp[2i8,(K)] is called the partial-wave S matrix. The reason for this terminology is that if we use Dirac's bra and ket notation to write I KZm) = Q P k l m , then taking in (4.31) and (4.36) = Qklm and working formally, without paying too much attention to the real mathematical meaning of these expressions, we get

+

(Kll'm' I S I Klm) = (kk')-' 6(k' - K ) 6,,, 6,*,

exp[2i 6,(k)].

In 97 we shall establish the following generalization of (4.34):

where (4.39)

Ao(k) = k2/2m.

These formulas can be recast in a more compact form if we introduce

V. Quantum Mechanical Scattering Theory

498

the measure po = p ( k ) x

p(w)

on a3, where B, E B1,

It is easy to verify (see Exercise 4.1) that, written in terms of the measure po , (4.38) becomes

(4.41)

(EH(B)+*)(r)= 1.i.m.

@c%9

IF@, 1,m) dPo(k, 1, m),

where $(k, 1, m) and @&(r) are arbitrary extensions of the functions $,(It) and @&L(r),respectively, outside the support of the measure po and to the whole of R 3 . In particular, for H = H , , we have

4.4.

EIGENFUNCTION EXPANSIONS FOR COMPLETE SETSOF OPERATORS

The free plane and spherical waves introduced in the preceding two sections set the pattern for both the mathematical and physical meaning, as well as the role, of free (or unperturbed) eigenfunction expansions in scattering theory*: in both cases we are dealing with integral kernels of unitary operators U appearing in Definition 5.2 of Chapter I V when executing the transition from the configuration representation space L2(R3) (namely the spectral representation space of position observables) to the spectral representation space L2(R",p ) of another complete set of observables, namely {PCx), P(v),PZ)} and {I P I, I L 1, L(z)},respectively. Indeed, in the first case L2(Rn, p ) is the momentum representation space L2(R3)of wave functions $(k) on which p("),p@),

* More abstract approaches based on rigged (Gel'fand and Vilenkin [1968]) or equipped (Berezanskii [1968, 19781) Hilbert spaces are feasible (see, e.g., Antoine [1969], PrugoveEki [1973b], Ascoli et al. [1978]), but on one hand a definitive treatment is still pending, whereas on the other hand the more elementary approach adopted in this section suffices in potential scattering and is also close to computational techniques actually used in practice.

499

4. Eigenfunction Expansions in Potential Scattering

and Pcz) act in accordance with (4.13), whereas in the second case L2([Wn,p ) equals L2([w3,po) since (4.43)

(I p I Gi(4 = JZ$z.m(k),

( ~ ( z ) a n ( =4 m$zm(4,

and for L' in (7.44), Chapter IV, (4.44)

(L'$h)&(k)= ZJzm(k),

L2 = L'(L'

+ 1).

We shall generalize now the above features of free plane and spherical waves into a concept of eigenfunction expansions that is applicable to all other choices of complete sets of observables in two-body as well as in multichannel scattering theory.

Definition 4.2. Let {Xl ,..., X,} and {Yl ,..., Y,} be two complete sets of operators in L2([w", p ) (in the sense of Definition 5.2 in Chapter IV). We shall say that @,(x) is an eigenfunction expansion for {Yl ,..., Y,) in the spectral representation space LZ([Wm,p) of {Xl ,..., X,> if there is a spectral representation space L2(W, V) for {Yl ,..., Y,} and a unitary transformation U of L2([wm, p ) onto L2([wm, V) such that

Clearly, the free Hamiltonian Ho is in potential scattering a function of the complete set { P )P), , P ) }as well as of {I P I, I L 1, L(z)},as seen from (4.14) and (4.39): Ho = I P 12/2m = Ao(l P I).

Hence, in general we can expect that the free Hamiltonian (or, in the language of perturbation theory, the unperturbed operator) shall be a function (4.48)

Ho = A ( Y , ,..., Y,)

of the complete set {Yl ,..., Y,}, so that by Theorem 5.3 in Chapter IV, (UHoU-lf)(y)= 4 Y ) f ( Y ) .

Consequently, by (4.46) we arrive at the following generalizations of

V. Quantum Mechanical Scattering Theory

500 (4.15) and (4.42):

(4.49) As partial isometries with final domains R, , the wave operators s2, can also be viewed as unitary transformations of M, onto R, treated as Hilbert spaces in their own right. If Yl ,..., Y , leave M, invariant, then so do Y?)

(4.50)

= Q+Y~Q**,

j = 1 ,...,n,

leave R, invariant, and from Theorem 5.1 in Chapter IV it can be easily inferred that { Yi*),..., Yi*)}constitute complete sets of operators in R* . By (2.4), (2.1 l), and Theorem 2.5 in Chapter IV,

(4.51)

H*

= A(Y?),

..., YF') = Q*A(Y, ,..., Y,)Q**

are the restrictions of H to R,. Thus, if eigenfunction expansions @:*)(x) of {Y:*),..., YA*)}in the spectral representation space L2(Rm,p) of {XI ,..., X,} exist, so that in accordance with Definition 4.2,

(4.52) f * ) ( y ) = (U(*')g)(y)= 1.i.m.

s

(4.53) g(x) = (~(*)-v(*))(x)= 1.i.m.

(4.54)

(*)

(*) (*)-1

( U yj U

di,

g E R* C L2(Rm, p)

(x)g(x) dp(x),

(*)*

1@?(x)g~(*)(y)

dv(y),

j

g')(Y) = Y d * ) ( Y ) ,

=

$*) E

l,..,

UM*

,

n,

where" V*)g = UQ,*g for all g E R, , then the argument leading to (4.49) in the present context yields

(4.55)

( ~ ~ * ( ~ ) g )= ( x1.i.m. )

s,,)

@%)

g'(*)(Y)W Y ) .

By setting above g = f, = Q,f, so that j ( * )= ing generalization of (4.25) and (4.41):

(4.56)

(E~*(B)~*)(X)

we arrive at the follow-

= 1.i.m.

In the notation introduced in (4.12)and (4.81), this condition ensures that (@:*) I g) = (@" I Q+* g>, so that in a sense (cf. Prugoveliki [1973b]) = (Q**+)LD~, where Q+*+ is the extension of Q* to a larger spacey+ 3 X .

501

4. Eigenfunction Expansions in Potential Scattering

Although in perturbation theory terminology GV(x) and @L*)(x) represent the eigenfunction expansions for the unperturbed and perturbed problem, respectively, motivated by their physical interpretation, the common names in physical literature are as follows: @Jx) is called a free wave, @L+)(x) an outgoing (or retarded) distorted wave, and @k-’(x) an incoming (or advanced) distorted wave.

4.5. GREEN’S OPERATORS AND GREEN FUNCTIONS There is an intimate relationship between the spectral measure EA(B) of a self-adjoint operator A and what in physical literature is called the Green’s operator

i.e., the negative value of the resolvent in resolvent set of A. Part of that connection will be investigated in $5.8, whereas in the present section we shall concentrate on those aspects in which eigenfunction expansions play a crucial role. To make our remarks immediately applicable to both the case of H , as well as H in potential scattering, let us be general and assume that A acts in the spectral representation space 2 = L2(1Wm, p) of a complete set {XI ,..., Xm}, which we can always decompose into the direct sum (4.57)

2

=S P A

@HCA,

#,A

=EA(Si)S,

S c A = EA(S,A)2.

The restriction A, of A to 2 , A is an operator with a pure point spectrum, and therefore the considerations of $5.1 in Chapter I11 apply to it, so that on 2,“ (4*58)

E’A(St) = EA(B

E”P(B) = EA(B)

sp”)

c c *A

=

IfA.a)(fA.a

1,

A€BnSPA a=l

where { f A , , : a = 1,..., n,} is an orthonormal basis in EA({A})2.Consequently, by (3.59)

Expressing the L2(1Wm, p) inner products on the right-hand side of the above relation in terms of integrals and introducing (4.60)

GAP@,x‘; 5)

= 1.i.m.

fA.a(x)f AZAZ1

I-’

’

V. Quantum Mechanical Scattering Theory

502

we can rewrite (4.59) in the form

if we assume that the orders of summation and integration can be interchanged. T o arrive at a similar formula for the continuous spectrum of A, we shall assume that the restriction A, of A to SoA is a functionF(Y, ,..., Y,) of a complete set {Yl ,..., Y,} that possesses an eigenfunction expansion QY(x) on L2(Rm,p). Thus, by Theorem 5.3 in Chapter IV

( W - 4-lu-lk)(Y)

= k(Y)/“

-F(Y)l,

and consequently according to (4.45) and (4.46) for f and g from a dense set 9(cf. Definition 4.1) inL2(Rm,p), we shall have (4.62)

(f I G A ( ~E”(s,A)g> ) -

s,.

d”(Y)“ - F(y)l-l

j

444 @,*(4g(x’).

dP(X)f*(X) @Ax) Wm

Assuming that the function (4.63)

exists and that orders of integration can be interchanged in (4.62), we finally obtain from (4.61)-(4.63): (4.64) (4.65)

(f I GA(5)g) = GA(x,x’;

I

~cL(x)~*(x4 ) 4 x 7 G&, x’;

5) = G+,(x,x’; 5)

5) g ( 4 ,

+ GA,(x,x’; 5).

The p x p-almost everywhere unique (see Exercise 4.2) function Ga(x, x’; Z;) that satisfies (4.64) for f,g €L2(Rm,p) is called the Green function of A on the spectral representation space L2(Rm,p). I n Theorem 5.1 we shall explicitly compute the Green function Go(r,r‘;9) of the free two-body relative motion Hamiltonian H , in the configuration representation space L2(R3). In physical literature Go(r,r’; Z;) is called the free Green function. As we shall see in 95.3, the full (or total) Green function G(r,r‘;Z;) corresponding to H can be computed from the second resolvent equation (3.61) rewritten in terms of Green functions [see (4.76)].

4. Eigenfunction Expansions in Potential Scattering

503

A closely related concept is that of retarded and advanced Green function GL+)(x,x’; A) and GL-)(x, x’; A), respectively, defined for h E SoA. If they exist, these two functions by definition have to satisfy the respective conditions

for at least a dense set of valuesf, g gL2([Wm,p).

4.6. LIPPMANN-SCHWINGER EQUATIONS FOR EIGENFUNCTION EXPANSIONS

By taking advantage of the fact that, as we shall establish in Theorem 5.1, the free Green function Go(r, r’; 0, 5 E C1- [0, a),is a known quantity in two-body potential scattering, and that the corresponding advanced and retarded Green functions Ghi)(r,r’;A), A E [0, a), exist, we derive from (3.34) the Lippmann-Schwinger equations (4.67)

@)(r)

=

+

Gr’(r, r‘; k2/2m)V(r’) @f)(r’)dr‘

Qk(r)

B3

for the distorted plane waves @k*)(r)introduced in p.2. As we shall see in $6.2, (4.67) provides a viable method of computing @Li)(r)for a wide class of short-range potentials. First of all, we note that by Exercise 2.7 (and more generally by Theorem 7.1 in §7), M, = L2(R3)ror the considered class of potentials, which includes all short-range potentials of practical interest (see Appendix 4.9 for long-range potentials). Hence, we can rewrite (3.34) in the form

V. Quantum Mechanical Scattering Theory

504

The key idea in the derivation of (4.67) is to show that

drg*(r)

J'

dr'Go(r, r'; k2/2m fi e ) V(r') &)(r')

dkf(k)

Ra

R3

for all functions g(r) E %b0([w3) and 3(k) E %O, C %b0(R3). V", consists, by definition, of all continuous functions whose compact support is disjoint from an exceptional set 9'; (assumed to be of Lebesgue measure $56.2 and 6.5) of values k E R3 at which @:*)(r) might not zero-see exist, and is such that for $(k) E %,O, (4.19) and (4.24) become +i(r) = (Q*+)(r)

=

s,.

aF)(r) $(k) dk.

Then, upon taking in (4.70) the limit E -P $0 we can argue that the outcome can equal (4.69) if and only if the square bracket in (4.69) equals the last integral in (4.70). Indeed, by (4.25) and Definition 3.4, for any a, b E R1, a < b, (4.71)

(g

1

Jab

GO@fic)vd,&Hf,)

=

lim 6 4 j=l

J R3

dr'Go(r, r'; Aj' fk) V(r')

drg*(r) R3

where the notable fact is that A,' E (Ajal , A,] so that, heuristically speaking, A,' and k2/2rn merge together in the limit 6 + 0. On the rigorous level, we have to assume that @:*)(r)is uniformly bounded in r when k is restricted to supp $, which is compact. This condition is certainly satisfied by Qk(r),and in 96 we shall establish that it is also satisfied by @:*)(r)for a large class of potentials." The potentials in that class are locally square integrable and of faster than r3decrease as r + co, so that considering the form (5.6) of the free Green function we see

* This will be achieved by solving the Lippmann-Schwinger equation (4.67), yet the reasoning is not circular. The procedure is rather analogous to, say, the establishing of power series solutions for some ordinary linear differential equations (see Ince [1956, Chapter XVI]), whereby one starts by assuming the existence of such solutions and upon computing the coefficients, one proves that the assumption was justified.

505

4. Eigenfunction Expansions in Potential Scattering

that the interchange in orders of integration in r’ and k can be carried out in (4.71) due to Fubini’s theorem (see Exercise 4.3). Furthermore, Lebesgue’s dominated convergence theorem can be invoked to establish that the right-hand sides in (4.70) and (4.71) are indeed equal, since upon taking the sum under the integral sign in (4.71) the resulting integrand is majorized in r, r’, k E R3 by the integrable function (4.72)

const. I g(r)l I V(r’)l I r - r‘ I-l I f(k)l exp[-(2m~)’/~I r - r‘ I].

Finally, upon inserting the outcome into the right-hand side of (4.68), the same theorem can be applied again (see Exercise 4.3). Obviously, the main ideas behind the derivation of (4.67) are not confined to position and momentum observables, but are equally well applicable to arbitrary eigenfunction expansions @Jx) (see Exercise 4.4), and in particular to partial waves, when the role of {Yl ,..., Y,} is played by {I P I, 1 L I, and that of {Xl ,..., X,} by the position }, corresponding to the spherical coordinates observables {I Q 1, 0, ,@ (Y, 0, $). If in this instance the potential is spherically symmetric, we easily arrive at the following partial wave Lz$prnann-Schwinger equations, (4.73)

@Eb,4 4 ) =

694)

@k&,

+

m

GF)’(r,Y‘;k2/2m)V(Y’)@Ei(r’,B,$)

Y”

dr’.

0

where GL*)z(r,r’) is given in (5.22). The solution-type Lippmann-Schwinger equations (3.41) in Hilbert space also have their counterparts for eigenfunction expansions. For example, duplicating the argument leading to (4.67) with (3.41) rather than (3.34) as a starting point, we obtain (4.74)

@(r)

= Qk(r)

+

JR3

G(*)(r, r’; k2/2m)V(r’) Qk(r‘)dr‘.

The only difference is that in the present derivation we have to make assumptions about the full retarded and advanced Green functions G(*)(r,r’;A), rather than about @L*)(r).In 45.4 we shall see, however, that these functions can be computed by means of the second resolvent equation (3.65), which can be written in the form (4.75)

(g

I W)f>= (g I G O ( 0 f )

+ (g I Go([)HlG(5)fh

f , g E A?.

Indeed, upon expressing these inner products on the spectral representation space of, in general, a complete set {XI ,..., X,}, and considering

V. Quantum Mechanical Scattering Theory

506

that (4.75) holds for all f(x), g(x) E '&bo(Rm), we infer that

(4.76)

G(x,x'; 1) = G0(x,x'; 1)

+ J G,(x,x"; S)[H?"G(x",x'; 5)] d p ( ~ " ) .

As we shall see in Theorem 5.6, in two-body potential scattering the equation (4.76) can be solved by the Fredholm method. Combined with (4.74) this method provides a means for the computation of distorted waves that is totally equivalent to the method based on solving the Lippmann-Schwinger equations (4.67). 4.7. THEON-SHELLT-MATRIX Upon introducing the spectral integral

(4.77)

T,

1

1 +*

=-

dAEpHIQ+

77 -m

€

( H , - h)2

+

€2

we see that (3.52) is equivalent to the simple statement that

(4.78)

T

= w-lim H+O

T, .

T o recast (4.77) in terms of eigenfunction expansions, we note that by (4.48) and by Theorems 2.3 and 5.3 in Chapter IV,

(4.79)

( E I E H U ( B l f )=

J

g ' * ( Y ) f ( Y )d 4 Y ) .

A-"B)

Upon generalizing the notation in (4.12), we write

(4.80)

Thus, proceeding formally, we see that (4.77) can be written as

507

4. Eigenfunction Expansions in Potential Scattering

The exact meaning of this expression follows from the next general result. Suppose that F(A) is a strongly continuous operatorC, that (HlzQU)(x)is p-square valued function for which IIF(h) (1 integrable in x E Rm, and that H1%Pp, €L2(Rm,p) represents a strongly continuous vector-valued function in y E S C R". If A ( y ) is continuous and &y) E %?bo((Wn) then

Lemma 4.1.

<

Proof. In view of the strong continuity of F(A(y)) and HIx@, ,

is continuous in y, so that the integral on the right-hand side of (4.84) certainly exists if i ( y )E %?bO(lWn). On the other hand, by (4.82) and Definition 3.4,

and this limit can be shown (see Exercise 4.5) to equal that same integral when we let a -+ - co and b -+ co. Q.E.D.

+

Under the restrictions imposed in 96 on V(r) (see also Exercise 4.3), the conditions of Lemma 4.1 are satisfied, so we can certainly write

- 9v whenever f(k)E %b0(R3). By (4.19) for f(k) E,%?~O(R~

1,

provided that (as shown in 96) @k++)(r) has the same continuity and

V. Quantum Mechanical Scattering Theory

508

asymptotic properties as Qk(r).Thus, under the same con-ditions,

since orders of integration can be interchanged by Fubini’s theorem if we assume the integrability of V(r) on R3, as that assures the existence of the following integral for k E R3 and k E R3 - Spy3 : (4.89)

(@k

IvI

=

s,.

Dk*(r) V(r) @$)(T)

dr.

The matrix-type notation on the left-hand side of (4.89) is a variant of the Dirac-type of notation that is in very wide use in physical literature, and the function of k, k E R3 given by (4.89) is referred to as a transition matrix or T-matrix. By using this expression and (4.88), and inserting the outcome in (4.87), we get (4.90)

where we have introduced the abbreviation &(A - A‘) = d { € / [ ( A

(4.91)

- A’)2

+ €”I>.

In (4.90) we can then explicitly take the limit E-+ +O whenever the T-matrix (4.89) is continuous in k E R3 and k’E R3 - YV3 so that the following lemma is applicable.

Lemma 4.2. If the functionf(x) is bounded on any finite interval in [0, 00) (or (- co, 01) and such that its limit f( +O) from the right [orf(-0) from the left] exists at the origin, and if S,(x)f(x) is integrable on [0, co) (or (- 00, 01) for any 0 < E e0 , then

+ +

(4.92)

<

lim ( 2 h )

€W+O

Sirn

[c/(x2

0

+ 291f ( x ) dx = &f( fO).

Proof. To prove the lemma for the first case of integrals over [0, choose for given r] > 0 some N(7) > 0 so large that

+a), b

509

4. Eigenfunction Expansions in Potential Scattering

and also such that whenever 0

< E < e0 , we

have

+ x 2 N(7). Consequently, for any 6 such that 0 < 6 < N(E)we have + <

S,(x) = +(x2

€2)

where M(7) = sup{lf(x)

Eo/7r(x2

€02),

I : 0 < x < N(7)). Since

[c/(x2

+ c2)] dx = arctan(b/r),

we obtain by using the generalized mean-value theorem of integral calculus

< <

<

< <

S} f Y(E, S) sup{f(x) : 0 x S}, so that where inf{f(x) : 0 x 6 ) - + f ( + O ) uniformly in E as 6 -+ +O. Thus, both terms on the right-hand side of the above inequality can be made smaller than 7 by choosing some appropriate 6 = S(r)), then making E / S ( ~ ) sufficiently small, and taking advantage of 2 arctan x approaching T as x -+ co. The proof for the case off(-0) runs along similar lines. Q.E.D. y(e,

+

We note that iff(.)

is actually continuous at the origin, then

where, following a widespread custom, we have employed Dirac’s S(x) symbol, albeit this so-called S-“function” is interpreted in mathematical literature as a generalized function or distribution, * whereas (4.93) is valid even when f ( x ) is not a so-called test-function. This notation has enabled us, however, to formally combine (4.89) and (4.90) by writing (4.94)

(g I Tf) =

k2 S g“*(k)a(= k2 T) (k I T I k ) J ( k ) d k d k , we

-

# See Dennery and Krzywicki [1967] for an elementary introduction, and Gel’fand and Vilenkin [1968] or Treves [196q for a systematic treatment.

510

V. Quantum Mechanical Scattering Theory

where, again according to widespread custom, we have introduced the following Dirac-type notation for the T-matrix (4.89): (4.95)

(k I T I k’)

= (@k

IvI

@s)) = (@k

I va+I @k’).

In a rigorous approach to deriving (4.94) we can make in (4.40) the transition to spherical coordinates for k , and afterwards apply (4.93) to the variable k’ = I k’ I when taking the limit E + $0. T h e easily computable outcome for g(k) E gb0((W3) and f(k) E %‘bO(R3 - Yv3) is (4.96)

(g 1 T f )

=m

J

dkg“*(k)kJ dw‘(KG I T I KG’)f(KG’),

RS

I; = k/K,

Qs

where k’ is the unit vector in direction of k , and it has the angular spherical coordinates w’ = ( O f , I)’). A comparison of (1.31) and (4.96) reveals that T(’)(K;w , w ’ )

(4.97)

=d

( K G I T I KG’)

where w and w’ are the angular spherical coordinates of k and k‘, respectively. Consequently, the formula (1.47) for the differential cross section u(p 6’) of particles of (relative to the target) momentum p scattering off a planar target in the direction of the unit vector of (with angular spherical coordinates w‘) assumes the form ---f

(4.98)

u(p + w’)

=

u(p2/2m, w ‘ )

= (244m2

1(pw’ I T I py.

Due to the relation (6.67) (to be established as part of Theorem 6.3) between the T matrix and the scattering amplitude fD(w‘) we arrive at the simple yet fundamental formula (4.99)

u(p + w’) =

If D ( w y 2 ,

w‘ =

(w,+’I.

A notable feature of the differential cross-section formula (4.98) is that it involves only T-matrix values (p’ I T I p) for which 1 p’ I = I p I. From the physical point of view this fact is taken to reflect energy conservation. Upon introducing in R6 the surface of points (p’, p) for which pt2/2m= p2/2m(which in physical literature is called the energy shell), it can be said that (4.98) involves only on-shell values of the T matrix.

T MATRIX AND F OPERATORS 4.8. THEOFF-SHELL The obvious approach to computing the T matrix for a given potential V(r) is to solve (e.g., by the methods discussed in $6) the Lippmann-

51 1

4. Eigenfunction Expansions in Potential Scattering

Schwinger equation (4.67) for the distorted plane wave @L+)(r) and then insert the result into (4.89). If, however, the full Green function G(r, r’; [) is known, then the solution-type Lippmann-Schwinger equation (4.74) can be used to obtain directly the desired solution from (4.89):

(4.100) (k I T I k)

=

!iTo(@k’ 1 v{1 + G[(k2/2m) + k]V}I @k)

G(+)(r,r’; kz/2m) V(r‘) ak(r’)dr‘]

.

Equation (4.100) suggests the introduction of the following F ( [ ) operators”

(4.101)

F(5)

=

V

+ VG([)V,

[E@’ - SH,

which consitute a family of in general unbounded operators in L2(R3) when 5 varies over the resolvent set of the total Hamiltonian H. When the free Green’s operator is applied to equation (4.101) from the left we obtain

(4.102)

GO(5)F ( 5 ) = [Go(<)

+ Go({) Jw5)1v

=

G(5)V,

where the last expression follows from the second resolvent equation (3.61) recast in terms of Green’s operators. We substitute this expression for G(OV into (4.101) and arrive at the relation

(4.103)

F(5)

=

v + vGo(5)F ( 5 ) ,

5 E @’

- SHY

known as the Lippmann-Schwinger equation for F([). Although the F([)operators are well defined in L2(Iw3),their primary importance stems from their close relation to the so-called off-shell T matrix

(4.104)

(P’I F ( 5 ) I P> = (3.I V + Y 5 ) V I 3)’

which, generally speaking, is a function of the real variables p, p’ E R3, and the complex variable 5 E @’ - SH. The interest of this quantity

* It is the custom in physical literature to refer to (4.101) as “the T operator,” albeit one is actually dealing with an operator-valued function. T o avoid notational confusion, authors who follow this custom (see, e.g., Taylor [1972], Amrein et al. [1977]) instead of dealing with the transition operator (3.50) work with the “R-operator” R = -2m‘T.

V. Quantum Mechanical Scattering Theory

512

lies in the possibility of deriving from (4.103) the integral equation (see Exercise 4.6)

where (p' I V I p) is proportional to the Fourier-Plancherel transform P(p - p') of V(r): (4.106) (p' I V I p)

=

(aD, I V I 3)= (21~)-~ 1.i.m.

V(r) exp[i(p - p')r] dr.

Thus, a strategy often followed in computing the on-shell T matrix (p' I T I p) is to solve (4.105) for the off-shell T-matrix (4.104), then set I p 1 = 1 p' I and 5 = (p2/2m) ie in that solution, and thus finally obtain the desired quantity by letting E-+ +O. This procedure is symbolized by the standard formal expression for the S matrix,

+

(4.107)

which, in fact, simply reflects the aforementioned computation of the on-shell T matrix (4.108)

the insertion of the outcome into (4.94), and finally the computation of the transition amplitude

using in the process the notationally expedient device of writing

Although easily avoidable as well as, strictly speaking, mathematically unjustified when used as before in conjunction with functions f(p), f(p) ~ L ( ~ ) ( (which, l w ~ ) in general, might display discontinuities or even singular behavior on sets of Lebesgue measure zero), the 6-"function" notation has achieved general acceptance in physical literature. One

4. Eigenfunction Expansions in Potential Scattering

513

other of the many illustrations of its usage can be found in thegeneralized optical theorem, which states that for I p [ = I p' I, (4.110) (P' I T I P>* - (P' I TI P> = 27ri

/

(p" I T I p'>* S

(- PI2 - c) (p" I T I p) dp". 2m

Pa

2m

Without the use of 8-symbolism, (4.1 10) can be written in terms of the scattering amplitude (4.20) as follows,

where w' = (O', 4') are angular spherical coordinates of p' = pw'. The relation (4.111) is a reflection of the unitarity of the S operator, as can be easily established (see Exercise 4.7) by observing that S*S

=

1 - 274T - T*) + 4r2T*T = (SS*)*,

and therefore that S*S = SS* = 1 if and only if T*T

(4.1 12)

= (T*

-

T)/2ni== TT*.

By setting p = p' in (4.1 11) we obtain the optical theorem

that relates the scattering amplitude fpm,(wo)in the forward direction oo to the total cross-section o(E) at energy E = pz/2m. Finally, it is interesting to note that if we use the solution-type Lippmann-Schwinger equation (4.74) in (4.90) and take (4.78) and (4.108) into consideration, we can then directly relate the transition operator T to the F ( 5 ) operators by means of a double spectral integral, (4.1 14)

T

= w-lim w-lim €++O

n++O

+m

d,E?

+m

J-m

Sc(h - p) 9-(h

+ iv) dfiE:,

by applying Lemma 4.1 to express the k and k integrations of (4.90) in terms of integrations with respect to the spectral function of H o . **4.9. APPENDIX: SCATTERING THEORY FOR LONG-RANGE

POTENTIALS We have mentioned in $2.7 that the standard time-dependent theory

V. Quantum Mechanical Scattering Theory

514

developed in $91-2 does not apply to long-range interactions corresponding in general to potentials of the form* (4.1 15)

VJr)

=

Vo(r)

+ y ~ - ~ ,4 <

01

< 1,

where V,(r) is some short-range potential of the kind to which the standard theory is applicable. T h e same remark is, of course, valid for the stationary theory. When we have V , G 0, a! = 1, and y = e1e2, we are dealing with the important case of the Coulomb interaction between two charges el and e 2 . Hence it is of both physical and mathematical interest to learn how the standard theory should be modified in order to cope with long-range cases.+ The rigorous time-dependent scattering theory for long-range potential is based on the observation (first made by Dollard [I9641 for the Coulomb case) that the modijied (or renormalixed) wawe operators (4.1 16)

= s-lim t+Tm

exp(iHt) exp{-i[EZ,t

+ D,(t; H,)])

exist for the following (asymptotically unique) choices of operatorvalued functions of H,, and t > 0

The most mathematically interesting feature of these operators is that they intertwine H , in (4.14) and H = Ho V, given in (4.115), so that (2.3)-(2.5) are satisfied if Q* is replaced by d + ,and perturbation theory can then be pursued in the ordinary manner. Physically, however, this fact does not suffice, but it turns out* that expectation values for

+

4

* The case of 0 < OL < requires iterative techniques involving derivatives of Vv(r)see Buslaev and Matveev [1970]. t See Amrein [1974] for a review of early rigorous results. For some more recent results see, e.g., Lavine [1973], Klein and Zinnes [1973], Gibson and Chandler [1974], Cattapan et al. [1975], Ikebe [1975], Narnhofer [1975], Semon and Taylor [1975, 19761, Gersten [1976], Hormander [1976], Masson and PrugoveEki [1976], Rejto [1976], Saenz and Zachary [1976], Zachary [1976], Zorbas [1976a,b, 19771, Alsholm [1977], Kitada [1977], Saito [1977], AIt et al. [1978], Lapicki and Losonsky [1979], and Rosenberg [19791. See Manoukian and PrugoveEki [1971] and PrugoveEki [1971, 19761 for proofs concerning position, momentum, and angular momentum observables, and Amrein et al. [1970], Corbett r19701, and Zachary [1972] for proofs based on the “algebra of observables” approach.

*

4. Eigenfunction Expansions in Potential Scattering

515

the modified asymptotic states (4.118)

approach in the limit t + f00 those of truly free asymptotic states of q t ) , ~ e x ( t )= eiHOtYex(0),

Y~X(O) = ~/.x(o) = &*Y(o),

so that the physical asymptotic conditions (2.46) and (2.47) are satisfied by these free states Yex(t). This in turn implies that all basic physical relationships that were studied in $1 can be recovered in the context of long-range interactions. The oscillatory behavior of exp[iD,(t; H,)] as a function of t and the existence of the strong limit in (4.116) implies that in the longrange case (4.1 19)

so that the Maller wave operators (2.1) have trivial initial domains M, = (0). This in turn implies that (see PrugoveEki and Zorbas [1973a, Section 2]), (4.120)

so that the Lippmann-Schwinger equations (3.34) are not satisfied. It is, of course, possible to reproduce the considerations of 593.1-3.2 adopting (4.1 16) as a starting point (see PrugoveEki [1971b], Chandler and Gibson [1974]), and thus arrive at a modified version of (3.34) corresponding to a replacement of the integrand of the spectral integral in (3.33) by 7

exp[i(H, - h

i I 7 1)t

+ iD,(t; H,)] dt - 1,

but from a computational point of view one runs into difficulties with explicitly taking the limit 77 ---t fO, due to the presence of oscillatory terms generated by exp[iD,(t; H,,)]. One way of eliminating these oscillatory terms (characteristic of long-range problems) is to introduce an asymptotically compensating operator 2 (see Matveev and Skriganov [1972], PrugoveEki and Zorbas [1973a, b]) in terms of which we can rewrite (4.116) as follows:

516

V. Quantum Mechanical Scattering Theory

Formally, we are then faced with the same kind of expressions as in the two-Hilbert space formulations of multichannel scattering theory described in 98.6, the difference being that the embedding operator J has to be replaced by the asymptotically compensating operator 2. Indeed, one can show (PrugoveEki and Zorbas [1973a, b]) that the equations (8.71) with 2 replacing J hold, and that they can be solved by iterative methods (Masson and PrugoveEki [19761). The preceding outline indicates that two-body scattering theory for long-range potentials can be brought in line with the short-range case. However, these results are actually of little practical interest, since due to screening effects (such as those for the Coulomb potential of a nucleus by the electron cloud around it) one can always replace y in (4.115) by a screening (or space cutofl) function y(r) that equals y on some neighborhood of the origin, but rapidly decreases to zero as I r I -+ 00. Of course, we can incorporate those particles that give rise to screening into the problem, but then one is dealing with a many-body problem, so that the phenomenological approach based on a screening function presents great computational advantages. Furthermore, it can be shown rigorously (see PrugoveEki and Zorbas [1973b, Section 71, Zorbas [1974]) that in its basic aspects the theory with a space cutoff merges into the exact theory as the cutoff is removed. A fortuitous aspect of two-body scattering for Coulomb potentials is that its eigenfunctions can be computed explicitly by solving (4.17) for V(r) = yr-l (see, e.g., Messiah [1962, Chapter IX, §§7-11]), and the addition in (4.1 15) of the short-range potential V,,(r) can be then treated as a perturbation of this pure Coulomb problem. Consequently, the main thrust of most of the work on scattering with long-range forces quoted in this section has been in the direction of multichannel scattering, the two-body case providing mainly a testing ground of the proposed ideas. **4.10.

APPENDIX:EIGENFUNCTIONS AND TRANSITION DENSITY MATRICES I N STATISTICAL MECHANICS

The general concepts and results of $54.4-4.7 can be adapted with ease to Liouville space. In fact, a Liouville space 9 ( H ) is itself a Hilbert space, and therefore from a purely mathematical point of view one might simply consider such constructs as eigenfunction expansions and T matrices directly in 9(2), without reference to H.However, such an oversimplified approach would ignore the basic differences between the formulas (1.14) and (8.69) in Chapter IV, which reflect a difference in the physical interpretation in 9(8) as opposed to Z.

4. Eigenfunction Expansions in Potential Scattering

517

Indeed, whereas for a wave packet #(x) it is I #(x) l2 that is interpreted as a probability density, for a density matrix p(x', x") = (x' I p I x") it is its "diagonal" values p(x, x), and not I p(x', x") 12, that play an analogous role. Consequently, in a physically correct approach to 2(#) we cannot altogether ignore 2. For example, in treating eigenfunctions related to H, = fi-l[H,, -1 [as defined by (8.51) in Chapter IVj we shall use as a starting point an eigenfunction expansion @ J x ) for a complete set of observables represented by operators Yl ,..., Y , in # and satisfying (4.45)-(4.48). The functions (4.122)

4 w * , w . ( x ' ,x") = @ f ( x ' )

then provide an eigenfunction expansion for the complete set of superoperators [Yi , .] and (Yi , .}, j = l,,.,, n, but these superoperators do not always possess a physical interpretation, although Yl ,..., Y , might. On the other hand, we have (4.123)

w'*z"4 w'*v" (x', x") = h-l[A(y') - A(y")]4V*,y.(x', x"),

so that the concepts and results of 9w.4 and 4.5 can be transferred with ease to H, and *yn,y*(x', x"). As mentioned at the end of $3.8, the methods of quantum scattering theory can be adapted to classical statistical mechanics for the study of transport phenomena. In particular, it has proven advantageous to study classical differential cross sections within a superoperator formalism inspired by the quantum approach.* In fact, through the use of the master Liouville space L& described in $8.9 of Chapter IV, it is possible to provide a unified treatment of both classical and quantum differential cross sections. The idea is to start with a density operator describing a state in which the incoming reduced particle is to be found within a finite cylinder of base B and length I (i.e., within afinite beam), its momentum probability distribution being a 6-like function 6,3(p - p,) peaked around p,. Proceeding very much as in $1.8, one then uses the T superoperator formula (3.89) to express the probability of that particle scattering in a potential (centered at the origin) within a cone C(+), which is then treated as in (1.40) by letting C(+) become infinitely +

*See, e.g., Miles and Dahler [1970], Coombe et nl. [1975], and Eu [1975].

+Thistreatment has been espoused in Sections 3 and 4 of PrugoveEki [1978c], where proofs of the subsequently outlined results can be found.

518

V. Quantum Mechanical Scattering Theory

narrow. The key difference between the treatment of $1.8 and the present case emerges, however, as one makes the transition to an infinite beam [by letting in the present case B and 1 go to infinity, so that S:(p - po) .--t 63(p - p,,)]. Indeed, whereas in $1.8 prior to taking this limit it had proven to be mandatory” to carry out an averaging over an infinite plane (reflecting a planar distribution of particles in the target), that is not at all the case in the present context. As a matter of fact, since the present derivation is applicable not only to the quantum but also the classical case, any such averaging would be mathematically ad hoc, since the well-known formula (see, e.g., Balescu [1975]) (4.124)

a,1(p -+ w’) = (b db d+)/do‘

for the classical differential cross section in terms of the impact parameter b can be derived without any kind of averaging. Furthermore, in the context of statistical mechanics, any such averaging would be also physically unjustified since in such realms of applicability as collisions within gases obviously no planar pattern can be claimed to exist. The resulting general formula for the (point-target) differential cross section is

where the transition density supermatrix

makes its appearance. The classical cross section (notationally represented by s = cl) formally corresponds to i(cl) taken to be a &function, and it indeed turns out to equal (4.124) upon adopting for Hic1)the interaction term in equation (8.74) of Chapter I V specialized to the case of a single particle. In the quantum case $@)is given by (8.80) and (8.86) in Chapter IV, and for very precise momentum measurements (i.e., very large values of s) us in (4.125) appears to be approximately equal to u in (4.98) under realistic assumptions about relative orders of magnitude of the basic parameters appearing in a scattering experiment (see PrugoveEki [1978c, Section 4]), but no exact equality has been established.

# This averaging procedure is common to all rigorous derivations of the formula u(p -+ w ) = /f,,(w)I2, and is not peculiar to the specific derivation in $1.8 (see, e.g., Messiah [1961, Chapter X, 5 9 and 61, Taylor [1972, Section 3-el and Newton [1979]).

519

4. Eigenfunction Expansions in Potential Scattering

EXERCISES 4.1. Show that the expressions on the right-hand sides of (4.38) and (4.41) are identical. 4.2. Show that if G>l)(x,x’; 5) and GT)(x, x’; 5 ) are two Green functions of A and p is finite, then GL1’(x, x’; 5 ) = G>’’(x, x’; 5 ) almost everywhere in x and x’ (with respect to the measure p x p). 4.3. In potential scattering G(r, r’; h f k ) and G(*)(r, r‘; A) are given by (5.6) and (5.17), respectively. Prove that if V(r) is square integrable and I V(r)l = O(l/r2+ro)for some e0 > 0, and if I @L*)(r)l MB(r)for all k E B, where M,(r) is integrable on R3 for any compact set B disjoint from the exceptional set 9’; mentioned in v . 2 , then provided J I r - r’ 1-l V(r’l) dr‘ const (cf. Excercise 6.2),

<

<

lim

W*O

/ t,bO*(r)Go(r, r‘;h + iq) V(r‘) @F)(r’)$(k) dr dr‘ dk / t,bO*(r)Gr’(r, r‘; A) V(r’) @lf’(r)$(k) dr dr‘ dk =

for #o(r) E @b(R3) and

&k)

E pb(R3)

with supp

$ n 9’:

= 0.

4.4. State conditions under which you can derive Lippmann-Schwinger equations from (3.34) for arbitrary eigenfunction expansions: @F’(x) = @,(x)

+

G ~ ) ( xx’; , A ( y ) )V(X‘)@F’(x’)dp(x’). Wm

4.5. Show that if the integration on the right-hand side of (4.84) is restricted to A-l((a, b ] ) , the outcome equals the limit in (4.86). 4.6.

Prove that if V(r) is square integrable on R3, then

exists [in the sense of (4.81) and Definition 4.21 and that (4.105) is satisfied almost everywhere in (p‘, p) E R6.

Note. If V(r) is also integrable, then its Fourier transform exists and is continuous, and therefore (4.105) holds whenever (k I F(5)I k) is continuous in k, k‘ E R3. 4.7.

Show that (4.112) implies (4.111) if (6.62) holds true.

520

V. Quantum Mechanical Scattering Theory

5. Green Functions in Potential Scattering 5.1.

THEFREEGREENFUNCTION

Potential quantum scattering is the special case of quantum scattering theory in which the underlying formalism is that of wave mechanics, and the interaction is determined by a potential. Proceeding as in 994.7-9, in applying the general methods of scattering theory to potential scattering of two particles we shall limit ourselves to systems of two spinless, nonidentical particles interacting via a potential V(r), where r = r2- r l , and r l , r2 are the position vectors of the two particles. This case exhibits all the essential features of all other cases. For example, the case of nonzero-spin particles can be treated essentially by the same methods as the zero-spin case when group-theoretical methods are applied. In the case of systems of two identical particles one must also take into account the presence of the Fermi-Dirac or Bose-Einstein statistics. From the practical point of view this means that in these last two cases the calculations have to be carried out in the respective subspaces of antisymmetric or symmetric functions of L2(R3),rather than in the space L2(R3)itself." Upon removal of the center-of-mass motion (see §1.7), the resulting subspaces of L2(W) containing the states of the reduced particle consist of wave functions of even or odd parity, respectively. In the absence of external fields and mutual interaction, the relative motion Hamiltonian Ho of a system of two spinless particles is the kinetic energy operator in the center of mass system that can be viewed also as the kinetic energy operator for a particle of reduced mass m [see (5.93) in Chapter 111. We recall from Theorem 7.3 of Chapter IV that Ho = T, is a self-adjoint operator for which

and that H , is the only self-adjoint operator having this property, On wave functions $(k) E U,9?,s CL2(R3)in the momentum space, H, acts in the following manner:

* See Rys [1965] and Taylor [1972, Chapter 221 for a general treatment of the scattering theory of two identical particles. Naturally, for particles of nonzero spin one has to work on direct sums of L2(R3)spaces [see (7.9) in Chapter IVl.

521

5. Green Functions in Potential Scattering

The above operator is obviously self-adjoint and, since it is a function of the momentum operators P), P ) ,P(s),it immediately follows that it has a pure continuous spectrum SHO

(5.3)

=

p co

=

[O, f w ) .

Thus, its resolvent (5.4)

is defined for all values of 5 E @I. Moreover, R0(5)is a bounded linear operator defined on the entire Hilbert space 2 = L2(R3)whenever Re 5 < 0 or I m # 0, i.e., when the argument arg of 5 is within the open interval (0, 27r) and for such values in scattering theory -R,,(5) is called the free Green's operator.

<

<

Theorem 5.1. The Green's operator Go(<)of the free Hamiltonian (5.2) is an integral operator (5.5)

(Go(5)+)(.) = -(Ro(O+)(r)

GO6-3 r'; 5) +(r')dr'

= L 3

<

for values of restricted to the range 0 < arg Go(r, r'; t), called the free Green function, is Go@,r'; 5)

where

e

=

-m

27r I r - r' I

t is the square root

< < 27r. The kernel

exp[i d/2m51r - r' I],

for which Im

2/2mL > 0.

Proof. According to (5.2) and Theorems 2.5-2.7 in Chapter IV, (5.7)

Assume that the wave function $(r) in the configuration space belongs to V$(Iw3). Then $(r) and all its derivatives have Fourier transforms

which by Theorem 4.5 of Chapter I11 are continuous and bounded, and therefore $(k) eikr is integrable* on R3.Since (k2/2m - 0-l is bounded

* Throughout this section, whenever we deal with integration, we have in mind integration with respect to the Lebesgue measure, except if otherwise explicitly stated.

522

V. Quantum Mechanical Scattering Theory

in k over R3 when 0

< arg 5 < 27r, the

inverse Fourier transform of

(k2/2m- [)-l$(k)

exists. By virtue of (5.7), (5.8), and Lebesgue’s dominated convergence theorem,

Due to the fact that +(r’) is of compact support, the function

<

is Lebesgue integrable on R3 x {k: I k I R} with respect to the variables r and k. Hence, we can apply Fubini’s theorem to interchange the order of integration in (5.9): (5.10)

(RO(1)W) 1 - (2.rr)3 --

JR3

dr’#(r’) lim R+w

I

k2

Ikl
dk

(K- 5)

-1

exp[ik(r - r’)].

T o carry out the above integration in k, let us write (for fixed r - r’) the variable k in spherical coordinates introduced in such a manner that k(r - r’) = kp = kp cos 8. p =

I t is very easy to carry out immediately the integrations in 0 and 0 4 277:

< <

k2 1 (x 1)

<8 <

T

-1

(5.11)

-

IklSR

=

eikPdk

I: dk /: dB s” d+ k2 k kp I, p(k2/2m 1)

sin B

0

R

=

-49r

sin

-

dk

k2 (- 2m =

-1

- 1)

-4m~lp

eikpCoSe

1- k2k fR

R

sin kp - 2m5

dk .

In the last of the above integrals we have to integrate with respect to k E [-R, +R] the function (5.12)

k eikD k sin kp k2 - 2m5 2i(k - 6 1 ) ( k

k e-iko

+6

4 )

+ 1/2m5)

2i(k - 2/27;;5)(k

523

5. Green Functions in Potential Scattering

and then let R - t 00. If we work with the complex variable k E C1, the above functions are analytic everywhere in the complex plane, except for two simple poles at f61/4, Im dt > 0.Let us close the contour of integration by semicircles in the upper-half and lower-half complex plane, respectively, when integrating on the real line the first and second function on the right-hand side of (5.12). Since the contributions from the respective integrations on the semicircles vanish in the limit of their radii R becoming infinite, we easily determine, by using Cauchy's integral formula and computing the respective residua: + ,

k sin kp 1 J k2 - 2m1 d k = - 2i

k &kr dk-2-2m[ 2i

f k2

-~ dk k e- i k2m1 p

Inserting this result in (5.11), we get from (5.10)

for all $ ~ q a R 3 ) .Since %?;(R3) is dense in L2(R3) (see Chapter 11, Theorem 5.6), while both R,(() and the integral operator with the kernel (m/277)l Y - Y' 1-l e x p ( i 4 m I Y - Y' I) are bounded linear operators defined on the entire Hilbert space L2(R3), we infer with the help of the extension principle (Chapter 111, Theorem 2.6) that (5.13) is true for all $ €L2(R3). Q.E.D. It follows from (5.6) that the free Green function has the following symmetry properties: (5.14)

Gdr, r'; 1) = Go*@,r'; [*),

(5.15)

Go@,r';

I ) = Go@', r; I )

for any ( E @' not on the positive real axis. We note that considered as a function on Rs, Go(r, r'; () is continuous and infinitely differentiable everywhere except on the set (5.16)

D,

= {(r,

r') : I r - r' I

which is of Lebesgue measure zero on

Iw6

= O},

(see Exercise 5.1). Further-

524

V. Quantum Mechanical Scattering Theory

more, (5.17)

GF)(r, r'; k2/2m) = =

hlGo[r, r'; (k2/2m)fie] [--m/(27r I r - r' I)] exp(f i k

I r - r' I)

also exists pointwise for any K >, 0 and all (r, r') E R6 - D,", and by Lebesgue's dominated convergence theorem (4.66) is certainly satisfied in the present context for all f(r), g(r) E Vb0(R3). Consequently the retarded and advancedfree Green functions GA+'(r, r'; A) and GA-)(r, r'; A) do exist for all A E S p = [O, +a).These functions inherit the symmetry property (5.14), and we have (5.18) for all A

Gr)(r, r'; A)

=

GF)(rf,r; A)

=

Gr)*(r,r'; A,

0, as is easily verified by the use of (5.17).

5.2. PARTIAL WAVEFREEGREENFUNCTIONS

As we have shown in $4.6, the free Green function Go(r, r'; C) is a key ingredient of the Lippmann-Schwinger equations (4.67) used in computing distorted plane waves Qk*)(r). In such computations one implicitly works with the complete sets of observables Q and P, as reflected by the fact that the plane waves Qk(r) are functions of the position variables r, and are parametrized in terms of the momentum variables k. We shall see in $7.4 that when the potential is central (i.e., spherically symmetric) it can be advantageous to work with the complete sets {I Q I, O Q , QQ} and {I P 1, I L I, instead. It is obvious that Ho commutes with I P I, I L I, and I,@), and in fact, in accordance with (4.39), Ho = Ao(l P 1, I L I,L("))= I P 12/2m. Since Ho has no point spectrum, by (4.42) and (4.63),

Using spherical coordinates (r, 8, +) for r, and taking into consideration (4.26) and (4.40), we arrive at the following result.

Theorem 5.2.

The free Green function equals a sum (convergent

in the mean), m

(5.20)

Go(r, r'; 1) =

C

+l

2 G:(r,

1=0 m=-1

r'; 1) Yzm(6,+) Yzm(e',+')*,

525

5. Green Functions in Potential Scattering

of the partial wave free Greenfunctions

-2miKjz(KY) -2miKjz(KY’)

hF)(KYf),

Y

hF)(KY),

Y

< Y‘ > Y’,

where K = (2rn5)1/2, Im K > 0. Upon noting that the integrand in (5.21) is an even function, the computation of the previous explicit expression for Gor(r,r’; 5) proceeds by contour integration and the use of the spherical Hankel functions h\lv2)(kr). Indeed (see, e.g., Butkov [1968]), as 112 I -+ co in the complex plane,

+

jz(kr‘)= $ [ h p ) ( k ~ ‘ ) h P ) ( k ~ ’ ) ] N

(2ikr’)-l{exp{i[kr’ - 4(17r)]} - exp{--i[kr’ - $(ZT)]},

so that upon inserting this result in the integral in (5.21) we see that for r i= r‘, jZ(kr)h\l)(kr‘)and h!2)(kr)jl(kr’) converge to zero as I k I -+ co in the upper-half and lower-half complex plane, respectively. Hence, proceeding as we did from (5.12) to (5.13), we close the contour of integration appropriately, and using the residuum theorem arrive at the first of the explicit expressions in (5.21) upon noting that j z ( - k r ) = (-l)Y[(kr) and hj2)(-kr) = (-l)%\’)(kr). For r > r‘ the roles of r and r’ in the above procedure are obviously reversed, and the second expression immediately follows. 0, we arrive at In (5.21) taking the limits Im 5 -+ f O for fixed Re 5

The two functions in (5.22) are obviously the advanced and retarded partial wave free Green functions. 5.3. FREDHOLM INTEGRAL EQUATIONS WITH HILBERT-SCHMIDT KERNELS One of the main significant features of the free Green function Go(r,r’;5) is that it provides an integral equation for the full Green function G(r,r‘; c), corresponding to the Schroedinger operator H , for which (5.23)

(H”

=

-(1/24

4 4 9 + W) W),

# E go

a

V. Quantum Mechanical Scattering Theory

526

This integral equation is the second resolvent integral equation (4.76), which in the present case becomes

This equation can be solved by the Fredholm method if its kernel K(r, r"; 5 )

(5.25)

=

Go(r, r"; 5 ) V(r")

is of Hilbert-Schmidt type.+ Before attacking the problem of solving (5.24), let us first review some basic facts about integral operators with Hilbert-Schmidt kernels.

Definition 5.1. The kernel K(a, a'), a, a' E Rm, of the integral operator K on L2(Rm), (5.26)

(@)(a)

=

wm

K(a, a') +(a') dma',

+ €L2(Rm),

is a Hilbert-Schmidt kernel (or L2 kernel) if 1 K(a, &')I2

is integrable on

R2m:

(5.27)

A Hilbert-Schmidt kernel which is continuous and such that K(a, a ) is integrable on Rm is called a trace-class kernel. The terminology for kernels of integral operators is related to the terminology for the corresponding operators in a straightforward fashion: integral operators with trace-class kernels are of trace class* with T r K = J K(a, a ) dma, and integral operators with HilbertSchmidt kernels are Hilbert-Schmidt operators (see Exercise 5.3). The Fredholm integral equation is (5.28)

where w is, in general, some complex number. If we search for solutions K(a, a') is the kernel of an integral operator on L2(Rm), and if f€L2(Rm), then the integral equation (5.28) can be written in operator form: +(a) which are Lebesgue square integrable, so that

(5.29)

+ =f+

wK+.

* A condensed and very lucid presentation of the Fredholm theory for Hilbert-Schmidt kernels (Lakernels) is given by Smithies [1965] in Sections 6.5-6.7.

527

5. Green Functions in Potential Scattering

Hence, (5.28) has a unique solution in L2(Rm) if and only if (1 - wK)-I exists and iff is in the domain of definition of (1 - wK)-'. For w # 0, ( I - wK1-l is related in a straightforward manner to the resolvent &(<) of the operator K: (1 - w K ) - l

1

=w

(-w1

-1

-K)

=

1

- -R, W

(-).1

W

If K(a, a') is a Hilbert-Schmidt kernel, then K is a Hilbert-Schmidt operator, and as such, K has a pure point spectrum. Hence, (1 - wK)-' will exist if and only if 1/w !cj SK. When this is the case, the domain of definition of (1 - wK)-' is the entire space L2(Rm). The Fredholm theory essentially provides a means for the computation of (1 - wK)-l from the kernel K(a, a'). The main results of this theory for L2 kernels are contained in Theorems 5.3 and 5.4. The reader interested in the proofs of these theorems is advised to consult Smithies [1965], Theorems 6.5.1-6.7.1.

Theorem 5.3.

(5.30)

dn =

Let K(a, a') be a Hilbert-Schmidt kernel and let us

-1n!

,pn

and the modijied first Fredholm minor of K(a, a') K(a, q)

(5.31)

..*

K(a7 a n )

(-1P Dn(a, a') = n!

I,

for n = 1,2,...; for n = 0, do = 1 and Do(a,a') = K(a, a'). In that case the series m

(5.32)

d(w) =

C dnwn n-0

is convergent for all complex values of o. Likewise, for any fixed

* A proof of this statement can be based on Mercer's theorem (see Riesz and Sz. Nagy [1955, Section 98]), or it can be deduced from (8.10) in Chapter IV.

V. Quantum Mechanical Scattering Theory

528 wE

C', the series W

(5.33)

D(a, a'; w)

=

1 D,(cr,

a')w"

"=O

is convergent in the mean [in L2(R2m)]for all w . Moreover, the function D(a, a'; w ) is a Hilbert-Schmidt kernel for any fixed complex w .

Theorem 5.4. If d(w) # 0 and ~ E L ~ ( R the ~ ) ,integral equation (5.28) has the unique Lebesgue square-integrable solution (5.34)

Theorem 5.5. The function d ( w ) vanishes if and only if eigenvalue of the integral operator K. 5.4.

w-'

is an

THEFULLGREENFUNCTION

We can apply the Fredholm theory to the integral equation (5.24) when its kernel (5.25) is of the Hilbert-Schmidt type. Under these circumstances, (5.24) is a special case of the integral equation (5.28), in which the variables a, a' E Rm become r, r" E R3, and w = 1. Moreover, we note that since in (5.6) ( is not on the positive real axis,

is square integrable in r or r' because according to Theorem 5.1 we have to choose the square root of ( for which I m & > 0. Hence, for any fixed r", (5.24) has a unique solution, provided that d(1) # 0 for the given kernel (5.25). T o establish that d( 1) is not zero for the kernel K(r, r'; () in (5.25), note that K(r, r'; () is the kernel of the closed operator

(5.36)

GO(5)Hl = (5 - H o ) - V - H0)-

According to Theorem 5.5, d(1) = 0 if and only if w = 1 is an eigenvalue of this operator, i.e., if there is a nonzero t,hl for which

< ( 5 - Ho)# I $1)

= ( ( 5 - HO)$I

G0(5)Hl$l)

=

Wl$ I #)

for all a,b E gH3 g H C 0 g H.1 However, the above relation implies that Ha,bl = (a,bl. Thus, as long as ( does not belong to the spectrum

5. Green Functions in Potential Scattering

529

of H, the Green function G(r, r'; 5) of H exists, and can be computed by means of (5.34) with o = 1 if K is Hilbert-Schmidt. Let us find out now the restrictions imposed on the potential V(r) by the demand that K(r, r'; C) be square integrable. Applying Fubini's Theorem, we find that (5.37)

I ' 1 K(r, r'; [)I2

dr dr'

=

R

I,dr'l ,V(r')12 5

drl Go(r,r'; <)lo.

R '

Using the expression (5.6) for Go(r,r'; 5) we easily compute that Im

qt > 0.

Thus, we get the relation

which shows that Go(r, r'; C) V(r') is for Im 5 # 0 a kernel of HilbertSchmidt type if and only if V(r) is square integrable on Rs. The main conclusions of the above discussion are contained in Theorem 5.6.

Theorem 5.6. If the potential V(r) is square integrable on R3, then the full Green function G(r, r'; [) of H [where H 1 -

(1/2m)d (5.40)

+ V(r)] exists for 1 q! SH and can be written in the form

G(r, r'; 5 ) = Go(r,r'; 6)

1

+-

D(r, r"; 1) Go(r",r'; 5) dr",

where d( 1; 5) and D(r, r"; 1) are obtained by taking (5.41)

K(r, r'; 1) = Go(r,r'; c)V(r')

as a kernel in (5.30)-(5.33).

5.5. FREDHOLM EXPANSION OF

THE

FULLGREENFUNCTION

The kernel D(r, r"; 1) of the integral operator in (5.40) is given by the infinite series m

(5.42)

D(r, r";5)

= K(r, r";5 )

+ C k ( r , r";11, n=l

V. Quantum Mechanical Scattering Theory

530

This would imply that the full Green function can be expanded in a series m

G(r, r'; 5)

(5.44)

Gn(r, r'; 5),

= n=O

where Go(r, r'; (5.45)

5) is the free Green function, and

Gn(r, r'; 5)

=

1 d(1; 5 ) I

DnP1(r,r"; 5) Go@",r'; 5 ) dr" 3

for n = 1, 2, ..., if the order of integration and summation can be interchanged. T o prove that this is indeed the case, we need Lemma 5.1.

Lemma 5.1. If K(a, a') is a kernel of Hilbert-Schmidt type, then the Hilbert-Schmidt norm

of the modified first Fredholm minor D,(a, a') satisfies the inequality. (5.47)

11 D, (I2

< n-n/2{e1/211 K 1 2}n+1,

n = 1, 2,...,

and the modified Fredholm determinant d, satisfies the inequality (5.48)

I dn I < {(e/n)1/211 K112}n,

n

=

1,2,-.,

where 11 K (I2 is the Hilbert-Schmidt norm of the integral operator K, with the kernel K(a, a'): (5.49)

(1 K (I2 =

,1

WZrn

I K(a,

r2

dma dma'

.

The proof of the above result can be found in the work of Smithies [1965, Section 6.51.

53I

5. Green Functions in Potential Scattering

Applying in (5.45) the Schwarz-Cauchy inequality at fixed r, r' E R3 and using afterwards (5.47), we get by (5.38)

Due to n-% this expression converges very rapidly to zero as n + so that we can state the following theorem.

+

CQ,

Theorem 5.7. If V(r) is square integrable on R3, then the full Green function G(r, r'; <), 5 # SH, is square integrable in r E R3 for almost all r' E R3, and the series (5.44) converges in the mean with respect to the variable r E R3 to G(r, r'; 5) for all such r' E R3. 5.6. SYMMETRY PROPERTIES OF THE FULL GREENFUNCTION The two symmetry properties (5.14) and (5.15) of the free Green function are inherited by the full Green function, i.e., for 5 4 SH

5) = G*(r, r'; [*), G(r, r'; 5) = G(r', r ; [)

(5.51)

G(r, r';

(5.52)

holds almost everywhere in RS. The first of these two relations is a straightforward consequence of the formulas (5.42) and (5.44) for G(r, r'; C), resulting from the fact that V(r') is real, and consequently K(r, r'; 4)

=

Go(r, r'; [) V(r')

=

Go*(r, r'; [*) V(r')

= K*(r,

r'; [*).

To prove (5.52) we have to resort to a lengthier argument. First of all, we have to note that the function r';

5)

=

I

D(r, r"; 1) Go(r", r';

5) dr"

P

is the kernel of a Hilbert-Schmidt operator. This follows from the fact that G,,(r", r'; 5) is the kernel of the bounded operator (5 - H0)-l, while D(r, r"; 1) is a kernel of the Hilbert-Schmidt type, since by

532

V. Quantum Mechanical Scattering Theory

(5.47) and Theorem 5.3 it is the limit in the mean of the HilbertSchmidt kernels (see Exercise 5.4)

Consequently, D(r, r"; 1) is the kernel of a Hilbert-Schmidt operator D(1). Thus, Sl(r, r'; 5) is the kernel of the operator D(1)(5 - Ho)-l, which is the product of a Hilbert-Schmidt operator and a bounded operator, and therefore is of Hilbert-Schmidt type (see Exercise 5.5). The function f*(r)g(r') is square integrable in R6 whenever f,g E L ~ ( WSince ). r1(r,r'; 5) is a kernel of Hilbert-Schmidt type, it is also square integrable in R6. Hence, the function f *(r) r1(r, r'; 5) g(r') is integrable in UP.Consequently, Fubini's theorem can be applied to infer that

It can be seen from (5.6) that a corresponding relation holds for Go(r,r'; c). Consequently, in view of (5.40), we have (5.53) -

.J,

dr G(r, r'; 5)f*(r).

On the other hand, the relation

in conjunction with (5.51) implies that

-

=

.J,

dr'g(r') J

dr G*(r', r; {*)f*(r) RS

I,, dr' g(r') I dr G(r', r; t;)f*(r). bP

533

5. Green Functions in Potential Scattering

Comparing (5.53) and (5.54), we see that

for all square-integrable functions g(r’). This is possible if and only if G(r, r‘; I)f*(r) dr =

Jd

G(r‘, r; t)f*(r) dr

for almost all r’ E R8. Since the above equality holds for all squareintegrable functions f(r), we infer that (5.52) holds almost everywhere in R8 x R8 = We. This completes the proof of (5.52).

*5.7.

APPENDIX:THESPECTRUM OF THE SCHROEDINGER OPERATOR

We have seen already that if K(r, r’; 6 ) is a Hilbert-Schmidt kernel the full Green function exists as long as 5 4 SH. This is certainly the case if I m 5 # 0. The following theorem shows that this will be also the case if 5 is real but negative, with the possible exception of a finite or at most countably infinite number of values.

Theorem 5.8. Suppose the potential V(r) is locally square integrable and V(r) -+ 0 as r + 00. Then the continuous spectrum Sf of the Schroedinger operator H 2 -(1/2m)A V(r) contains no negative values; moreover, there are only at most countably many negative eigenvalues, all of finite multiplicity, # and having no negative number as an accumulation point. In order to prove the above theorem, we need a number of auxiliary results. Since these results are of interest in themselves, we shall present them as lemmas and theorems. We recall from $3 that if H and H , are any two operators (not necessarily self-adjoint) with identical domains of definition and 5 belongs to their resolvent sets, then the resolvent R(5) of H can be expressed in terms of H I = H - H , and of the resolvent Ro(5)of H, by making use of the infinite series in (3.68), provided that this series converges in some sense. Lemma 5.2 represents, to some extent, a converse to this result, valid under the additional assumption that

+

* Recall that the multiplicity of an eigenvalue is the number of linearly independent eigenvectors corresponding to that eigenvalue. Local (square) integrability means (square) integrability over arbitrary bounded Bore1 sets.

534

V. Quantum Mechanical Scattering Theory

HIR,([) has a bound less than one; in this context, it should be noted that when g H= g H, the operator HIRo([)is defined everywhere on &' since R,([) is dekned everywhere on .# and it maps any f E 2 into R,({)f E gH0 F g H 1so , that HIRo([)f is defined.

Lemma 5.2. Suppose that H and Ho are two operators (not necessarily self-adjoint) with identical domains of definition BH3 gH0 , and that the complex number { belongs to the resolvent set of H , . If )IH,Ro([)(l < 1, Hl = H - Ho , then [ belongs also to the resolvent set of H,and n

(5.55)

c

R(5) = Ro(5)

(-1)k(~l%(5))k*

k-0

Proof. Since 11 HIRo([)ll < I, the uniform limit in (5.55) exists (see Exercise 5.7). For any f~ g Hwe have

=% M H O

- 5 ) f - Ro(5)HlRO(S)(HO - 5 ) f +

+ (-1)"Ro(5) *..RO(C)(HO- 5)f + %(5) Hlf - Ro(5) HlRO(5) Hlf + + (-l)nRo(5) Ro(5)Hlf = f + (-l)nRo(5) Ro(5)

.**

H l

H l

*.I

*.-

H l *..

H1fn*+m f.

This shows that if A ( [ ) denotes the operator on the right-hand side of the equality (5.59, then we have A([)(H - [) f = f for all f E g H . Since B,., = gH0and Ro([) has the range aH0, the operator (H - 5) Ro([) is defined everywhere on 2. Consequently, we can write

Thus, we also have (H - 5) A([) = 1, i.e., A ( [ ) is identical to ( H - [)-' = R([). Q.E.D.

If A is an unbounded self-adjoint operator, then its spectrum S A is an unbounded set. However, the set S A C R1 could be still bounded from

5. Green Functions in Potential Scattering

535

above or from below, in which case we shall say that A is bounded from above or from below, respectively. I n the case where A is bounded from below, we denote by mA the greatest lower bound of the set SA: mA = inf A. A

d

We note in passing that A is bounded from below if and only if the set

{ ~ ~ f ~A~f >-: zf E
0)

is bounded from below, with the same greatest lower bound (see Exercise 5.8).

Theorem 5.9. Suppose H and Ho are two self-adjoint operators with identical domains of definition, that Ho is bounded from below, and that for some 0 \< a < I, b 0,

>

for all f (5.57)

E gH0 3

g H Then . H is also bounded from below, and b mH 2 mHo- max 11 b --a'

+

mH0

11-

Proof. We shall prove the main statement of the theorem by showing that for any real b (5.58) 5<mHo-max -u,b+almH~II,

11

we have 11 HlRo(5)11< 1, and consequently, according to Lemma 5.2, such a 5 E R1 belongs to the resolvent set of H. This means that the spectrum of H contains only such points which do not satisfy (5.58); i.e., it is bounded from below and its greatest lower bound mHsatisfies (5.57). To prove that (1 H,Ro(5)11 < 1, we use (5.56) to derive the inequality

+

(5.59)

II HlRO(5)Il < 41 ~ O ~ O ( 5 ) I l bll RO(5)ll.

Since S H o C [mHo,

+ a),and therefore for any f E &' and real 5 < mHo,

V. Quantum Mechanical Scattering Theory

536

we conclude that

Combining the above two inequalities with (5.59), we obtain (5.60)

Since it is easily seen that the expression on the right-hand side of the inequality (5.60) is smaller than one when 5 satisfies (5.58), it follows from Lemma 5.2 that such 5 is in the resolvent set of H. Consequently, H is bounded from below by the expression on the right-hand side of (5.57). Q.E.D.

+

We recall from $7 in Chapter IV that when H = T, V , is the Schroedinger operator with a potential satisfying the conditions of (Chapter IV, Theorem 7.5) Theorem 7.4 in Chapter IV, then gH= gH0 if Ho = T, is defined by (7.15) in Chapter IV and (5.56) is satisfied by any 0 < a < 1. Hence, we can state Theorem 5.10.

Theorem 5.10.

(5.61)

T h e n-body Schroedinger operator H

(W(r,...,rn)

= w.1 3 . . * ,

r,)

$0.1

,.**,

=

H,

+ V,

rnh

corresponding to a potential V(r, ,..., r,) satisfying the conditions of Theorem 7.4 in Chapter IV, is bounded from below. As a by-product of the above theorem and (5.57), we can obtain estimates on the lower bound of an n-body Schroedinger operator H by computing the constants a and b, as was done in the proof of Theorem 7.4 in Chapter IV. A single-particle Schroedinger operator H with a potential satisfying the requirements of Theorem 5.8 also satisfies the conditions of Theorem 7.4 in Chapter IV. Hence, such a Hamiltonian H is bounded from below.

537

5. Green Functions in Potential Scattering

The second resolvent equation ( H - A)-1

(5.62)

= ( H , - A)-1

- ( H , - A)-lV(H

- A)-1

holds for any value of h in the resolvent set, and therefore, in particular, for any real X < m , . The desired result about the spectrum of H, stated in Theorem 5.8, can be then obtained from Lemma 5.3, which will be derived with the help of the next theorem, which implicitly introduces the following classification of the spectrum of a self-adjoint operator.

Definition 5.2. A point A, in the spectrum SA of a self-adjoint operator A is a limit point of SA if it is either an eigenvalue of infinite multiplicity or an accumulation point of SA. The set Sf of all limit points of S A is called the essential spectrum of A, and its complement SAd -- SA - Sf , i.e:, the set of all isolated eigenvalues of finite multiplicity, is called the discrete spectrum of A. *Theorem 5.1 1 ( Weyl's criterion for limit points). A real number A, is a limit point of the spectrum of a self-adjoint operator A if and only if there is a sequence fi,fa,... E g Awhich converges weakly to the zero vector, and is such that 11 f, )I = 1 and (5.63)

lim ll(A - h0)fn I1 = 0.

n++m

Proof. Suppose that 1) f, 11 = 1, that w-lim,++oofn = 0, and that (5.63)is true. For an arbitrary open interval (a, ,b,) containing A, we can write -~

f

1' n =

fw ( A -

~ Effn I I 11'

+ (a, - hg)' yo41Effn 11' z (b, - h0)' Il(1 - Ea,)fn 11' + (a, - h0)' I1 E:Jn 1.' 2 (b, - 4)' Jrndl1E f f n 11'

--m

b0

Now, the expression on the left-hand side of the above inequality approaches zero in the limit n --t +co. Hence, we have lim I1 Eofn I1 = ;jymIl(1 - E t ) f , II

n-r+-m

and consequently lim II EA((a,

n++m

Y

b0I)fn

11'

= n++m lim ll(E{

- G 0 ) f n 1'

= 0,

538

V. Quantum Mechanical Scattering Theory

This implies that the subspace EA((uo,b , ] ) 2 onto which EA((uo,b,]) projects cannot be finite dimensional; in fact, if this subspace were of finite dimension N, we could select an orthonormal basis {el ,..., eN} in it, and write

11

3

bO1)fn

/I2

=

Ifn>12

+ + ”*

I(eN lfn>I2

+

0,

since limn,+oo (el I fn) = ... = limn,+m (eN I f,) = 0 due to the weak convergence to zero of f, , f a ,.... Thus, we conclude that A, is either an accumulation point of S A or an eigenvalue with an infinite-dimensional characteristic subspace, i.e., that A,, is a limit point of SA. Conversely, if A,, is a limit point of the spectrum of A and I, 3 I2 3 ... are open intervals containing A,, , then each .@(I,) is the projector on an infinite-dimensional subspace of 2. Hence, we can choose an infinite orthonormal system {el ,e2 ,...} such that EA(I,) en = e, , n = 1, 2,... . Since we have

ll(A - &)en 112

=

J

I,

( A - &I2

4 Efen 112

< 11, l2 II en 112

+

0

when these intervals were so chosen that their lengths 1 I, I shrink to zero, and since limn,+, (e, 1 f ) = 0 for any f E 2 because of Bessel’s inequality, we see that the three conditions of the theorem are satisfied by this sequence el , e2 ,... E QA . Q.E.D.

*Lemma 5.3. If A and K are self-adjoint bounded operators on 2‘ and K is also completely continuous, then essential spectra of A and A K coincide.

+

Proof. If A, is a limit point of the spectrum of A, according to Theorem 5.11 we can find a sequence f, ,fi ,... E 2 such that 11 f, (1 = 1, w-limn,+oo fn = 0 and limn++, ll(A - A,) f, I1 = 0. Since f, ,f 2 ,... is a bounded sequence and K is completely continuous, it must contain a subsequence g, ,g, ,... such that Kg, , Kg, ,... converges strongly to some vector h E 2. However, since w-lim,,+- g, = 0, we infer that 11 h (1, = lim,,,, (h I Kg,) = limn,+, (K*h I g,) = 0. Hence, limn++mll(A K - A,) g, 11 = 0, which in conjunction with 1) g, 1) = 1 and w-lim,,+,g, = 0 implies, by virtue of Theorem 5.11, that A, is a limit point of the spectrum of A, = A K. Conversely, if A, is a limit point of the spectrum of A, ,then, according to the same argument applied to (-K), we deduce that A, is also a limit point of the spectrum of A = A, - K . Q.E.D.

+

+

We can return now to the proof of Theorem 5.8, and observe first that since the spectrum of Ho is [0, +a),the spectrum of ( H , - A)-1,

539

5. Green Functions in Potential Scattering

h

< 0, is [0, -l/h],

as seen from the relation

E R J I ) ( B ) = EHyF-yB)),

F(h') = (h' - A)-1.

The operator (A - Ho)-l V is an integral operator with the kernel K(r, r'; A) = G,(r, r'; A)V(r'). T o prove that the operator K(A) is n and Vn(r) = 0 completely continuous, we set V,(r) = V(r) for Y for Y > n, and note that due to the local square integrability of V(r) the operator &(A) with kernel G,(r, r'; A)V,(r') is Hilbert-Schmidt. Since obviously

<

II W) - K(4ll < Il(A - 4 - l II sup I @)I r,n

-+

0, n -,03,

we deduce from Theorem 8.3 in Chapter IV that K(A)is indeed compact. For A < mH , the operator (H - A)-l is bounded, and consequently, by Theorem 8.l(b) of Chapter IV, ( H , - A)-lV(H - A)-l is also a completely continuous operator. Hence, according to (5.62) and Lemma 5.3, the limit points of the spectrum of (H - A)-l are the same as the limit points of the spectrum of ( H , - A)-l, i.e., they constitute the interval [0, -l/A]. This implies that the limit points of the spectrum of H constitute the interval [0, a), i.e., H has no negative limit points. This means that the continuous spectrum of H is contained in [0, a), that any negative eigenvalues of H must be of finite multiplicity, and that these eigenvalues have no accumulation point on the negative real axis. Thus, Theorem 5.8 is established. Theorem 5.8 deals with the point spectrum of the Schroedinger operator on the negative real axis. For the more restricted (but still very large) class of potentials V(r) for which there is an R, > 0 such that for all Y 2 R, the potential V(r) is continuous and I Y V(r)l const, it can be shown (see Kato [1959, Theorem 11) that the Schroedinger operator has no positive point spectrum, i.e., no positive eigenvalues." The reader should now recall that these results on the point spectrum of the Schroedinger operator are well illustrated by the case of the hydrogen atom treated in 97 of Chapter 11. I n that case we were dealing with a point spectrum of the form St = {-a/n2 : n = 1,2,...}, where a is a constant characteristic of the hydrogen atom.

+

+

<

BETWEEN RESOLVENTS *5.8. APPENDIX:RELATIONS AND SPECTRAL FUNCTIONS

We know already that if the spectral function Eh of any self-adjoint operator H is given, then its resolvent R(5) = (H - <)-l can be

* See Iorio [1978] for recent results. General surveys can be found in Jorgens and Weidmann [1973], Reed and Simon [1978].

V. Quantum Mechanical Scattering Theory

540

computed by means of the formula ( f I R(5)g) =

(5.64)

1, q 1

d < f I EAg).

It is not so obvious, however, that the converse of this statement is also true-namely, that the spectral function E, of H can be computed from the resolvent I?([) (i.e., from the Green function of H, if it exists). To show that EA can always be computed from R([),we need the following lemma.

Lemma 5.4. If the function F(h) has a continuous derivative F'(h) on [a, b], and is Riemann-Stieltjes integrable on [a,b] with respect to .(A), then

[ F(X)d,p(A) b

(5.65)

b

= F(b) a(b) - F ( a ) .(a) -

J F'(h) .(A)

dh,

a

where the integral on the right-hand side of the above relation is a Riemann integral.

Proof. Taking hk' = hk in the defining formula (3.18) and recalling that A, = a and A, = b, we get

According to the mean value theorem of differential calculus, F(hk) = F ( h k - 1 )

+

F '(& c-l)(A k

- 'k-l)?

h-1

<

"b-1

<

*

Consequently, we have (5.66)

F(hk) k=l

-

F(hk) a ( A k - l ) k=l

Since F'(h) is uniformly continuous on [a,b],

and the expression in (5.66) converges to the right-hand side of (5.65).

Q.E.D.

541

5. Green Functions in Potential Scattering

Let us apply the result of the above lemma to the integral in (5.64). For Im 5 # 0 this integral can be considered to be a Stieltjes integral, and since (A - 0-l is continuously differentiable, we get

Letting a -+ - 00 and b ---t (5.67)

+m

+

1 x-5 d ( f I &g)

-m

00,

=

we arrive at the result

jCm EAg) dh, (h - 5)z ( f '

-m

Im 5 # 0,

from which we can derive Theorem 5.12. T h e o r e m 5.12. For any fixed f,g E S,the function (f I R(5)g) is analytic at any complex 5 with Im 5 # 0, and for A, A,

<

(5.68)


(EAp

+

EAp-O)g)

-

(f I ( E A ~ + ' A l - O k )

where the above integrations should be carried out along any piecewise smooth curves which do not intersect the real axis.

Proof. According to (5.67) the complex derivative

exists at any point 5 E C1 with I m 5 # 0 since the above improper Riemann integral converges absolutely and uniformly in any closed neighborhood of 5 which is disjoint from the real axis. Hence, ( f I R(5)g ) is analytic in the upper and in the lower complex plane. Due to the analyticity of (f I R(5)g) the integrals in (5.68) are independent of path. If we carry out the integration along straight lines and use (5.64) and (5.67), we get

(equation continues)

V. Quantum Mechanical Scattering Theory

542

The integrand in the last of the above integrals is Lebesgue integrable in s and h on the set [A1 ,&J x W1, since it is majorized by a function integrable on that set:

Hence, the order of integration in s and A can be reversed and, after carrying out the integration in s, we obtain

Upon splitting the interval of integration into (- a,Ai) and [Ai,

+ a),i = 1, 2, whereupon Lemma 4.2 can be applied separately to the first and second term in the above braces, we arrive at the relation lim IE- (f I

t++O

EA,-d)

+ (f I

which is identical to (5.68).

EA1-Og)

=(

fI

-
Q.E.D.

That the above theorem provides an important means of computing the spectral measure of a self-adjoint operator from its Green function.

EXERCISES 5.1. Prove that the set DG0defined in (5.16) is a Bore1 set of Lebesgue measure zero in Re. 5.2. Assuming that JRSn I K(a, a')J2&(a) dp(a') < +CO, show that = JRn K(a, a')f(a') dp(a') on the adjoint K* of the operator (Kf)(a) L2(Rn, p) is also an integral operator with the kernel (K*)(a,a') = K*(a', a), and that the kernel (K*K)(a,a') of K*K is equal to J@K*(a", a) K(a", a') +(a"). 5.3. Prove that any integral operator K on L2(Rn,p), which has an La kernel K(a, a'), is a Hilbert-Schmidt operator, and that Tr(K*K) = R"

5.4.

I K(a, a')lz+(a)

dp(a').

Give the reason why the limit in the mean K(a, a') of a sequence

543

6. Distorted Plane Waves in Potential Scattering

of Hilbert-Schmidt kernels KJa, a’), n++m lim I R s m

I K(a, a‘) - &(a,

a’)I2+(a) &(a’) = 0

is a kernel of Hilbert-Schmidt type.

5.5. Show that if A is Hilbert-Schmidt operator in a separable Hilbert space and B is a bounded operator then BA is a Hilbert-Schmidt operator.

5.6. A, , A, ,... xz=, 11 A, 11 < + Let

be a sequence of bounded operators for which

co. Prove that the uniform limit

c A, n

A

=

u-lim n++m

k=l

exists and is a bounded operator with

< k=l c 11 m

11 A 11 5.7.

11.

Use the results of Exercise 5.6 to show that

c ckAk n

u-lim n++m

exists if

(1 A 11 < 1 and 1 ck I

k=l

< M for all k = 1, 2, ... .

5.8. Prove that the self-adjoint operator A is bounded from below if and only if ( f l Af) 2 mAIlfl12 for a l l f E : g A .

6. Distorted Plane Waves in Potential Scattering 6.1. POTENTIALS OF ROLLNIK CLASS In $5.5 we have learned that for square integrable potentials the equation (5.24) could be solved by the Fredholm method, and on first sight it might appear that exactly the same method is applicable to the full advanced and retarded Green functions G(*)(r,r’; A), A E Sf , that are required in the solution-type Lippmann-Schwinger equations (4.74). Indeed, as discussed in $5.8, under reasonable conditions on V(r) (which from now on we shall take for granted) we have

(6.1)

S ~ :SC

=

[0,00),

s i n [o, 00)

and formally taking in (5.24) and (5.25) the

o, limit Im 5- fO =

at fixed

V. Quantum Mechanical Scattering Theory

544

h = Re 5 E SoH we obtain (6.2)

G(*)(r,r'; A)

=

Gr)(r,r'; A)

+

K(*)(r,r";A) G(*)(r",r';A) dr", I '

K(*)(r,r'; A) = Gr)(r,r'; A) V(r').

(6.3)

Unfortunately, the kernels in (6.3) are not Hilbert-Schmidt since

1 K(*)(r,r'; A)I2

ma I V(r')12 b2 Ir -rill2

=-

obviously is not integrable in r over R3. The Lippmann-Schwinger equation (4.67), which shares the same kernel with (6.2), (6.5)

+

@F)(r)= ak(r)

K(*'(r, r';k2/2m)@ii)(r')dr' ws

runs into exactly the same problem. The most frequently employed" way out of this impasse consists of factorizing V(r) as I V(r) 11/2V1/2(r),where vl/2(r) =

I

I V(r)11/2sgn V(r) 7 exp(-r2), 7 > 0,

if V(r) # 0, if V(r) = 0,

and then multiplying (6.2) or (6.5) from the left by W2(r).For example, in case of (6.5), upon setting (6.7)

&(r)

=

Vl/2(r)Ok(r),

@)(r)

=

V1/2(r)@f)(r),

the resulting equations can be written in the form (6.8)

$?)(r)

= 6k(r)

+

g(*)(r,r'; k2/2m)@)(TI)

dr'

wa

where the new kernels (6.9)

P ( r , r'; A)

=

V1/2(r)

GF)(~, r';A) I V(r1)11/2

are Hilbert-Schmidt for the class of potentials specified in the following definition. Apparently, this factorization method was first discovered by Rollnik [1956], and then independently rediscovered by many others (Schwartz [1960], Grossmann and Wu [1962], Scadron et al. [1964], etc.). For the sake of simplicity, in (6.11) and (6.12) we shall assume that V(r) # 0 almost everywhere in Rs.

545

6. Distorted Plane Waves in Potential Scattering

Definition 6.1. A real function V(r) is of Rollnik class if it is Bore1 measurable on R3 and if its Rollnik norm 11 VllR exists, where (6.10)

We easily see that if V(r) in (6.9) is of Rollnik class then Hilbert-Schmidt :

R(*)(A)are

-

A frequently encountered type of potential V(r) of Rollnik class is one that is continuous, except possibly for a divergent behavior V(r) ra, 0 < a < 1, as r + +0, and bounded by r-2--Eo, e0 > 0, as r -+co (see Exercise 6.6). 6.2. FREDHOLM SERIES EXPRESSIONS FOR DISTORTED PLANEWAVES Another great advantage of (6.8) over (6.5) is that the inhomogeneous term 6k(r)in (6.8) is square integrable if V(r) is integrable on R3 : (6.12)

I/ & 1,

I Gk(r)I2dr

=

=(

2 ~ ) JR3 - ~ I V(r)l dr

= (2m)-s

11 VII, .

088

Hence the Fredholm theory becomes immediately applicable, and we can state the following theorem.

Theorem 6.1. some e0 > 0 (6.13)

Suppose that V(r) is of Rollnik class and that” for V(r)

=

O(T-~*O), r - +

00.

Then the modified Fredholm determinants dA*)(A) and minors &*)(r, r’; A) of l?(*)(r, r’; A) in (6.9) [as given by (5.30) and (5.31), respectively] exist, and the infinite series (6.14)

d(*)(A) =

2 df)(A) 7Z=O

converge and are different from zero for all A E (0, co), with the possible

* Recall that f(r) = O(g(r)) when I r I +UJ

If09 < C I g(r)l for all I r I > R.

if there are constants C and R such that

V. Quantum Mechanical Scattering Theory

546

exception of a compact set Yvl of Lebesgue measure zero for which [0, +a)- YV1 C S O H . Furthermore, for all k outside the (compact) exceptional set SpYs

(6.15)

= {k :(k2/2m)E y;’}

of Lebesgue measure zero in R3, the Lippmann-Schwinger equations (6.5) have the unique solution

2%)

(r, r’;

x) 2m

W2(r’) Qk(r’) dr’,

The fact that Yvl (and therefore also Yv3) is compact and of measure zero is established in Appendix 6.7, whereas the rest of the statements are an immediate consequence of Theorems 5.3 and 5.4 since V(r) is integrable on R3 so that by (6.12) 11 d j k 11 < 00. Indeed, due to (6.13) V(r) is integrable outside some sphere of radius R around the origin, whereas inside that sphere its integrability follows by Theorems 3.9 and 3.13 in Chapter I1 from the existence of 11 V IJR. Theorem 6.1 establishes that the Lippmann-Schwinger equations (4.67) possess for almost all k E R3 unique solutions, but it does not establish that these solutions are the distorted plane waves for which (4.25) is true. Indeed, (4.25) has been assumed in the course of deriving (4.67), and now we have to prove that this assumption was justified. The following lemma provides the first of many steps in that direction.

Lemma 6.1. If V(r) = O ( r 3 - ~ ois) locally square integrable and of Rollnik class then (6.18)

&(k)

=

(@?) I (G)

@i*)*(r) #(r)dr

= 1.i.m. J-R.

exist for any k E R3 - Yv3provided @k*)(r)are the solutions (6.16) of (6.5), and if B C (0, co) is a compact set disjoint from Yvl then

+

547

6. Distorted Plane Waves in Potential Scattering

Proof. T o prove the lemma, we first have to make a few remarks which are also of independent interest. Comparing (5.5) with (5.10) we see that for fixed r E W3,

Consequently, by Theorem 4.6 in Chapter 111, for any given r E W3,

where 1.i.m. could be dropped, since by (5.6) Go(r, r'; 5) is obviously integrable in r' over W3 at fixed r E W3.Since by (5.52) and Theorem 5.7, G(r, -;5) E L ~ ( Wfor ) almost all r E W3, by analogy with (6.21) we can define

for

5 E C1- S". Hence by (5.24) and (5.25),

(6.23)

and upon introducing by analogy with (6.7) &(r; 5) = VYr) @k(r;51,

(6.24)

we see that (6.23) is equivalent to

Since K(r, r'; 5) is obviously Hilbert-Schmidt, we get analogy with (6.16): (6.27)

(6.28)

Qk(r;5)

=

+

ak(r)

B(r, r'; 5) 6k(r')dr',

V-1'2(r)

@>

,A

1.3

comF .ete

V. Quantum Mechanical Scattering Theory

548

If we now define by analogy with (6.18) $(k; [)

(6.29)

= 1.i.m.

Ok*(r; [*) +(r) dr,

we see from (6.22), (5.51), and (5.52) that

Consequently, we obtain from the first resolvent equation (3.62) (6.31)

<*I

[G(O - G(5*)1*>

= (5*

*

- 5)

Thus the connection of &(k) in (6.18) with the spectral function of H finally becomes evident as we consider that by Theorem 5.12 we have

when we take into account that EnH= E&, for A Yvl since no point in [0, CQ) - Yvl belongs to SpH. Indeed, if H+, = A,+, for some A, 2 0 then E g HC g V , so that V(r) I +,(r) l2 is integrable, and therefore its absolute value is integrable (see Chapter 11, Definition 3.5); consequently, V112+€L2(R3) and

+,

R(+)()b)( W 2 * )

=

W2(Ao - H0)-1V*

=

V1/2(A0 - H0)-1(Ao - H,)*

=

W$h,

which implies that (1 - E(+)(&))-l does not exist, and therefore by Theorem 5.4, d(&) = 0, i.e., A, E Yvl. On the formal level, (6.19) with B = [a, b] C (0, 00) follows from (6.32) and Lemma 4.2 by noting that (6.33)

Op'(r)

= &i)*(r)=

lim 4jk(r;(k2/2m)

€++O

+ k).

and consequently for k 4 YV3 (6.34)

&(k) =

(k2/2m) fi~).

The rigorous proof is, however, considerably lengthier since one has to justify the interchange of orders of integration in A and k in (6.32) and the application of Lemma 4.2 under the double integral

6. Distorted Plane Waves in Potential Scattering

sign with an integrand 1 $(k; h

+ i r ) l2 that is r-dependent.

see, this can be achieved by proving that

549

As we shall

I $(k; 5) l2 is uniformly bounded

k and 1: as well as continuous in 5 in the complex half-planes Im 5 2 0 or Im 5 < 0, respectively, wherever d ( [ )# 0, so that Fubini’s

in

theorem and Lebesgue’s dominated convergence theorems can be applied in a routine manner. Expanding the determinant in (5.31) along the first row or first column we get, respectively,

(6.35)

D,(a, a‘) = dnK(a, a‘)

+

Dn-l(a, 011) K(a1, a’)d”q L m

= dnK(a,a‘)

+

s,-

K(a, 011) Dn-l(al , a’)d”a1.

Combining these two relations we obtain

In the case of (6.26)-(6.28) this inequality yields

where due to (5.47) and (5.48) we can choose the posit? constants C, to be independent of r, r’, and 5, and such that C = Cn=,,C, < 00. Consequently, by (6.27),

550

V. Quantum Mechanical Scattering Theory

Due to the conditions imposed on the potential, PJr; 5) is locally square integrable and bounded at infinity (see Exercise 6.5). Since (5.47) yields the uniform convergence and therefore, due to the continuity of d(5) for Im >, 0 or Im 0, also the continuity of &((), this boundedness is uniform when 5 varies over any compact set not containing zeros of d(5).Hence, the procedure described earlier leading to (6.19) for B = [u, b] can be applied if $(r) E VbOD(R3). The outcome can be then extended to arbitrary $ eL2(R3)by taking the limits in the mean appearing in (6.18). The general case of compact B C (0, 00) disjoint from YV1 can be reduced to the above special case by using Theorems 1.3 and 1.7 in Chapter 11. Q.E.D.

<

6.3.

<<

ASYMPTOTIC COMPLETENESS AND EQUALITY

THE

GENERALIZED PARSEVAL'S

We recall from Theorem 5.8 that locally square integrable potentials which satisfy (6.13) give rise to a Hamiltonian H whose continuous spectrum contains no negative values, so that by Theorem 6.1 (6.40)

[O, a)- Y"1c S,H

c [O,a).

Let us introduce for each eigenvalue h E SPH an orthonormal basis in the proper subspace of that eigenvalue (which by Theorem 5.8 is of finite multiplicity if A < 0). Then for any f,g €L2(IW3), (@A,Y}

(6.41) (6.42)

(f I EH(SpH)g>= AES,C"

3 *(A Y ) = (f I @ A , A

Cf *(A,

Y ) &!(A,

Y),

v

!('A,

Y) = <@A,Y I g>.

Lemma 6.1 suggests that for the continuous spectrum we might expect a similar relation, so that the following generalized Pursevul's equality [cf. (4.15) in Chapter I] might hold: (6.43)

= (f I EH(SDH)g>+ (f I EH(ScH)g)

The nontrivial part of equation (6.43) is the last expression, which follows from the next theorem.

Theorem 6.2. Suppose that V(r) is locally square integrable, of Rollnik class, and O(r-3-Eo)at infinity. Then the continuous spectrum

55 1

6. Distorted Plane Waves in Potential Scattering

of H satisfies (6.40), and for any Borel set B C SoH,

(6.45)

&(k)

= (OF)I $) = 1.i.m.

@r)*(r)$(r) dr, JR.

for all $ €L2(IW3).Furthermore, for any f,g €L2(IW3), (6.46)

Proof. When B is compact and disjoint from sPvl, (6.46) is obtained from (6.19) by using formula (5.13) in Chapter 111. To arrive at (6.44) for such B we choosef(r) E%~”’(R~),so that by

(6.38) (6.47)

( 2 ~ ) ’ ’I ~@f)(r)f(r)I < [I

+FV(r;k2/2m)l I f(r)l

is integrable in (r, k) over IW3 x A;l(B). Hence, for arbitrary g €L2(IW3), the reversal in order of integrations in (6.48)

is justified by Fubini’s theorem, and we conclude that

If we interpret “1.i.m.” in (6.44) for any Borel set B C SCH as a procedure in which the integration is first performed over sets (6.50)

A;’(B n B,)

= {k : A,(k) = (k2/2m)E B

n B,},

with B, compact and disjoint from YV1, and then the limit in the mean is taken for B, C B, C - - -[0, + 00) - YV1, we finally arrive at the general result (6.46). Q.E.D. In view of (6.45), the formula (6.46) can be written in the symbolic

V. Quantum Mechanical Scattering Theory

552

notation introduced in (4.82) as follows, (6.51)

and therefore (6.41)-(6.43) implies that

Thus, Equation (6.52) is intimately related to the following notion of asymptotic completeness.

Definition 6.2. A quantum mechanical theory on S with given free and total Hamiltonians Ho and H is said to be asymptotically complete if any interacting state Y ( t ) orthogonal to the all bound states of the theory is a scattering state, i.e., it has incoming or outgoing (physical) asymptotic states Y e x ( t ) ,ex = in, out. T h e wave operators Q*.of a theory are said to be complete if and only if the theory is asymptotically complete. In $1.4 we denoted by %‘La’ the closed subspace spanned by all bound states in the Hilbert space S(2) of a two-particle system. In $1.7, upon eliminating from the problem the center of mass motion, we were left with the Hilbert space #’(l) of the reduced particle whose bound states spanned the closed subspace 86’’ I n. the present context of wave mechanics for spinless particles we have &(l) = L2(R3),so that (6.53) Hence the theory is asymptotically complete (and 9, are complete) if and only if” (6.54)

R+

= L$c(R3)

EH(SCH)L2(IWR).

On first sight it might appear that by establishing (6.43) or, equivalently, (6.52), we have already established the completeness of 9, for

* Motivated by primarily mathematical considerations, many authors follow the custom of calling S* complete if and only if the projector onto R* coincides with EH(S5) rather than EH(ScH),where Sfc is the absolutely continuous spectrum of H. As a rule they then proceed to prove (see, e.g., Amrein et al. [1977]) that the singularly continuous spectrum S$ = SCH - Sg is empty, so that in practical terms their definition coincides with Definition 6.2, which is dictated by physical rather than mathematical circumstances (see, e.g., Schechter [1978]).

553

6. Distorted Plane Waves in Potential Scattering

the class of potentials specified in Theorem 6.2. The trouble is, however, that @i*)(r) in (6.45) and (6.52) appear only as solutions of the LippmannSchwinger equations (6.5), rather than as eigenfunction expansions that are related to 52* via (4.25). Having already proven in $4.6 that (4.25) implies (6.5), we still have to complete the circle by proving that (6.5) in turn implies (4.25) under the conditions of Theorem 6.2 (and thus, incidently, establish asymptotic completeness in two-body potential scattering). This final task will be carried out in the context of proving Theorem 7.2 in $7 by first establishing in Lemma 7.2 the existence of operators W, for which (4.25) holds with y5, = W,y5 if we adopt the solutions (6.16) of (6.5) in the role of @L*)(r),and then, at long last, settling the issue in $7.3 by proving that W, = 52,. In the meantime, we shall derive some additional properties of @i*)(r) that are of great interest physically as well as mathematically.

6.4. THESCATTERING AMPLITUDE There are two distinct methods of defining the scattering amplitude in potential scattering: by means of the T matrix [i.e., by adopting formula (6.67), which we shall derive later, as a definition], or in terms of the asymptotic behavior of @L*)(r)as I r 1 -+00. We shall adopt the second alternative since it is in that context that the scattering amplitude makes its first appearance in most textbooks on quantum mechanics.

Definition 6.3. The function f k ( e , $ ) is called the scattering amplitude of the incoming distorted plane wave @L+)(r)in the direction (O,$) if for a given k E R3 - YV3 and (0, $) E 52, there is an E > 0 such that (6.55)

as r --t

vk(r) = (243'2[@k)(r) - @k(r)] = [fk(e, 00,

$h)/Y]

eikr

+ o(Y-'-')

where (r, B,+) are the spherical coordinates of r E R3.

Theorem 6,3. If V(r) = O(r-3-€0), r--t co, is square integrable and of Rollnik class then vk(r)is locally square integrable at each fixed k E R3 - Spy3. Furthermore, for each compact set B disjoint from YV3 and for a certain R, > 0 there are constants Cl(B, R) and C,(B, R) such that for all k E B,

554

where (6.58)

V. Quantum Mechanical Scattering Theory E

=

q,(l

fk(e,+)

+

E,,)-~

=-

and

m(27r)1/2 JR8 exp [-ik

I):

(rt *

V(r') Qp)(rl)dr'.

Proof. The conditions imposed on V(r) in the present theorem are the same ones as in Lemma 6.1. Consequently, vk(r) satisfies (6.56) on account of (6.38). Let us introduce the function (6.59)

m wk(r)= - -(eikr 27r

+ vk(r))V(r) = - - m ( 2 ~ ) ~ / ~ V@i+)(r) (r)

in terms of which by (6.3), (6.5), and (5.17), (6.60)

We note that vk(r) is certainly uniformly bounded at infinity for k E B (see Exercise 6.7), so that there is an RE > 0 independent of k E B such that (6.61)

I Wk(r)I

< const I v(r)l~

I I 2 RE

*

Let us write now (6.60) in the form

According to (6.61) there is a constant M such that for suitably large R (6.63)

Consequently, for such values of R

By selecting, for fixed r, the z1 coordinate axis in the direction of the vector r, and working in spherical coordinates, we arrive at the

6. Distorted Plane Waves in Potential Scattering

< 47rM [-

555

1

Furthermore, using Taylor’s formula, we get for rl/r -+ +0, exp(ik1 r - r1 I) = exp (ikr [l - r2

1 Ir-r1I

=

1 ; [l

+0

(?)I.

Consequently, we can write as rl/r --t +O (6.65)

exp(ik1 r - r1 I) lr-rll -

r

exp (-ik

-)r -r r1

I)$(

exp [ikr 0

[1 + 0.I):(

Let us take in (6.62) R = rl-8, with some fixed 6 < 1, and then choose r so large that (6.64) becomes valid. Inserting (6.65) in IR’,we obtain, after noting that for the selected R we have Rlr = r8,

Now, by virtue of (6.63), we also have for all k E B

and thus, uniformly in k E B,

V. Quantum Mechanical Scattering Theory

556

On the other hand, for the chosen value of

R,(6.64) yields

so that (with asymptotic contants independent of

1

k E B),

1

1

+ 0 (7) + 0 (m +) 0 (x) *

Setting in the above estimate 6 (6.57). Q.E.D.

=

eO/(l

+

eO)

< e0 , we

finally obtain

The main significance of the above theorem lies in the fact that it provides us with the explicit formula (6.58) for the scattering amplitude fk(O, 6).This formula requires, however, a full knowledge of @i+)(r). On the other hand, even in the case where @i++)(r) is not known, we can still obtain an approximation of fk(e,c$) by using (6.16) and neglecting the second term. Then we obtain the first Born approximation of the scattering amplitude: (6.66)

fk(e, 4) M =

m

jRs exp (-ik

r r' 7) v(r')exp(ikr') dr'

-m(27~)~/~P(kF - k).

Higher Born approximations can be obtained by an iteration method presented in the next section By denoting with w the unit vector with angular spherical coordinates (e,c$) and consideringfk(0, c$) as a functionfk(w) of w rather than (0, +), we immediately note the intimate relationship between the scattering amplitude (6.58) and the T matrix (4.95): (6.67)

fk(w)

= -(277)2m

L

@zm(r') V(r'> @?)(TI) dr'

@p))

=

-(277)2m(@km I V I

-

-(2~)~m(kwI T I k).

Written in this form, fk(w) exhibits a remarkable symmetry (intimately related to the time invariance of the Schroedinger equation): (6.68)

fkdw)

=f - k r ~ ( - ~ ) .

557

6. Distorted Plane Waves in Potential Scattering

Indeed, using (6.5) to eliminate the plane wave @,*(r)= @p--L(r) from (6.67) we obtain

s..

(6.69)

f k ( w )= - ( 2 ~ ) ~ r n &)(r’)

V(r’) @zm(rt)dr‘

+ 2 m 2jRdr’@k)(r’) V(r’) a

Under the conditions of Theorem 6.2 the integrand of the previous double integral is integrable in (r, r’) on R6 (see Exercise 6.2, and note that V(r) is integrable, whereas @i+)(r) is bounded in r E UP) so that Fubini’s theorem can be applied. Hence the right-hand side of (6.69) stays unchanged if we replace k and w with -w and -k, respectively. The relations (3.50), (4.96), and (6.67) between S operator, transition operator, T matrix, and scattering amplitude show that the action of the scattering operator on a wave packet $(k) = $(jlzw) can be expressed directly in terms of the scattering amplitude:

(S$)-(k)

(6.70)

= $(k)

+ ( i k / 2 ~ rJ) fkm,(k)$(kw‘)dw’. Qs

This represents yet another rigorous version of the formal relation (4.107). 6.5.

THEBORN SERIES

The simplest method of solving an operator equation such as (5.29) is by repeated iteration, which then yields $

(6.71)

= (1

+ wK + w2K2+ ... + w ” K n + *..)f,

provided that this infinite series converges in some sense. Clearly, the expression in between parentheses is nothing else than the series for (1 - wK)-l (cf. Exercise 7.9 in Chapter IV), and we shall obtain convergence in operator bound if 1 w I 11 K 11 < 1 (see Exercise 5.7). Furthermore, if K is Hilbert-Schmidt and I w I 11 K )I2 < 1, then this convergence in operator bound and is also convergent in the Hilbert-Schmidt norm (recall that 11 K 11 11 K \I2). Since for operators with Hilbert-Schmidt kernels K(a, a‘) such as those appearing in (5.28) we have

<

(6.72)

11 K 1;

=(K

, ,J

I K ) 2 = Tr(K*K) =

I K(a, a’)I2dmadma’,

558

V. Quantum Mechanical Scattering Theory

this convergence becomes (see Exercise 5.4) convergence in the mean on L2(RZm): N

( a , a‘) = 1.i.m.

(6.73)

N+X

1 w nK n( a, a‘), n=O

These remarks can be applied in particular to ( 6 4 , so that

+ 1 1 (l?(*)n)(r, r‘; k2/2m)$,(r’) m

(6.75)

$p)(r)

= Gk(r)

n=l

dr‘.

~3

Upon eliminating from both sides of (6.75) the common factor W 2 ( r ) , we get a series solution of the Lippmann-Schwinger equations (6.5): (6.76)

@f)(r)

= Gk(r)

+ -1 GF)(r, r’;k2/2m) V(r’) ak(r’) dr‘ w3

+J

dr,Gr)(r, r,; k2/2m) V(r,)

w3

x

s

dr‘GF)(rl, r’; k2/2m) V(r’) Gk(r’)

+

* a * .

This is the Born series for distorted plane waves, which inserted in

(6.58) provides the Born series for the scattering amplitude, whose first term we had encountered in (6.66). Working with the on-shell T matrix (4.95) instead of the scattering amplitude, we can write this Born series in the symbolic form m

(6.77)

(k I T I k)

=

2 (k I V(G?)(h)V)” n=O

1 k),

X

= k2/2m = k2/2m,

in which the nth term equals

-

Jw3

drak*(r)V(r)

s

w3

dr’(G?)(k2/2m)V)”(r, r‘) Gk,(r’).

559

V. Quantum Mechanical Scattering Theory

Theorem 6.4. If V(r) = O(r-3-fo), r + 03, is locally square integrable and 11 VII, < 03, then there is an energy value A, such that the Born series (6.77) for the on-shell T matrix converges uniformly in k and k at all higher energies h = k2/2m 2 A , . Furthermore, if 11 VII, < 27r/m then the series (6.77) converges uniformly in k, k E R3 at all energies, and the exceptional set 9,l is empty.

However, by specializing (6.72) we get from (6.9), (6.82)

11 R(+yA))*R(+)(A)ll; = Tr{[(B(+)(A))*R(+)(A)]’}

dr’ dr” 1 V(r‘) V(r”)l

x

11Gk)*(r, r‘;

By (5.17) the term in braces equals

A) Gk’(r, r”; A)

r

I V(r)l dr .

560

V. Quantum Mechanical Scattering Theory

In view of the Riemann-Lebesgue lemma" we conclude that upon introducing spherical coordinates, the radial integral in r E [0, co) approaches zero as k -+ m for any fixed k , k", and (O,+). Since for all K E [0, m), the integrand in (6.83) is majorized by an integrable and &independent function that is square integrable in k', k", and (0,d) (see Exercise 6.4), we deduce from Lebesgue's dominated convergence theorem that (6.82) approaches zero as A -+a, so that there is a A, for which (6.81) holds true whenever A 2 A,. Taking into account that by (6.79) and (6.80) for any such A,

n=O

?&=O

we see that the first part of the theorem is true. According to (6.11), (6.79), and (6.80), we also have I
(6.85)

m+'(V)" I k')l < (27T3II VIIl II R(+)(W < (2m)-311

Ill(m

11

llR/2T)",

so that (6.77) converges uniformly and absolutely for all k and k on the energy shell if 11 V [IR < 2.rr/m. The same inequality also implies that

(6.86)

exists for all A E [0, + co), and therefore by Theorem 5.5, d(A) # 0 at all energies, so that Spy1 is empty. Q.E.D. 6.6.

DISTORTED PLANE WAVES AS SOLUTIONS SCHROEDINGER EQUATION

OF THE

For the complete identification of @L*)(r)as distorted plane waves, we have to establish that @;+)(r)is a solution of the Schroedinger equation with the eigennumber k2/2m. Theorem 6.6 provides a set of precise criteria under which this is true.

Theorem 6.5. Suppose that the potential V(r) is continuous in the neighborhood of a point r E R8, and that it satisfies the conditions

* See Lemma 4.1 in Chapter 111, and note that its conclusion remains valid if we change variables of integration by setting x = h(y), where h'(y) exists and is positive (or negative) almost everywhere.

6. Distorted Plane Waves in Potential Scattering

561

imposed on it in Theorem 6.2. Then the distorted plane waves @&*)(r) satisfy the time-independent Schroedinger differential equation (6.87)

at that point r E R3, and for any k E R3 - 9’; . Before proceeding with the proof of the above theorem, let us make a few remarks. For the proof of this theorem we need, essentially, only to know that V(r) and @f)(r) are continuous in some neighborhood of r, and that

The restrictions on @i*)(r) are fulfilled automatically if V(r) satisfies the conditions of Theorem 6.2 and the restrictions imposed on it in the present theorem. It can happen, however, that @i*)(r) satisfies these restrictions even when V(r) does not obey all the conditions of Theorem 6.2 (see 57.2). Hence, the validity of Theorem 6.5 is somewhat independent of the validity of Theorem 6.2. The proof of Theorem 6.5 is straightforward, and it essentially consists in applying the Laplacian A to the right-hand side of equation (6.5) and computing the result. This computation is easily carried out with the help of Lemma 6.2.

Lemma 6.2. Supposef(r) = O(Y-~-~O) for some e0 > 0, and that there is a neighborhood JV of the point r E Rain which f ( r ) is continuous and the function (6.88)

is well defined by the above integral. Then g(r) is twice differentiable at that point in each of the variables x, y , and z, and (6.89)

Proof.

(-WW

= k2g(r)

+bf(r)*

Let S be a neighborhood of r and write

We shall compute separately the two terms on the right-hand side of the above identity.

V. Quantum Mechanical Scattering Theory

562

Straightforward differentiation shows that (6.90) (6.91)

a exp(ik(r - r' I) Ir- I

(x

- x') exp(ik1 r - r' I) (ik I r - r' 12

-

a2 exp(ik1 r - r' I) ax2 I r - r'I

On any bounded domain which does not contain some closed neighborhood of r, we have If(r')l

< const If (r'l Ir-r'I

'

Now, the function on the right-hand side of the above inequality is integrable on account of the existence of the integral in (6.88) and the fact that if a function is integrable then it is also absolutely integrable. Hence, we conclude that f(r) is locally integrable. It is easy to infer from (6.90) and (6.91) that for sufficiently large R, > 0 and all r, from some open neighborhood S C N of T

for n = 1,2. Thus, we deduce that atl exp(ik1 r - r' I) I r - r' I f(r') dr',

ax.

n = 1'2,

exist for r E S. Consequently, a/ax and iP/aXa can be carried under the integral sign when applied to the right-hand side of (6.88) (see Exercise 6.8). Since the same obviously holds for the variables y and LI, and since by (6.91),

6. Distorted Plane Waves in Potential Scattering

563

we obtain (6.93) exp(ik1 r - r' I) f (r')dr' I r - rI I

-A JRa-s

= k2

Jp-,

exp(ik(r - rr I) I r - r' I f (r')dr'.

Let us choose (6.94)

S

I x1 - x 1 < a, Iyl

= {rl:

-y

I < a, I z1- z I < a}.

If a > 0 is sufficiently small, then the generalized mean-value theorem of the integral calculus can be applied to obtain

where I yo - y I < a and I zo - z [ < a. Using (6.90) and the mean value theorem of integral calculus, we get

where I jj I < a and I 5 I < a. This shows that, due to the continuity o f f ( r ) in M , by taking the limit a --f 0 and using (6.90) we shall get in (6.95) a contribution of the form

To compute the above limit, we must compute the following integral explicitly: exp[iR(a2 Yl2 212)1/21 dy1 &,

+ +

564

V. Quantum Mechanical Scattering Theory

where Y, is obtained by applying again the generalized mean-value theorem to the first of the above two integrals, and it satisfies the inequality I yav I < a d z . Hence, the limit (6.96) is equal to -(4n/3)f(r). Due to the symmetry of the considered expressions under the exchange of x, y, and z,when S is the box in (6.94), we have (6.97)

lim A a+O

r r' I) I, exp(ik1 f (r') dr Ir-r'I -

a2

= 3 limwo ax2

J

exp(ik1 r - r' I) I r - r' I f(r') dr'

,

= 47rf(r).

Hence, the desired result in (6.89) is obtained by combining (6.97) Q.E.D. with the result (6.93) in which we let a -+ +O. In order to be able to apply the above lemma to derive (6.87) from (6.5), we have to establish that (6.98)

f(r)

m

= --

2n

V(r)@f)(r) = 0

(T)1

,

c0

> 0,

and that the above function is continuous in some neighborhood of the considered point r E R3. Since I @i*)(r)l= 0(1) for k 6 9';(see Exercise 6.7), it immediately follows that (6.98) is true. The continuity of @i*)(r)is an easy consequence of (6.5) (see Exercise 6.9). Hence, in computing the effect of applying the Laplacian A to the right-hand side of (6.5), we can use Lemma 6.2; then (6.87) easily follows.

*6.7. APPENDIX:ANALYTIC OPERATOR-VALUED FUNCTIONS We have seen in $5.8 that (f I R,([)g) is an analytic function on the resolvent set C1- SA of a self-adjoint operator for all f,g E 2. This categorizes R,([) as an analytic operator-valued function. The general theory of such functions is based on the following definition.

Definition 6.4. The vector-valued function f (5) E i@ defined on an open domain D of the complex plane C1is said to be analytic at E D if

c0

(6.99)

+ 4)-f(Ol/41

f'(0 = y$$ff(S

exists in some neighborhood of

5,.

Lemma 6.3. Iff([) is analytic at lothen (6.100)

f'(50) =

s&{rf(lo

+4

- f ( 5 0 ) 1 / 4

565

6. Distorted Plane Waves in Potential Scattering

Proof. Let V be a circle centered at [, that lies within an open neighborhood where (6.99) exists. Since obviously (g If([) ) is for any fixed g E % an analytic complex-valued function in that neighborhood, we can apply Cauchy’s integral formula from complex analysis and write (6.101)

(g

+ 4) -f(5011/41 - f ’ ( b o D = (274-l J, [(4)-’([5 +41-’

I (lf(5,

(50

- [(5 - 50)-”(g

- (5 - 50)-l)

lf(5))d5

+

if [, A [ lies within the interior of the disk enclosed by V. Furthermore, I (g If([) I is continuous and therefore bounded on the closed and bounded (and therefore compact) set V for each g E 8. By Lemma 2.2 in Chapter I11 (generalized from sequences to arbitrary sets) I ( f ( [ ) I g ) I is therefore bounded uniformly in g E i.e., l(g

lf(5)>1

< Cllg II,

5

Hence the absolute value of the right-hand side of (6.101) is majorized by

where the integral can be as small as desired by choosing I A [ I sufficiently small. Consequently, the same is true of the absolute value of the lefthand side of (6.101), since it is majorized by (6.102) for all g €*. Hence, (6.100) follows. Q.E.D.

Definition 6.5. Let A([), 5 E D, be bounded operators defined everywhere in 8.The operator-valued function A([) is said to be analytic on the open domain D C C1if A([)f is an analytic vector-valued function at all [ E D for any given f E 2. Theorem 6.6. If A([) is an analytic operator-valued function on the open connected domain D in C1and A([)is a completely continuous operator for each 5 E D, then [l - A([)]-’either does not exist at any [ ED, or it exists at all [ E D with the possible exception of a subset Y o which , has no accumulation points in D, in which case [I - A([)]-’ is analytic on D - 9,. Proof. Since A([,) is completely continuous for any given toE D,

V. Quantum Mechanical Scattering Theory

566

according to Theorem 8.4 in Chapter IV, there is a finite-rank operator

c I eL> N

=

AN

I

k=l

such that 11 A, - A((,) 11 < 4. On the other hand, the analyticity of A ( ( )at C0 implies" the existence of a neighborhood D , of 5, in which 11 A(() - A(5,) 11 < +,SO that I] A ( ( )- A, 11 < 1 on Dco. Consequently (see Chapter IV, Exercise 7.9 and Theorem 8.2), (6.103)

exists as a compact operator for all 5 E Dco, and it obviously defines an analytic operator-valued function. Furthermore, since upon introducing the compact operators (6.104)

AN([)

+

-A([)

= AN(l

AN)-1

N

I ekl) &(ek I (1 - A(5)

= k=l

+

AN)-1,

we can obviously claim the validity of the operator identity 1-

=

[I - AN(5)](1 -

+

t EDCo

AN),

9

we conclude that 1 - A(() has an inverse for ( E Dco if and only if 1 - A N ( ( )has an inverse. By (6.104) the operator A N ( ( ) is of finite rank and therefore (AN(()- 1)-1 does not exist if and only if there is a nonzero f E 2 for which

c N

f

=AN(5)f =

hi(ei

+

I (1 - A([)

i=l

AN)-V)

%'.

Since el' ,..., eN' is an orthonormal system, the previous equation is equivalent to the following system of linear algebraic equations: (6.105) N

(el

If)

-

c h,(ei I (I

k=l

+ AN)-1e;)(eL

- A([)

If)

= 0,

i = 1,..., N .

* The detailed argument requires the generalization of Lemma 3.3 to Banach spaces, and then the treatment of A(()as a function in the C*-algebra %(* viewed as a Banach space (see Dunford and Schwartz [1957]).

567

6. Distorted Plane Waves in Potential Scattering

In turn, this system has a nontrivial solution for ( e l if and only if the corresponding determinant (6.106)

a([)

= det{aik - &<ei

I (1 - 4 1 )

+

41-l

If),

i

=

1, ..., N ,

e,l>)

vanishes. Since, however, (6.106) is analytic on Dc0due to the analyticity of (6.103), a([) cannot vanish on any set in Dcothat has accumulation points in Dc0without vanishing identically on Dco. Thus the theorem is established in D c 0 ,and by standard methods used in analytic continuations, its validity can be therefore ascertained on all of D. Q.E.D. The family I?([) of Hilbert-Schmidt operators with kernels (6.26) is obviously analytic in the upper as well as lower open complex halfplanes, so that Theorem 6.6 is applicable to it. However, in solving (6.5), we were interested in the existence of (1 - I?([))-l when I'm [ + &O, rather than for Im [ # 0. Hence we need the following variant of Theorem 6.6.

Theorem 6.7. If A ( [ )is a compact operator-valued function that is analytic whenever I m [ > 0 (or Im [ < 0) and uniformly continuous for Im [ 2 0 (or Im [ 0, respectively), then either (1 - A([))-ldoes not exist anywhere on the domain of definition of A([), or it exists everywhere on the real axis with the possible exception of a closed set Yoof real values of which has Lebesgue measure zero in R1. The proof of the Theorem 6.7 proceeds in exactly the same manner as that of Theorem 6.6 until (6.106) is reached. Then, in ascertaining the properties of the set 9, on which a([) in (6.106) vanishes, we can apply a theorem* from complex analysis to reach the desired conclusion. By the argument used in the first part of the proof of Theorem 6.4 we know that there is a A, > 0 such that (1 - I?([))-, exists if 1 [ 1 > Av . This tells us, first of all, that the set Yvl of real points at which (1 - I?([))-l does not exist is bounded, and second, that due to Theorem 6.7 it has to be closed and of Lebesgue measure zero.

<

EXERCISES 6.1. Prove that the integral in r' of the function I r' I-2-fo over the region {r': I r' I 2 Rl} exists when R, > 0 and

I r - r' 1-l > 0, and

E,

* That theorem states that if S(5) is any function that is analytic in the open upper (or lower) complex half-plane and continuous in the closed upper (or lower) complex half-plane, then the set Yo of points on the real axis at which S ( 0 vanishes is closed and of Lebesgue measure zero if 8(0is not identically equal to zero (see, e.g., Lemma 8.21 in Amrein et al. [1971).

568

V. Quantum Mechanical Scattering Theory

that when r-+ +a,it behaves as O(l/rco) if 0 if E , > 1. 6.2.

< E,

< 1, and as O ( ~ / Y )

Prove that if the potential V(r) obeys the restriction of Lemma

6.1, then the following integral is continuous in r E R3 and

where the constant C is independent of r.

6.3. Show that if the potential V(r) = O(r-2-co)satisfies the remaining conditions of Lemma 6.1, then as r -+ co

and that the above integral defines a locally integrable function of r. O) the condi6.4. Show that if the potential V(r) = O ( Y - ~ - ~satisfies tions of Lemma 6.1, then as rl -+ 00 for fixed r E R3,

and the above function is locally square integrable in r1 and r.

6.5. Prove that if the potential V(r) satisfies the conditions of Lemma 6.1, then the integrals appearing on the right-hand side of (6.39) exist, and that Fv(r;5 ) is locally square integrable and bounded at infinity in r whenever J(5) # 0.

6.6.

Show that if the potential V(r) is locally square integrable, C,Y-~-~O,E, > 0, for r 2 R,, then the function V(r) I r - r' V(rp)is integrable on R6 if it is integrable on the set {(r, r'): Y R, , I' R,}. and

I V(r)I

<

<

<

6.7. Prove that when the potential V(r) satisfies the conditions of Lemma 6.1 then the function vk(r) defined by (6.55) is uniformly bounded (i.e., I wk(r)l const) for r 2 R, and k assuming values from a compact set Do containing no points for which k2/2mE 9'; .

<

7. Wave and Scattering Operators in Potential Scattering

569

6.8. Let h(r, u ) be a function with a continuous (in r and u ) partial derivative hu(r,u ) = ah(r, u)/au, and assume that for I 2 Ro

where p(r) is integrable on {r: r 2 Ro}. Show that if hu(r,u)f(r) and f(r) are integrable over the sphere {r: r < Ro}, then d

du 6.9.

1,.

h(r, u)f(r) dr

=

I

w8

hu(r,u)f(r) dr.

Show that the function g(r) ==

r J exp(ik1 Ir

-

- r' I)

1

f (r') dr'

is continuous in r iff(r) is locally square integrable and integrable on Rs.

6.10. Show that the family %; of all functions in %g(R2) with supports disjoint from the set 9'; (mentioned in Theorem 6.1) is dense in L2(R3).

7. Wave and Scattering Operators in Potential Scattering 7.1.

THEEXISTENCE OF STRONG ASYMPTOTIC STATES

I n the study of a particular scattering model it is crucial to determine from the start the magnitude of the family of strong asymptotic states. In other words, we have to establish the size of the initial domain M, of

In,

*

In potential scattering, under very mild restrictions imposed on the potential, this question is answered by Theorem 7.1.

Theorem 7.1. Suppose the potential V(r) is locally square integrable and that for some fixed E > 0

V. Quantum Mechanical Scattering Theory

570

Then the initial domain M, of the wave operators Q, corresponding to H , 1-( 1/2m)A and H 2 -( 1/2m)A V(r) coincides with the entire Hilbert space L2( R3). The proof of this theorem is based on Theorem 2.8, and it will be carried out in several stages. It is important, however, to verify before starting with the proof that in the present case H is indeed self-adjoint. This self-adjointness of H is a consequence of the restrictions imposed on V(r), which has to be locally square integrable and bounded at infinity to satisfy these restrictions. Consequently, -( 1/2m)A V(r) determines a unique self-adjoint operator by virtue of Theorem 7.5 in Chapter IV. The starting point of the proof of the above theorem consists in choosing an adequate domain g1 which satisfies the conditions of Theorem 2.8. We shall prove now that we can choose for 9,the linear manifold spanned by all functions

+

+

$Jk)

(74

= exp[-(1/2m)

k2 - ikp],

P E Rs,

where p varies over all vectors in UP. As a straightforward consequence of (5.2) we obtain

2m

k2 - ikp).

Hence, by virtue of Theorems 7.3 and 7.5 in Chapter IV we have* exp(-itHo) $, E gH0 A g H,and consequently exp(-itH,) BlC BHo n9,. The second condition imposed on g1in Theorem 2.8 is that when t, -P - 00 and t, + +a,the strong limits of Wl(tl , tz)$ , exist. We have already pointed out, at the end of $2, that the existence of the integrals (2.41) and (2.42) is sufficient for the fulfillment of this condition, In view of this fact, we shall settle the question by showing that

for all

p E R3.

Note that TS= Ho and Ts+

Vs G H .

57 1

7. Wave and Scattering Operators in Potential Scattering

By taking the Fourier inverse transform of both sides of the relation (7.3), we obtain for any real S I(exP(-~~~o)lClp)(r)I = (24-312

IJ

$,,(Is) RS

exp [-it

k2 2m + ikr] dk

1

m3/2

I 1 + it 13'2 exp [- 2 i + t 2 1 r - p I-1/2+8 = m812 I it 11+8 I1I2-'exp +

1%

[-

(r -

2

1 + t 2

1

*

If we now take 0 < 6 < 1/2, so that 1/2 - 6 > 0, we get, by multiplying both sides of the above inequality by V(r),

where C is a constant which majorizes the function

for all values of r, p, and t. If we choose S so that in addition 6 < E , then by virtue of (7.1) the function on the right-hand side of (7.5) is square integrable. Consequently, we can write

where Cp is constant with respect to the variable t E R 1 . Thus, (7.4) holds and the second condition of Theorem 2.8 is fulfilled. This establishes that the closure a, of the linear manifold Bl spanned by all functions (7.2) is contained in M, .

V. Quantum Mechanical Scattering Theory

572

T o conclude the proof of Theorem 7.1, we still have to show that and thus establish that M, =L2(Rs). T o arrive at this result we need the following lemma, which is a straightforward generalization, to arbitrarily many dimensions, of a theorem by Wiener.

a,=L2(R8),

*Lemma 7.1. Let us denote by XIthe set on which the Lebesgue square-integrable function f (x), x E Rn, vanishes. The Lebesgue measure of X, is zero if and only if the linear manifold (PI) spanned by the family S,of all functions exp[ip x ] f ( x ) , obtained by varying p over all vectors p E Rn, is dense in L2(Rn).

-

Proof. We set out to prove that, given any g E L2(Rn) and E > 0, we can find vectors p, ,...)p, E Rn and complex numbers a, ,..., a, such that

I,. I

8

g(x) - f ( x )

U-1

We note first that for any E (7.7)

, I &)la

B

1 a,ed2"yIt dnx < 166.

> 0 we can satisfy the inequality

dnx

< c,

B'

=

Rn - B,

with a compact Bore1 set B, by choosing that set sufficiently large. Let us define the functions

for any positive integer M; note that hM(x)is almost everywhere well defined, sincef(x) vanishes only on a set of measure zero. It is easy to see that by virtue of Theorem 3.10 in Chapter I1

Consequently, we can find an M for which

I . Wave and Scattering Operators in Potential Scattering

573

Since (7.7) obviously implies that JB,

I h&)f(x)l2 d"x < E?

we easily obtain from this inequality and (7.8), by using in the process the triangle inequality in L2(Rn),

Let L be a box L

= {x:

--E

< xi < 1, i = l,..., n}

which contains B, and let gl(x) be a function equal to xB(x)hM(x)in L and periodic in each of the variables x1 ,..., x, , with the period 21. For sufficiently large I we have

I X B ( 4 h M ( x ) f ( x ) - gz(x)f(X)l2d"x = kl,

....k,=-m

II

xB(x)

hM(x)f(xl

L

+ 21k1

,..a,

xtz

+ 2 k ) I 2dnx

< M 2 IL' If(x)12dnx < c, where L' = Rn - L, and the prime on the summation sign indicates that the sum does not include the term with k, = = k, = 0. Combining (7.9) and the above inequality we obtain

-..

by making use of the triangle inequality. Let us expand g l ( x ) in a Fourier series,

where k = (k, ,..., k,) and the convergence of the above series is in

V. Quantum Mechanical Scattering Theory

574

the mean. A straightforward computation yields, for any positive integers N and No < N, +N

k,,

k * x

....kn--N

where the summation C" extends over all k = (k, ,..., k,) in which I k, I > No for at least one value of j = 1, ..., n. Due to the convergence of C I ck 18, the second sum on the right-hand side of the above inequality can be made arbitrarily small by choosing No large enough, while the first sum can be then made arbitrarily small by taking N sufficiently larger than No . Hence, the sequence of functions N k,.

....kn--N N

=

1,2, ...,

converges in the mean to zero when N +a.Thus, it follows from Theorem 4.5 of Chapter I1 that there is a subsequence frJ,(x),fNJxj,.,., which converges to zero for almost all values of x E R". Now we can apply Lemma 3.1 of Chapter IV to infer that --f

after noting that the use of this lemma is justified by virtue of the inequality IfN(x)f(x)l

< {I gdx)1 + M > l f ( x ) l .

In fact, the above inequality follows from the estimate

(equation continues)

575

I . Wave and Scattering Operators in Potential Scattering

the estimation of the last integral in the above relation could be carried out due to the formula sin2[Nr(x, - x,')/21] sin2[+, - x,')/21] '

N

kj=-N

which can be proven by mathematical induction in N. Since (7.1 1) is true, we can choose one N, = N for which

x (I -

w)

I

ck exp (ir7) k*x dnx < E .

Combining (7.10) and (7.12), we immediately obtain, by means of the triangle inequality,

.--(1 - -!-$!-)

I

c,exp ( i kr *7x ) dnx < 16r.

Since the above inequality becomes identical to (7.6) after an appropriate change of symbols, we have proven the statement that the linear manifold (Sl spanned ) by Sl is dense in L2(R8). Conversely, if (Sl is ) dense in L2(R3),then the measure of Jv; is zero. In fact, if this were not so, then by choosing g(x) = xJc,(x) we would have

so that (7.6) would not be satisfied for all and p , p8 E Rn. Q.E.D.

,...,

E

> 0 by any

a,

,...,a, E C1

I f f ( p) is the Fourier-Plancherel transform off(x), then it is easy to see that the Fourier-Plancherel transform of f ( x ) e * p u . Z is f(p - p J .

V. Quantum Mechanical Scattering Theory

576

The function f ( p - p.) is usually called the translation of f (p) by the amount p , . By taking the Fourier-Plancherel transform of the integrand in (7.6) we obtain

Hence, we conclude that the linear manifold spanned by all translations of f ( p ) is dense in L2(Rn). This statement constitutes Wiener's theorem on the closure of translations. Let us return now to the proof of Theorem 7.1. If we set

then (7.2) assumes the form

qP(k)=f(k)

cikP.

Sincef(k) is different from zero everywhere on R3, we deduce from Lemma 7.1 that the linear manifold g1spanned by all function (7.2) is dense in L2(R3).Consequently, =L2(R3), and since we already know that $Bl C M, , we finally obtain M, 3 L2(R3). This concludes the proof of Theorem 7.1. We note that this theorem guarantees that M, = L2(R3) for all almost everywhere continuous potentials V(r) which decrease at infinity faster than the Coulomb potential. However, at this stage we cannot say anything about the unitarity of the S operator in potential scattering. Only when additional restrictions on the potential are imposed, as required in the theorems of the next section, will we be able to prove that in potential scattering the scattering operator S is unitary.

a,

7.2.

THECOMPLETENESS OF

THE M 0 L L E R

WAVE

OPEFtATORS

The next theorem will relate the Moller wave operators to distorted plane waves, providing at the same time the answers to all the basic questions regarding R, , M, , and the unitarity of the scattering operator S in potential scattering.

Theorem 7.2. Suppose the potential V(r) is locally square integrable and of Rollnik class, and that (7.13)

7. The Formalism of Wave Mechanics

577

for some E , > 0. Then the initial domain M, of the M ~ l l e rwave operators Q, is L2(R3),SZ, are complete [i.e., the final domains R, of Q, are both identical to LLc(R3)= EH(Sf)L2(R3)]and the S operator is unitary on M, =L2(R3); furthermore, (7.14)

(Q++)(r)= 1.i.m.

@F)(r)$(k) dk, L a

where $(k) is the Fourier-Plancherel transform of #(r), (7.15)

$(k)

= (21~)-~/~ 1.i.m.

a

+(r) e-ikr dr,

and @k.)(r) are the outgoing distorted plane waves, while @L-)(r)= @$)*(r)are the incoming distorted plane waves given as solutions (6.16) of (6.5). The validity of statements which are essentially equivalent to the assertions made in the above theorem about M,, R,, S, and SZ, has been taken for granted by physicists from the very beginning of quantum scattering theory. I n proving the above theorem, we confirm that this faith was justified only to the extent that the potential satisfies the conditions imposed on it at the beginning of the theorem. Consequently, it is important to realize that these conditions are essential only to the extent of making applicable Theorems 6.1 and 6.2 on distorted plane waves. T o avoid getting into a maze of technical details we have opted for (7.13)from the beginning of our systematic study of @L*)(r)in 96, but in fact (7.13)can be relaxed to

I V(r)l

(7.16)

= 0(r2-€),E

> 0.

However, in that case one has to base (6.7)-(6.9) on factorizations Vo I V 1l-B with t9 E f) that, in general, might* not equal 4. Various other approaches exist (such as those of Ikebe [1960]based on Banach space techniques) that can achieve essentially the same result, but clearly, as demonstrated by a counterexample of Pearson [1975],the previous unitarity of the S operator does not hold for arbitrary shortrange potentials. In case the potential is spherically symmetric, i.e., V(r) = Vo(r), the above conditions can be further relaxed by replacing (7.16) with

(a,

(7.17)

I v&)l

=

1 0 (F) >

E

> 0,

* See, e.g., Amrein et al. [1977],pp. 372-373, or Reed and Simon [1979], p. 353.

V. Quantum Mechanical Scattering Theory

578

and making the additional requirement that r K(Y)is locally integrable on [0, +a)(see Green and Lanford [1960]). Since the assertions of Theorem 7.2 are not true for the Coulomb potential, it is clear that for spherically symmetric potentials one cannot hope to do any better by further relaxing (7.17). However, in the general case, it is an open question whether one could relax (7.16) any further. We shall prove Theorem 7.2 in a few stages. The first step consists in verifying that the integral operator in the right-hand side of the equation (7.14) is a well-defined mathematical entity.

*Lemma 7.2. If @*)(r) are the solutions (6.16) of (6.5) with a potential satisfying the conditions of Theorem 7.2, then there are unique operators W+and W- defined by (W*+)(r)= 1.i.m.

(7.18)

k-9

s..

@r)(r) &(k) dk

for arbitrary sequences &(k), &(k),..., E %?: converging in the mean to

$ E L2(R3),and these operators satisfy the relations W*W**

(7.19)

(7.20)

= EH(Sf),

W**EH(B)= EHo(B) W**,

B E @.

Proof. We shall prove the lemma only for W , , since the corresponding results for W- easily follow from the relation @k-)(r)= @%) *(r). First of all, we note that under the present restrictions on the potential, Theorem 6.3 holds. It follows that for any given bounded closed set B which is disjoint from the exceptional set 9'; defined in (6.15) we can find a locally square integrable and bounded at infinity function C,(r) such that (7.21)

I @?'(r)l

=

1(24-3/2[edkr

+ q.(r)ll < C,(r)

as long as k assumes only values from B. Let us consider the bilinear form (7.22) in t,h and $', where (7.23)

(+ I +'I =

f $*w$+wdk R

579

I . Wave and Scattering Operators in Potential Scattering

I

@p’*(r)$’(r)dr. w’ With the help of Theorem 6.2 we show that this bilinear form is bounded:

(7.24‘)

$+’(k) =

Consequently (see Chapter 111, Exercise 2.5), there is a unique bounded linear operator W+satisfying the relation (7.25)

w+*I *‘) = (* I t)

for all a+4, $’ eL2(R3). Let us choose now $(k) E ‘if: and $’(r) ~Vrip(R~),where %: is the family of all functions in %,“(R3) with supports disjoint from Yv3. For such $(k), “l.i.m.JJ in (7.18) can be dropped, and (7.22) can be written in the form

Since (7.21) is satisfied by IPk(r), we immediately find that for such choices of $‘(r) and $(k) the function $*(k)IPi+)*(r)#’(r)is integrable in Re. Consequently, by Fubini’s theorem, we can interchange the order of integration in (7.26), thus arriving at the result

Since %rip(R3) is dense in L2(R3), equality (7.27) holds for all $’(r) E %,“(R3) if and only if (7.28)

(W+*)(r)=

I,W ( r ) $(k) dk

almost everywhere. Hence, (7.18) is established for $ E %: . Moreover, due to the fact that is dense in L2(R3) (see Exercise 6.10) the general validity of (7.18) for W+ and arbitrary I ) EL~(W)is an immediate consequence of (7.28). In order to express W+* as an integral operator, let us take in (7.22) $(r) E %c(R3) and write for compact B, -+ R3 - 9’;

I

(+ I W+*f) = (W++ I #’) = ( 2 ~ ) -n-m lim ~ / ~ dk$+’(k) n

,1

rikr+(r) &I*. P’

V. Quantum Mechanical Scattering Theory

580

Since, under the present conditions, the function #*(r) e
(# I W+*+') = ( 2 ~ ) - lim ~ / ~ d r #*(r) 12-m

JR3

11 dkr$+'(k)dkl . B"

Now, the type of reasoning earlier applied to (7.27) leads us to the conclusion that (7.29)

(W+*#)(r) = ( 2 ~ ) -1.i.m. ~/~

1 eikr$+(k) dk R3

for arbitrary $+(A) given by (6.45). Combining (6.48), (7.18) and (7.29), we obtain (W+W+*#)(r)

=

[W+(W+*#)](r)

= 1.i.m.

=

1.i.m.

1w8Ok'(r)(W+*#)"

I,. Ok)(r) $+(k) d k

(k)dk

= (EH&)#)(r), H

where the last step follows from Theorem 6.2, and in particular from (6.44). Hence (7.19) is established. Finally, combining (6.44)-(6.46) with (7.29), we easily derive (W+*EH(B)+)(r)= ( 2 ~ ) -1.i.m. ~/~ =(

1., edkr(EH(B)#);(k)dk elkr$+(k) dk,

2 ~ ) -1.i.m. ~/~ 'ka/2meB

( E ~ O ( BW+*#)(r) )

= (24-312

eikr(W+*#)" (k) dk

1.i.m. 'ka/2meB

=(

2 ~ ) - 1.i.m. ~/~

eikr$+(k) dk. 'ka/2rnEB

This establishes (7.20). *7,3,

Q.E.D.

PROOF OF ASYMPTOTIC COMPLETENESS

We are now ready to carry through the proof of Theorem 7.2 to its completion by showing that W , = Q, . Let us adapt the relation (2.37) to the present case, setting t, = 0,

58 1

I . Wave and Scattering Operators in Potential Scattering

t,

= 7,

(7.30)

and f = $ to obtain ((D(T)- 1)I) I g ) = i

IT((ePiHot+ I He-*Htg)

- (Hoe-*HotI) I e-"tg))

dt

0

for any g E g H . Since the restrictions imposed at present on the potential meet the demands of Theorem 7.5 in Chapter IV, we infer that if $ E BHo, then t,b E g V .Hence, for $ E gH0 , ( H - Ho)e-iHo$h= Ve-iHof$, and the relation (7.30) assumes the form (7.31)

(~(711)I g>

= (I) I g>

+ i J' (e'Htve-iHot 0

I) I g> dt.

The above relation can be extended to arbitrary g €L2(R3).Setting g = W-4, 4 E L ~ ( R ~ and ) , using the relation W-*eiHf = e*Hof W-*, which is a direct consequence of (7.20) (see also Exercise 2.1), we obtain

Since it can be readily shown (see Exercise 7.1) that

we easily arrive at the result (7.33)

(4 I W-*Q-I)>

% ;) w - 4 I

= =

Q(T)+>

(4 1 W-*I)) - !j%i

I+-

e-Gt(+ I efHotW-*Ve-fHot I)) dt.

0

It should be emphasized that, in this context, all the integrals over the infinite interval [0, +a) are improper Riemann integrals, and not Lebesgue integrals. We can calculate the integrand in the above relation by using (7.29) and (4.8), (7.34)

(+ I efHotW-*Ve-iH ot I)) =

Iw3$*(k)

ei(ka12m)t(W-* Ve-iHot

I))" (k)dk

(equation continues)

V. Quantum Mechanical Scattering Theory

582 =

/

c$*(k)ei(k*l2m)t(Ve-iHot I):

(k) dk.

R8

We resort now to the relation (6.45), which for +(r) E %'g(Rs)yields ( Ve-iHot#)l(k) =

(7.35)

I

@-'*(r)V(r)(e-iHot#)(r\ dr.

R8

We note that the limit in the mean in (6.45) has been dropped in the above relation, since the integrand in (7.35) is integrable on R3. To arrive at this conclusion, we recall (7.21) and note that ( c i H o f +)(r) is uniformly bounded for all t E R1 (see Exercise 7.2). By combining (7.34) and (7.35) we obtain (7.36)

(+ I exp(iHot)W-* V exp( -iHot)#) =

I,. dk p ( k ) I

dr @~-'*(r)V(r)(exp[--i(H,- k2/2m)t]#)(r).

R

Let us show now that when +(r) E %;(R3) disjoint from 9'; we have (7.37)

1

@;-'*(r)V(r)(exp[--i(H,

and B is compact and

- k2/2m)t]#)(r)dr

1<

Co,

where C, is a constant independent of k E B and t E R1. Indeed, since (exp[-i(Ho - k2/2m)t]+)(r) is uniformly bounded in r, k E R3 and t E R1 (see Exercise 7.2), by using (7.21) we get

1 @i-'*(r) V(r)(exp[--i(H, - k2/2m)t]#)(r)Jdr

< const

thus establishing the validity of (7.37). Let us insert (7.36) in (7.33) and choose&k) E %g(R3), thus obtaining

where

I

+W

(7.39)

If(+,#)

=i

0

x

dt e-'$

I,.

dkc$*(k)

dr &'*(r) V(r)(exp[-i(Ho

- k2/2m)t]#)(r).

I . Wave and Scattering Operators in Potential Scattering

583

Using (7.37), we easily infer that if supp 4 C B the above integrand is integrable in t E [0, co) and k E R3. Consequently, the reversal of the order of integration in t and k is justified by Fubini's theorem. Hence, we have

+

(7.40) If( $,#) = i

(7.41)

j dk$+(k) R'

dt (8'; 1 exp[-i(Ho - k2/2m - ie)t]+>, 0

F#r) = xB(k)@(i)(r)V(r).

Using Theorem 3.1 of Chapter IV, we infer that

Consequently, we easily compute

If we combine (5.5) and (5.6) with the above result, we get

gj lj

If($,*) = n6

w8

exp[i(k2 - 2mi~)l'~ I r - r' I] IT-r')

x V(r') @L-)(r') dr'

I*

$+(k) +(r) dr dk.

Inserting this result in (7.38), and noting that

584

V. Quantum Mechanical Scattering Theory

and that the absolute value of the integrand in the above limit is majorized by I r - r‘ 1-l I V(r’)l C,(r‘) when kEB, we arrive at the following result: (7.43)

(4 I W-*Q-#>

=w-4

I #> -

I,

x V(r’)@L-)(r’)dr’

1 r - r‘ I) I - I

exp(-ik [JP8

rl

*

1 $*(k) #(r) dr dk.

From (7.18) we obtain

where in view of (7.21) the second equality follows from Fubini’s theorem. Substituting the expression for (W-4 I $) from (7.44) in (7.43), we obtain

x

1,s

exp(-ik I r - r‘ I) V(r’) @i-)(r‘)dr’ I r - r’l

I*

by using in the process the Lippmann-Schwinger equation (6.5) for (r). In the above relation $(r) E Vg(R3) and $(k) E % ;‘ . Since %?(UP)is dense in L2(R3), it follows that W-*O- = 1. Consequently, by virtue of (7.19) and the fact that ER- < EH(Sf),we conclude that

Completely analogous considerations lead to the conclusion that W+ = Q+ . Hence, we have thus established the representation (7.14) for the wave operators SZ, in potential two-body scattering for arbitrary $ E L ~ ( WThis ) . shows that in the present case M, = L2(Rs). Moreover, in view of (7.19), we have also established that R+ = R- =LiJR3)). Consequently, the proof of Theorem 7.2 has been completed.

7. Wave and Scattering Operators in Potential Scattering

7.4.

585

PHASESHIFTS FOR SCATTERING IN CENTRAL POTENTIALS

is a We had mentioned in 97.8 of Chapter IV that {I P I, I L I, I.(”)) complete set of observables next in importance to the position and momentum complete sets of observables. In 94.3 we had briefly outlined how the stationary scattering theory can be formulated in terms of partial (or spherical) waves constituting an eigenfunction expansion for {I P 1, I L I,L(”)).We now propose to prove by a very general argument that indeed for spherically symmetric potentials

(7.45)

V(r)

=

Vo(r),

r

=

I r I,

the S operator can be expressed in terms of the phase shifts 6,(K) mentioned in 94.3. We already know that Ho in (4.14) or (5.1) and (5.2) commutes with {I P I, I L I, L‘”)). Obviously, the interaction Hamiltonian H , = V does not commute in general with I P 1, but it does commute with I L I and L(“)(in the sense of Definition 1.2 in Chapter IV) when (7.45) is satisfied. Indeed, in that case V = Vo(lQ I), and by Theorems 2.5 and 2.6 in Chapter IV, all we have to prove is that the spectral measure EIQWof I Q I, (E~a’(B)!4(r, 4 $)

(7.46)

=

Xk)

6’ $1,

&!9

commutes with the spectral measures of I L I and L(“)-a fact that is implicit in the relation (7.43) of Chapter IV. Since H , and V do not commute, we cannot argue that I L I and commute with H = H , V (in the sense of their spectral measures commuting) simply because they commute with H , and V. This result, however, can be deduced from the following theorem by Trotter [I 9591, which finds many other applications in quantum mechanics.

+

+ A,

*Lemma 7.3. If A, and A, are self-adjoint operators and A, is self-adjoint on 93 = gA1 n aA8, then Trotter’s product formula (7.47)

ei(Al+Az)t

= S-lim(eiAlt/n e i A n+m

holds true for any t

E

t

1n

0

W.

Proof. By Theorem 3.1 in Chapter IV and Exercise 5.6 in Chapter

I11 we have for any f E 9, s-lim 7-l(eiA1T e%Tf -f) T+o

= s-lim eiAlT~-l(eiA,Tf 7’0

+ s-lim 7-0

T-l(eW7f

-f) -f) (equationcontinues)

V. Quantum Mechanical Scattering Theory

586

= i(AJ

+Azf) -f).

= s-lim T-l(ei(A,+A&f 7-0

Consequently, if we set M(7) = 7-1 (e*(A,+A& ’

&As+),

then 11 M(7)fII -+ 0 as 7 + 0 for all f e 9. T o prove that as 7 -+ 0, 11 M(7)fII converges to zero uniformly on a dense set of vectorsfe 9,we shall employ Lemma 2.2 from Chapter 111. Hence we introduce in 9 the new inner product

(f I g>9 = ((A, + A2)f I (A, + A&> + (f I g>,

(7.48)

+

and we note that since A, A, is self-adjoint (and therefore closed) on 9, the linear set 9 is a Hilbert space with respect to this inner product. Furthermore, the linear functionals

L ( f=)(M(7)g I W.)f>, f €9,Ilg 119 = 1,

TE

[-I,

+1],

are bounded inf, g E 9 (with inner product (-I -)9),so that by Lemma 2.2 in Chapter 111 (which can be easily extended from sequences to arbitrary sets of linear functionals), we conclude that

I

dT,,(f)l

<

1 ‘

3

llf119 = Ilg 1 1 9=

1,

7

[-I,

+I].

Since obviously a similar constant C, can be found if T E IW’ - [- 1, we can state that

IIM(7)fll < Cllfll9,

f E 9 ,

+11,

TER1,

so that in accordance with (1.16) in Lemma 1.1,” M(7)f 11 + 0 uniformly i n f E 9 , Ilfll9 = 1, as 7-0. T o apply this result we write Il{exp[i(A,

+ Az)t/nIn)f - [exp(iA,t/n) exp(iA2t/n)lnfll

n-1

1 [exp(iA,t/n) exp(iA,t/n)]j

+ Az)t/n] - exp(iA,t/n) exp(iA,t/n)} x {exp[i(A, + A,)t/n])”-j-l I1 x {exp[i(A,

f

(equation continues)

587

I . Wave and Scattering Operators in Potential Scattering

and note that as n + m (and therefore t / n + 0), the last of the above norms converges to zero uniformly in u E [- I t I, I t I] since I(exp [i(A, A&] f 119 = (1 f for any fixed f E 9. Consequently, (7.47) is established when applied to f E 9. Since 9 is dense in 2 and the operator on the left-hand side of (7.47) is bounded and defined everywhere in the extension principle can be invoked to immediately extend this result to all f E %. Q.E.D.

+

+

Since by Theorems 7.3-7.5 in Chapter IV we have that QH = gH0 C gV, Lemma 7.3 can be applied if V(r) is locally square integrable and bounded at infinity, so that (7.49)

exp(iHt,) exp(iL%,)

= s-lim[exp(iH,t,/n) n+m

= s-lim n+m

exp(iVt1/n)lne~p(iL(~)t,)

exp(iL(”tz)[exp(iH~t1/n) exp(iVt,/n)]n

= exp(iLW,)

exp(iHt,),

and similarly for I L 1. Hence by Theorem 6.2 in Chapter IV, we conclude that I L I andL(”)commute with H.

Theorem 7.3. If V(r) = O(r-l-€), r + m, is spherically symmetric and locally square integrable, then the S operator is a function of {P, I L I, L(“)), (7.50)

s = Fs(l P I, L‘,L(”),

(7.51)

F s ( k 44

IL I

= [L’(L‘

+l)y,

= exp[2iW)I,

where 6,(k) are unique modulo T real functions that do not depend on m.

Proof. Under the present conditions on V(r), our earlier considerations leading to (7.50) are valid, so that I L I and L(”)commute with both H , and H. Consequently, 1 L I and commute also with the M ~ l l e rwave operators, since, e.g., (7.52)

V. Quantum Mechanical Scattering Theory

588

for any B E @. Since S = Q-*Q+, we also have (7.53)

[E"G)(B),S ]

=

[EILI(B),S ]

On the other hand, since I P in Chapter IV, (7.54)

I

=

B E 91.

= 0,

(2mH0)1/2,we have by Theorem 2.3

E'P'(B) = EHO(F-'(B)),

F(X) = (2mX)1/2,

so that according to Theorem 2.3, I P I also commutes with S. Therefore, by Theorem 5.6 in Chapter IV the S operator must be a function of the complete set {I P I, I L I, L(*)},i.e., (7.51) is satisfied. T o show that this function is of the form (7.55)

Fs(k, I, m) = exp[2iS(k, I, m)],

S(k, I, m) = S*(k, I, m),

we note that by Theorem 7.1, M, = L2(R3),and that [as we had remarked when discussing (7.17) in $7.21, R, = R- , so that by Theorem 2.5 S is unitary on L2(R3).Consequently, by Theorems 2.6 and 2.7 in Chapter IV, (7.56)

F,*(k, I, m)Fs(k, I, m) = 1

almost everywhere on R3 with respect to the measure po introduced in $4.3. Thus, (7.55) must be satisfied, the functions S(K, I, m) being obviously Po-almost everywhere unique modulo rr. The fact that S(K, I, m) does not depend on m, and that therefore (7.55) reduces to (7.5 I), follows from a general group-theoretical theorem". Q.E.D. We note that Theorem 7.3 establishes the existence of the phase shifts S , ( K ) without any reference to distorted spherical waves. Therefore, as yet we have not proven that the functions S , ( K ) appearing in (7.51) coincide with those in (4.36). However, this result follows from the further considerations of $7.6. In the meantime, we shall establish in $7.5 that (7.50) and (7.51) have a generalization of sorts valid even when V(r) is not central.

PHASE-SHIFT FORMULA FOR 7.5. THEGENERAL OPERATOR

THE

SCATTERING

Let us introduce in the space L2(Qe,pa) defined in $7 of Chapter I1 [see (7.16) of Chapter 111 operators analogous to (3.50) [c.f. also (4.97)

* The

Wigner-Eckart theorem (see, e.g., Miller [1972], Section 7.10).

589

I . Wave and Scattering Operators in Potential Scattering

and (6.6711: S(k) = 1 - 2&T(k),

(7.57)

s,

-k (7.58) (T(k)u)(w)= (2Pl2

f(km')(W)

k E [0, CO),

u ( 4 dw',

u EL2(% 9 Ps).

The unitarity of the S operator imposes the condition that S(k) should be unitary in L2(S2,,pa) for almost all k 2 0: (7.59)

S*(k) S(k) = S(k) S*(k) = 1.

As a matter of fact, for arbitrary

u E L ~ ( Qps) , , and p €L2(SZ,,pr) [see (7.16) of Chapter 111 it easily follows from (4.96), (6.67), and (7.58) that for (p * u)(k,o) = p ( k ) u ( w )

(7.60)

(S*Sf

*

N k , w)

= p(k)(S*(k)

S(k)u)(w).

Consequently, we have for arbitrary u, w E L2(sZ,,ps)

This means that (S(k)u, I S(k)uj) = 6, at almost all k E [0, 00) for any given i, j = 1, 2,..., if u1 ,u2 ,..., is an orthonormal basis in L2(sZ,, ps). Since the set {(i,j):i, j = 1, 2, ..., } is countable, this implies that S*(k)S(k) = 1 for almost all k E [0, 00). A similar argument establishes that S(R) S*(k) = 1 also almost everywhere in [0, 00). The unitarity of the operators S(k), 0 \< k < +00, enables us to derive Theorem 7.4.

Theorem 7,4, Suppose the potential Y(r)satisfiesall the conditions imposed on it in Theorem 7.2. Then for almost all k E [0, a),there is an orthonormal basis {wl(k), w2(k),...I in L2(sZ,, ps) such that

+

n

m

(7.61)

S(k) =

2 I w,(k)) eQiSJK)(wv(k)J = s-lim 2 I wv(k))e2fSv(k)(wv(k)I, v=l

n++m

v=l

590

V. Quantum Mechanical Scattering Theory

where the numbers 6,(k), 6,(k),... are real and such that W

Tr[T*(k) T(K)] = C I sin 6,(K)I2

(7.62)

< +a.

-1

Proof. The kernel T ( l ) ( kw, ; 0 ' ) = - ( 2 ? ~ ) - ~kfh.(o) of the operator T(K) is square integrable in (w, w') on SZ, x rR, with respect to the measure ps x ps (see Exercise 7.3). Hence, the operator T(k) on L2(sZ,,p,) is of the Hilbert-Schmidt type" and therefore completely continuous. Since S(k) is unitary for almost all Iz E [0, a),we can apply Theorem 6.1 of Chapter I11 and write (7.63)

S(K) = r e i AdEA(K), 0

where the above integral converges in the uniform sense of (6.1) in Chapter 111. We shall prove now that the complete continuity of the operator T(k) implies that the spectral function EA(k)is constant in A except for an at most countable number of discontinuities Aio)(k), Aio)(Iz),...,where EAp)(k)# EAP)-,(k). Suppose to the contrary that there is a point A, E ( 0 , 2 ? ~ such ) that EAo(k)= EA-o(k) but E,(k) # E,(k) for any a < A. < b such that a, b E (0,24.Then we can choose a sequence A1 ,A, ,... + A,, which is monotonically increasing or decreasing, and such that EA,(k)# EAn+,(k). To be specific, let us say that A, < A, < < A,, and let us choose - EA,(Iz))fn= f, vectors f, ,f2 ,... E S which are such that (EAn+,(k) and l f , 1) = 1. We shall prove that no subsequence of T ( k ) f l , T ( k ) f ,,... is convergent, thus contradicting the complete continuity of T(k). In fact, it is easily seen that T ( k ) f , IT ( k ) f , for m # n (see Exercise 7.4), and consequently

+

- V ) f112 ,= I1 W f T n /I2 II W f n 112. On the other hand, I( T ( k ) f , 11 2 C > 0 for all n = 1,2,...,

II W

f

m

1 2inf

4+3

n=1.2..

.. 1 sin A,, ,1 > 0.

* This can be easily inferred by setting p,(B) p , to

= 0 for B C Rz - Q, , thus extending Rz,and then applying the result of Exercise 5.3 and of Theorem 8.6 in Chapter IV.

591

7. Wave and Scattering Operators in Potential Scattering

Thus, no subsequence of T ( k ) f , , T(k)f, ,... is a Cauchy sequence, which is impossible if T(k) is of Hilbert-Schmidt type. Hence, EA(k) can increase only in a discontinuous manner. Since La(QE ,pE) is separable, there can be at most countably many such points hio)(k),hao)(k),... of discontinuity. Thus, (7.63) becomes

where it is easily seen that the convergence of the series is in the strong sense, but not necessarily in the uniform sense (think of the case S(k) = 1, when the convergence is not uniform). If we introduce an orthonormal basis in each one of the mutually orthogonal subspaces on which E,;)k)(k)- EA;)(k)-o(k),n = 1, 2,..., project, we obtain an orthonormal basis in P(QE ,pE) by taking the union of all these bases. After adequately labeling the elements of these bases, and replacing @(k) with the adequately labeled &(k) = &&?(k), we obtain (7.61). The corresponding expression for T(k) is (7.64)

1

1 "

1 I wu(k))e*"v(k)sin a,,@)

T(k) = -(S(k)- 1) = 2m. 71 r

(wu(k)l.

l

Thus, we have .

m

(7.65)

and since T(k) is a Hilbert-Schmidt operator, (7.62) follows. Q.E.D. We have seen in the preceeding section that when the potential V(r) is spherically symmetric, physicists call the numbers &(k) phase shifts. We can apply this terminology (see also Ikebe [1965]) to the general case, when V(r) is not necessarily spherically symmetric. Such a name for &(K) seems appropriate, since the absence of interaction is characterized by vanishing phase shifts 6,(K) = 0; in fact, when there is no interaction we have S = 1, and therefore

Furthermore, if wu(k)= Y$ for all K [see (7.76)-(7.78)].

E [0,

co) then 6,(k) equals 6,(k)

If we expand f(k,,,(w) in the orthonormal basis {wl(K), wz(k),...} and

V. Quantum Mechanical Scattering Theory

592

compare the Fourier coefficients of this expansion with (7.64), we get m

(7.66)

w,(k; w) eisJk)sin G,(k)w,*(k;w').

f(k,*,(w)= --4?TK-l v=l

Hence, the total scattering cross section a(k2/2m) at a given energy k2/2m can be expressed conveniently in terms of phase shifts:

Naturally, both formulas (7.66) and (7.67) are unambiguous only if T(R;w', w ) and wv(R;w') are continuous functions of w', so that (1.47) is valid.

7.6. PARTIAL-WAVE ANALYSIS FOR SPHERICALLY SYMMETRIC POTENTIALS If the potential V(r) is spherically symmetric, i.e., if (7.45) is satisfied, then the functions wv(K;w) can be easily computed. Using the well-known Bauer's formula (see Exercise 7.5), m

(7.68)

e x p ( h ) = exp(ik cos 0) =

+ l)jz(k)PZ(cose),

iz(21 z-0

and combining it with the addition theorem for spherical harmonics,*

(7.69) we obtain upon setting Ylm(w)= Ylm(w), m

(7.70)

exp[ikr(w * w')]

= 4v

I

c c ilj,(kr) Y l m (wY&(w'), )

l=O m=-1

where UJ = (6, 4) and o1 = (el, +1) are spherical coordinates which correspond to the unit vectors w E R3 and w1 E R3, respectively. Let us insert the above series in the integral in (6.58) definingf,(w). Due to the uniform convergence of the series for w, w' E IR, (see Exercise 7.5)

* See, e.g.,

Messiah [1962], p. 496.

7. Wave and Scattering Operators in Potential Scattering

593

we can integrate term by term, thus arriving at the following relation:

c c 4dYZm(w) m

fk(w) = d % m

(7.71)

2

L O +-I

dwlY&(-wl) &)(rwl).

From the Lippmann-Schwinger equations (6.5) we immediately see that when V(r) is spherically symmetric @;+)(r)is a function of only I , K, and cos 8 = r k/rk:

-

(7.72)

c ~ e). s

C D + ) ( ~ )=

Hence, by carrying out a change of variables in which w1 is replaced by spherical coordinates of r with respect to k, we obtain by (7.18) in Chapter I1 (7.73)

1

Y,*,(-wl) @P)(rol)dw,

0s

for m # 0

Consequently, (7.71) can be written in the form

where the convergence is in the mean on Q, x Q, , and (7.75)

a@)

=

dG Kmiz

lm

I

+1

dr r2V0(r)j2(Kr) du @+)(r, u ) Pl(u). -1

Thus, we have arrived at the following expansion for S(k):

cc m

(7.76)

S(K) = 1 - 2ni T(K) =

2

I Ylm)(l - 2 n i a ~ ( K ) ) <1.y ~ ~

2=0 m=-2

Comparison with (7.61) yields the following explicit expression for the phase shifts:

V. Quantum Mechanical Scattering Theory

594

The reader can compare the above results with the considerations in Section 7.4, and easily arrive at the conclusion that (7.78)

S,(k) = e2i8r(k),

K E [0,co), 1 = 0,1, 2,...,

with S,(k) given by (7.75) and (7.77) is indeed the S matrix in partial wave representation, whose existence has been established in Theorem

7.3. The great importance of the above formula for the S matrix lies in the fact that in conjunction with (7.75) and (7.77) it enables us to analytically continue S,(K)in the complex planes of the variables I and K. The properties of the resulting complex S matrix S(K, I) provide the foundations of a very fruitful theory of the S matrix, which has become the fundamental stepping-stone of much of the theoretical physics in the 1950's and 1960's. Since this book is concerned exclusively with functional-analytic methods, this topic is beyond its scope, but the reader is urged to consult the References for literature dealing with this very important subject."

EXERCISES 7.1. Show that if f ( t ) is continuous and the improper Riemann integral

exists, then the improper Riemann integral

+Z, = Jo e - d f ( t ) dt exists for every c

> 0, and limC++,Ic= I,.

7.2. Show that for any given $(r) E W2(R3) there is a constant Co such that J(e-iHo6 +)(r)l C, for all r E RS and all t E R1.

<

A mathematically rigorous treatment of analyticity properties of the S matrix in potential scattering is given in the work by de Alfaro and Regge [1965], but most other textbooks on scattering theory also treat this subject at an adequate level of mathematical rigor-whereas the same certainly cannot be claimed when it comes to the functional analysis aspects of scattering theory,

I . Wave and Scattering Operators in Potential Scattering

7.3.

Prove that for K

E

[0,

+

595

with k2/2m4 9; the integral

00)

exists, provided V(r) satifies the prerequisites in Theorem 6.4 and

T y K ; w, 7.4.

w’) =

-(2

4 - 2 k f k a , (w)

-

,i = 1 , 2

Suppose [E,*,(k) - EAi(K)] fd

= fi

(4 ,A,’] n (A2 &’I 7

= 0,

,A,,

where E,(K) is the spectral function of S(K), and A, Prove that T ( k ) f , 1T(K)f,.

7.5.

Al’, A,’

E (0,27r).

Prove that for any 01 E R1

and that the convergence is uniform in u E [-1, 7,6.

for

Show that for any /3

> 0 and

+1].

any roE R3

if #,o(r)is a wave packet with a Fourier transform of the form

&o(k) = exp

[-

B2 k2 - ikro] 2m

Use this result to show that for any bounded Bore1 set B

11 EQ(B)e--iH t d, 112 O

7.7.

Any Schroedinger-picture free state #:(r) = (e-“Ot

corresponding to 1+5:

E L2(W3),displays

&,

#3(r),

a tendency of “spreading out”

V. Quantum Mechanical Scattering Theory

596

in space as t + -a. This conclusion can be reached by establishing the evanescence of the wave packet $$(r)from any bounded Borel set B:

Combine the result of Exercise 7.6 with Lemma 1.1 and Lemma 7.1 to prove that the above limit is indeed equal to zero. 7.8. Prove that any interacting state packet,

4t(r) = W H4o)(r>, t

40

$t

represented by a wave

EL2(W

which has an incoming (outgoing) asymptotic state $p($:ut)evanesces from any bounded Borel set B in the configuration space when t + --co ( t + +a).

8. Fundamental Concepts in Multichannel Scattering Theory 8.1.

THECONCEPT

OF

CHANNEL

T h e outcome of the mutual interaction of three or more particles can lead to results which are qualitively different from the possibilities encouhtered in two-particle interactions. Indeed, a system of two isolated particles can either be found forever in a bound state, or the two particles are free before and after collision. However, in the case of three particles 6,, 6, , and 6,, there already are other possibilities in addition to the two alternatives of 6, , 6, , 6, becoming eventually free or constituting a bound system {61-62-63}. For example, it is possible that 6, and 6, will stay bound forever, thus building a new system 6,-6, ,while 6, becomes free, etc. Each one of these possibilities determines a particular outgoing arrangement channel of the scattering process between G,, G,, and 6,. In general, an arrangement channel (or clustering) in the scattering process of n particles 6, , 6,,..., 6, is a particular partitioning of the into a number of nontrivial subsets, called clusters. set {G, , 6, ,..., 6%} When all the particles in each cluster are actually bounded together either before or after collision with particles in other clusters, then we say that the clusters constitute fragments of the collision process. Clearly, we can talk about incoming channels and outgoing channels, depending on

8. Fundamental Concepts in Multichannel Scattering Theory

597

whether the particular channel we are considering refers to the system before or after the scattering had taken place. In the case where some of the particles in the system (6, , 6, ,..., Gn} are identical, not all partitions of the set {G, , 6, ,..., Gn}will represent distinct arrangement channels. This is due to the presence of BoseEinstein or Fermi-Dirac statistics, respectively (see Chapter IV, 94) which rules that identical particles are indistinguishable. For example, if 6, , 6, , and 6, are identical, then the channel (6,-G, , 6,) in which B, and 6, are bound, while G3 is free, is identical to the channel {G, , 62-63) in which 6, is free and 6, is bound to 6,. As a matter of fact, in this example both above arrangement channels are also identical to {G, , 61-63} so that we have only three distinct arrangement channels: , 62-63}, and {GI 9 6 2 3 63}* If each fragment 9 is in an eigenstate of a set 0, of commuting observables which are constants of motion in that particular arrangement channel, then the system is said to be in the channel determined by . eigenvalues of the observables in the family In practice, it is often convenient to take an 0, containing the internal (binding) energy of the fragment, its internal angular momentum (called the spin of the fragment), etc. If the internal energy of each fragment 9has been prescribed, then we call a channel determined by the given internal energy eigenvalues an energy channel.” In the case of many-body (three or more particles) scattering we can distinguish between elastic and inelastic scattering’ by looking at the distribution of energy among fragments before and after the scattering has taken place. Elastic scattering is, by definition, the scattering process in which all the fragments and their internal energies are preserved; for example, if the initial energy channel consists of a particle 6, being scattered from a bound system 6,-6, of two particles, then the scattering is elastic if after the scattering has taken place the particles 6, and 6, are still in a bound state of the same energy as the initial bound state, i.e., if no kinetic energy has been transferred from 6, to the internal

us@,

We depart somewhat from conventional terminology, in which a channel with respect to some of the above-specified observables is simply called a channel. However, since there does not seem to be complete agreement among different authors on the precise contents of the set 0, when a channel is defined, the above general approach seems quite desirable. + The distinction can be made also in the case of two-body scattering in an external field. In fact, the external field describes the interaction of the two particles with one or more additional particles which are not explicitly included in the system, but are effectively represented in an approximative manner by the “external” field. In that case, each one of the particles in the system can be bound to the field. However, it is clear that two-body scattering in an external field is, in fact, many-body scattering.

V. Quantum Mechanical Scattering Theory

598

energy of 6 2 - 6 3 . Naturally, any scattering which is not elastic is called inelastic. Suppose all the fragments in an energy channel are in their internal energy ground states and that AE is the smallest energy gap between the internal energies of these fragments and the energetically nearest nonground eigenstate of their respective internal energies. Then, obviously, no inelastic scattering can take place as long as there is not sufficient kinetic energy to be transfered to the K,th fragment to raise it to an “excited” state of internal energy Ei:t AE. This amount of energy is then called the threshold of the inelastic process in that particular energy channel. If in an inelastic scattering the fragments themselves are decomposed and new fragments result as an outcome of the scattering process, then we talk of a rearrangement collision; for instance, in the above example this is the case when a bound state of GI and 6, is formed, while 6, goes free after the interaction has taken place. It should be immediately realized that a rearrangement collision cannot take place at any distribution of energy between the different fragments. For example, in an energy channel of {G1,G2-G3} a minimal energy equal to the internal binding energy of G2-G3 is necessary for the decomposition of the fragment +G3 occurring in the initial state. This energy is then the threshold energy for a rearrangement collision.

+

8.2. CHANNEL HAMILTONIANS AND WAVEOPERATORS We shall illustrate the mathematical counterparts of some of the physical concepts introduced above on the case of three-particle potential scattering. For this purpose we consider three distinct particles which interact with one another via two-body forces. Here, the statement that we are dealing with two-body force means the total potential can be written in the form (8.1)

Wl ,r,

r3)= V12h - r 2 )

+

v23(rZ

- r3)

+

v13(r1

- r3),

and therefore each one of the particles interacts with each other separately. In addition, we simplify our considerations by assuming that these forces are of finite ranges pij ,i.e., Vi,(r), i < j = 1, 2,3, vanishes for t 2 pi5 . The total Hamiltonian of the above system is given by the Schroedinger operator (8.2)

= HO

+ + + VlZ

v2,

V13

Y

599

8. Fundamental Concepts in Multichannel Scattering Theory

where Ho is the free Schroedinger operator

and Vij , i < j , is the potential energy operator between the ith and the jth particle, (8.4)

(Vdj#)@l

Y

r2 Y r3)

=

V i h i - rd

Jm.1 9

r2

Y

r3).

Naturally, these operators act in the Hilbert space L2(R9). The arrangement channel {G,, G,, G,} in which all three particles are free is characterized by the fact that all these particles are so distant from one another as to be outside the ranges of the forces with which they interact on one another. Hence, for such distances V(rl , r2, r3)vanishes and the Hamiltonian H(1.2.3)of the system is given effectively by the operator H , , which is said to be the Hamiltonian of that particular arrangement channel. In the arrangement channel {G1, G2-6,}, the particles 6, and 6, are within the range ~ 2 3of their mutual interaction forces, while 6, is outside the range p12 from G,, and outside the range pi3 from 6,. Consequently, in this case V(rl , r2,r3) becomes equal to Vg3(r2 - r3), and the Hamiltonian of the arrangement channel is H(1,S-S) = H, V23. We see that in the present situation we are dealing with a much more intricate framework then in two-body scattering: instead of a single “free” Hamiltonian, we have a different Hamiltonian for each one of the five arrangement channels. We can write the Hamiltonian of each one of the arrangement channels as the sum of the Hamiltonians of each one of the fragments in that channel. Thus, for example,

+

(8.5)

= H(l)

H(l.2-3)

+

*(2-3)

2

denotes the kinetic energy operator of the ith where, in general, particle, so that in the momentum representation, (8.6)

(%#)“

(PI Y Pa 9 P3)

= Pi2/%

&Pl

9

P2 9 PSI,

i = 1,293,

and H(2-3)describes the system {G2-G3};we can write H(2--3) in the form (8.7)

where (8.8)

H(2-3)

H(2,3)

= H(2,3)

+

V23

9

is the kinetic energy of the fragment {6,-G3}, 4 2 . 3

= H(2)

+

4 3 )

9

V. Quantum Mechanical Scattering Theory

600

and V,, describes the interaction between the two particles in the fragment. When the system in some arrangement a and there is no interaction or negligible interaction between the fragments, then the state Y ( t ) = e - V P ( O ) of the system is described quite well by the states

provided they are a-channel strong asymptotic states in the sense that” lim 11 Y(t)- Y:x(t)11

(8.10)

t-rTW

= 0,

where “ex” stands for “in” ( t -+ -a)or “out” ( t -+ +a), depending on whether the channel is an incoming or outgoing channel, respectively, In two-body scattering we could formulate a general consistent timedependent scattering theory for any two self-adjoint operators H a n d H,, , However, in multichannel scattering one has to be careful and establish from the very beginning whether the channel description is indeed consistent, since it is not apriori clear that some interacting state might not have two or more distinct asymptotic states satisfying (8.10). I n order to be able to find the conditions under which such ambiguities in the asymptotic description do not occur, we have to understand more fully the structure of the arrangement channel Hamiltonian H a . Let us consider the arrangement channel {GI, 6 2 - 6 3 } in the scattering of the three particles 6, , 6, , and G3 interacting via the two-body potential (8.1). The fragment 6 2 - 6 3 is asymptotically in a bound state (8.1 1)

Y(2-3)(t)

= exd-i1’(2-3)t)

y(2-3)(0)

EL2(R6)

of 6, and 6,. Let us assume that Y(2-3)(0)is represented by the wave function (8.12)

where according to $5 of Chapter 11, tjY describes the state of the center of mass of the system G2-G3 , while yY(r3 - r,) describes the state of the two particles in relation to one another. The two particles are in a bound

* In fact, the remarks made in $2.6 still apply, i.e., the use of norm topology is too restrictive to apply when long-range forces are present-only expectation values of observables actually being of importance.

8. Fundamental Concepts in Multichannel Scattering Theory

601

state when Y(r3- r2)is an eigenvector (8.13)

= E(2-3)#",

H$!3)#"

< O,

E(2-3)

of the internal energy operator 3

int

(8.14)

H(2-3)

-

1

A f

--

2m23

v23

Y

where A is the Laplacian in the variable r = r3 - r2.Naturally, the most general Y(2-3)(0) has to be an element of the closed subspace of L2(R6) spanned by all such vectors (8.12), corresponding to all eigenvectors $" of H&$) . When 6,-6, is asymptotically in the state Y ( 2 - 3 ) ( t ) EL,(W) and Gl is asymptotically in the state Y(l)(t),then the entire system is in the state (8.15)

Yyt)

=

Y(&) 0 YJ(Z&)

ELZ(R9).

If Y(2-3)(0) is of the form (8.12) and $" is represented by the eigenvector in (8.13), then it is easily seen that (8.16)

H(1,z-3)

Yex(0)= (H(1) = (H(l)

where

The quantity

+ HgFS) + Hi:!,)) yex(0)

+

@253)

+

E(2-3))

y"(o>t

is the kinetic energy operator of the fragment 6,-6,,

I E(2-3)I is called the internal energy of the fragment

6,-63.

Naturally, it is often desirable to consider energy channels with respect to such internal energy operators. This is possible since in general, as well as in the above particular case, these internal energy operators commute with the arrangement channel Hamiltonian, and therefore they are constants of motion for the clusters in the given arrangement channel. Let us assume that {G1, G2-63) is an incoming energy channel, and that the kinetic energy part of $(r2,.r3)has a sharp distribution I $'(P23)12 of momentum, i.e., that the incoming momentum of G2-6, is prepared very accurately around some value P ! 3 . We can choose the system of reference in such a manner that P:3 M 0, and consequently

V. Quantum Mechanical Scattering Theory

602

Hence, we see that in this case the incoming particle 6, must have an amount of kinetic energy at least equal to I E ( 2 - 3 ) I in order to be able to impart it to 6,-6, in the collision process and break up that fragment, thus giving rise to a rearrangement collision. Consequently, in this case -E(,-,) represents the threshold energy for a rearrangement collision of the system {G1 , 6, , G,} in the given incoming energy channel. The reader can easily verify that the Hamiltonian Ha of any of the above arrangement channels of the system {G1 , 6, , G3}is the sum of a kinetic energy part H,kinof the center of mass of the fragment, and of an internal energy part H&lnt.For example, in the arrangement channel {61

9

62-63},

(8.19)

in the arrangement channel {G, , G2, G3},

in the bound-state channel {61-62-63}, Hk1i2-3)is the kinetic energy operator of the center of mass of the entire system, and

(8.21)

It is obvious that the kinetic energy operators of all channels commute with one another since they are all functions of the compatible momentum observables of the three particles. We also know from $5 of Chapter I1 that each center-of-mass kinetic energy operator commutes with the corresponding internal energy operator. However, in general, the Hamiltonians for different arrangement channels do not commute among themselves. This is easily seen to be so, for example, in the case of {G,, G,, G3} and {G1-G2, S,} when the channel Hamiltonians are H, and H = H, V,, , respectively. Following the guidelines set up by the above example, we postulate that in any multichannel scattering process a unique channel Hamiltonian Ha is attached to every arrangement channel a. Let us denote by Mt) the sets of all vectorsf which are not eigenvectors of Ha and for which the respective strong limits

-+

exist. By applying Lemma 1.2, we immediately conclude that ):M

are

8. Fundamental Concepts in Multichannel Scattering Theory

603

closed subspaces of 2 which might contain only the zero vector, as often happens in practice (see Exercises 8.1 and 8.2). Hence, we can introduce the projectors EMt) onto Mk) and define the M011er wave operators SZ:) for the channel a by the formula (8.22) The above formulas are obviously straightforward generalizations of (2.1). Hence, as mathematical objects, 9:)will have for each channel a! the same properties as SZ,. In particular S:) are partially isometric operators with initial domains MZ) and final domains RZ), which coincide with the ranges of these operators. The intertwining properties (2.3)-(2.5) (8.23) (8.24) (8.25)

f & . )

ettH,

- ettHan'u) f ,

tER*,

a!.)E ~ ~ ( =B E) ~ ( B~) 2, )

Qk' H, f = HQtY,

B E @,

fE9 H a ,

and the relations (2.10) and (2.11) =E

My,

(8.26)

Ok)*Q!f)

(8.27)

Qk' fit)*= ERy

9

will hold in the present case. Theorem 3.1 is also still valid, so that

Moreover, the above two formulas can be taken to be the starting point of the time-independent approach to multichannel scattering (see Section 8.6). 8.3.

THEUNIQUENESS OF CHANNEL STRONG ASYMPTOTIC STATES

Take now f E Mk) = MY) n MI") and write f* = SZt'f. Since

V. Quantum Mechanical Scattering Theory

604

we see that e - i H Q f f is the incoming asymptotic state of the interacting state e - * H t f y ) , and the outgoing asymptotic state of the interacting state e-ifft f i_”). The questions now arises, however, whether e-iH-tf is the only incoming(outgoing)asymptoticstate of e-*Hff?)( e - i H t f Y ) ;namely, there Fertainly cannot be another incoming asymptotic state riHQf g of f in the same arrangement channel, but, in general, there could be some other incoming asymptotic state e-iHaf g of r i H t f $ 4in some other arrangement channel # a. It was pointed out earlier (in the three-body case) that we cannot expect that such a uniqueness of asymptotic states would hold for any arrangement channel Hamiltonians H, picked up at random from the family of self-adjoint operators in YY. T o prove such a uniqueness of asymptotic states we have to restrict the families of candidates for arrangement channel Hamiltonians by requiring that such operators obey certain conditions, which are dictated by the physical situation at hand. Extrapolating from the example considered earlier of three-body scattering, it seems sensible to require that every channel Hamiltonian H , on MY) and on M?) be the sum (8.30)

H,

= Htin

+ HPt

of a center-of-mass kinetic energy part Htin and an internal energy part HPt; here H:in is the operator which represents the sum of the kinetic energies of the centers of mass of all clusters, while the operator HPt represents the sum of the internal energies of all clusters in the arrangement channel a. Judging from the wave mechanical three-body problem, it is also reasonable to postulate that Hint has a pure point spectrum and commutes with Hfin, while H5in has only a continuous spectrum, and that all kinetic energy operators H:in commute among themselves. When the channel Hamiltonians have the above indicated structure we can prove that incoming and outgoing (strong) asymptotic states of any interacting state are uniquely determined by that interacting state, and consequently the description of the scattering experiment in terms of the asymptotic behavior of the system is completely unambiguous. Let us first understand better the problem of uniqueness in mathematical terms. We have seen in (8.19) that if Y(t)= riHf Yois an interacting state, and if Yo= Qy’Y?, then Yin(t) = !PAn is an incoming asymptotic state of Y(t). If Yp(t)= exp(--iH,d) ?Pinwere another incoming asymptotic state of Y(t),then we would have Yo= Qy’)Yp.Thus, Yin(t)could be a state different from Y?(t) if and only if a # a‘ and Yo = Qy)YAn = Qy‘)Y?, i-e., if Yo belonged to both r g H l r t

8. Fundamental Concepts in Multichannel Scattering Theory

605

ranges RY’ and RY” of Szt’ and a?’),respectively. Consequently, the necessary and sufficient condition which has to be fulfilled in order to have a unique asymptotic state corresponding to an interacting state is that the different ranges RY’ should have no state vectors in common. Naturally, a similar conclusion holds for the ranges R!?, which are also required to have no state vectors in common. The two ranges RY) and RY), a # a‘, will have no state vectors in common if, in particular, they are orthogonal to one another; in fact, in that case RY) and RY’I have in common only the zero vector which is not a state vector. The following theorem shows that if the channel Hamiltonians have the earlier mentioned structure, which is reflected in (8.30), then R): 1 RY’) for all a # a’ and R!? 1R1_B“for all /3 # p’. Consequently, under those conditions every interacting state has at most one incoming and one outgoing (strong) asymptotic state.

Theorem 8.1. Suppose that for a given system the arrangementchannel Hamiltonians H a , acting in the separable Hilbert space %‘, are of the form (8.30), where all Hfin commute among themselves, while each Hint has a pure point spectrum when restricted to Mg’ and it commutes with the corresponding Hfin. If a and /? are two distinct arrangement channels and Hfin- Htin has no point spectrum, then R$) 1RY’ and R?’ 1RE’.

Proof. It follows from the basic definitions of wave operators that (8.31)

Since H:nt has a pure point spectrum on Mg), the integral in the speccan be written as a sum: tral decomposition of

(8.32)

here the summation extends over all eigenvalues of

Illntcorresponding

V. Quantum Mechanical Scattering Theory

606

to eigenvectors in Mg), and Ea({A}) is the projector onto the eigenspace corresponding to the eigenvalue A. It should be observed that in this context it is indeed possible to sum over all eigenvalues A on account of the fact that there can be at most countably many eigenvalues of a selfadjoint operator in a separable Hilbert space. We immediately get from (8.32), by using in the process the commutativity of H:in and H,int,

where the above sum, if infinite, converges in the strong sense. Since a similar relation can be derived for EM? etHat, and since H,kinand Hih commute, we can write for any f E MT) andg E Mg)

wherefa, = E(@)({A'}) f and ga = E(a)({A})g. Let us take any two vectorsf E ME) and g E M!? which have only a finite number of nonzero componentsh. and g A ,respectively. Then, in view of the fact that (f I etH@'eA8a'g) is continuous and therefore integrable in t on any finite interval, we have

(8.34)

J

T

(f I eiHst e - i H a t g ) dt

c c J (f,.I exp[i(H;in T

=

A'

0

+

A

+ A' - H:ln

- ~ ) tga) ]

dt.

0

Since Hiin A' - Hfin - A is a self-adjoint operator, we can apply the mean ergodic theorem (see Theorem 8.3 in Appendix 8.9 to this section) to each term in the sum of (8.34). According to this theorem

8. Fundamental Concepts in Multichannel Scattering Theory

607

where the above inner product is zero because the projector onto the characteristic subspace of HBkin - HFin corresponding to the eigenvalue A - A‘ has to be zero on account of the assumption that Htin - Htin has no eigenvalues. By using (8.31), we obtain (see also Exercise 8.3)

If f E Mf) and g E ME) have only a finite number of nonvanishing componentsf,, and gA, respectively, then in view of (8.34) and (8.35), we conclude that (Ql_8lfl SZP’g) = 0 for all such f and g. Since the set of all such vectorsf E MY) and g E MP) is dense in Mf’ and MP),respectively, it follows that the same is true for all f, g E #. This establishes the assertion that R!? 1RF’ for 01 # p. Q.E.D. A similar argument yields that RY) 1 RY) for a # p. The above theorem is valid under the restriction that H:in - Hiin should have no point spectrum when a # 8. It should be noted that this condition is certainly satisfied if HFinand Htinare two distinct functions of the momentum operators of the n particles partaking in the scattering process. The reader can easily convince himself that such is the case, for example, in potential scattering, so that Theorem 8.1 is always applicable in that case.

8.4. INTERCHANNEL SCATTERING OPERATORS In multichannel scattering we are faced with the possibility that a system prepared in some given arrangement channel does not have to end up in the same arrangement channel. Hence, obviously, no single scattering operator S could describe all the transitions which the system can undergo. Instead, to any pair (/?,a) consisting of an incoming arrangement channel a and an outgoing arrangement channel p we have to attach a different scattering operator,

(8.36)

s,,

#=

Q!*Q

+(a),

describing the transitions between these two channels. In fact, if Y(t)has the incoming asymptotic state Yp(t)= e-iHJ Y?(O),then the transition

V. Quantum Mechanical Scattering Theory

608

amplitude to the outgoing state ?Pyt((t) = riHJ !PTUt(O) of free fragments in channel is (8.37)

lim

t++m

(vt((t) I Y(t))

= lim (e-iHBt q ' ( 0 ) I edH' Y(0))

= (Q!

t++m

T t ( 0 ) 1 Y(0)) = (SZY YYt(0) I

= < v t < 0 I>s,u

SZy Y?(O))

PW>.

From (8.23) we easily obtain for all t E R'

s,,

(8.38)

p u t

=Q )!

*

~ ( u eiHat )

+

- eiHBt ~ -( 6* )Q(u) +

=

* eiHt Q?)

s,, .

= edH@

We can also derive in similar manner

One of the most important properties of the single-channel scattering operator is its unitarity. To establish the presence of some analogous property for multichannel scattering we have to consider simultaneously the entire array of all interchannel scattering operators. Then, using (8.27) we obtain (8.40)

C S,$Sy6 = C (f2~)*SZ(y))(SZ?)*SZ(6) +) Y

Y

In the case of single channel scattering the identity of R+ and R is S on M, (see Theorem 2.5). A reasonable generalization of that condition to the multichannel case would be to require that

a necessary and sufficient condition for the unitarity of

(8.41)

where we tacitly assume that the conditions of Theorem 8.1 are satisfied, and therefore RY) I RY' and RE) I RE' for a # ,f3. The assumption (8.41) implies that

8. Fundamental Concepts in Multichannel Scattering Theory

609

since Sz?) is a partial isometry with final domain RT'. Hence, (8.40) yields (8.42)

where 6,, = 1 for a = /3 and 6, = 0 for a # /3. A similar procedure leads to the conclusion that (8.43)

when (8.41) is satisfied. 8.5.

THEEXISTENCE OF STRONG ASYMPTOTIC STATES IN

n-PARTICLE POTENTIAL SCATTERING

Let us consider the wave mechanical case of n particles without spin interacting via two-body forces, so that

Let us choose an arrangement channel a containing the fragments

Sf), ...,SinJ.The free motion of each one of these fragments FLkis) described by the Hamiltonian (8.45)

where the summation is over the indices of the particles in that fragment. The channel Hamiltonian is (8.46)

Ha = H:)

+ ... + H?).

Take RLk)to be the center-of-mass position vector of the kth fragment, and let p:k' stand for some choice of internal motion coordinates, so that

V. Quantum Mechanical Scattering Theory

610

(RLk),pLk)) completely describes the positions of all the particles in the fragment. Then proceeding in the manner indicated in $5 of Chapter 11, we can write (8.47)

Hf)

=

-(1/2Mf')

ARt)

+ Hf)lnt,

Mf'

mi 9

= i E 3 y

where MLk)is the mass of the fragment FLk)and HLk)int is the Hamiltonian of the internal motion of the particles in that fragment. If $~:?(p:~)),k = 1,..., n, , represent any eigenvectors of HLk)int,then riHJ with Y,(O) represented by

x(O),

(8.48)

+,(rl ,... r,)

(k)

Ug

=

(k) (R, )

k=l

+,

(k) (k)

(P,

1,

'@,..., y?) EL~(R'),

is a state in which all fragments in the channel a are moving independently. Obviously, the closed linear manifold N'")spanned by all the functions of the form (8.48) coincides with the set of all free state vectors in the channel a.

Theorem 8.2. (Hack's Theorem). Suppose the functions V4j(r), i <j , i, j = 1,...,n, are Lebesgue square integrable on R3. Then the

?:) initial domains M;) 5 M): of the Moller channel wave operators L coincide with the closed linear subspace of L2(R3") spanned by all functions a,b, in (8.48) obtained by letting vary over all eigenvectors of H:k)intand Y$ over all of L2(R3)for k = 1, ..., n, . I n order to avoid very involved notation, we shall prove Theorem 8.2 for the special case of the arrangement channel {GI,G&3} in threebody scattering. I n this case

+Ak)

where mZ3is the reduced mass of the second fragment Gg-G3 and (8.50)

rz3= rz - r 3 ,

R

=

m2rz -k

%r3

m2+%

*

The space N(,) of free state vectors in this channel is the closed linear manifold spanned by all the functions (8.51)

+a(ri >

ra > r3) = yP(ri) Y?(R)

+!)
611

8. Fundamental Concepts in Multichannel Scattering Theory

corresponding to all Yjl) ,Y:2)€L2(R3) and all eigenvectors

t,bi2 of)

On account of Wiener's theorem (see Lemma 7.1) the linear set spanned by all functions (8.51) with YLk),k = 1, 2, represented by functions which are inverse Fourier transforms of Gaussian wave packets exp [-(p - P ~ ) ~k] ,= 1,2, is dense in N(') when p1 and p2 are allowed to vary over R3. For such functions we can easily compute

Hence, according to Theorem 2.8, the strong limits

exist for all P EN(u)if the improper Riemann integral

1'- [I( +

(8.53)

v12

v13)

4 11 dt

--a3

is convergent for all t,b of the above indicated form. Now, using (8.52) we get the following kind of estimate:

II

V12 e-fHut =

I,.

112 v122(r1

- r2) [4 (1

+ (&)"]"" exp[--a,(r1

-

w21

(equation continues)

V. Quantum Mechanical Scattering Theory

612

Hence, we conclude that (8.53) converges, and consequently N(a)=

MY) = ME) = Mt).

With the existence of asymptotic states established, the next question of general interest is asymptotic completeness. If Hbdenotes the closed subspace of bounded states of a general n-particle system described in the Hilbert space @ then in accordance with Definition 6.2, the theory is said to be asymptotically complete if (8.41) is satisfied and 2

(8.54)

=

R@2%.

In multichannel potential scattering theory there are at the present time proofs of asymptotic completeness in the three-body case for large classes of short-range interactions (Faddeev [19651, Thomas [1975]), as well as for special instances of n-body (n 4) scattering (Hagedorn [1978], Sigal [1978a, b]), but the general situation is as yet far from being as satisfactory as in the two-body case.

>

**8.6.

OF MULTICHANNEL TWO-HILBERT SPACEFORMULATION SCATTERING THEORY

We have seen in $8.4 that in multichannel scattering theory we cannot express all relevant physical information in terms of a single scattering operator acting in the Hilbert space 2‘ of the system, but that we have to deal rather with operators S,, of the form (8.36) for each pair (/3, a) consisting of an incoming and an outgoing arrangement channel a and /3, respectively. We can, however, represent all these interchannel scattering operators in terms of a single operator SJ from a new Hilbert space of incoming asymptotic states Hi” into another new Hilbert space %Out of outgoing asymptotic states if we build the direct sums (Ekstein [19561) (8.55)

2 i n =@ a

MJ“),

2out -@

M‘B’

B

over all incoming and outgoing arrangement channels a and /3, respectively. We shall refer to Yinand H o u t as the in- and out-asymptotic Hilbert space, respectively.

8. Fundamental Concepts in Multichannel Scattering Theory

613

Let us therefore denote by J* the identification (or injection) operators which, by definition, assign to the vectors (8.56)

f

=

O f uE XIn,

g

=

@ gB E &Out, 6

U

the respective vectors fu and &gp from Z . Naturally, the J* operators are not in general invertible, but they are bounded transformations (in the sense of Definition 1.5 in Chapter 111) from Zex, ex = in, out, into Z. Therefore they have adjoints J*t, which, by definition, satisfy" (8.57)

for all f~ Zexand h E % where (. I .),, , ex = in, out, represent the inner products in the asymptotic Hilbert space S e x constructed in accordance with Theorem 6.1 in Chapter 11. It is a routine matter to extend Theorem 2.7 of Chapter I11 to bounded transformations from one Hilbert space into another and thus conclude that J*' are bounded transformations from Z into %ex. It is also easy to establish that the strong limits (8.58)

exist as bounded operators from Sex into % and that in fact (8.60)

Q+J= @ szy,

LLJ = @ szi"'. B

01

Therefore their adjoints (8.61)

(sz+J)t

=

c Qy*,

(Q-J)t =

c szis'*, 6

OL

also exist as bounded operators, but from the Hilbert space Z of the system into the asymptotic Hilbert spaces Zinand &Out, respectively, so that (8.62)

SJ = (sz-J)+sz+J

is a bounded operator from Zininto #Out. *: In general, if A is a bounded linear transformation from XIinto X2(see Chapter 111, Definition 1.5) and (. 1 - ) i , i = 1, 2, denote the respective inner products, then by definition (Atg If)l = (g I Af)z, f E g E SZThe existence and uniqueness of At is proven exactly as in the special case where XI = Hz (cf. Chapter 111, Theorem 2.7).

.

V. Quantum Mechanical Scattering Theory

614

Let us denote by E k ) and E?) the projectors onto M t ) and ME), respectively, when these two spaces are viewed as subspaces of .%'in and S o U t , respectively. Then by (8.60)-(8.62), (8.63)

EF'SJEt) = EF)QF)*J-t J+Qt)EJ"),

so that the above operator is essentially" equivalent to S o , . In fact, we have forf and g given by (8.56),

(8.64)

( E F I~s J ~ J " ' f ) o u=t (ge I s o u f a > ,

so that SJ incorporates information about all interchannel scattering operators Sap. In Theorem 8.2 we have seen that in potential scattering MY) = M Y = Mlp) for all arrangement channels, so that we have a single asymptotic Hilbert space S e xand a single identification operator 1: (8.65)

X i n = %out

= %ex

7

I+ = J-

=

J.

Moreover, (8.41) is satisfied as part of asymptotic completeness. As easily seen, this implies that SJ is unitary on the Hilbert space .%'*I appearing in (8.65). Proceeding under the assumption that (8.65) is true, we can make the transition to stationary two-Hilbert space scattering theory by following exactly the same pattern as in $3 in the one-Hilbert space context. Thus we introduce the Bochner integrals (8.66)

=

iTm e b t eiHtJe-Cfext

+

dt,

and then routinely establish that the following results are true (see PrugoveEki [1973a, Theorems 2.1-2.41): (8.67)

(8.69)

Yf = .Q*Y '

= s-Iim €++O

QLY,

EH(B).Q*J= Q+"EHe"(B),

YJ E xex,

B E @,

There is an obvious conceptual distinction between MF' as a subspace of #, and the space M:' of all vectors in % .?, which have only the component f a different from zero. To avoid too cumbersome a notation we shall, however, ignore this distinction.

8. Fundamental Concepts in Multichannel Scattering Theory

(8.70) (8.71)

JtY*

=

Y+

615

(A - Hex fic)-'(JtH - HexJt) dAEAHy*,

Y* = ]Y + si-li+y f-, (A +m

- H fk)-'(H] - ]Hex)

+m

d A e e xJtH (

-

HexJt) ( H - h;z

+

dAEfexY.

€2

Furthermore, the spectral-integral expressions (8.72) for the TJ operator can be related to a F J ( [ )operator analogous to the F ( 5 ) operators introduced in $4.8 for the two-body case. For example, the first set of formulas (8.72) yields (see Chandler and Gibson [1973]): (8.73)

TJ = w-lim w-lim r++O

s++o

s,

+m

+m

J-,

dAEfex

S,(A - p) P ( p

+ i7)d,Efex,

We note that if #ex = # and J = 1 (so that HeX = Ho), then (8.70)-(8.73) reduce to (3.34), (3.41), (3.52), and (4.1 14), respectively. Clearly, (8.70) and (8.71) can be justifiably called two-HiZbert space Lippmann-Schwinger equations. These equations remain valid even in the presence of long-range forces under the proviso that asymptotically compensating operators 2,[analogous to 2 in (4.121)] are incorporated* into J. 8.7. MULTICHANNEL EIGENFUNCTION EXPANSIONS AND T-MATRICES In its fundamental aspects the approach to the formulation and treatment of eigenfunction expansions is the same in the n-body case as it has been in the two-body case, but the level of complexity increases drastically with increasing values of n. Furthermore, the Fredholm

* For details see PrugoveEki [1973a]. A variant of this approach has been extensively studied by Gibson and Chandler [1974] and Chandler and Gibson [1974, 1977, 19781.

V. Quantum Mechanical Scattering Theory

616

method of 96, used extensively in the n = 2 case, becomes inapplicable in its original form when n 2 3. T o formulate the concept of eigenfunction expansions in the multichannel case we have to choose in accordance with Definition 4.2 for each arrangement channel a complete sets of observables {Xp), Xf), ...} and {Yp), Y k ) ...} , that are best suited for that particular channel. Among the X(,)-observables, the standard choices always include the center-of-mass observables QL;: for each of the k = 1,..., n, clusters [whose position variables in the corresponding spectral representation spaces for complete sets of observables, which incorporate Q:::, we have denoted in (8.48) by RAk)]. The corresponding center-of-mass momentum observables P i : [whose variables we shall denote by PP)] are always included among the Y(,)-observables, with the internal energies for each fragment being the next standard choice. Thus, in accordance with (8.48) the corresponding eigenfunction expansion consists of functions (in physical literature usually called stationary states) of the form (8.74) R,

= (Ry,...,R?’),

P,

= (P:)

,...,P?)),

where p l k ) and cobk) are variables associated with the rest of the complete set of observables for the kth cluster. For example, in case of the arrangement channel {Gl, G, - G3) [with which the equations (8.5)-(8.8) and (8.11)-(8.17) are associated], p1 and w1 are redundant for the first cluster which consists only of G,, whereas for the second cluster r23= r2 - r3 play the role of p, and E(2-3)is incorporated among the variables in w , [to be very specific, if P, - P3 is an electron-proton pair, and the spin observables are ignored, then in accordance with the results of 97.7 in Chapter 11, we could choose w, = ( I , m,n)]. The presence in (8.74) of the plane waves (8.75)

@p$)(Rf)) = ( 2 ~ ) ” ’ ~e~p(iPp’R!’),

k

=

1,..., n, ,

tells us that, as in the two-body case, are not elements of MIp). However, in general, if we make in (8.73) the transition to the configuration representation by expressing (for spinless particles) R, and pa in terms of r1 ,..., r, [cf. (5.10) in Chapter I1 for a specific example] so that ,,@ ,,‘: becomes a function dpL,w,of r1 ,..., r, E R3, then as a

617

8. Fundamental Concepts in Multichannel Scattering Theory

function of P, and

w,,

(@e,u,

[ +) = 1.i.m.

IRD

(rl ,..., r,) #(rl ,..., r,) dr,

@a)* p~*wm

dr, ,

is in general a well-defined element of

for any 1,5 E Mca) CL2(R3”) [where N , denotes the total number of Y(,)-observables and vfco) is the measure over the spectral representation space for the complete set of observables related to the internal degrees of freedom in the Kth cluster]. Using the symbolic notation in (4.81), we can express the fact that, for all channels a, @k:,w, constitutes an eigenfunction expansion in Mlp) by the shorthand statement

;J I

(8.76)

dV,(P,

@ : * u , >

2

%)(@P:*u,

n dP, nu

dV,(P,

9

w,) =

(k)

I (k)

=

lex,

(k)

dv, (% >,

k=l

where 1”” denotes the identity operator in sex. Likewise, the Green’s operator Gex([) for the asymptotic Hamiltonian Hex [given by (8.59), but with Hin = H O U t ] can be expressed as (8.77)

Gex(5) = (1 - Hex)-l = C

I5 I

dV,(P,

@&,)

9

I

%>(@&,

+A 3 ’

- x”,*=1[(P52M3

where AAk) is the internal energy of the Kth cluster in the a-channel, and it is in general a function of wik). Hence, the Green “function” for the channel Hamiltonian H, is in general a complicated entity, @k!*,,(R,

(8.78) GJR,

9

Po;

R,’, f,‘;

5) =

9

Pa>

dV,(P, %) @P;*(R:, P:) 9

+

9

5 - g 3 P Y S / 2 M ? ) A,(k)(%(k) )I

which might turn out to be well-defined only in the sense of distributions rather than as an ordinary function. Naturally, the explicit use of the theory of distributions can be avoided once it is realized that in general the integrations in PAk) and wik), K = 1,...,n, , have to be

V. Quantum Mechanical Scattering Theory

618

executed after the integration in R, ,p a , Rat, and pat have been performed upon taking inner products with suitable wave packets. These complications make the task of reproducing the derivation of 994.6 and 4.7 in the multichannel context generally unfeasible without the introduction of new techniques. True, on a purely formalistic level, it would appear that (8.70) unambiguously yields

(8.80)

However, the fact that Va'.cp~~,wa is not square-integrable even in the presence of two-body forces with integrable potentials invalidates the technique used in the two-body case. A further warning that something is seriously amiss with purely formal extrapolations from the two-body case is provided by the observation that due to the presence of different arrangement channels, (8.79) does not have in general unique solutions* even in the case where Theorem 8.2 assures us that SZz) exist on nontrivial initial domains M:), and are unique. All the previous observations are applicable to multichannel T matrices, to whose derivation we turn next. *This can be easily established on a heuristic level by using the second resolvent equation to write for p # a,

since @b?L,w,6 is not an eigenfunction of H a , and therefore the first term on the right-hand side of the previous equation vanishes. Thus, : :(, :@ : u@$:,:),~ is another solution of (8.79) for any a E and any 9, # a (see Lippmann [1956] and Epstein [1957] for further details).

c'

+

619

8. Fundamental Concepts in Multichannel Scattering Theory

According to (8.72), we have for f E Mk) and g E ML8’,

where in accordance with the notation in (4.81) and (4.89) we have written (8.86)

fP, w,) 9

= 1.i.m.

=

(@g&I f>

=

1.i.m. ~ R 3 n 6 k ~,..., ~ ur,)f(r, ~ ( r l ,..., r,) dr,

- 1 -

dr,

/6$iu,8(rl,..., r,) V8(r1,..., r,) 6&!kl(rl ,...,r,) dr,

,

*-.

dr,,

with remaining symbols being defined along similar lines. We note that whereas, e.g., @&., was defined in (8.74) as a function of the centerof-mass position variables RLk)and additional internal motion variables pkk) for each cluster k in the arrangement channel 01, &p,,u, is a function of the position variables r1 ,..., r, of each of the individual particles GI,..., 6, in the system-the transition from one set of variables to the other resulting from equations such as those in 95.2 of Chapter 11, relating two such sets of variables. Naturally, if 01 = /3 then one can use in (8.87) the cluster variables RLk)and pkk), making sure, however, to also express V , in term of these variables. It should be recalled now that, whereas in our treatment of the twobody case we had already eliminated in the initial stages of our con-

620

V. Quantum Mechanical Scattering Theory

siderations the center-of-mass variables, so that we have been working the Hilbert space X(l)(see $1.7) of the reduced particle, in the present n-body case, n 2 3, that reduction has not been performed since, in general, it does not prove to be as advantageous as in the n = 2 case. However, if there are no external forces [and therefore V as well as each V , do not depend on the center-of-mass variables R of the entire system; cf. Exercises 5.1 and 5.3 in Chapter 111, we can always easily dispose of three integrations in (8.87) by introducing R as one set of integration variables. Indeed, we only have to note that, , ,@ $: in (8.74) contains the product of exponential functions (8.75), from which the factor (8.88)

@I.“;,.,

can be extracted, and the same holds true for Hence, formally speaking, the R-“integration” gives rise to a 63-“function,” so that we can write

where the two bona jide functions of P,,w,, Pi, w i appearing on the right-hand sides of (8.89) and (8.90) are called, respectively the “post” and “prior” versions of the on-shell T matrix for the multichannel collision process in question. Naturally, the appearance of the a3“functions” merely signifies that in deriving (8.84) and (8.85), the orders of all the P, and R, integrations could not be actually reversed, so that, for example, the 1.i.m. in (8.87) had to be interpreted in the context of the last integral in (8.84): in that integral the integration in R, and Pain fact have to be restricted to compact sets in order to justify the reversal of integrations in R, and P, , and upon letting those sets tend to R3%, Fourier theory (e.g., Theorem 4.5 in Chapter 111) has to be invoked in relation to the R-variables [the procedure being reminiscent to that leading from (5.9) to (5.10)]. Thus, the use of 63-“functions”

621

8. Fundamental Concepts in Multichannel Scattering Theory

could be avoided, but in physical literature their explicit appearance is favored due to their providing a graphic illustration of the conservation of the total momentum for the entire system. In this same spirit of notational expediency, it is customary to combine the first operator relation in (8.72), i.e.,

sJ= lex - 2 d P ,

(8.91)

with (8.82), (8.84), (8.85), (8.89), and (8.90) into a so-called S-matrix formula that generalizes (4.107): (8*92)


wfl’

=

,a,

I sJI p~

9

wa>

n 63(Pp’‘ na

- P?’) SWa~,,,

k=l

Clearly, the first term on the right-hand side represents a symbolic variant of the formula (8.76) for leX,and either one of the two versions of the on-shell T matrix in (8.89) and (8.90), respectively, can be used.

**8.8.

MULTICHANNEL BORNAPPROXIMATIONS AND FADDEEV EQUATIONS

A “prior” version of the T matrix could have been introduced in the two-body potential scattering as well, but nothing new would have been gained due to the relationships (8.93)

@k*(r)= @-k(T),

&’*(r>

= di’(r>,

and to the presence of a single arrangement channel and therefore of a single interaction channel Hamiltonian, namely V . That this is not so in the multichannel case becomes clear as soon as we consider extrapolating the Born approximation method to the n-body case with n 3. Indeed, by repeatedly iterating the Lippmann-Schwinger equation (8.79) we obtain a Born series for retarded stationary states,

>

(8.94)

622

V. Quantum Mechanical Scattering Theory

and by the same token, also a Born series for advanced stationary states,

Inserting these results in (8.89) and (8.90), respectively, we obtain “post” and “prior” Born series for the corresponding “post” and “prior” versions of the on-shell T matrix. We immediately see that in rearrangement collisions already the first post and prior Born approximations are going to be markedly different even on the energy shell, namely

so that computationally the two versions of the Born series can lead to significantly different approximation schemes. Despite its wide usage in atomic and molecular physics, there is almost no knowledge of the convergence properties of the general Born series (8.94) and (8.95). The main difficulty stems from the fact that the Fredholm method of $96.1-6.9 is no longer applicable already in the three-body case. This can be easily verified, for example, for the channel 01 = {GI, 6, , G3} where, assuming two-body forces, we have by (8.2) and (8.80),

It is convenient to work in the momentum representation so that assuming that Vij(r) EL(,)(R~)we obtain

,,} and noting that by using center-of-mass coordinates for { 6 $S

Hence, G,(()V, = (t; - H,,)-1V(1,2,3}equals the sum in i <j =

8. Fundamental Concepts in Multichannel Scattering Theory

623

1, 2, 3 of the following three operators:

The question is whether upon making the transition in (8.102) to the variables pl, pz , p3 [through the use of (8.lOl)l and then eliminating the center of mass motion of the three particles by factoring the Hilbert space iW3)= L2((ag)of the entire system into a tensor product Xi1)@ 3F2) [representing an extension of (1.26) to the n = 3 case], we are going to end up with Hilbert-Schmidt operators. Now, working in the center-of-mass frame amounts to holding (8.103)

p

= P1+

P2

+ P3

= pii

+P k

fixed, and by Theorem 8.12 in Chapter IV and Exercise 5.3, the resulting operator in 3Y2)will be Hilbert-Schmidt if and only if it is an integral operator with square-integrable kernel. We see, however, from (8.102) that this will not be the case except if pij(p)= 0 almost everywhere. If G(5)and Go(5)denote the Green’s operators of H and Ho ,respectively, in the present three-body case, then by Theorem 3.2,

G(t)= GO(%)+ G0(%)(V12+ ‘13

(8.104)

+

G(%),

‘23)

and our earlier conclusion implies that we cannot apply at present the Fredholm method as we had been able to do in the two-body case due to Theorem 5.7. T o circumvent this difficulty, Faddeev [1961a] rewrote (8.104) in the form (8.105)

@(5) = GO(t”j’,, + VjkGO(5) 9:m Go({), i # i # k, where Yfa(5) is the three-body counterpart of (4.101) corresponding to incoming and outgoing arrangement channel 01 = {Gl, 6, , G3}. Indeed, in view of (3.61) and (4.102),

c m%)Go(%)[‘ + VGO(5) 9:m Go(%) 3

(8.106)

=

=

Go(%)‘G(%),

i=l

and we recover (8.104) from (8.105)-the whole procedure is reversible. We note that Sf*({) can be also viewed as a component of the Y J ( { )

624

V. Quantum Mechanical Scattering Theory

in (8.73),

Yta([) = E1"9;'([) EP',

(8.107)

Mk'

=

Mt),

for the incoming and outgoing arrangement channel a = (6, , 6, , G3}. Following Faddeev [1961a, b, 19631, we can proceed by using the two-body problem to solve the three body one. Hence we introduce in the 9-operators for the arrangement channel { G $ ,Gj- a,}, (8.108)

-%c(5)

= Vj,

+ Vj,Gj,(C)Vj,

Gjk(5) = (5

9

- Ho - Vj,)-',

These F-operators correspond to the physical process in which 6, and 6 k collide, whereas 6, is ever present but never interacts with either Gj or 6 k . If we now write (8.106) in the form (8.109) G'"(5) = Go(5)fl:i)(5) Go(5),

fl:i)(S) =

Vj,

+ Vj,Go(5) 9 ( 5 ) ,

then in view of (4.103), (8.110)

ria([)= V12 + VI, + V ~+S (V12 + V1a + V28) =

G0(5)fl;a(t)

fl2tS)+ 92)(S)+ e?(S).

According to one of the second resolvent equations we have (8.1 11)

Go(() = GdS) - GdS) V&o(S),

and substituting this result into the defining expression of Fz)([) we get by (8.108), (8.1 12)

fl::)(S> =

q, + v,,Gjk(5)[fl,u(5)- Vj,G0(5) fl:a(5)1

= Vj, =

+ Vj,Gj,(5)[fl,a(S) - fl:;)(5) + VjJ

q,cr;> + Vj,Gj,(5)[qu(5)

-9 3 0 1 .

Finally, taking into consideration that by (8.108) and (4.102), (8.1 13)

Go(%),

VjljKGiK(5)=

we arrive at the Faddeev equations for 9;-operators: (8.114)

r:i)(5)= ?,(
+ qk(5) Go(a)[r:i)(S) + 5:35)1, i#j#

k

=

1,2,3.

8. Fundamental Concepts in Multichannel Scattering Theory

625

The operators fz'(()are of no direct physical significance, but if we can compute them, then by (8.110) we have computed Y E ' ( ( ) , and in turn by (8.105) and (8.106) we have also computed the full Green's operator. In fact, by (8.1 13), (8.115)

Gjk(5) - Go([)= Go([)VjlcGjn(S) = Go([)%(O

GO([),

so that the combination of (8.109) and (8.114) yields Faddeev equations directly for the components G(i)(()of the full Green's operator (8.105): (8.116)

+ Go([)%(5)[G'j'(5) + G'"(5)].

G'i'(5)= Gjk(5) - Go([)

In turn, proceeding heuristically, by (8.105) we can write a counterpart of the solution-type Lippmann-Schwinger equations for the advanced and retarded eigenfunction expansions in the a = {GI, B, , G3}channel, (8.117)

@k:(*)

=

flim icG(A, f ie) f++o

@k:

i=l

(8.118)

a:$$:)

= &lim icG'"(A, r++O

& ie) @lph ,

Pa = (pl ,pa ,pa),

and thus derive from (8.116) Faddeev equations for these advanced and retarded stationary states: (8.119)

@{$it) = @!'$EL- @x + Go(Aafi0)9&i,

fiO)[@{$!:)

(8.120)

@{$EL= flim ieGjk(Aafic) @? , f+M

C (pi2/2mi).

+ @::]g)]

r

A,

=

i=1

In view of the developments in $6.2, it is obvious that @{$j$: are in fact the distorted plane waves for the two-body scattering between Bj and Gk,in which, however, the (free) center-of-mass motion has not been removed, and which in addition incorporates the presence of the (free) particle B4:

Faddeev [1965] has shown that for a large class of potentials Vtj(ri - rj), the operators Go(()fjk(() in (8.114) and (8.116) are completely continuous at positive energies Re ( 2 0, and that in fact

626

V. Quantum Mechanical Scattering Theory

the fifth powers of these operators are compact at all energies, so that the solutions of these equations can proceed by already well-established mathematical methods. All these statements stay true for any pair of incoming and outgoing arrangement channels 01 and b, respectively. In fact, in general,

and the derivation of the Faddeev equations for all pairs of incoming and outgoing arrangement channels can proceed simultaneously through the use of the two-Hilbert space formalism of 98.6. Yakubowskii [1967] has extended the Faddeev technique in its basic aspects to the n-body problem for n 2 4, so that in gross outlines, mathematically exact and yet computationally implementable methods are available for the treatment of nonrelativistic scattering of n (point) particles interacting via two-body short-range potentials.

MEANERGODIC THEOREM *8.9. APPENDIX:VON NEUMANN’S The following theorem was first derived by von Neumann [1932] in order to prove the quasi ergodic hypothesis of classical Hamiltonian mechanics, and it was used by us to obtain (8.35).

Theorem 8.3. Suppose A is a self-adjoint operator in the Hilbert space c%? and (8.122)

U, = eiAt =

IR1 dE, . eitA

Then for any f, g E X (8.123)

irrespective of the mode in which t , - t, tends to infinity. We note that there is a unique bounded linear operator B ( t , , ts) (a Bochner integral) for which the relation (8.124)

627

8. Fundamental Concepts in Multichannel Scattering Theory

holds for allf, g E &? (see Exercise 8.7). Hence, (8.123) states that E({O>) =

t$,l@&

Wl

3

tz).

In order to prove Theorem 8.3, we write (8.125)

(f I B(t1 tz)g) = (f I Wl ,tz) E({O))g) Y

+ (f I B(t1

9

t2)

Em1 - {OHg).

From (8.122) we get

(f I UtE({O))g) = J

w’

eita

d
=

(f I E({O))g),

and consequently, we have (8.126)

(fIW1

Y

tz) E({O))g) = (f l E({OHg).

Thus, we see from (8.125) and (8.126) that (8.123) is true if and only if (8.127)

lim

tn-t,++oO

(f 1 B ( t l , t,)h)

= 0,

h

= E(R1- {O))g.

We shall prove that (8.127) holds by showing that

converges to zero when t , - tl Since Ufl* = U-ti , we get

+

+

OC).

After introducing in the above integral the new variables t and s = t’ t”, we arrive at the following relations:

+

=

t” - t’

V. Quantum Mechanical Scattering Theory

628

Since the integral

obviously exists, we can apply Fubini's theorem to interchange the order of integration in t and A. Thus, we obtain, after carrying out the integration in t ,

sin &(tz- tl)h

)

11 EAh 112*

Let us split the domain R1 of integration in three parts (-GO, -71, and (7, GO), and majorize the integrand by 1 on (-v, +q] and by ( i ( t z - tl)q)-z on the other two intervals. Then we arrive at the estimate

(-v, +TI,

+

By choosing 7 small enough, we can make the first term on the right-hand side of the above inequality smaller than z/2 for any a priori given z > 0; this is due to the fact that lim ll(E,

v-++ 0

-

E-,)h I1 = 11 E({O})h11

= 0.

Then, for such a value of 7,the second term in (8.60) can be made smaller than z/2 by choosing I t , - t , I sufficiently large. Hence lim

t,-t,++m

11 B(tl ,t,)h I/

= 0.

Thus, (8.127) holds, and therefore (8:123) is true.

EXERCISES 8.1. (a) How many arrangement channels can there be in the scattering process of three distinct particles ? (b) How many arrangement channels have actually been observed in the scattering of a protonp, a neutron n, and an electron e, i.e., for which channels 01 do we have M t ' = (0) ?

629

8. Fundamental Concepts in Multichannel Scattering Theory

(c) Give the conventional names of the fragments in all experimentally realizable arrangement channels in the scattering of n, p , and e interacting by means of nuclear and Coulomb forces. 8.2. (a) Count the arrangement channels in four-particle scattering in which there are only two pairs of distinct particles. (b) How many of these channels are not empty in the scattering of two protrons and two neutrons (as far as our present experimental knowledge extends) ?

8.3. Suppose that f ( t ) is a continuous function for t and that liml-,,+mf(t) = a exists. Show that

E

[0,

+coo),

8.4. Consider multichannel scattering with a finite number of arrangement channels a. Show that the operator

S‘ = CQ$’Q?’* (I

commutes with e i X t and with EH(B). Remark. This operator was introduced by Jauch [1958b] as a candidate for a scattering operator in multichannel scattering. However, its inadequacy for this role is reflected in the fact that its knowledge is not sufficient to compute transition probabilities for scattering processes in which we have transitions between distinct arrangement channels (see also Exercise 8.6). 8.5. Show that if RY’ addition (8.41) holds, then

1 RY’

S’*S’

=

and R?’ S’S’*

1 RE’

for a # /?,and if in

= ER

8.6. Show that if Y J t )and Y+(t)have the respective strong asymptotic states Y!?(t)= e-iHJ Yi”(0)and Yyt(t) = e-iHsf Yyt(0), then the transition amplitude (Y+(O) I Y-(O))= (Yyt(0) I S,,Y?(O)) can be written in the form (Y+(O)I Y-(O)) = ( V t ( 0 )I QJB’*S’Qjo)Yf”(O)) -

I

= (YYt(0) Ql“’*S’Q?)!p

(O)),

provided that the conditions stipulated in the preceding exercise are satisfied, and that MY’ = M‘6’ - .

630

V. Quantum Mechanical Scatering Theory

8,7. Prove that for any family U,of unitary operators defined by (8.122) there is a unique bounded operator B(t, , tz) which satisfies (8.124) for all f,g E 2.

References for Further Study Quantum scattering theory is the subject of a multitude of textbooks and monographs, and its basic aspects are routinely treated in just about every single textbook on quantum mechanics. That being so, we shall mention only a few monographs especially well suited as supplementary reading for this chapter. Newton [I 9661 treats all the principal aspects of the subject (concentrating, however, exclusively on physical ideas), and also provides an extensive bibliography covering most of the relevant material published by the mid-1960s. Taylor [1972] gives a very lucid exposition that incorporates as well as enhances the physicist’s traditional approach to the subject, and he also supplies a mathematically careful presentation of complex-analysis methods in scattering theory. De Alfaro and Regge [1965] treat this last subject in detail, and are doing full justice to its mathematical aspects. Amrein, Jauch, and Sinha [1977] give a mathematically rigorous presentation of most of the basic aspects of quantum scattering theory, whereas Reed and Simon [1979] concentrate on an array of topics of special mathematical interest. Mathematically closely related to the subject of scattering theory is perturbation theory for linear operators (see Kato [1966], Reed and Simon [1978]), as well as the theory of the structure of the spectra of Schroedinger operators (see Jorgens and Weidmann [19731). Titchmarsh [1962] provides a detailed treatment of the theory of eigenfunction expansions for second-order differential operators. A general treatment of the subject is supplied by Berezanskir [1968, 19781, and his second book covers many topics of direct interest in quantum mechanics. Some of the early papers laying the mathematical foundations of quantum scattering theory are those by Kodaira [1949]. Povzner [1953, 19551, Jauch [1958a, b], Jauch and Zinnes [1959], Hack [1959], Zhislin [1960], Ikebe [1960, 19651, Green and Lanford [1960], Faddeev [1961a, b]. Jordan [1962a, b]. Grossmann and Wu [1962], Scadron et al. [1964], Hunziker [1964, 19651, and van Winter [1964, 19651. Since the mid-I960s, the number of mathematical papers on scaffering theory has been escalating at an everincreasing rate. Hence the reader desiring to familiarize himself with more recent research in this area is well advised to consult published conference proceedings on the subject (such as those edited by La Vita and Marchand [1974], Nuttal [1978], and de Santo, Saenz, and Zachary [1980]), in addition to the references quoted in the appropriate context in $53.8, 4.9, 4.10, 5.7, 8.6, and 8.8.

Hints and Solutions to Exercises

CHAPTER I 1.3. I n order to prove that W'(rW1) is infinite dimensional note that all the polynomials x", n = 1, 2,..., belong to Yo(R1) and are linearly independent.

1.5. Vsis the vector subspace of V which is a subspace of any other vector subspace containing S. 2.2. I n order to prove that Yy2,(R1)is closed under the operation of vector addition in Yo(R1), apply the triangle inequality (Definition 2.2, point 4) on Yo([-a, +a]), and then let a -+ 00 in order to obtain

+

in case thatf(x), g(x) E qzl(R1).

2.3. Apply the Schwarz-Cauchy inequality on Yo([-., +a]), and then let a -+ co in order to prove that the integral

f *(x) d x ) dx -W

is convergent. 3.1. I n checking the transitivity condition make use of the triangle inequality. 631

632 3.2.

Hints and Solutions to Exercises

Employ the triangle inequality to obtain 451

9

52)

G

451

and then reverse the roles of

9

771)

+

4771

9

72)

+ 4%

Y

62),

el , 5, and v1 , r], .

3.3.

Apply the triangle inequality first on

3.5.

I d(e,r])- d ( & , r])l

e, 5,

and then on ( , r ] ,

r]

5.

< d(f, en)according to Exercise 3.3.

4.7. If S is countable then S x S is also countable. If we have a countable number of countable sets S, , S, ,..., their union S, u S, u *.. is also a countable set. 4.8. Show this for Z2(n) by specializing the methods employed in proving Theorem 4.3. 4.9. In proving that [S] =(s), show that the closure linear space (S)is also a linear space.

4-10,

5.2.

Derive I(f I g> - (f, I gn)l

d II g II Ilf - f, II

cs, of a

+ Ilfn I1 II g - g, 1 .

In order to prove that the above integral converges for any

f,g E %$)(R1) use the same hint as in solving Exercise 2.2. 5.5.

Note that the distance d ( f , g ) of any two distinct vectors - g 11 = 4 2 . Let S = {h, ,h, ,...} be a sequence dense in X. We can map T in S by assigning to each f E T a vector h(f ) E S such that d ( f , h) = [If - h 11 < 1 / 4 2 . This mapping between T and a subset of S is one to one, since 11 h ( f ) - h(g)ll > 0 forf # g.

f,g E T is Ilf

CHAPTER I1 1.1. Consider R and S to be the subsets of 3 = R u S. If T’ denotes the complement of T C Z with respect to Z, then

(RUS)-(R

S) = (R uS )n(R

S)’.

By applying Lemma 1.1 we get ( R U S ) n ( R A S)’

= (R n(R

S)’)U ( S n ( R A 5’)’).

Further applications of Lemmas 1.1 and 1.2 yield Rn(RAS)’=RnS=Sn(SAR)’

Chapter I1

633

1.3. Prove first that if R, S E W,then R - S and R A S are in 9. Then use the result of Exercise 1.1 to show that R n S E W. 1.4.

Use precisely the same method as in proving Theorem 1.2.

1.5.

Prove that

and use the results of Exercise 3.2 and Lemma 3.2. 3.4.

We can write

where Sl(.3=,) is the set of all rational (irrational) numbers inside [0, 11. = 0 because !Ill Compute the integral from (3.7) by noting that is a countable union of all sets {r}, r E !Ill, for which p\l)({r}) = 0, . while 31= [0, 11 Note that the Riemann integral J i f ( x ) dx is not defined. 3.5.

Note that if

634

Hints and Solutions to Exercises

we can write sl(()

3.8.

and s2(() in the form

Note that for a real function f ( x ) = f + ( x ) -f-(x)

+

while

If (41=f + ( 4 f - ( 4 . 2 0 for 5 E R.

3.9.

Note that!(()

4.1.

In proving transitivity, note that for any functions f ( x ) , g(x),

h(4, 4.2.

- g(5)

{ x : f ( x >f h(x)) c Wf(4

To show that 1) f 11

zg(x>>" { x :g ( x ) f h(x)H*

> 0 for f f 0, use

Lemma 4.1.

4.4. Prove first that Ci is countable by noting that % x x 3 (2n times) is countable since % is countable. Then show that Ds is countable by using the fact that % x '3 x Ci is countable, and that S, u S, u is denumerable whenever S , , S, ,... are denumerable.

A S) = So I xR(x) - ,ys(x)l dp(x).

4.5.

Note that p ( R

4.6.

Show that the functions for x # y for x = y ,

corresponding to all real-number y , constitute an uncountable orthonormal system in L2([w1,p). Note that for this measure p, the Hilbert space L2(I,p) is nonseparable even when we choose I to be a finite interval. 4.7. Prove first the statement that C& is a Boolean algebra of subsets of Rn by using the same technique as in proving Theorem 1.3.

a,,".

4.8. It is obvious that &?t,oC gn,and consequently du(g&J C Conversely, = {I n Q: I E Yn}C du(B&J and therefore grin = d f f ( & n ) C dff(Bt,.J. In fact, I n 52 E do((at, for ,)any I E P since I n Q can be written as a countable union of intervals contained in sd. For example, for integer k > 0 and finite I divide each one of the edges of I into k equal parts and take the union S, of all thus obtained subintervals of I which are also contained in Q; then, in view of the fact that Q is open, we obviously have (Jkm_l S, = I n Q.

635

Chapter 11

5.1. If we denote by Fik the force with which the ith particle acts on the kth one, note that

2 F,k

F(.8 . = 0.

= mifk,

i=l

Employ afterwards the action-reaction principle (Newton's third law), according to which F,, = - F k i , in order to derive that F = 0.

+

5.2. Note that since a, I $ l2 = (a,$*) $ +*a,$ is square integrable and continuous, we deal with Riemann integration and by theorems of the calculus we have (justify)

Choose a finite domain D of R3n of the form D, x ... x D, where D, ,..., D, are spheres centered at the origin, and denote with s k the boundary of Dk C R3. By using (5.5) and Green's theorem, derive

The above integrals along the boundaries s k vanish, when each s k expands to infinity in all directions, provided I +Vk+ I = O(rk-.'), ffk

> 2.

prove that ar,/aX = e,. 5.3. We have aV/aX = &(V,V)(ar,/aX); Since similar results are valid for aV/aY and aV/aZ, one can write n V,V = VkV = Fk = Fik (see the notation in Exercise 5.1). Since Fik = -Fki by the action-reaction principle, one gets V,V = 0.

-x =;,

xi=,

6.2.

Show that the mapping

(f,,...,fn) + ( f k , ,...,fk,),

f, E 8, , provides an isomorphism between 8, @ &;c, @ @ 8 k . s

f1

E 81,...'

@ 8 , and

6*3* Verify that the mapping (fi (fz , f 3 ) ) --t (fi ,f2 ,f3), fi E 81v f2E g2,f 3 E g3,provides an isomorphism between 8,@ (g2@ 4) and 9

81

082 08 8

*

+

6.6. ImQ(f g) = 0 implies that Im(f I g) Im Q(f ig) = 0 implies that Re(f I g) = Re(g If).

+

6.9.

=

-Im(g

Note that for any f,fl E X , , g, g, E X z, IlfOg-flOgllI


+llf1O(g-g1)ll.

If)

and

636

Hints and Solutions to Exercises

If g E A?, and g, ,g, ,... E V, converges in norm to g, then to some limit 6 E A?, ; 2 is independent of the chosen sequence converging tog. T h e mapping g --t j of A?, into X 2 is linear and preserves the inner product. For every h E X 2, there is a sequence h, , h, ,... E S2converging to h, and consequently h is the image of the limit h of h, , h, ,... E v,, where hk is that element of V, which is mapped into h k . 6.10.

t1,t, ,... converges in norm

7.3. Use the ratio test which states that a series if 1imk+" I a k + l / a k I < 1.

C,"=,,a k converges

7.4. In order to show that A is not a Hermitian operator, take f ( r , 6,$) everywhere twice continuously differentiable, e.g., f ( r ) = e-9, and take g(r) = (l/r) crs.

7.6.

Prove by induction that for m


thus showing that (lk I 1,) = 0 for k < n. Use the above result in the first step of the following derivation:

=-I

1 " pne-odp n!

=

1,

which shows that the Laguerre functions are normalized in L2([0, 7.8.

The substitution x

+ a)].

yields

= e-p

'f*(-lnx)

g(-lnx)

dx.

This shows that the mapping

+

defines a unitary transformation of L2([0, a))onto L2([0, 11). Since by Lemma 7.4 the family of all polynomials p ( x ) = a, a,x a,x* is dense in L2([0,l]), the family of all functions e-o/2p(e-o) = e-0/2(uo

+ a,

e-0

+

+ + an e-"O)

+ +

631

Chapter 111

+

is dense in L2([0, 00)). Each e-p12p(e-p) can be approximated arbitrarily well in the mean by a function of the form bolo(P)

+ blll(P) + + *.*

b&*(P)7

since it follows from the last result of Exercise 7.7 (by taking an appropriate s) that each e--p12 e - k p can be approximated arbitrarily well in the mean by a linear combination of Laguerre functions. 7.9. The orthonormality can be deduced from the formula (see Chapter I, Exercise 5.4) dn h,(u) = (-1)n eu*/z -(e-ua) dun

by the methods employed in solving Exercise 7.6. T o prove completeness, write for any f E L2(W)

fi(4 = t [ f ( 4*f(-41*

f ( 4=f+W+f-(u),

Since we have, after the substitution u = dj,

=

JW

(p-’l”*(~~))*(p-’/*g*(

0

43)dP,

the mapping +

P-l’Y*(G),

f ( 4E L Z ( W ,

is a unitary transformation of the subspaces L:(R1) of even and odd functions, respectively, onto L2([0, 00)). The completeness of ho(u), hl(u),... in L2(R1)can be deduced from the completeness of Zo(p), Zl(p), ... in L2([0, 00)).

+

+

CHAPTER I11 1.2. Prove first that (Kf)(x) is continuous on I for any given f ( x ) , f E L2(I),and therefore that Kf is square integrable in I. Show next that if fl(x) = f 2 ( x ) almost everywhere, then ( K f ) ( x )= (Kf2)(x) almost everywhere. Then check the linearity of the above mapping. 1.3. that

Derive, by applying the Schwarz-Cauchy inequality in L2(I),

Hints and Solutions to Exercises

638

By using Fubini's theorem (Chapter 11, Theorem 3.13) derive from the above inequality 2

1.5. A

Apply B to a sequence of nonnegative functions f,(x) ..., with support inside [ - l / n , + l / n ] , for which

E

6P,"(R1),

= 1,2,

llfn 112 = and show that

-m

lfn(x)I,dx

=

1,

11 Bf, 11 -t co when n --+ co.

2.5. Note that for fixed f,4, defined by +,(g) = (f I g) is a continuous linear functional. Use Riesz' theorem (Theorem 2.3) to establish that for each f E A? there is a vector A ( f ) such that (fI g) = #,(g) = ( A ( f ) I g), and check that the mappingf-t A ( f ) is a linear operator on A?. Use the boundedness and Hermiticity of (flg) to prove the boundedness (by taking g = Af) and Hermiticity of A.

If E projects on M, dim M = m, choose {el ,..., em} and ,..., en} to be orthonormal systems spanning M and M I , respectively.

3.3. {em+,

4.1. If dim A? = n, introduce an orthonormal basis {el ,..., en} in A?. Prove that { Vel ,..., Ve,} is also an orthonormal basis, and, therefore, that 9, = A?. 4.2. Use the procedure employed in proving Theorem 5.6 in Chapter 11. The difference between that case and the present case is that Theorem 5.6 applies to I = R', while at present I = [a, b], and the norm to be taken is l l f l l = J: If(x)l dx. 4.3. Write the equivalent of (4.8) for the general case, and apply Fubini's theorem (Chapter 11, Theorem 3.13) in order to be able to proceed in the same fashion as for n = 1. For instance,

1: dx, 1' dx, ,... / -U

=

+U

dx, f (u

sinhx, +x)X1

--(I

sink, ju dx, 1:

/

X1

approaches zero when h -+

+U

dx2

sinhx, x,

-

dX,f(U

+

X)

-U

00,

due to Lemma 4.1.

sink, x2

sink,,, xln

- -

639

Chapter 111

4.4. If f , , f 2 )...E g A is a Cauchy sequence, then, due to the isometry of A , the sequence Af, , Af2 ,... is a Cauchy sequence also. Hence, if g, g2 ,... E g Aconverges to g E g B, then Ag, , Ag, )... has a strong limit. The isometry of B can be deduced from the isometry of A and the continuity of B. )

5.2. According to Exercise 5.1, the eigenvalue problem for A is equivalent to solving n

(Aik

-

0

*k

k=l

in h and x1 ,..., x, . The eigenvalues h are the roots of the characteristic polynomial of 11 Alk 11) i.e., they satisfy det 11 A,, - AS,,

11

= 0,

thus being the roots a polynomial of the nth order, which are at most n in number. Explain why the vectors f = xlel xne,, corresponding to all the solutions x1 ,..., x, of the matrix eigenvalue equation, span X .

+ +

5.3. Note that 11 A 11 is a norm in %(#) (see Exercise 1.6), and, therefore (Chapter I, Theorem 3.1), every sequence convergent in the norm is a Cauchy sequence. 5.5.

Note that

for any g E 2.

5.6. Write AB note that

I I W - 4)B(4)fll < ll(A while II AW(B - B(t))fII 5.7.

-

4 4 ) Bfll

+ I1 4w - B(t))fll

< c ll(B - B N ) f II.

See Chapter I, Exercise 1.4.

5.8. Use the continuity from above and below of the measures pt(B) = ( f I E ( B ) f ) , f E X , and the relation

1 (f I (W) m4J)f)I -

valid whenever B, C B or B, 3 B.

= II(E(B) - E(B,))f1I2,

Hints and Solutions to Exercises

640 5.9.

< (1 A (1 from

First derive that (1 A * (1

II A*fl12

=

(f I AA*f) G llfll I1 AA*fll

< II A II l l f l l

II A*fll,

and then reverse the roles of A and A* in the above derivation. 5.10. Recall that the operator bound 11 A 11 has all the properties of a norm. I n proving the last of the assumed relations use the result of Exercise 5.9. 5.11. Note that 11 em+l - em 11 = 1/2, so that Cauchy's criterion is not satisfied. On the other hand, lim(f I e,) = 0 for any f E H , since x:=ll(f 1 e,)I2 11f )I2 by Lemma 4.1 of Chapter I.

<

6.1. Note that (f I g) = (f 1 Ag) is a positive-definite bilinear form; apply Theorem 6.4 of Chapter 11. 6.2. The problem can be reduced to showing that if C 2 0 and C 0, where C = C*,gC= if, then C = 0. This follows from the fact that (f 1 Cf) 2 0 and (f I Cf) 0 imply (f I Cf) = 0, f E X , and from the generalized Schwarz-Cauchy inequality of Exercise 6.1.

<

6.3.

<

Note that for any f, g E if

(f I &) = (f I S =

~

FAng) = n-m lim

(f I Ang)

lim (Anf I g> = ( 4I g>* n+ m

6.5. It is sufficient to consider the case of strictly positive polynomials p(eiw) > 0, y E [0, 27~1,since: the more general case can be reduced to the above case by adding to the polynomial an E > 0, proving the existence of a polynomial qc(x) for which I qC(x)l2= p ( x ) E, and letting E --t 0. In case that p(eiw)= ck eikQ> 0,the polynomial P(z) = xnp(z) = ck-,zk has no zeros on the unit circle I z I = 1. Moreover, we have

&,

+

x:=-n

P(z) = [(z*)2n P( l/z*)]*

which shows that to each root a of P(z) in the interior of the unit circle corresponds a root 1/a* in the exterior of the unit circle. Thus, P(z) has the roots a1 ,..., a,, l/al*,..., l/an*and

n n

P(z) = c'

(z - cyk)(Z - 1/cyk*).

k=l

Therefore

n n

p(eiv) = c

k=l

(eiq

- ak)(e-iq - (Yk*),

Chapter IV

641

where necessarily c

> 0. Hence, one can take n

fl

q(eiq) = 6

(eiq

- ak).

k=l

6.6.

For each h E I, denote by k(X) the greatest integer for which ukc\)(h)

-

<

such an integer exists since u,(h) = l i q + mv,(h). Denote by I(h) the neighborhood of h consisting of all points x for which WA)(X)

- un(X)

< I/n;

I(h) is nondegenerate since uk(A)(F)and v ( x ) are continuous. The family of all I(h) constitutes a covering of I. By Borel’s covering theorem, any covering [in our case {I(X)}] of a finite interval I has a finite subset [denote it by I(&),.,.,I(h,) in the present case] which also covers I. The smallest of the number A(&), ..., K(h,) is the K, we are seeking. 6.7. Start withf(() and g(() positive, and prove the statement by going to the original definition of integration and approximating these functions by simple functions. Proceed then to more complex cases by retracing the definitions of integration in Chapter 11, $3. 6.8. Note that Up(U) = p(U)U for any polynomial p ( U ) in U, and then follow the construction of EA in Lemmas 6.1-6.3, using the fact that A s-lim Bk = (s-lim B,)A. 6.10,

Take g

=

Af in (6.25), consider Exercise 6.7, and note that

CHAPTER IV 1.1.

Note that for any f,g E &‘


= =

I,

1

d
I A,g).

R’

1.2. The commutativity of the spectral functions can be derived from the commutativity of the Cayley transforms Vk =

(A, -i ) ( A k

+ i)-’

642

Hints and Solutions to Exercises

by noting first that [pl(Vl), p2(V2)]= 0 for any polynomials r(6.10) of Chapter 1111, and then retracing the construction of the spectral function Elk) (Chapter 111, Lemmas 6.1, 6.3) to obtain [E::', E::)] = 0. 1.3. It is straightforward to verify that the statement is true for simple functions f ( x ) . Follow, step by step, the generalization of the concept of integral contained in Definitions 3.4 and 3.5 of Chapter 11, to generalize the result to arbitrary p,-integrable functions f ( x ) . 1.4.

Taking in Exercise 1.3 p(x) = I $(x)12 one derives

so that gQk coincides with the domain of the self-adjoint operator with the spectral measure W ) ( B ) .Furthermore, applying the decomposition (5.13) of Chapter I11 to the complex measure (& I Eck)(B)t,b2), one reduces it to a linear combination of measures of the form ($ I E(k)(B)$) to which the result of Exercise 1.3 can be applied to obtain

s,.

xk#I*(x)

d<#,

#a@) dnx =

I E'nk'$2>.

JR'

1.5. Verify that the statement is true for Borel sets B in Iwn of the x B,, and therefore also in the Boolean algebra form B = B, x In generated by such sets. According to Theorem 5.5 of Chapter 111, this implies that the two measures coincide on an,which is the Boolean o algebra generated by dn.

1.6.

Verify that U-'EA(B)U

=

E(B) is such that (f I Alg) =

J A d( f I E d ) for any f E % and g E gAl , and that (cf. p. 225)

2.1. Retrace Definitions 3.4 and 3.5 of integrals, as given in 93 of Chapter 11. 2.2. Note that SA1 x x SA- is a closed set in Rn, and consequently it is Borel measurable, so that the integration in (2.3) can be limited to

S"I x

-.. x

SAn.

643

Chapter IV

Use the result of Theorem 2.4 to show that (f 1 F , F g ) = for all f, g E 2 f .

2.5.

(f I F2F&)

2.6. Show first that A is linear. g Ais dense since the set of vectors f for which (EA,- EA1) f = f for some real A, < A, is contained' in QA (because then the above integral converges for all g E Z ) and is dense in X . Prove that A * 2 A and then show that gA* C g Aby duplicating the method used in proving Theorem 2.6.

2.7. Use the result of Exercise 1.3 to establish that g A= 9F(01,...,os) and that (4, I F(Q, ,...,Qn) #2) = (#, I A#,) for all #, E Z and all *Z

E g A

-

+

At) - +(x, t)]. Accord3.1. Write f (x, t) = s-lim~,,,(l/At)[#(x, t ing to the mean-value theorem of differential calculus for 0 el, 0, 1

<

<

=

I Re #&, t + 8, A t ) + i Im &(x, t + 8, A t ) -f ( x , jR*"

t)I2danx.

If we consider, when letting A t + 0, a sequence (At), , (At), ,...+ 0, then by definition 6((At),) ---t 0 for n -+ 00. Use Theorem 4.5 of Chapter I1 to derive

+

almost everywhere, where (At),',

(At), ,... . Since by continuity

Mx,t ) = n+a lim [ Re#t(x, t the result #, = f follows. 3.2.

(At)2',... is a subsequence of (At),

+ 4 ( 4 ) + i Im #&, + 8 , ( A t M ] , t

Apply Parseval's formula in Theorem 4.6 of Chapter I to derive W

(AB),, = (A*ei I Be,)

W

(A*ei I ef)(ej I Be,)

=

=

1 A,B,, .

i=l

j=1

3.4.

,

Employ the identity

1

At

(e(i/fi)Ho(t+dt)

u(t + A t , 0) - e(i/fi)Hot u(t

- e(i/fi)tlo(t+dt)

9

+ A t , 0)- U(t,0 ) wx At

m>

0))

644

Hints and Solutions to Exercises

and consult the methods employed in proving Theorem 3.2 to show that in the limit when A t + 0 the above vector-valued function of t converges strongly to

3.5. Write Rn(A,t) 2.6 and 2.7,

=

eiAt- CE==o ( ~ A t ) ~ ./ kAccording ! to Theorems

Since it is easily shown by using Taylor's formula that (for fixed t) I R,(A, t)l +0 uniformly in h E [-a, +a] when n + 00, it follows that

II R,(4 t>fII 0. +

3.6. First note that [(Hot)"$](r) vanishes outside B, for all t E R1. Hence, if the expansion were correct and the series were convergent in some sense, then we would have e-(i/')Hof $ vanish outsideB, at all times t. Since e-(il')Hof is unitary, this would imply that

for all t 4.1.

E

R1.

For a vector of the formf, @

@ f, we have

O A,)(fl O .-.0fn)lt = It Alfl111 II Anfn ltn < II A1 I1 llfl 111 *.. II An II llfn lln = II A, I1 It An It llfl O - * * Ofn II.

ll(4 O

-**

**-

.**

---

Extend the above inequality to Hl@ @ Hnby using an orthonormal basis of vectors eit) @ .--0):le (see Chapter 11, Theorem 6.10). 4.2. Use Theorem 6.10 in Chapter 11, and take advantage of the manner in which HP'" and SPA"are constructed from the vectors (4.5) and (4.6), respectively.

645

Chapter IV

4.3.

Note that

(ul..c..

fil

Ofin

0

.in)

=

c..

I

....kn)

T ( k l ).**) km)

( 1 .... (il.

( k l . . .kn)

n(kl ,**.,

@

k ) g k ,

Qgk,,)

(k1.

(fi1

Igk1)l

“‘

(finI gk,).)

= O,

in)

since the sum by which each ~ ( k ,..., , k,) is multiplied on the right-hand side has the same value for each (k,,..., A,). 4.5.

Resort to Definition 4.1.

4.9. Show that U , @ 0U, is isometric on 2,’0, 0,H,‘, due to the isometry of U , ,..., U,. T o prove that the range of 2,‘ is mapped onto U , 0 0 U, is Z2, note that 2,’0, 0, Sl” 0, 0,S,”, and that the algebraic tensor product is dense in the Hilbert tensor product. 5.1. Prove first the statement forL2(Rn, p) by following the procedure used in proving Theorem 5.6 of Chapter 11; the only difference in the present case is that, since the boundary of an interval is not necessarily of p-measure zero, the first part of the proof should be carried out not only for semiclosed finite intervals of the form (5.28) in Chapter 11, but also for any finite closed interval (take all combinations of closed and open end points). Then prove the statement for any L2(Rn, p) by using the definition of symmetric and antisymmetric tensor products. 5.3. Let U , be the unitary transformation of 2”onto L2(Rnu, pJ which is such that A:‘” = UvA:)U;l, where A’:) is the multiplication operator in Definition 5.2. Then U = U, 0 --.0 Urn(see Exercise 4.9) onto is a unitary transformation of $j1 0

sm

La(Rnl,Pl)0 ..*

having the property that @)(&) =

0L2(RL, Pm),

A:(”)= U&)U-l, where @

4:)

... 4:) ... 0$6:)) @ ... @ (x:)Ip;) 0 @ @

@

.-*

&).

According toTheorem 6.9 of Chapter 11, there is a unitary transformation ff ofL2(Rnl, pl) 0 oLZ(~12.1, pm) onto L2(Rn1+...+nm, p1 x * - . x pm) is mapped into such that +il) 0 0

):L+

4:) ...)&:

EL2(~nl+”‘+%,

x

... x

Hints and Solutions to Exercises

646

The operator ( 0 U ) A p )(oU)-l is the canonical form of L2(IWnl+-.+~, x *.. x pm). 5.4.

A:) in

Apply Theorem 2.7 to answer the second part of the exercise.

5.5. Use the spectral theorem, and in particular (6.24)of Chapter 111, to find the domains of AE(R) and E(R)A,and afterwards make use of Theorem 2.7 to derive that (g I A E ( R ) f ) = (g I E ( R ) A f ) for all f E 9”andg E 2. 6.3. Note that it is sufficient to prove the statement for to = 0. Using Theorems 2.2 and 2.4, one derives (cf. (3.11)) Il(eiAt

- l)fll2 =

w

I eiAt - 1 12 d(fl E A A ~ ) , j

E

2,

where I eiAf - 1 I converges pointwise.to zero when t --+ 0 and is bounded by 2. Hence, Lebesgue’s bounded convergence theorem (Lemma 3.1) can be applied. 6.4. Note that since Uf(Ut0)-l= V - 4 , it is sufficient to prove the statement for to = 0. Use the decomposition (6.1) of Chapter 111, to establish that

Il(vt- l)fllz < lim I edt”‘ - 1 1’

<

- EA,-,)fllz

k

if t is chosen sufficiently small so that I e“” - 1 I Hence, 11 U f - 1 11 E for all such t.

<

<

E

t2

Ilfll*

for all p E [0,2~].

6.5. Show that commutes with E, by proving that U(277) commutes with E n . Then check that all the requirements of Definition 5.5 in Chapter I11 are satisfied. 6.6. Approximatef(h) within [-n, n], within the limits of accuracy I/n,by a trigonometric polynomial of period 29. When n --P 00, pl(X), pz(X),... converges tof(h). Iff(h) = x,(h), where I is finite, note that x,(X) can be approximated pointwise arbitrarily well by continuous functions of compact support. 6.8. Choose a finite interval I. The function x , ( x ) p ( x ) is square integrable on IWn. Select a monotonically decreasing sequence of continuous functions h,(x), hz(x),... converging pointwise to x,(x). Then one obtains j’1 F(x)l 1 h,(h) - x,(x)l I p(h)l dx +0, i.e., j’F(x)a(x) p ( x ) dx = 0 for any continuous function F(x) of compact support. Since V;(W) is dense in L2(W), one concludes that x,(x) p ( x ) = 0 almost everywhere.

647

Chapter IV

Consequently, if Il ,I2 ,... are finite disjoint intervals and have p ( x ) = EnxI,(x) p ( x ) = 0 almost everywhere.

U I,

=

Rn, we

6.9. Show first that (edQo"$)(x)= ei"%,h(x) by proving that (2.3) is satisfied when n = 1, A , = Q,(EQ(B)$)(x) = xB(x)$(x). Then use the relation exp(iPu) = exp( -iU;lQU,u) = U;' exp( 4 Q u ) U , , to establish that (eipu$)(x) = $(x fiu).

+

7.1.

Observe that if (. I -),' denotes the inner product in X,f and

7.2,

Apply to

(. I -)," denotes the inner product in X,", then

P)

C (fi' I gi')if iA=l

(fj"

I gi")j"

the theorem on the absolute convergence of double series [Randolph, 1968, Sec. 3-1 1, Theorem 3, p. 1621, which states that Cz,=l a,, converges absolutely if and only if CT==, a,, = b, and C& b, converge absolutely, and in that case C&=, a,, = CE16, . 7+3+ Note that C& (fk that the range of U is &2.

I gk)k

=

C g l (ukfkI Ukg& , and

show

7.4. Start by proving the assertions for appropriately symmetrized functions of the form (7.6), and then, using the fact that the set of such functions is dense in the considered subspace, extend the result to the entire subspace. 7.5.

Show that the spectral representation space L2(R4,p) of

{Q(*), Q ( v ) ,Q ( Z ) , S"} is also a spectral representation space of {P"), P(v),FZ), S"} by showing that

( ~ ( 9 w ( P Pl 2 Ps > 4

P, Ps 4, with similar formulas for P(v)and P(z),where $ is the Fourier-Plancherel 9

9

= Pl&Pl

transform of $ at each fixed s = -u,...,

9

9

9

+u.

7.6, Obviously A*** 2 A*. Iff e gA*** and f* = A***f, we must and , therefore also for all have (f* I g ) = (f I A**g) for all g E g A * * g E g A .Hence, f E g A * .

648

7.7.

Hints and Solutions to Exercises

Compute first

Then observe that the Fourier transform of P(r, ,..., rn)fo(rl,..., rn)is P(-(i/A)V, ,..., -(i/A)Vn)fO(pl ,..., pn)in case thatf, is continuous and of faster than polynomial decrease at infinity (these conditions are obviously satisfied byf, = exp[( -l/2fi)(r12 rn2)]).

+ +

7.8. Introduce the new variables r = r1 - r2, p = r1 apply Theorem 3.13 of Chapter I1 to write

Then treat the integration in r separately for r 7.9.

+ r2, and

< R and r > R.

Establish that

converges uniformly, and that m

m

(1 + A )

2 (-A)* (2(-A)k)(l + A ) =

k-0

=

I.

k-0

8.1. An infinite orthonormal sequence {el, e2 ,...} of vectors from M is bounded but it has no convergent subsequence since (1 en, - en 11 = 42 for m # n. 8.2. If &,E s: then EA(I) # 0 for every interval containing h,. Since EA({A,}) = 0, this implies that every neighborhood contains a point of the spectrum. I n fact, if some closed neighborhood I. of A, does not contain any points of the spectrum besides A,, then for every h €1, we could find a subinterval I, of I,, such that E‘(IA) = 0. Use Borel’s covering theorem to establish the existence of a finite number IAl ,..., IA- of such intervals, with the property that I, = IAl u * * . v IA, , and, consequently, EA(I,) \< E”(IAx) ... E”(IAn)= 0.

+ +

649

Chapter V

8.3. If A,, A, ,... are the eigenvalues of A, it follows from the spectral theorem (see Chapter 111, Theorem 6.4) that R

A

=

hJP((hk}).

u-lim n-m

k=l

8.5. Assume that the Bore1 set B, of negative numbers with = sup{h : A E B,} < 0 is a subset of the spectrum of A. Then EA(B,) # 0, i.e., EA(B,)f, = fo for somef, # 0. Show that A,

8.6. Build an orthonormal basis in 2 by selecting orthonormal bases in E([O, co))Xand E((-co, 0 ) ) X and taking their union. Note that the unconditional convergence of the series in (8.5) implies its absolute convergence, from which the existence of Tr(A(+) A(-)) can be deduced.

+

8.7. Apply the results of Exercise 5.10 in Chapter I11 to {A,B,} and then to {(A,$,) C,}.

CHAPTER V 1.1. In deriving lr(t)lsets Ir(t)l= I rex vext 1 large enough,

+

1 r(t)l

- t I vex

rexVex

+-

I vex I

I =p ( t )

t

I vex I -+ rexvex/lvex 1, note that if one qex(t) ---t 0. In fact, if I t I is

+ vex(t), then

+ (I rex I2 + 2trexvex + t2 I vex 12)”*

- t I vex I

for t + Too.

1.2. Recall that for the given potential the general solution for a r(t) which is not a bound state is a hyperbola or parabola. If r(to) = ro is the minimum of r ( t) = I r(t)l, then &/dt 5 0 for t 5 t o . Hence, r(t) has an inverse in (- co,to] and in [to , +GO).Since 4m,v2

+ l / r = &n,(f2 + r2@)+ l/r = const

and

r2d = const,

650

Hints and Solutions to Exercises

we have

where b and c are constants. Consequently, setting 1

and

a =

w’

we obtain

+

Using the above result and recalling that r(t) ---t co when t + 7 co, we get r(t)/t --+ ( 2 b / w ~ ~ )so l /that ~ , I vex l2 = 2b/m0 according to Exercise 1.1. Show that r(t) - t I vex 1 = r(t) - t ( 2 b / w ~ , ) l /diverges, ~ instead of converging to rexvex/Ivex 1, which should happen (according to Exercise 1.1) if asymptotic states exist in the sense of (1.5) and (1.6). 1.3. Use Fubini’s theorem to derive from (K*g that K* is an integral operator of the above form.

If)

=

(g I Kf)

1.4. It follows from Theorem 1.1 that Y(0) = Q+Yin(0) and Y(0) = Q-YJout(0). Show that Q-*Y(O) = S Z - * S Z - Y o U t ( 0 ) = Y O ” t ( 0 ) [see (2.1011. This implies that Yout(0)= Q-*Q+Yin(0)= SYin(0).

1.5. The computation will be the same as in the case treated in the text, except that the integration in P cannot be eliminated in (1.38), but rather must be retained to the end of the calculation.

1.7.

Note that

1.8.

Note that E(”)(B)commutes with Hi2),and consequently

I E(,)(B)~ e x ( t ) >=
(~ex(t)

Y~x(o)>

for any B E 9Y3;then insert A = E(”)(B)and Yo(t)= exp(-iHh2)t)YeX(0) in Exercise 1.7.

65 1

Chapter V

1.9.

Consider the case of &(k)

=

= exp

(-E

h,(k)

h,(k)$(k), where (cf. p. 218) k2 2m

-t)

Taking inverse Fourier transforms of both sides of the first relation appearing in this exercise leads to

s,.

(exp(-iHot) +,)(r) = ( 2 ~ ) - ~ / ’ exp{i[kr - (k2/2m)t]}&(k) dk =(

1

2 ~ ) - ~ dk exp{i[kr - (k2/2m)t]}h,(k) ws

Since the above integrand is integrable on can be applied: (exp(-iHot) +,)(r)

=(

exp(-ikr’) +(r‘) dr‘. w8

lw8

[w6

in (k,r), Fubini’s theorem

dkexp{i[k(r - r’) - (k2/2m)t]}h,(k).

2 ~ ) - ~ dr’+(r’) w8

After integrating in the (6, +) spherical coordinates of k, thus arriving at sin k 1 r - r‘ I

+ i)(k2/2m)t] Ir-r‘I 2m ~ + i = - 1 Jrnexp (- -k2t+ 2m +i

4~ lOmexp[-(e

+l

E

a=-1

a

k dk

I r - r’I k) dk,

0

and then letting E -+ +0, one obtains the desired relation. In fact, the limit on the right-hand side of the preceding relation can be taken under the integral sign by applying Lebesgue’s bounded convergence theorem (Chapter IV, Lemma 3.1) to infer that for t,4 €L1(IW3)nL2(R3)

IR, 1 1 exp{i[k(r - r’)

!js I (cl(r’)l

1

- (k2/2m)t]) 1 1 - h,(k)( dk dr’ = 0.

RS

On the left-hand side, the continuity of the operator e-iHof implies that s-lim e-%t €++O

+,

=

4.

1.10. Since it is quite obvious that U;” and UI2)map L2(R3) onto itself, it is only necessary to check that these operators are isometries; for example, setting P = mr/t, one gets

652

Hints and Solutions to Exercises

+

Apply Uj1)Ui2)to integrable functions +(r), €L2(R3), and use the result of Exercise 1.9 to write that almost everywhere in R3 (u,‘“u~)+>(r) =

exp[im(r2/2t)1(~,(2)+)-(mr/t)

= (m/it)s/2exp[ im(r2/2t)]1 / ( 2 4 3 / 2

J

exp[-i(mr/t) r‘] R3

x exp[ im(r’2/2f)] +(r’)dr‘ = (m/2~rit)3/~

exp[i(m/2t)(r - r‘)7+(i) dr’ ws = (exp(-iHot)+)(r).

Since the family of all such functions is dense in L2(R3),and since both e-iHol and U11’U12)are bounded operators, the above relation can be extended by operator continuity to all of L2(R3). 1.I 1.

Note that due to the unitarity of UI1)

-

-

J,. I exp[ im(r2/2t)]- 1 l2 I +(r)izdr.

When t + m, the above integral converges to zero by Lebesgue’s bounded convergence theorem (Chapter IV, Lemma 3.1). 1.12. By taking in Exercise 1.7 Yo(t)= c i H O r +, Y(t)= Ull)+, and A equal to the projector on C ( + ) i.e., , (A+)(r) = xc,+,(r)+(r), one gets

The substitutions k

=

Fmr/t yield

2.1. Take a bounded interval I C [-a, +a], and approximate xr(h) in the mean on [-a, + a ] by a trigonometric polynomial p,(X) = C,“=-, uk exp[(nik/a)h]. If n

Pn(4

=

c

k=-n

Qk

exp[iM441,

653

Chapter V

one easily derives pn(A1)C= Cp,(A,). Since by Theorems 2.5-2.7 in Chapter IV $,(Aj)+ E A y ( I u IN),I , = {A 20j : A E I , j = & l , f 2 , ...}; letting a+ GO so that EAf(I,)+0, one obtains EA1(I)C = CEAz(I).This result can be easily extended from finite intervals I to arbitrary Bore1 sets B E 9i'l (see Chapter 111, Theorem 5.6).

+

2.2. Use the method employed in proving Theorem 4.1 of Chapter 111. 2,3.

Prove that (1 - E)f = 0 by establishing that

(f If) = (Ef I E f ) + ((1 - E l f I (1 - E)f). 2.4.

Use (2.16), (2.20), and Theorem 2.7 of Chapter IV.

2.5. Note that the bounded operators A, = J[-,,+,I A dEA leave M, = M n E A ( [ - N , +N])% invariant. Hence, any polynomial p(A,) leaves M, invariant and therefore E A ~ ( B=) EA(B), B C [ - N , + N ] , leaves M, invariant. Since M, -+ M when N + co we get EA(B)f€M for f E M, and EA(B)g 1M for g 1M, i.e., EMEA(B)= EM. Consequently, EA(B)EM= EM* = EMEA(B).

2.6.

Note that Ef commutes with ei*' and therefore

3.1. The argument runs along the same lines as the proof of the corresponding statements about Q-, and Q+, . 3.2.

Note that

W+€ - Q+.(Wl12 = J < IKQ, E

3.3.

m

e-Y(Q,

- Q,(s))fII

- Q€(W I QWf)dt E Jm

dt.

Use Definitions 3.1-3.5 of Chapter 11.

3.4. Note that the linearity property is shared by every RiemannStieltjes sum in the sequences whose limits define these integrals. 3.5. Establish these properties for the corresponding RiemannStieltjes sums in (3.18) and (3.20), and take the limit. 3.6, Prove first that if F(A) and F(A, A,) are nonnegative, then the corresponding Riemann-Stieltjes sums are nonnegative, and therefore the corresponding integrals are also nonnegative. Then set F = Fl - F, and apply the linearity property (see Exercise 3.4).

654

Hints and Solutions to Exercises

3.7. Prove the statement first for proper Riemann-Stieltjes integrals by establishing it for Riemann-Stieltjes sums and then taking the limit; afterwards, generalize the result to improper Riemann-Stieltjes integrals.

<

3.8. If m F(A) one can derive

< M on [a, b ] , then using the result of Exercise 3.6,

Consider the nontrivial case when u(u) < u(b). If m and M are the minimum and the maximum of F(A) in [a, b], respectively, then F(A) assumes all the values between m and M on [a, b]; hence, there is a point A' at which

I

b

F(h') =

F(h) dAU(h)/S) dAu(h).

0

0

For the existence part, prove that the limit (3.17) exists and is independent of the chosen sequence of subdivisions {A(k)} by showing that if { A ( k J ) } is a finer subdivision than {A(k)} [with A(k*o) = A'k)], then

can be made arbitrarily small for all sufficiently fine subdivisions { A ( k ) ) [here , ! ( k J ) denotes the point preceding A(kJ) in the subdivision { A ( k * l ) } ] . To obtain this result, exploit the uniform continuity of F(A) in [a, b], according to which for given E > 0 we have

I F(h"k')

- F(h""."')l

<e

for all sufficiently fine subdivisions {A(k)}. Hence, the above difference is

3.10. Note that (g I ei(a-g)tf) is continuous and therefore Bore1 measurable in t , and that

11 ei(a-t)tfll dt 0


Im< +co etlmt dt

0

655

Chapter V

for I m 5 < 0. Hence, e i ( A - c ) l is Bochner integrable. Using the spectral theorem, we obtain for any g E SP

where the order of integration could be reversed by Fubini’s theorem. 4.1. Use Fubini’s theorem to carry out the integration in (4.41) successively, first with respect to p ( k ) and then with respect to p(w). I n view of the definition of p(0)in (4.40),we obtain the desired result by applying Theorem 3.1 1 of Chapter I1 with R, = {(I, m)}.

4.2. Let B, and B- be the Borel sets in Wmon which G(l)(x,x‘; 5) G(2)(x, x‘; 5 ) is positive or negative, respectively. For any Borel sets B, , B, E grnof finite measure we get, using Fubini’s theorem,

i = 1,2. Hence, if B,

x B, C B, , we infer (see Chapter 11, Lemma 4.1) from (Gt)(x, x’;

5 ) - GF)(x, x’; 5)) d(p x

p) = 0

J’BlxBt

that B, x B, is of p x p measure zero. Since by the definition in Theorem 2.4, Chapter 11,of products of measure, we get from Theorem 1.7 and (2.26) of Chapter I1 m

(p

x p)(B*) = sup

c

m

(p

k=l

we thus deduce that ( p x p)(Bf) = 0. 4.3.

Note that

BI”’ x B y CB*] ,

x p)(BI“) x B?)): k=l

656

Hints and Solutions to Exercises

where the function on the left-hand side of the above inequality converges pointwise to zero when y + fO, while the function on the righthand side is integrable on Rs in r, r' and k. The desired result is then obtained by applying Lemma 3.1 of Chapter IV. 4.4. then

If xj(y) denotes the characteristic function of L I - ~ ( ( A ~ ,Aj]) -~ g'*(Y)(@W I wv,')f>

The integrand on the right-hand side of this equality converges pointwise to the integrand on the right-hand side of (4.84). Lebesgue's dominated convergence theorem can be applied since on supp &v), 11 Hlz@,11 C; , and consequently,

<

I ( @ v I ~ l F ( h w< l l l f f l Z @ V l l IIF(4')fll < WIlfll. 4.5.

Write (4.103) in terms of matrix elements with f, g E gH0 C 9", (g

I Y(5)f> = + (Vg I (5 - 4J-l Y'(Of>

and express both sides of this equation in the momentum representation, using (4.82):

Since V(r)Qk(r) EL(~)([W~) if V(r) EL(~)([W~), we have

657

Chapter V

Inserting these results in the preceding equation, and noting that (v@k

[I -k

vG*({)]v@k"

I @k') I @kf)

1

= (@k = (@k*

I

vI

@k,)

=

(k I

+ vG(6)v 1

v I k'),

@k')

=

( k I r(6)

I k'),

we arrive at the result that the left and right hand sides of (4.105) multiplied by g"*(k)J(k) and integrated over R6 have to be equal at least for all f,g" E Wb0(R3), and therefore (by the same argument as in Exercise 4.2), that (4.105) has to be satisfied almost everywhere in R6. 4.6. Write (4.1 13) in terms of matrix elements for f, g €L2(R3), use (4.96), and apply the argument of Exercise 4.2. 5.1. The functionf(r, r') = I r - r' I is continuous and therefore Borel measurable in Rs (Chapter 11, Theorem 3.1). Since DGo= f - l ( { O } ) , it follows that DGois a Borel set. T o establish that its Lebesgue measure p?(DG0) is zero, note that

and write the above integral as an iterated integral by applying Fubini's theorem. 5.2. Since K(a, a') as well asf(a)g(a') are square integrable in R2" with respect to p x p, the function K(a, a')f(a)g(a') is integrable in R2". Hence, Fubini's theorem can be applied in the following derivation:

which shows that K* has the kernel K*(a', a). For arbitrary f E L2(R", p ) , the function

Hints and Solutions to Exercises

658

is p-square integrable in an E Rn, and consequently

exists. Thus, we can apply Tonelli's and Fubini's theorems in succession to infer that

=

IP' j dp(a')f(a')

dp(01")K*(a",a) K(a",a').

tan

5.3. According to Theorem 8.6 of Chapter IV, K is a HilbertSchmidt operator if and only if K*K is of trace class. In order to establish that K*K is of trace class, it is sufficient to establish that m

m

Tr(K*K)=

c

k=l

<#k

I K*K#k>

=

c 11

K#k

(I2 < +O0

k=l

is some orthonormal basis {& , a,h2 ,...}. In view of the fact that the function K(a, a') is an element of L(2)(R2n,p x p) and that {I,+ @ $tk*; i, k = 1,2,...} is an orthonormal basis in L2(R2n, p x p) (see Chapter 11, Theorems 6.9 and 6.N) we obtain

In deriving the above equation we have used in the first step Parseval's p x p), in the second step the unconditional converrelation in L2(1w2n, gence of a double series with positive terms, as well as Fubini's theorem, and in the third step Parseval's relation in L2(Rn,p).

5.4, Apply the Riesz-Fisher theorem (Chapter 11, Theorem 4.4) to conclude that K(a, a') is square integrable on R2m.

659

Chapter V

5.5. The operator BA is completely continuous by Chapter IV, Theorem 8.1. Suppose that A = A* and that { e l , e, ,...} is an orthonormal basis for which Ae, = &e,. Then

which implies that BA is a Hilbert-Schmidt operator. i A , , where In general, if A # A*, then we can write A = A , A, = A,* and A , = A,*. Then BA is a Hilbert-Schmidt operator since BA, and BA, are Hilbert-Schmidt operators by the above, and a family of Hilbert-Schmidt operators is a linear (normed) space (see Theorem 8.10 in Chapter IV).

+

5.6.

Note that for any f E H

can be made arbitrarily small for sufficiently large n and all m Hence, Af

=

Akf

s-lim n++m

> n.

k=l

exists for all f E &', and obviously defines a linear operator on 2. Moreover, since by letting below n -+ +GO we see that

we conclude that A is bounded, and the desired estimate is obtained for By the same procedure we get

11 A 11.

when n -+ +GO. 5.7.

Note that

Hints and Solutions to Exercises

660 5.8.

Since mA is a lower bound of

SA,

Conversely, if (f I Af) 2 y 11 f1I2, then y must be a lower bound of I n fact, if that were not so, then we would have A,, E S A for some A. < y. This means that we could find for e0 = #(y - A,,) a nonzero eo])fo = fo . Consequently, vector foE g Asuch that EA([Ao- eo , A. we would have SA.

+

which contradicts the initial assumption that (fo 1 A , , ) 2 y 1) fo112. 6.1. Introduce spherical coordinates (r', 0, +), orienting the z axis along the vector r. Then

I,.,,

dr'

I r' 12+'01 r

= 2..j

6.2.

fm

Rl

[rz

- r'

I

+ 2rr' + (r')Z]l'z - [rz - 2w' + ( Y ' ) ~ ] ~ dr' /~

Let R, be such that

r(r')l+€o

1 V(r)) < Co for all r 2 R , . Divide

Y~+€O

Chapter V

661

the domain of integration into two parts, so that

and note that

Constants C, and C, can be chosen which are such that (see Exercise

6.1)

for all r E R3. The continuity can be proven as in Exercise 6.9.

6.3. Since V(r) is square integrable for Y, the Schware-Cauchy inequality) for Y > R,

= const

1

I r - rav Is

< R, ,we obtain (by using

=0

($);

<

here r,, , yaV R,, is a vector (whose existence follows from the mean-value theorem of the integral calculus) for which I r - r, assumes its average when integrated in r1 over the sphere {r,: Y, R,}. Since I V(rl)l C,Y~’-~O for Y, 2 R, ,we easily carry out the following computation by orienting the z1axis in the direction of r:

<

<

It is immediately seen that the above integral is O ( l / ~ l + ~ owhen ) 0 < E , < 1. For E , >, 1, the integral on the right-hand side of the above equalities does not converge when the lower limit R,/r is replaced

Hints and Solutions to Exercises

662 by zero. However, then

T o prove the local integrability in r of the function under conis square integrable sideration, note that the function I V(rl)I I r - rl R < +a,rl E R3}; this follows (by Tonelli’s on the set {(r, TI): r theorem, Chapter 11, Theorem 3.14) from the existence of the iterated integral

<

Hence, the desired result follows by Fubini’s theorem. 6.4.

By using the Schwarz-Cauchy inequality we obtain

Thus, the present case is reduced to the case considered in Exercise 6.3. 6.5. Combine the results of Exercises 6.2 and 6.3; note also that V(r) is square integrable.

6.6. Since V(r) is locally square integrable it is measurable on R3 (see Chapter 11, Theorem 4.1). Since I r - r’ V(r) satisfies I r - r’ I V(r)l C, I r - r‘ )-2r--2-Eofor r 2 R, , it is integrable on the set {r: r R,} due to the integrability of I r - r’ /--2r-z-~~ on that set (see Chapter 11, Theorem 3.9). By Fubini’s theorem the function

>

<

is locally integrable in r‘, and majorized by const I r’ J-l--ro for I r’ I 2 R (see Exercise 6.3). Hence, it is integrable on the set {r’: r’ R,}. The integrability of I V(r)I I r - r’ I V(r‘)l on R6 now follows by Tonelli’s theorem from the existence of the iterated integral.

>

663

Chapter V

6.7.

According to (6.29)

I vk(r)I d FV(G k2/2m)Use the result of Exercise 6.5 and the continuity of d(k2/2m) to prove that Fv(r;k2/2m) const for all r R, and all k E Do .

<

6.8.

>

By the mean-value theorem of differential calculus AI -

1

Au - d~ 18. =

hu(r, u RS

For given

E

[h(r, u

+ 4 - h(r, u)lf(r) dr

+ 0 Au)f(r) dr,

> 0, choose R, > R,

0

< 0 < 1.

so that

Since hJr, u ) is uniformly continuous over the region ((r,u R, , I du I const}, we can choose du so small that

Y

<

<

I hu(r, u

+ du):

+ 0Au) - hu(r, u)l If(r)l dr < 42.

Hence, we have

for sufficiently small I d u 6.9,

For given

E

1.

> 0, choose a so small that

for all r1 from some closed neighborhood .N of r. Then select R, > 0 so large that for all those r1 ,

Since 1 r1 - r'

I-, exp(ik 1 r1 - r' I) is uniformly continuous for rl E N

Hints and Solutions to Exercises

664 and r' E So = {r': I r' - r I 2 a,

I,/

I'

< Ro}, we can obtain

exp(ik1 r - r' I) - exp(ik1 r1 - r' 1 r - rll I r1 - r' I

I)

I If(r')l

dr'

< f3

for all sufficiently small values of I r1 - r I. Combining these three results, we easily obtain I g(rl) - g(r)l < E for such values of I r1 - r I. 6.10. Since 9'; is a closed set of measure zero, it is possible to construct a sequence hl(r), h2(r),... of infinitly many times differentiable functions with supports disjoint from 9'; , which is such that J I h,(r) 1 12 dr --+ 0. An example of such a sequence would consist of the functions h,(r) = 0 for r E 9';, h,(r) = 1 for r $ Pvat a distance d(r) > l/n from 9';, and h,(r) = exp(-n2 d2(r)[n2d2(r) - 11-l) for r $9'; and d(r) l/n. Since fn(r) = f(r) h,(r) E %: for any f(r) E %g(R3), and %;(R3) is dense in L2(R3), so is %; .

<

7.1.

Setting F(u) =

IU

f ( t ) dt,

u

2 0,

0

and integrating by parts, we obtain

=E

/

+a

e-etF(t)dt. 0

This establishes the existence of I, for E > 0. We note that the above limit for 01 --+ +a, exists since limt++mF(t)= I , exists [and consequently lim,,+,e-eaF(or) = 01, and since from the continuity of F(t) it follows that I F(t)I Cofor 0 t < a,,so that e-cfF(t) dt exists. Using a method already employed in proving Theorem 3.1, we write

<

For any given 3

> 0,

select

<

01

>0

+

so that

01

< (3/2)(C0+ I I. I),

Chapter V

665

and consequently

For such a fixed a > 0, choose E for all u 2 a. Hence

> 0 so small that I F(u/r) - I, I < ~ / 2

and consequently

< r + j + m2< L2+ L T ) .

IIc-I0I

O

a

We note that the proof would be much more straightforward if If(t)l were integrable on [0, +a).However, in the present case we deal with an improper Riemann integral and not with a Lebesgue integral, and the convergence of the improper Riemann integral of f ( t ) does not imply the convergence of the improper Riemann integral of

If (t)l. 7.2.

Note that since $(k) is integrable, (243’2l(exp(-iHot)’,lr)(r)I exp[-i(tk2/2m - kr)] $(k) dk

1 < I I $(k)I

dk.

R3

7.3.

Using (6.58) and (6.38) we get (see also Excercise 6.5)

=

1 6 2 const

IJ’,

I V(r‘)l (1 3

+ Fy(r‘; k2/2m)dr‘

Hence, the square integrability of T(l)(K;0, and Fubini’s theorems.

0’)

follows by Tonelli’s

Hints and Solutions to Exercises

666

7.5. The Legendre polynomials Pl(u)are the solutions of a SturmLiouville problem (see Chapter 11, Theorem 7.5). Hence, the expansion '

-

-

$ (21 + 1) Pl(u)1"

2 1=0

Pl(u)eiuudu

-1

is a Sturm-Liouville expansion, and as such it converges uniformly (see Titchmarsh [1962, Theorem 1.91). T o calculate the coefficients in this series, expand eiuu in a power series and show first that l"um

Pl(u)du

-1

+

+

is zero unless 1 m is even. For the case of even 1 m, the above integral can be computed by using standard relations obeyed by Legendre polynomials (see Butkov [1968, Chapter 9, in particular Problem 171). 7.6.

Note that as in the case of (7.2)

(e-iHat&,)(r) = (27r-3/2

exp[-(P2

+ it)(k2/2m)+ ik(r - r,,)]d k

JRS

and carry out the integration in k E R3.

667

Chapter V

7.7. Note that the required result has been established for &(r) = dl,(r) in Exercise 7.6. Now, according to Lemma 7.1, the set {& : roE R3} is dense in L2(R3).Since 11 EQ(B)c i H o 6 (1 < 1, we can apply Lemma 1.1 to reach the desired conclusion for arbitrary a,bo €L2(R3). 7.8.

Since 11 # t - #Fx

11

-+

0 when t

+

Fco and

the desired result follows from Exercise 7.7. 8.1. (a) Four channels: (n,p, e), (n-p, e), (n-e, p), and (p-e, n), plus the bound states (n-p-e). (b) (n-e) has not been observed so that Mipin-e) = {0} as far as present experimental knowledge goes. (c) (p-e) = hydrogen atom, (p-n) = deuteron, (p-n-e) = deuterium.

8 2 , ( 4 Eight channels: ( p , p , n, n), ( p ,p , n-4, (n,n,p-p), (n,p , .-PI, (n-p, n-p), (p-p, n-n), (n,n-p-p) and (p, p-n-n) plus the bound states

(n-n-p-p). (b) ( p , p , n, n), all four particles are free; ( n , p , n-p)

= neutron, protron, and deuteron; (n-p, n-p) = two deuterons; ( p, p-n-n) = protron and triton ( p-n-n); (n,p-p-n) = neutron and the ionized helium isotope (3He)2+of 3He; (n - n- p - p) = alpha particle (4He)2+.

8.3. Sincef(t) is continuous and has a limit at +a,it is bounded; M for all 0 t < +a.For given E < 0 choose N(E) i.e., If(t)l so that I f ( t ) - a I < ~ / for 2 all t 2 N(E).

<

<

Then

we have Hence, for all T 2 T ( E )= ~M[N(E)/E],

I (+) p 8.4,

t)

dt - a

I<

Use the relations (8.23) and (8.24).

e.

668

8.5. (8.43). 8.6.

Hints and Solutions to Exercises

Follow the line of argument employed in proving (8.42) and

Note that

since Qy)*Q!/? = EMY),while Q ~ ) * S Z ~ = ) 0 for # y due to the fact that the initial domain RY) of Qy)* is orthogonal to the final domain RY' of A similar argument leads to the conclusion that

a?).

is a bounded bilinear form, and apply the result of Exercise 2.5 in Chapter 111.

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Index

A Accumulation point, 27 Additive class, see Algebra, Boolean Algebra associative, 174 Boolean, 58,60 commutative, 175 generated by observables, 276 maxhally Abelian, 328 Almost everywhere, 90 Asymptotic completeness, 552 Asymptotic condition physical, 455 strong, 418 Asymptotic Hilbert space, 612 Asymptotic states in classical mechanics, 417 incoming, 418,455 in interaction picture, 474 modified, 514 Mgller, 418 in multichannel scattering, 600 outgoing, 418,455 physical, 457 strong, 418 Axioms, 2 for soin and statistics, 309

B Banach space, 30 Bauer’s formula, 592 Bessel’s inequality, 38 Bilinear form, 137 Bore1 set, 62 Born approximations, 556 Born series, 557,621 Born’s correspondence rule, 262,266 Bose- Einstein statistics, 307 Bosom, 353 Bound state, 49 energy eigenvalues of, 49 Bounded variation, 465

C

Canonical commutation relations, 329 irreducible representation, 341 reducible representation, 341 representation of, 331 Canonical form, see Operators; Spectral representation space of operators Channel arrangement, 5% concept of, 5%, 597

67.9

680

Index

Channel (Cont.) energy, 597 incoming, 5% outgoing, 5% Characteristic function, 81 Characteristic subspace, 227 Characteristic value, see Eigenvalue Characteristic vector, see Eigenvector Closed graph theorem, 210 Closure, 27 Cluster of particles, 5% Cluster point, see Accumulation point Collision, rearrangement, 598 Completion, 27 Confidencefunction, 410 Configuration representation, 352 Confluent hypergeometric equation, 165 Confluent hypergeometric function, 165 Copenhagen school, see Measurement Correspondence rule, 3,259 Coulomb interaction, 164 Cross section differential, 426,436,510 total, 426,513 Cyclic vector, 365

multiplicity of, 534 nondegenerate, 49 Eigenvalue equation, 47 Eigenvector, 226 Energy shell, 510,620 Ensemble, state of, 393 Equal a priori probabilities, principle of, 394 Equation of motion, 3 Equivalence class, 28 physical, 293 relation, 27 Euclidean spaces, 18 complete orthonormal system in, see Euclidean spaces, orthonormal basis in isomorphism of, 23 orthogonal basis in, 36 tensor products of, 137 unitary equivalent, see Euclidean spaces, isomorphism of Evanescence, 301 Expectation value, 263,277, see also Mean value theorem Extension principle, 188 F

D

Decompositionof identity, see Spectral function Deficiency index, 223 Deficiency subspace, 223 Dense embedding, 3 1 Dense set, 27 Density matrix, 402 Density operator, 391 Diffraction effect, 7 Dirac delta function, 509 Direct sum of Euclidean spaces, 133 Distance function, see Metric Dynamical law, 2

E Eigenfunction, 491,498 Eigenfunction expansion, 499 Eigennumber, 491 Eigenvalue, 48,226 degenerate, 49

Faddeev equations, 624,625 Fatou's lemma, 287 Fermi- Dirac statistics, 307 Fermions, 353 Field, 58, see also Alegebra, Boolean Fourier coefficient, 38 Fragments, 5% internal energy of, 597 Fredholm determinant, modified, 527 first minor of, 527 Fredholm theory, 526 Free states, 418 Fubini's theorem, 96 Function appropriate symmetrization of, 312 Riemann integrable, 90 spectral, 235 square integrable, 101 total variation of, 467 Functional, see also Vector spaces bounded, 182 continuous, 182 linear, 182

Index

68 1 G

Gelfand triples, see Hilbert spaces, rigged Green’s function, 501 advanced, 503 free advanced, 524 free retarded, 524 full, 502, 528 series expansion of, 529 partial wave, 525 retarded, 503 symmetry properties of, 53 1 Green’s operator, 501 Ground state. 168

H Hack‘s theorem, 610 Hahn’s theorem, 237 Hamiltonian, 286 of arrangement channel, 599 eigenfunction expansion for, 500 internal free, 442 time-dependent, 292,372 Harmonic oscillator, 55 Heisenberg’s equation, 3,294 Heisenberg’s picture, 293 Heisenberg’s uncertainty principle, 8 Hellinger and Toeplitz theorem, 195 Hilbert spaces, 30,32,33,41,109,220 equipped, 498 extended, 186 Hilbert tensor products of, 144,145 isometric transformation of, 212 rigged, 186

I

Idempotent, 200 Identification operator, 613 Incident flux, 426 Indicial equation, 165 Inertial frame, 4 Inner product, 18 Integral Bochner, 480 Lebesgue, 90,114

Riemann- Stieltjes, 463 improper, 464 cross-iterated, 464 spectral, 483 time-ordered, 372 Interacting state, 420 Interaction picture, 298 Intertwining property, see Wave operators Isometric operators and transformations, 212

Isomorphism of Euclidean spaces, 23

K Kato’s theorem, 367 Kernel Hlbert-Schmidt, 526 L*, see Kernel Hilbert-Schmidt trace class, 526 Kummer’s series, 165

L Laboratory frame of reference, 119 laguerre functions, 170 Laguerre polynomials, 170 generating function of, 171 Lebesgue bounded (dominated) convergence theorem, 287 Lebesgue intergral, see Integral Lebesgue monotone convergence theorem, 92

Legendre functions, associated, 152 Legendre polynomials, associated, 167 Limit strong, see Operators, sequences uniform, see Operators, sequences weak, see Operators, sequences Limit point, 537, see also Accumulation point Weyl’s criterion for, 537 Linear manifold, see Vector spaces Linear operator, see Operators Linear space, see Vector spaces Linear subspace, see Vector subspace Liouville equation, 397 Liouville space, 405

682

Index

Lippmann- Schwinger equations for eigenfunctions. 503,618 in Hilbert space, 470 perturbation solution of, 456 solution-type, 472 forY(l),511 two-Hilbert space, 615

N Neighborhood, 27 Norm, 20 completeness in, 30 convergence in, 30 Hilbert- Schmidt, 384 Rollnik, 545 trace norm, 386

M

Mapping Bore1 measurable, 1 15 isometric, 27 isomorphism, 175 Matrix mechanics, 297 Mean value theorem, 490 Measurable space, 67 simple, 81 Measure, 58,67 complex, 232 continuous, 240 extension of, 70 finite, 67,79,232 inner, 79 Lebesgue, 67,78,79, 114 Lebesgue- Stieltjes, 79 lower variation of, 239 outer, 71 positive operator-valued, 406 probability, 68 signed, 236,237 spectral, 231 upper variation, 239 Measure spaces, 76 Cartesian product of, 77 Measurement determinative, 6,262,424 Born’s correspondence rule for, 262 preparatory, 6,264,266 theory of, 4,262,264 Metric, 25 Metric space, 25 complete, 26 completion, 27 densely embedded, 27 Mplller wave operators, 421,603 Momentum representation, 354 Monotone class of sets, 63 Monotonic sequence of operators, 242

0

Observables, 2 compatible, 260,262 complete set of, 316,498 expectation value, 277 fundamental, 270 Operators adjoint, 188 antilinear, 173, see also Transformations, antilinear asymptotically compensating, 515 bound of, 178 bounded, 242 from above, 535 from below, 535 relative to, 360 canonical form, 379 canonically conjugate, 329 closed, 191 compact, 375 complete set of, 315 completely continuous, 375 core, 366 density, see Statistical operator essentially self-adjoint, 358 extension, 186 finite-rank, 377 function of, 272 graph of, 191 Hermitian, 127 Hilbert- Schmidt, 383 hypermaximal, see Self-adjoint operators linear, 173, 186, 192, see also Transformations, linear closure of, 355 matrix representation of, 175, 176 monotone sequences of, 242 positive definite, 381 projection, see Projector

683

Index relative bound, 361 restriction, 186 self-adjoint, see Self-adjoint operators sequences strong limit of, 230,231 uniform limit of, 230 weak limit of, 230,231 symmetric, 192 Optical theorem, 513 generalized, 513 Orthogonal system, 21 Orthonormal basis, 36 P

Parseval’s relation, 38 Partially isometric operators, 440 Pauli’s exclusion principle, 168,307 Phase shifts, 497,589,593 Phase space, 2 Physical equivalence, 293 Physical interpretation, see Measurement, theory of Planck constant, 46 Plane waves advanced distorted, 495 distorted, 494 free, 491 incoming distorted, 495 outgoing distorted, 495 retarded distorted, see Plane waves, outgoing distorted Potential central, 149 long-range, 514 spherically symmetric, 149 Pre-Hilbert space, 18 Principle of indistinguishabilityof identical particles, 307 Probability, 46 amptitude, see Transition, amplitude current, 410,411 frequency interpretation, 47 Projector, 197

Q Quantum number azimuthal, 168

magnetic, 168 principal, 168 Quotient space, 138

R R operator, 511 Radon- Nykodim theoreom, 269 Rellich’s theorem, 366 Resolution of identity, 235 Resolvent equations f i t , 478 second, 478 Resolvent of operators, 475 Resolvent set, 475 Riemann-Lebesgue lemma, 216 Riemann-Stieltjes sums, 464 Riesz- Fischer theorem, 105 Riesz theorem, 184 Ring, Boolean, 66 Rodrigues’ formula, 152 Rollnik class, 545 Rollnik norm, 545

S

S matrix, 512, see also Scattering, matrix partial-wave, 497

S operator, 423,438 for multichannel scattering, 607,613 perturbation expansion of, 450 unitarity of, 442 Scalar product, see Inner product Scattering amplitude, 494,553,556 elastic, 597 experiments, 416,425 inelastic, 598 matrix, 512 operator, see S operator state, 421 superoperator, 487 theory stationary, 458 three-body, 622 time dependent, 438 time independent, see Scattering, theory, stationary Schmidt procedure, see Vectors

684 Schroedingerequation, 3,46,48, 122,332 Schroedinger operator, 355 essential self-adjointness of, 358, 368 form, 122 n-body, 536 Schroedingerpicture, 291 Schwarz- Cauchy inequality, 19,20 Self-adjoint operators, 192 commuting, 258,259,261 spectral measure of, 250 Sequence Cauchy, 26 contracting, see Sequence, monotonically decreasing convergence in mean, 105 expanding, see Sequence, monotonically increasing fundamental, see also Sequence, Cauchy in mean, 104 monotonically decreasing, 63 monotonically increasing, 63,209 of operators, see Operators, sequences Sesquilinear forms, see Bilinear form Set function, 58 continuous, 68 measurable, 80 monotone, 79 subtractive, 79 Sets additive class of, see Ring, Boolean Bore1measurable, 80 closed, 27 Lebesgue measurable, 80 measurable, 65,7 1,74 monotone class of, 63 partially ordered, 203 Shift operator, 213 Spectral decomposition of operators, 242, 252

Spectral family, see Spectral function Spectral function, 235 Spectral integrals, 483 Spectral measure, 231 Spectralrepresentation space of operators, 315

Spectrum of operators absolutely continuous, 438 continuous, 253,475 discrete, 537 essential, 537 point, 49,227,253,475

Index residual, 475 simple, 315,447 singularly continuous, 552 Spherical wave, outgoing free, 4% Spin measurement of, 9 Stem- Gerlach experiment, 9 State vectors, 301 States, 2 Statistical operator, 269 Stem- Gerlach experiment, see Spin Stochastic phase space, 410 Stochastic value, 410 Stone’s theorem, 335 Subspaces orthogonal complements of, 199 orthogonal sums of, 199,200 Superoperator, 398 Super selection rules, 301, 302

T T matrix, 508,620, see also Transition, matrix y o p e r a t o n , 511 Tensor product, 303 algebraic, 141 antisymmetric, 306 HiIbert, 144 symmetric, 306 Threshold, 598 Time evolution, 287,292 Tonelli’s theorem, 98 Trace of an operator, 380 class, 380 Transform Cayley, 219,223 Fourier, 2 18 Fourier-Plancherel, 219, 224 Transformations antilinear, 172 bounded, 178 continuous, 178 isometric, 212 linear, 172 semilinear, see Transformations, antilinear unitary, 212 of vector spaces, 172, see also Operators Transformer, 398

685

Index Transition amplitude, 472 matrix, 506,510 operator, 423,474,510 probability, 268,472 superoperator, 488 Translation, 576 Trotter’s product formula, 585

U Unitary equivalence, 332, see also Isomorphism of Euclidean spaces Unitary operator, 212 Unitary space, see Euclidean spaces V Vector basis, 15 Vector spaces, 11,36 algebraic tensor products of, 141 conjugate, see Vector spaces, dual dimension of, 14 dual, 183 functional on, 182 inner product, see Euclidean spaces isomorphism of, 16 Vector subspace, 16 Vectors bra, 186,190 ket, 186,190 linearly independent, 14 normalization of, 21 orthogonal, 21 orthogonal system of, 21

orthonormalization procedure of Schmidt (or Gram-Schmidt), 22 span of, 15,36 von Neumann’s equation, 3% von Neumann’s theorem, 342 W Wave advanced distorted, see Wave, incoming distorted free, 491,4% incoming distorted, 495,4% outgoing distorted, 495,4% retarded distorted, see Wave, outgoing distorted Wave function, 45,120 Wave operators channel, 603 complete, 552 existence of, 438 incoming, 423 intertwining properties of, 439,603 modified, 514 outgoing, 423 renormalized, 5 14 Wave packet, evanescence of, 5% Wave superoperators, 487 Weierstrass approximation theorem, 157

Weyl relations, 333

Y Yukawa interaction, 365

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