Mathematical Concepts of Quantum Mechanics (Universitext)

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Universitext

Springer Berlin Heidelberg New York Hong Kong London Milan Paris Tokyo

Stephen J. Gustafson Israel Michael Sigal

Mathematical Concepts of Quantum Mechanics

Springer

Stephen J. Gustafson University of British Columbia Department of Mathematics Vancouver, BC V6T 1Z2 Canada e-mail: [email protected]

Israel Michael Sigal University of Toronto Department of Mathematics Toronto, ON MSG 3C3 Canada e-mail: [email protected]

Cataloging-in-Publication Data applied for A catalog record for this book is available from the Library of Congress. Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at http://dnb.ddb.de

Mathematics Subject Classification (2000): 81S, 47A, 46Nso

ISBN 3-540-44160-3 Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. Springer-Verlag Berlin Heidelberg New York a member of BertelsmannSpringer Science+Business Media GmbH http://www.springer.de ©

Springer-Verlag Berlin Heidelberg 2003

The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: design & production GmbH, Heidelberg Typeset by the authors using a Springer ID'EX macro package Printed on acid-free paper 46/3142db- 5 43 2 1 0

Preface

The first fifteen chapters of these lectures (omitting four to six chapters each year) cover a one term course taken by a mixed group of senior undergraduate and junior graduate students specializing either in mathematics or physics. Typically, the mathematics students have some background in advanced analysis, while the physics students have had introductory quantum mechanics. To satisfy such a disparate audience, we decided to select material which is interesting from the viewpoint of modern theoretical physics, and which illustrates an interplay of ideas from various fields of mathematics such as operator theory, probability, differential equations, and differential geometry. Given our time constraint, we have often pursued mathematical content at the expense of rigor. However, wherever we have sacrificed the latter, we have tried to explain whether the result is an established fact, or, mathematically speaking, a conjecture, and in the former case, how a given argument can be made rigorous. The present book retains these features. Prerequisites for this book are introductory real analysis (notions of vector space, scalar product, norm, convergence, Fourier transform) and complex analysis, the theory of Lebesgue integration, and elementary differential equations. These topics are typically covered by the third year in mathematics departments. The first and third topics are also familiar to physics undergraduates. Those unfamiliar with Lebesgue integration can think about Lebesgue integrals as if they were Riemann integrals. This said, the pace of the book is not a leisurely one and requires, at least for beginners, some amount of work. Even in dealing with mathematics students we have found it useful, if not necessary, to review basic mathematical notions such as the spectrum of an operator, and the Gateaux or variational derivative, which we needed for the course. Moreover, to make the book relatively self-contained, we recall and sometimes discuss the basic notions mentioned above. As a result, the text is interspersed with mathematical supplements which occupy in total about a third of the material. A mathematically sophisticated reader can skim through them, or skip them altogether, and concentrate on physical applications. On the other hand, readers familiar with the physical content of quantum mechanics, and who would like to enhance their mathematics,

VI

Preface

could concentrate on those detours and consider the physics chapters as an application of the mathematics in a familiar setting. Though we tried to increase the complexity of the material gradually, we were not always successful, and first in Chapter 8, and then in Chapter 13, there is a leap in the level of sophistication required from the reader. This book consists of fifteen main chapters and one supplementary chapter, Chapter 16. The latter chapter is more technical than the preceding material. We did not include many standard topics which are well-covered elsewhere. These topics are referenced in Chapter 17, where we also give some comments on the literature and further reading. Acknowledgment: The authors are grateful to W. Hunziker, Yu. Ovchinnikov, and especially J. Frohlich and V. Buslaev for useful discussions, and to J. Feldman, G.-M. Graf, 1. Herbst, L. Jonsson, E. Lieb, B. Simon and F. Ting for reading parts of the manuscript and making useful remarks. The second author acknowledges his debt to his many collaborators, and especially to V. Bach, J. Frohlich, Yu. Ovchinnikov, and A. Soffer.

Vancouver /Toronto, Sept. 2002

Stephen Gustafson Israel Michael Sigal

Table of Contents

1

Physical Background ..................................... 1.1 The Double-Slit Experiment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Wave Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 State Space. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 The Schrodinger Equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Mathematical Supplement: Operators on Hilbert Spaces. . . . .

1

1 4 4 5 6

2

Dynamics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 13 2.1 Conservation of Probability. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 13 2.2 Existence of Dynamics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 14 2.3 The Free Propagator.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 18 2.4 Mathematical Supplement: Operator Adjoints. . . . . . . . . . . . .. 19 2.5 Mathematical Supplement: the Fourier Transform. . . . . . . . .. 22 2.5.1 Definition of the Fourier Transform. . . . . . . . . . . . . . . .. 22 2.5.2 Properties of the Fourier Transform. . . . . . . . . . . . . . . .. 22 2.5.3 Functions of the Derivative. . . . . . . . . . . . . . . . . . . . . . .. 24

3

Observables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 3.1 Mean Values and the Momentum Operator. . . .. . ... . . .. . .. 3.2 Observables............................................ 3.3 The Heisenberg Representation. . . . . . . . . . . . . . . . . . . . . . . . . .. 3.4 Quantization........................................... 3.5 Pseudo differential Operators. . . . . . . . . . . . . . . . . . . . . . . . . . . ..

4

The 4.1 4.2 4.3

5

Spectral Theory. . .. . . .. . . ... . . . . . .. .. . .. . . . . .. . . . . . . .. . .. 5.1 The Spectrum of an Operator. . . . . . . . . . . . . . . . . . . . . . . . . . .. 5.2 Functions of Operators and the Spectral Mapping Theorem.. 5.3 Applications to Schrodinger Operators. . . . . . . . . . . . . . . . . . .. 5.4 Spectrum and Evolution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 5.5 Variational Characterization of Eigenvalues . . . . . . . . . . . . . . ..

25 25 26 27 27 29

Uncertainty Principle. . . . .. .. . . . . . . . .... .... .. .. . . ... 31 The Heisenberg Uncertainty Principle. . . . . . . . . . . . . . . . . . . .. 31 A Refined Uncertainty Principle. . . . . . . . . . . . . . . . . . . . . . . . .. 32 Application: Stability of Hydrogen. . . . . . . . . . . . . . . . . . . . . . .. 33

35 35 39 41 46 48

VIII

Table of Contents 5.6 5.7

Number of Bound States. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 53 Mathematical Supplement: Integral Operators. . . . . . . . . . . . .. 58

6

Scattering States ......................................... 6.1 Short-range Interactions: p, > 1. . . . . . . . . . . . . . . . . . . . . . . . . .. 6.2 Long-range Interactions: p, :s; 1 . . . . . . . . . . . . . . . . . . . . . . . . . .. 6.3 Existence of Wave Operators. . . . . . . . . . . . . . . . . . . . . . . . . . . ..

61 62 64 65

7

Special Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 7.1 The Infinite Well. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 7.2 The Torus. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 7.3 A Potential Step ....................................... 7.4 The Square Well ....................................... 7.5 The Harmonic Oscillator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 7.6 A Particle on a Sphere. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 7.7 The Hydrogen Atom.. . ... .. . . .. . . .. . .... . . . . . . . . .. . . . .. 7.8 A Particle in an External EM Field. . . . . . . . . . . . . . . . . . . . . ..

67 67 68 68 69 71 74 74 77

8

Many-particle Systems. .. . ... .. .. . . .. . .. . . . . . . . . .. . . . . ... 8.1 Quantization of a Many-particle System. . . . . . . . . . . . . . . . . .. 8.2 Separation of the Centre-of-mass Motion .................. 8.3 Break-ups............................................. 8.4 The HVZ Theorem ............................. , . . .. . .. 8.5 Intra- vs. Inter-cluster Motion. . . . . . . . . . . . . . . . . . . . . . . . . . .. 8.6 Existence of Bound States for Atoms and Molecules. . . . . . . .. 8.7 Scattering States. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 8.8 Mathematical Supplement: Tensor Products ...............

79 79 83 85 86 88 90 91 93

9

Density Matrices ...................... . . . . . . . . . . . . . . . . . .. 9.1 Introduction........................................... 9.2 States and Dynamics. . .. . .. . . . .. . . .. .... . . . . . . . .. .. . . ... 9.3 Open Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 9.4 The Thermodynamic Limit. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 9.5 Equilibrium States. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 9.6 The T ----> 0 Limit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 9.7 Example: a System of Harmonic Oscillators ................ 9.8 A Particle Coupled to a Reservoir ........................ 9.9 Quantum Systems ...................................... 9.10 Problems .............................................. 9.11 Hilbert Space Approach ................................. 9.12 Appendix: the Ideal Bose Gas ............................ 9.13 Appendix: Bose-Einstein Condensation .................... 9.14 Mathematical Supplement: the Trace, and Trace Class Operators .............................................

95 95 95 96 98 99 99 100 102 102 103 103 104 109 112

Table of Contents

10 The 10.1 10.2 10.3

IX

Feynman Path Integral .............................. The Feynman Path Integral. ............................. Generalizations of the Path Integral ...................... Mathematical Supplement: the Trotter Product Formula ....

115 115 118 119

Quasi-classical Analysis ................................... 11.1 Quasi-classical Asymptotics of the Propagator .............. 11.2 Quasi-classical Asymptotics of Green's Function ............ 11.2.1 Appendix ....................................... 11.3 Bohr-Sommerfeld Semi-classical Quantization .............. 11.4 Quasi-classical Asymptotics for the Ground State Energy .... 11.5 Mathematical Supplement: Operator Determinants .........

121 122 125 127 128 130 132

12 Mathematical Supplement: the Calculus of Variations .... 12.1 Functionals ............................................ 12.2 The First Variation and Critical Points .................... 12.3 Constrained Variational Problems ........................ 12.4 The Second Variation ................................... 12.5 Conjugate Points and Jacobi Fields ....................... 12.6 The Action of the Critical Path .......................... 12.7 Appendix: Connection to Geodesics .......................

135 135 137 141 142 143 147 150

13 Resonances ............................................... 13.1 Tunneling and Resonances ............................... 13.2 The Free Resonance Energy .............................. 13.3 Instantons ............................................. 13.4 Positive Temperatures .................................. 13.5 Pre-exponential Factor for the Bounce .................... 13.6 Contribution of the Zero-mode ........................... 13.7 Bohr-Sommerfeld Quantization for Resonances .............

153 153 155 157 159 161 162 163

14 Introduction to Quantum Field Theory ................... 14.1 The Place of QFT ...................................... 14.1.1 Physical Theories ................................. 14.1.2 The Principle of Minimal Action ................... 14.2 Klein-Gordon Theory as a Hamiltonian System ............. 14.2.1 The Legendre Transform .......................... 14.2.2 Hamiltonians .................................... 14.2.3 Poisson Brackets ................................. 14.2.4 Hamilton's Equations ............................. 14.3 Maxwell's Equations as a Hamiltonian System ............. 14.4 Quantization of the Klein-Gordon and Maxwell Equations ... 14.4.1 The Quantization Procedure ....................... 14.4.2 Creation and Annihilation Operators ............... 14.4.3 Wick Ordering ...................................

167 167 168 168 169 169 170 170 171 173 174 175 179 181

11

X

Table of Contents 14.4.4 Quantizing Maxwell's Equations .................... Fock Space ............................................ Generalized Free Theory ................................ Interactions ............................................ Quadratic Approximation ............................... 14.8.1 Further Discussion ................................

182 183 186 186 189 195

15 Quantum Electrodynamics of Non-relativistic Particles: the Theory of Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.1 The Hamiltonian ....................................... 15.2 Perturbation Set-up .................................... 15.3 Results ................................................ 15.4 Mathematical Supplements .............................. 15.4.1 Spectral Projections .............................. 15.4.2 Projecting-out Procedure ..........................

197 197 200 202 205 205 206

14.5 14.6 14.7 14.8

16 Supplement: Renormalization Group ..................... 16.1 The Decimation Map ................................... 16.2 Relative Bounds ........................................ 16.3 Elimination of Particle and High Photon Energy Degrees of Freedom ............................................... 16.4 Generalized Normal Form of Operators on Fock Space ...... 16.5 The Hamiltonian Ho(c, z) ............................... 16.6 A Banach Space of Operators ............................ 16.7 Rescaling .............................................. 16.8 The Renormalization Map ............................... 16.9 Linearized Flow ........................................ 16.lOCentral-stable Manifold for RG and Spectra of Hamiltonians. 16.11Appendix .............................................

207 207 210 211 216 218 221 223 225 226 230 234

17 Comments on Missing Topics, Literature, and Further Reading .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 Index ......................................................... 245

1 Physical Background

In this introductory chapter, we present a very brief overview of the basic structure of quantum mechanics, and touch on the physical motivation for the theory. A detailed mathematical discussion of quantum mechanics is the focus of the subsequent chapters.

1.1 The Double-Slit Experiment Suppose a stream of electrons is fired at a shield in which two narrow slits have been cut (see Fig. 1.1.) On the other side of the shield is a detector screen. ~ shield ~

1

~ electron ~ gun ~

~

1)'s 1I

screen

~

Fig. 1.1. Experimental set-up. Each electron that passes through the shield hits the detector screen at some point, and these points of contact are recorded. Pictured in Fig. 1.2 and Fig. 1.3 are the intensity distributions observed on the screen when either of the slits is blocked.

S. J. Gustafson et al., Mathematical Concepts of Quantum Mechanics © Springer-Verlag Berlin Heidelberg 2003

2

1 Physical Background

-7

t

PI (brightness)

-7 -7 -7 -7

T Fig.1.2. First slit blocked.

-7 -7

1

-7 -7 -7 Fig.1.3. Second slit blocked. When both slits are open, the observed intensity distribution is shown in Fig. 1.4.

1.1 The Double-Slit Experiment

3

1

I

T Fig. 1.4. Both slits open. Remarkably, this is not the sum of the previous two distributions; i.e.,

P

-I- PI + P2 . We make some observations based on this experiment.

1. We cannot predict exactly where a given electron will hit the screen, we can only determine the distribution of locations. 2. The intensity pattern (called an interference pattern) we observe when both slits are open is similar to the pattern we see when a wave propagates through the slits: the intensity observed when waves EI and E2 (the waves here are represented by complex numbers encoding the amplitude and phase) originating at each slit are combined is proportional to lEI + E212 -I- IEI12 + IE212 (see Fig. 1.5).

Fig.1.5. Wave interference. We can draw some conclusions based on these observations.

4

1 Physical Background

1. Matter behaves in a random way. 2. Matter exhibits wave-like properties. In other words, the behaviour of individual electrons is intrinsically random, and this randomness propagates according to laws of wave mechanics. These observations form a central part of the paradigm shift introduced by the theory of quantum mechanics.

1.2 Wave Functions In quantum mechanics, the state of a particle is described by a complexvalued function of position and time, 1/J(x, t), x E ]R3, t E lR. This is called a wave function (or state vector). Here]Rd denotes d-dimensional Euclidean space, ]R = ]Rl, and a vector x E ]Rd can be written in coordinates as x = (Xl, ... ,Xd) with Xj E lR. In light of the above discussion, the wave function should have the following properties. 1. [1/J(', t) [2 is the probability distribution for the particle's position. That is, the probability that a particle is in the region Q C ]R3 at time t is [1/J(x, t)[2dx. Thus we require the normalization In~3 [1/J(x, t)[2dx = 1. 2. 1/J satisfies some sort of wave equation.

In

For example, in the double-slit experiment, if 1/Jl gives the state beyond the shield with the first slit closed, and 1/J2 gives the state beyond the shield with the second slit closed, then 1/J = 1/Jl + 1/J2 describes the state with both slits open. The interference pattern observed in the latter case reflects the fact that [1/J[2 =I- [1/Jl[2 + [1/J2[2.

1.3 State Space The space of all possible states of the particle at a given time is called the state space. For us, the state space of a particle will usually be the squareintegrable functions:

(we can impose the normalization condition as needed). This is a vector space, and has an inner-product given by

(1/J, ¢):=

r 7[;(x)¢(x)dx

JIR3

(in fact, it is a "Hilbert space" - see Section 1.5)

1.4 The Schrodinger Equation

5

1.4 The Schrodinger Equation We now give a motivation for the equation which governs the evolution of a particle's wave function. This is the celebrated Schrodinger equation. Our equation should satisfy certain physically sensible properties. 1. The state 'I)!(', to) at time t = to should determine the state 'I)!(', t) for all later times t > to (causality). 2. If 'I)! and ¢ are evolutions of states, then o.'l)! + (3¢ (a, (3 constants) should also describe the evolution of a state (the superposition principle). 3. In "everyday situations," quantum mechanics should be close to the classical mechanics we are used to (the correspondence principle). The first requirement means that 'I)! should satisfy an equation which is firstorder in time, namely (1.1) for some operator A, acting on the state space. The second requirement implies that A must be a linear operator. We use the third requirement in order to find the correct form of A. We first recall that one of the fundamental equations of classical mechanics is first-order in time. It is the Hamilton-Jacobi equation, f) f)t

S

=

-h(x, V x S )

(1.2)

where h(x, k) = ~ + V(x) is the classical Hamiltonian junction, V is the potential, m is the mass, and S(x, t) is the classical action. This equation, in turn, is similar to the eikonal equation,

which is a high-frequency approximation of the wave equation for u = ae i ¢. We make an analogy between the passage from the wave equation to the eikonal equation (that is, from wave optics to geometric optics) and the passage from quantum mechanics to classical mechanics. Thus, we require that Equation (1.1) (whose form we are seeking) has solutions of the form 'I)! (x, t)

= a(x, t)eiS(x,t)/li

where n is some very small constant, with S satisfying equation (1.2). Assuming a, S, and their derivatives are of order one in n, then to the leading order in n, 'I)! satisfies the equation

n

2 f) in~'I)!(x, t) = --i1x'l)!(x, t)

ut

2m

+ V(x)'I)!(x, t).

(1.3)

6

1 Physical Background

This equation is of the desired form (1.1). In fact it is the correct equation, and is called the Schrodinger equation. It can be written as (1.4) where the linear operator H, given by

IH1jJ := -~L11jJ + V1jJ I is called a Schrodinger operator. The operator L1 = ~~=1 8J is the Laplacian (in spatial dimension 3), and the function (and multiplication operator) V is the potential. The small constant fi is Planck's constant; it is one of the fundamental constants in nature. For the record, its value is roughly fi ~ 6.6255

X

10- 27 erg sec.

Example 1.1 Here are just a few examples of potentials.

1. 2. 3. 4. 5.

Free motion: V == o. A wall: V == 0 on one side, V == 00 on the other (meaning 1jJ == 0 here). The double-slit experiment: V == 00 on the shield, and V == 0 elsewhere. The Coulomb potential: V(x) = -a/lxl (describes a hydrogen atom). The harmonic oscillator: V(x) = m~2IxI2.

We will analyze some of these examples, and others, in Chapter 7.

1.5 Mathematical Supplement: Operators on Hilbert Spaces We have seen in this chapter that the space of quantum-mechanical states of a system is a vector space with an inner-product (in fact a Hilbert space). We saw also that an operator (a Schrodinger operator) on this space enters the basic equation (the Schrodinger equation) governing the evolution of states. In fact, the theory of operators on a Hilbert space provides the basic mathematical framework of quantum mechanics. This section describes some aspects of operator theory that are essential to a study of quantum mechanics. Let H be a (complex) vector space. We assume H is endowed with an inner-product, (., -J. Recall that this means the map

(, ):HxH--4C satisfies the properties

1.5 Mathematical Supplement: Operators on Hilbert Spaces

7

1. linearity (in the second argument):

(v, aw

+ (3z)

a(v, w)

=

2. conjugate symmetry: (w, v) = (v, w) 3. positive definiteness: (v, v) > 0 for v

=I=-

+ (3(v, z)

0

for any v, w, z E H and a, (3 E C. It follows that the map given by

Ilvll

:=

I . I : H --+

[0,(0)

(v, v)1/2

is a norm on H. Recall that this means that for any v, w E H and a E C, 1.

2. 3.

lIavll = lalllvil Ilv + wll : : : Ilvll + Ilwll Ilvll > 0 for v =I=- o.

(the triangle inequality)

If H is complete in this norm (that is, all Cauchy sequences converge), then H is called a Hilbert space. Our main example of a Hilbert space is the L2-space

with the inner-product

(7/J, ¢):=

r {;¢.

JlR d

Here, and in what follows, we use the notation f for flRd f for flR d f(x)dx. Another important example of a Hilbert space is the Sobolev space of order n, n = 1,2, ... :

Here a is a multi-index: a = (a1, ... , ad), aj a non-negative integer, and lal := 2::=1 aj. The expression 8<>7/J denotes the partial derivative 8::; ... 8:::7/J of order lal. In other words, Hn(JRd) is the space of functions all of whose derivatives up to order n lie in L 2 (JRd). The inner-product that makes Hn(JR d) into a Hilbert space is

(7/J, ¢) Hn

:=

L

(8<>7/J,8<>¢)

1<>I:Sn

where (.,.) is the L2 inner-product defined above. We recall here two frequently used facts about Hilbert spaces (see, eg., [Fo] or [RSI] for proofs). Proposition 1.2 (Cauchy-Schwarz inequality) For v, w E H, a Hilbert space,

I(v, w)1 : : : Ilvllllwll·

8

1 Physical Background

Recall that a set {V n } C 1i, n = 1, 2, ... is called orthonormal if II Vn I = 1 for all nand (vn' V m ! = for n f- m. It is a complete orthonormal set (or basis) if the collection of finite linear combinations of the v n ' s is dense in 1i. Recall that for a subset D c 1i to be dense in 1i means that given any v E 1i and f > 0, there exists wED such that Ilv - wll < f.

°

Proposition 1.3 (Parseval relation) Suppose {vn } C 1i is a complete orthonormal set. Then for any w E 1i,

n

In this chapter, the main objects of study are linear operators on 1i (often just called operators), which are maps, A, from 1i to itself, satisfying the linearity property A( o:v + (Jw) = o:Av + (JAw for v, w E 1i, 0:, (J E
A: D(A) ----; 1i. An example of a dense subset of L 2(]Rd) is CQ"(]Rd) , the infinitelydifferentiable functions with compact support. (Recall, the support of a function f is the closure of the set where it is non-zero: supp(f) = {x E ]Rd I f (x) f- O}. Thus a function with compact support vanishes outside of some ball in ]Rd. For Dc ]Rd, CQ"(D) denotes the infinitely differentiable functions with support contained in D.) Here are some simple examples of linear operators, A, on the Hilbert space L2(]Rd). In each case, we can simply choose D(A) to be the obvious domain D(A) := {'Ij; E L 2(]Rd) I A'Ij; E L2(]Rd)}. 1. The identity map 2. Multiplication by a coordinate

(i.e. (Xj'lj;)(x) = Xj'lj;(x)) 3. Multiplication by a continuous function V : ]Rd

V: 'Ij;

f----+

V'Ij;

(again meaning (V'Ij;)(x) = V(x)'Ij;(x)). 4. The momentum operator (differentiation)

----;

1.5 Mathematical Supplement: Operators on Hilbert Spaces

9

5. The Laplacian d

Ll: 1f'!

f-4

La;1f'! j=l

6. A Schrodinger operator

7. An integral operator

K: 1f'!

f-4

J

K(·, y)1f'!(y)dy

J

(i.e. (K1f'!)(x) = K(x, y)1f'!(y)dy). The function K : JRd x JRd called the integral kernel of the operator K.

---+

C is

The domain of the first example is obviously the whole space L 2 (JRd). The domain of the last example depends on the form of the integral kernel, K. The domains of the other examples are easily seen to be dense, since they contain COO (JRd) (assuming V(x) is a locally bounded function). If the potential function, V(x), is bounded, then the largest domain on which the Schrodinger operator, H, is defined, namely D(H) := {1f'! E L 2(JRd) I H1f'! E L2(JRd)}, is the Sobolev space of order 2, H2(JRd). Remark 1.4 If the kernel K is allowed to be a distribution (a generalized function), then the last example above contains all the previous ones as special cases.

It is useful in operator theory to single out those operators with the property of boundedness (which is equivalent to continuity). Definition 1.5 An operator A on H is bounded if

IIAII

:=

sup

{'if;E'H 111'if;11=1}

IIA1f'!1I <

00.

(1.5)

In fact, the expression (1.5) defines a norm which makes the space B(H) of bounded operators on H into a complete normed vector space (a Banach space) As we will see, the bounded operators are, in some respects, much easier to deal with than the unbounded operators. However, since some of the most important operators in quantum mechanics are unbounded, we will need to study both. Often we can prove a uniform bound for an operator A on a dense domain, D. The next lemma shows that in this case, A can be extended to a bounded operator. Lemma 1.6 If an operator A satisfies IIA1f'!1I :::; CII1f'!1I (with C independent of 1f'!) for 1f'! in a dense domain Dc H, then it extends to a bounded operator (also denoted A) on all H, satisfying the same bound: IIA1f'!1I :::; CII1f'!1I for 1f'! E H.

10

1 Physical Background

Proof For any u E 1i, there is a sequence {un} n -+ 00 (by the density of D). Then the relation

C

D such that Un

-+

u as

shows that {Au n } is a Cauchy sequence, so AU n -+ v, for some v E 1i (by completeness of 1i), and we set Au := v. This extends A to a bounded operator on all of 1i (with the same bound, C). 0 The next important notion is that of the inverse operator. Given an operator A on a Hilbert space 1i, an operator B is called the inverse of A if D(B) = Ran(A), D(A) = Ran(B), and

AB =

lIRan(B).

Here

Ran(A)

:=

{Au I u

E

D(A)}.

It follows from this definition that there can be at most one inverse of an operator A. The inverse of A is denoted A-I. Put differently, finding the inverse of an operator A is equivalent to solving the equation Au = f for all

f

E

Ran(A).

A convenient criterion for an operator A to have an inverse is that A be one-to-one: that is, Au = 0 ===} u = o. (1.6) The operator A is said to be invertible if A has a bounded inverse. Thus A is invertible if in addition to being one-to-one, A is onto: that is,

Ran(A)

=

1i.

(1.7)

Problem 1.7 Show that (1.6) and (1.7) imply that A is invertible.

The following result provides a widely used criterion for establishing the invertibility of an operator. Theorem 1.8 Assume the operator A is invertible, and the operator B is bounded and satisfies IIBII < IIA- 1 11- 1 . Then the operator A+B, defined on D(A + B) = D(A) is invertible.

This theorem follows from the relation A+B = A(l+A-l B) and the exercise below. Problem 1.9

1. Show that the series 00

called a Neumann series, is absolutely convergent (i.e. L~=o 00) and provides the inverse of the operator 1 + A-I B.

II( -A-l B)n I <

1.5 Mathematical Supplement: Operators on Hilbert Spaces

11

2. Show that if operators A and C are invertible, then so is AC, with (AC)-I = C- I A-I. We conclude this section with a useful definition. Definition 1.10 The commutator, [A, B], of two bounded operators A and B is the operator defined by

[A, B] := AB - BA.

Defining the commutator of two operators when one of them is unbounded requires caution, due to domain considerations. Given this warning, we will often deal with commutators of unbounded operators formally without giving them a second thought.

2 Dynamics

We recall that the evolution of the wave function, 'l/J, for a particle in a potential, V, is determined by the Schrodinger equation:

in~~ = where

H'l/J

(2.1)

n2

H= --.:1+ V

2m is the appropriate Schrodinger operator. We supplement equation (2.1) with the initial condition (2.2)

where'l/Jo E L2. The problem of solving (2.1)- (2.2) is called an initial value problem or a Cauchy problem. The purpose of this chapter is to investigate the existence, and a basic property, of solutions of the Schrodinger equation. Both the existence and the property we have in mind do not depend on the particular form of the operator H, but rather follow from a basic property - self-adjointness. This property, which is rather subtle, is defined and discussed in the mathematical supplement at the end of this chapter. For the moment, we just mention that self-adjointness is a strengthening of a much simpler property - symmetry. An operator A on a Hilbert space H is symmetric if for all 'l/J, ¢ E D(A),

(A'l/J, ¢)

=

('l/J, A¢).

2.1 Conservation of Probability Since we interpret I'l/J(x, t)12 at a given instant in time as a probability distribution, we require (2.3) at all times, t. If (2.3) holds, we say that probability is conserved. Theorem 2.1 Under the evolution given by (2.1) with 'l/Jo E D(H), probability is conserved iff H is symmetric.

S. J. Gustafson et al., Mathematical Concepts of Quantum Mechanics © Springer-Verlag Berlin Heidelberg 2003

14

2 Dynamics

Proof For 7/; solving the Cauchy problem (2.1)-(2.2), we compute d

.

dt (7/;, 7/;) = (7/;,7/;)

=

.

+ (7/;, 7/;) =

i~ [(7/;, H7/;) -

1

(in H 7/;, 7/;)

1

+ (7/;, inH7/;)

(H7/;, 7/;)]

(here, and often below, we use the notation ;p to denote fJ7/;/fJt). This is zero for all times iff (H¢,¢) = (¢,H¢) for all ¢ E D(H) (for the "only if" part, take 7/;0 = ¢). This, in turn, holds iff H is a symmetric operator. The latter fact follows from a version of the polarization identity,

whose proof is left as an exercise below. 0 Problem 2.2 Prove (2.4).

2.2 Existence of Dynamics Definition 2.3 We say the dynamics exist ifthe Cauchy problem (2.1)- (2.2) has a unique solution which conserves probability. Theorem 2.4 The dynamics exist iff H is self-adjoint .. We will prove only the implication which is important for us, namely that self-adjointness of H implies the existence of dynamics (for a proof of the converse statement see [RSI]). Proof that H self-adjoint ===} existence of dynamics: We begin by discussing the exponential of an operator. For a bounded operator, A, we can define the operator e A through the power series

which converges absolutely since

~ IIAnl1 < ~ IIAlln = e11A11 < 00.

~ n=O

n!

-~ n=O

n!

Problem 2.5 Find e- iH for the 2 x 2 matrices (operators on the Hilbert space <e 2 )

2.2 Existence of Dynamics

15

For an unbounded self-adjoint operator A, it is still possible to define the bounded operator eiA . We will do this by approximating A by bounded operators. By Theorem 2.21, the operators

A.x

:=

~A2[(A + iA)-l + (A -

iA)-l]

are well-defined and bounded for A > O. The operators A.x approximate A in the sense that A.x'¢ -+ A'¢ as A -+ 00 for all '¢ E D(A 2 ). Indeed, using the estimate II(A ± iA)-lll ::; ~, which is established in Theorem 2.21, we compute

as A -+ 00. Since A.x is bounded, we can define the exponential eiA ,\ by power series as above. We will show now that the family {e iA '\, A > O} is a Cauchy family, in the sense that

(2.6)

as A, A' -+ 00 for all '¢ E D(A2). To prove this fact, we represent the operator inside the norm as an integral of a derivative:

eiA,\I _ eiA'\ =

r ~eisA,\I ei(l-s)A,\ds. 1

io as

(2.7)

Problem 2.6 Using the power series representation, show that if A is a bounded operator, then 1.

a .A .A .A _e"S = iAe"S = e"S iA

(2.8)

as

2. if A is self-adjoint, then eiA is an isometry: (2.9) (this is a special case of Theorem 2.9 below). Problem 2.7 Show that A.x is self-adjoint.

Using (2.8) and (2.9) in (2.7), we find (noting that A.x and Av commute)

II(e iAV -

eiA,\)'¢11 = 1111 eisA,\Iei(l-s)A,\i(Av ::; 11 IleisA,\I ei(1-s)A,\i(A v

A.x)'¢dsll

- A.x),¢llds

= 1 111 (A.x - Av),¢llds = II(A.x' - A.x),¢II·

(2.10)

16

2 Dynamics

(the inequality used in the first step can be proved by writing the integral as a limit of Riemann sums and using the triangle inequality - see [FoD. Due to (2.5), relation (2.6) follows. The Cauchy property (2.6) shows that for any 'IjJ E D(A2), the vectors eiA.x 'IjJ converge to some element of the Hilbert space as >. ----; 00. Thus we can define eiA'IjJ:= lim eiA.x'IjJ (2.11) )..->00

for 'IjJ E D(A 2). It follows from (2.9) that II eiA'IjJ II :::; 11'ljJ11 for all 'IjJ in D(A2) (which can be shown to be dense). Thus, as in Lemma 1.6, we can extend this definition of eiA to all 'IjJ E 1i. This defines the exponential eiA function for any self-adjoint operator A. If A is self-adjoint, the exponential e iA belongs to an important class of operators which we introduce and discuss now. Definition 2.8 An operator U is called unitary if UU* = U*U = 1. Unitary operators are isometries, as follows from (U'IjJ,U~=('IjJ,U*U~=('IjJ,~,

(2.12)

and in particular 11U'ljJ11 = 11'ljJ11. This also implies that unitary operators are bounded operators, with norm equal to one. Another property of a unitary operator is its invertibility: U is invertible with inverse U* (this is just a re-statement of the definition). We have the following important result. Theorem 2.9 If A is self-adjoint, then e iA is unitary. Proof We start with an exercise:

Problem 2.10 Using the power series representation of eiA , prove the theorem for bounded A. Given the result of this exercise, we prove the theorem for unbounded A. By the definition (2.11), we have

(eiA'IjJ, eiA ¢)

= Jim (eiA.x'IjJ, eiA.x¢) = ('IjJ, ¢) ",->00

by (2.12). Hence (eiA)*e iA = 1. Similarly, one can show that (e iA )* = e- iA . Then applying the above argument to e- iA gives eiA(e iA )* = (e-iA)*e- iA = 1. Thus eiA is unitary. 0 The following simple example illustrates Theorem 2.9. Example 2.11 If ¢ : lRd

----;

lR is continuous, then the bounded operator

2.2 Existence of Dynamics

17

is easily checked to be unitary on L 2(JRd) (just note that U* is multiplication by e- i
where HA := ~A2[(H + iA)-l

+ (H -

iA)-l] are bounded operators.

Problem 2.12 Using the power series definition of the exponential, show that

a e-iH)..t/nn/.

at

_

'/-'0 -

-

Yii H A e-iH)..t/nn/.,/-,0·

Using this result and formally bringing the differentiation into the limit, we obtain, for ¢ E D(H),

in~(rI. e-iHt/n'lj; ) = in~ lim (ri. e-iH)..t/nn/. ) = in lim (ri. ~e-iH)..t/nn/. ) at ,/-"

0

at A-+CXJ ,/-"

'/-'0

A-+CXJ ,/-"

at

'/-'0

= ,lim (¢, HAe-iH)..t/n'lj;o) = lim (HA¢, e-iH)..t/n'lj;o) A--+OO

/\...-...+CXJ

= (H¢, e-iHt/n'lj;o). This exchange of limits is readily justified (the reader is invited to supply the details). Furthermore, if'lj;o E D(H), one can show that e-iHt/n'lj;o E D(H), and therefore in%te-iHt/n'lj;o = He-iHt/n'lj;o. So 'lj;(t) satisfies (2.1). Clearly U(O) = 1, so that (2.2) holds. Finally, we note that 'lj;(t) is the unique solution of (2.1 )-(2.2). Indeed, if there are two solutions, their difference, ~, solves (2.1) with ~It=o = O. By symmetry of H, 11~(t)11 = II~(O)II = 0 for all t, hence ~=O. 0 The operator family U(t) := e- iHt / n is called the propagator or evolution operator. It enjoys the important group property

U(t)U(s) = U(t for t, s

E

+ s)

(2.13)

R To prove this property, we again use representation (2.11).

Problem 2.13 Show that e-iH)..t/n has the group property (2.13). (Hint: since H A is a bounded operator, use the power series representation of

e-iH)..t/n).

Given this result, we proceed as in the proof of Theorem 2.9: for any 'lj;, ¢ E 'ft,

18

2 Dynamics (7jJ, U(t)U(s)¢)

=

(U(t)*7jJ, U(s)¢)

=

}~~ (e iH>.t/fi7jJ, e-iH>.S/fi¢)

= Jim (7jJ, e-iH>.(Hs)/fi¢;) = (7jJ, U(t + s)¢) /<---+00

which proves (2.13). As we have seen, if H is self-adjoint, the propagator U(t) = e- iHt / fi exists, and is unitary for all t. Since it also satisfies (2.13), U(t) is an example of a one-parameter unitary group. We conclude this section by emphasizing that for the Schrodinger equation formulation of quantum mechanics to make sense, the Schrodinger operator must be self-adjoint. We will address the important question of which potential functions, V, give rise to self-adjoint operators H = + V in Chapter 5.

:::..:1

2.3 The Free Propagator In this section we investigate the solution of Schrodinger's equation in the absence of a potential. That means we want to study the free propagator U(t) = eiHot/fi with Ho := The tool for doing this is the Fourier transform, whose definition and properties are reviewed at the end of this chapter. Let g(k) = e- a l;12 (a Gaussian), with Re(a) ~ o. Then setting p := -iff\! and using the result of Problem 2.24, we have

:::..:1.

g(p)7jJ(x)

=

(27Tan?)-d/2

J

e-

1~:~J2 7jJ(y)dy.

(2.14) Taking a = ;,\ here, we obtain an expression for the Schrodinger evolution operator e-iHot/fi for the Hamiltonian of a free particle, Ho = . (e-· Hot / fi 7jJ)(x)

=

(27Titit) -d/2

--:;;;:-

r

:::..:1:

irnlx-yl 2

J~d e~7jJ(y)dy.

(2.15)

One immediate consequence of this formula is the pointwise decay (in time) of solutions of the free Schrodinger equation with integrable initial data: (2.16) As another consequence, we make a connection between the free Schrodinger evolution, and the classical evolution of a free particle. Using the relation Ix - Yl2 = Ixl 2- 2x· y + lyl2, we obtain

2.4 Mathematical Supplement: Operator Adjoints

19

(2.17) where (as usual) :J; denotes the Fourier transform of 'IjJ. One can show that if ;j;( k) is localized near ko E ~d, then so is ;j;t (k) for large t, and therefore the right hand side of (2.17) is localized near the point

Xo = vot,

where

Vo

=

ko/m,

i.e., near the classical trajectory of the free particle with momentum k o.

2.4 Mathematical Supplement: Operator Adjoints In quantum mechanics, operators which are self-adjoint playa central role. The present section is devoted to a discussion of this class of operators, and to the broader class of symmetric operators. All operators A are assumed to be defined on a dense domain D(A). Definition 2.14 The adjoint of an operator A on a Hilbert space H, is the operator A * satisfying (A*'IjJ, ¢) = ('IjJ, A¢) (2.18)

for all ¢

E

D(A), for'IjJ in the domain

D(A*) := {'IjJ

E

H 11('IjJ, A¢)I ::::: C,pII¢11 for some constant C,p (independent of ¢), V¢ E D(A)}.

It is left as an exercise to show this definition makes sense. Problem 2.15 Show that equation (2.18) defines a unique linear operator A* on D(A*) (hint: use the "Riesz lemma" - see, eg., [RSI]). Definition 2.16 An operator A is symmetric if

(A'IjJ, ¢) = ('IjJ, A¢) for all 'IjJ, ¢ in D(A). Definition 2.17 An operator A is self-adjoint if A = A*.

20

2 Dynamics

By definition, every self-adjoint operator is symmetric (this follows from relation (2.18)). Indeed, an operator A is self-adjoint iff it is symmetric and D(A) = D(A*). However, not every symmetric operator is self-adjoint. Nor can every symmetric operator be extended uniquely to a larger domain on which it is self-adjoint. For example, the Schrodinger operator A := -.1 c /1 X 12 with c > 1/4 is symmetric on the domain (~3 \ {O}) (the infinitely differentiable functions supported away from the origin), but does not have a unique self-adjoint extension (see [RSH]). It is usually much easier to show that a given operator is symmetric than to show that it is self-adjoint, since the latter question involves additional domain considerations.

Co

Problem 2.18 Referring to our earlier list of examples of operators on L2(~d), show that the following operators are symmetric.

1. 2. 3. 4. 5.

The The The The The

multiplication operator V (if V(x) is real-valued). differentiation operator, Pj (hint: integrate by parts). Laplacian, .1. Schrodinger operator, H (again if V is real). integral operator J( (if K(y,x) = K(x,y)).

In fact, Pj and .1 are actually self-adjoint operators on L2(~d), as is V if it is a "nice enough" function (we will be more specific in Section 5.3). To be more precise, there exist domains for these operators, containing CO(~d), for which they are self-adjoint (and, moreover, they are the unique self-adjoint extensions from Co(~d)). For example, D(.1) = H2(~d). The subtleties surrounding domains and the question of self-adjointness are absent for bounded operators, as Lemma 1.6 demonstrates. Lemma 2.19 If A is bounded and symmetric, it is self-adjoint.

Proof. Since A is bounded, by Lemma 1.6 we may assume D(A)

= Ji.

Since

1(1f!, A¢) 1::; 111f!IIIIAIIII¢II, for 1f!, ¢ E Ji, we have D(A*) self-adjoint. 0

= Ji, and so D(A) = D(A*). Hence A is

We will, however, have to deal with unbounded operators. As an example, consider the following: Problem 2.20 Show that the operators Pj := -ifWj and T := unbounded on L2(~d).

-!!:n .1 are

For what follows, we need the following important result: Theorem 2.21 If A is a self-adjoint operator, then for any z E C with Im(z) =I- 0, the operator A - zl has a bounded inverse, and this inverse satisfies (2.19)

2.4 Mathematical Supplement: Operator Adjoints

21

Proof. Suppose Im(z) i=- O. Then we compute, for 'I/J E D(A),

II(A - zl)'l/J11 2 = ((A - zl)'I/J, (A - zl)'I/J) = IIA'l/J112 - 2Re(z)('I/J, A'I/J) + Iz1211'I/J112

(2.20)

so

II(A - zl)'l/J11 2 2: IIA'l/J112 - 2IRe(z)III'l/JIIIIA'l/J11 + Iz1211'I/J112 = (1IA'l/JII-IRe(z)III'l/JII)2 + IIm(z)1211'I/J112. Hence

(2.21 )

II(A - zl)'l/JII 2: IIm(z)III'l/JII· This shows that the operator A - zl has an inverse (A - Zl)-l : Ran(A - zl)

----+

1i

(here Ran(A) := {'I/J E 1i I -::J¢ E D(A), 'I/J = A¢} denotes the range of A) and this inverse satisfies (2.19) for any ¢ E Ran(A-zl) (just take ¢:= (A-zl)'I/J in (2.21)). Moreover, if ¢ -.l Ran(A - zl), then (¢, (A - zl)'I/J) = 0 for all 'I/J E D(A). Hence by Definition 2.14, ¢ E D(A*) = D(A). Therefore 0= (¢, (A - zl)'I/J) = ((A - Zl)¢, 'I/J)

for all 'I/J E D(A). Since D(A) is dense, this implies (A - Zl)¢ = 0 and therefore, by (2.21), ¢ = O. Hence Ran(A - zl) is dense. Consequently, by Lemma 1.6, the operator (A - z1)-l can be extended to a bounded operator on the entire space 1i, with the estimate (2.19) valid for all ¢ E 1i. 0 We conclude this section with a useful definition. Definition 2.22 A self-adjoint operator A is called positive (denoted A > 0) if ('I/J,A'I/J) > 0

for all 'I/J E D(A), 'I/J i=- O. Similarly, we may define non-negative, negative, and non-positive operators. Problem 2.23 Show that the operator -d on L 2(JRd) is positive (we can take D(-d) = H 2 (JRd)). Hint: integrate by parts (equivalently, use the divergence theorem) assuming that 'I/J E D{3 := {'I/J E C 2 (JR d) I I8a 'I/J (x) I ~ Ca (1 + Ixl)-{3Va, lal ~ 2} for some f3 > d/2, Then use the fact that D{3 is dense in H 2(JRd) to extend the inequality to all 'I/J E H 2(JRd). If A is a positive operator, then we can define the operator e- A in a way similar to our definition of eiA above. We take e- A := lim).-+oo e- A ", where A). = (A + >..)-1 >"A, a family of bounded operators.

22

2 Dynamics

2.5 Mathematical Supplement: the Fourier Transform The Fourier transform is a useful tool in many areas of mathematics and physics. The purpose of the present chapter is to review the properties of the Fourier transform, and to discuss the important role it plays in quantum mechanics. 2.5.1 Definition of the Fourier Transform The Fourier transform is a map, F, which sends a function 'ljJ : JRd another function {j; : JRd -+
{j;(k)

:=

(27rfn- d / 2

r

JIRd

-+

e-ik.x/n'ljJ(x)dx

(it is convenient for quantum mechanics to introduce Planck's constant, fi, into the Fourier transform). We first observe that {j; = F'ljJ is well-defined if 'ljJ is an integrable function ('ljJ E Ll(JRd ), meaning fIRd 1'ljJ(x)ldx < 00), and that F acts as a linear operator on such functions. In the following exercise, the reader is asked to compute a few Fourier transforms. Problem 2.24 Show that under F _ Ixl 2

alkl2

•

(ha)d/2 e- -2- (Re(a) > 0). Hmt: try d = 1 first - complete the square in the exponent and move the contour of integration in the complex plane. 2. e-2h-xoA-lx f---+ fid/2(detA)1/2 e-!k.Ak (A a positive d x d matrix). Hint: diagonalize and use the previous result. 1. e ~

f---+

-v'b/1l. 2 Ixl

3. ~e Ixl f---+ (lkl 2 + b)-l (b > 0, d = 3). Hint: use spherical coordinates. Alternatively, see Problem 2.26 below. In the first example, if Re(a) > 0 then the function on the left is in Ll(JRd), and the Fourier transform is well-defined. However, we can extend this result to Re(a) = 0, in which case the integral is convergent, but not absolutely convergent. 2.5.2 Properties of the Fourier Transform The utility of the Fourier transform derives from the following properties. 1. The Plancherel theorem: F is a unitary map from L 2(JRd) to itself (note

that initially the Fourier transform is defined only for integrable (Ll (JRd)) functions - the statement here is that the Fourier transform extends from L1(JRd) n L2(JR d ) to a unitary map on L 2(JRd)).

2.5 Mathematical Supplement: the Fourier Transform

23

2. The inversion formula: the adjoint F* of F on L 2(JR.d) is given by the map 'ljJ f---4 1f where

1f(x) := (27fn)-d/2

r eiX.k/I'i'ljJ(k)dk

JJfII.d

(and by the Plancherel theorem, this is also the inverse, F- 1 ). For the next four statements, suppose 'ljJ, ¢ E CO"(JR.d).

-in~(k) = k(/;(k). x'ljJ(k) = in\lk(/;(k). ¢'ljJ = (27fn) -d/2¢ * (/;. ~ = (27fn)d/2¢(/;.

3. 4. 5. 6.

Here

(f

* g)(x):=

r

f(y)g(x - y)dy JJfII.d is the convolution of f and g. The last four properties can be loosely summarized by saying that the Fourier transform exchanges differentiation and coordinate multiplication, and products and convolutions.

Proof. The proof of Property 1 is somewhat technical and we just sketch it here (see, eg, [Fo] for details). In particular, we will show that Ilfll = lIill. Suppose f E CO", and let C e be the cube of side length 2/ E centred at the origin. Choose E small enough so that the support of f is contained in C e • One can show that {Ek := (E/2)d/2 eik.x/1'i IkE En7fZd } is an orthonormal basis of the Hilbert space of functions in L2(Ce ) satisfying periodic boundary conditions. Thus by the Parseval equation (Proposition 1.3),

as

E

--+

O.

Problem 2.25 Show that {Ek} is an orthonormal set. We will prove Property 3, and we leave the proofs of the other properties as exercises. Integrating by parts, we have

-in\l'ljJ(k)

= (27fn)-d/2

J J

= k· (27fn)-d/2 D

e-ix.k/I'i. (-in\l)'ljJ(x)dx e-iX.k/I'i'ljJ(x)dx = k(/;(k).

24

2 Dynamics

Problem 2.26

1. Show that for b > 0 and d

= 3, under F-l,

(hint: use spherical coordinates, then contour deformation and residue theory). 2. Show that under F- 1 ,

Here 5 is the Dirac delta function - not really a function, but a distribution - characterized by the property f(x)5(x-a)dx = f(a). The exponential function on the right hand side is called a plane wave.

J

2.5.3 Functions of the Derivative

As an application, we show how the Fourier transform can be used to define functions of the derivative operator. Recall our notation p := -iti'v. Motivated by Property 3 ofthe Fourier transform, we define an operator g(p) (for "sufficiently nice" functions g) on L2 (JRd) as follows. Definition 2.27 ~(k) := g(k)(/;(k) or, equivalently, g(p)'I/J := (27rn)-d/2 g * 'I/J.

Let us look at a few examples. Example 2.28 1. If g(k) = k, then by Property 3 of the Fourier transform, the above definition gives us back g(p) = p (so at least our definition makes some sense).

2. Now suppose g(k)

= Ik1 2 . Then ~(k) = IkI 2 (/;.

Problem 2.29 Show that

-n2 iJ.'I/J = IkI 2 (/;.

Thus we have Ipl2 = -n2 iJ.. Extending this example, we can define g(p) when 9 is a polynomial "with our bare hands" . It is easy to see that this definition coincides with the one above. 3. Let g(k) = (2~ Ikl 2 + A)-I, A > 0, and d = 3. Then due to Problem 2.26, we have

((Ho

+ A)

-1

'I/J)(x)

=

1m

- v"2ffi>: Ix-yl IX - YI 'I/J(y)dy.

melt

2 ~2 7rn

IR3

(2.22)

3 Observables

Observables are the quantities that can be experimentally measured in a given physical framework. In this chapter, we discuss the observables of quantum mechanics, as well as the notion of "quantizing" a classical theory.

3.1 Mean Values and the Momentum Operator We recall that in quantum mechanics, the state of a particle at time t is described by a wave function 7/J(x, t). The probability distribution for the position, x, of the particle, is 17/J(·, t)12. Thus the mean value of the position at time t is given by J xl7/J(x, t)j2dx (note that this is a vector in JR3). If we define the coordinate multiplication operator

Xj : 7/J(x)

f---t

Xj7/J(x)

then the mean value of the lh component of the coordinate x in the state 7/J is (7/J, Xj7/J). Using the fact that 7/J(x, t) obeys the Schrodinger equation i~ = H7/J, we compute

d

.

dt (7/J,Xj7/J) = (7/J,Xj7/J)

.

+ (7/J,Xj7/J) =

1

(iIi H 7/J,Xj7/J)

1

+ (7/J,Xj iIi H 7/J)

i i i

= (7/J, Yi Hx j7/J) - (7/J,XjYi H 7/J)

=

(7/J, Yi[H,xj]7/J)

(where recall [A, B] := AB - BA is the commutator of A and B). Since H =- K Ll + V ' and 2m Ll(x7/J) = xLl7/J + 2V7/J, we find leading to

S. J. Gustafson et al., Mathematical Concepts of Quantum Mechanics © Springer-Verlag Berlin Heidelberg 2003

26

3 Observables

As before, we denote the operator -in\! j by Pj. As well, we denote the mean value (7jJ, A7jJ) of an operator A in the state 7jJ by (A),p. Then the above becomes

Im~ (Xj),p

= (pj),p

I

(3.1)

which is reminiscent of the definition of the classical momentum. We call the operator p the momentum operator. In fact, Pj is a self-adjoint operator on £2(JR3). (As usual, the precise statement is that there is a domain on which Pj is self-adjoint. Here the domain is just D(pj) = {7jJ E £2 (JR d) I a~ 7jJ E J

£2(JRd).)

Using the Fourier transform, we compute the mean value of the momentum operator

This, and similar computations, show that 1-0(k)12 is the probability distribution for the particle momentum.

3.2 Observables Definition 3.1 An observable is a self-adjoint operator on the state space £2(JR3). We have already seen a few examples of observables, including the position operators, Xj, the momentum operators, Pj and the Hamiltonian operator, H

11,2

1

2m

2m

= --L1 + V = -lpl2 + V

(here IpI2 = 2:~=1 p;). Another example (with an obvious classical analogue) is furnished by the angular momentum operators, £j = (x x pk The reader is invited to derive the following equation for the evolution of the mean value of an observable.

Problem 3.2 Check that for any observable, A, we have

We would like to use this result on the momentum operator. Simple computations give [L1,p] = 0 and [V,p] = in\!V, so that *[H,p] = -\!V and hence

(3.2)

3.4 Quantization

27

This is a quantum mechanical mean-value version of Newton's equation of classical mechanics. Or, if we include Equation (3.1), we have a quantum analogue of the classical Hamilton equations. Since obviously [H, H] = 0, we also have d/dt(H),p = 0, which is the mean-value version of the conservation of energy.

3.3 The Heisenberg Representation The framework outlined up to this point is called the Schrodinger representation of quantum mechanics. Chronologically, quantum mechanics was first formulated in the Heisenberg representation, which we now describe. For an observable A, define

A(t)

:= eitH / 1i Ae- itH / Ii .

Let 7/J be the solution of Schrodinger's equation with initial condition 7/Jo: 7/J = e- itH / li 7/Jo. Since e- itH / 1i is unitary, we have (by a simple computation which is left as an exercise)

(A),p

(A(t)),po·

(3.3)

Problem 3.3 1. Prove equation (3.3). 2. Prove that

d

i

dtA(t) = /i[H,A(t)]. This last equation is called the Heisenberg equation for the time evolution of the observable A. In particular, taking x and p for A, we obtain the quantum analogue of the Hamilton equations of classical mechanics:

and

mi:(t) = p(t)

(3.4)

p(t) = -V'V(x(t)).

(3.5)

In the Heisenberg representation, then, the state is fixed (at 7/Jo), and the observables evolve according to the Heisenberg equation. Of course, the Schrodinger and Heisenberg representations are completely equivalent (by a unitary transformation).

3.4 Quantization We now describe a procedure for passing from classical mechanics to quantum mechanics, called quantization. We start with the Hamiltonian formulation of classical mechanics, where the basic objects are as follows:

28

3 Observables

1. The phase space (or state space) lR~ x lR%

equipped with the Poisson bracket, {-, .}. The latter is defined on pairs of functions on lR~ x lR% by

~ (Of og

{j, g} = ~

ok·

j=l

J

of Og)

axJ - ax·J ok·J

.

2. The Hamiltonian, h(x, k), a real function on lR~ x lR% (which gives the energy of the classical system). Then the classical dynamics of the system is given by Hamilton's equations: {h,x}

j; =

k=

{h,k}.

The corresponding fundamental objects in quantum mechanics are the following: 1. The state space L2(lR~), and the commutator,

acting on L2(lR~). 2. A Schrodinger operator, H state space L2(lR3 ).

=

*[.,'J,

of two operators

h(x,p) (the Hamiltonian) acting on the

= Ikl2/2m + V(x), the corresponding quantum Hamiltonian is the Schrodinger operator

If the classical Hamiltonian function is h(x, k)

H

Ipl2

n2

m

2m

= h(x,p) = -2 + V(x) = --.1 + V(x).

Recall that the dynamics of the quantum system can be described by the Heisenberg equations .

i

x = n[H,x]

.

p

i

= n[H,p]

(which are the same as (3.4) and (3.5)). We note that for classical mechanics

and for quantum mechanics

These are called the canonical commutation relations. The functions x and k are called canonical variables of classical mechanics, and the operators x and p, canonical operators of quantum mechanics. The following table provides a summary of the classical mechanical 0 b jects and their quantized counterparts:

3.5 Pseudodifferential Operators

QM

CM

Object state space evolution of state observable

29

lR 3x x lR 3k and L2(lR~) and Poisson bracket commutator path in phase space self-adjoint operator real function on state space on state space result of measuring deterministic probabilistic observable object determining Hamiltonian Hamiltonian (Schrodinger) dynamics function operator x and k operators canonical coordinates x (mult.) and p

3.5 Pseudodifferential Operators The correspondence between classical observables and quantum observables is a subtle one. For canonical variables, the variable x is mapped to the operator of multiplication by x, and the variable k is mapped to the operator p = -ifi'il x . Continuing in this way, we should have a function f(x) mapping to an operator of multiplication by f(x), and a function g(k) mapping to an operator g (p). However, the following simple case shows the ambiguity of this correspondence. The function X· k = k· x could, for example, be mapped into any of the following distinct operators:

x·p,

p·x,

1

2"(x. p + p .x).

In this case, the ambiguity is resolved by requiring that the resulting operator be self-adjoint. But in the general situation

f(x, k)

-4

f(x,p),

showing that this condition resolves the ambiguity and that it can be implemented in terms of a mathematical formula requires some work. In mathematics, operators obtained by a certain quantization rule from functions f(x, k) satisfying certain estimates are called pseudodifferential operators. Differential operators with smooth coefficients, as well as certain integral and singular integral operators are examples of pseudo differential operators.

4 The Uncertainty Principle

One of the fundamental implications of quantum theory is the uncertainty principle - that is, the fact that certain pairs of physical quantities cannot be measured simultaneously with arbitrary accuracy. In this chapter, we establish precise mathematical statements of the uncertainty principle.

4.1 The Heisenberg Uncertainty Principle We consider a particle in a state 'IjJ and think of the observables x and p as random variables with probability distributions i'IjJi 2 and i¢i 2 respectively. Recall that the means of Xj and Pj in the state 'IjJ (E D(xj) nD(pj)) are (Xj),p and (Pj),p, respectively. The dispersion of Xj in the state 'IjJ is

and the dispersion of Pj is

Theorem 4.1 (The Heisenberg uncertainty principle) For any state 'IjJ E D(xj) n D(pj), ,1x .,1p.

n

> -. 2

J -

J

Proof The basic ingredient is the commutation relation i

-n[P,x]=1

*

(this is a matrix equation, meaning [pj, Xk] = Ojk). For notational simplicity, we assume (x),p = (p),p = o. Note that for two self-adjoint operators A, B, and 'IjJ E D(A) n D(B), (i[A, B]),p = -2Im(A'IjJ, B'IjJ).

So assuming 'IjJ is normalized

(ii'IjJii =

1), and 'IjJ

E

D(xj)

S. J. Gustafson et al., Mathematical Concepts of Quantum Mechanics © Springer-Verlag Berlin Heidelberg 2003

(4.1)

n D(pj), we obtain

32

4 The Uncertainty Principle

1

i

2

2

= ('Ij;,'Ij;) = ('Ij;, n[pj,Xj]'Ij;) = -n1m(Pj'lj;,Xj'lj;) ::::: nl(Pj'lj;,xj'lj;)1 2

::::: n11pj'lj;llllxj'lj;ll

=

2

n C,1Pj)(.1Xj).

This does it. 0

4.2 A Refined Uncertainty Principle The following result is related to the Heisenberg uncertainty principle.

Theorem 4.2 (Refined uncertainty principle) On L2(JR3),

-.1 ~

1

41x1 2 '

Recall that for operators A, B, we write A ~ 0 if ('Ij;, A'Ij;) ~ 0 for all 'Ij; E D(A), and we write A ~ B if A-B ~ O. So by the above statement, we mean ('Ij;, (-.1 - 41;1 2 )'Ij;) ~ 0 for 'Ij; in an appropriate dense subspace of D( -.1). We will prove it for 'Ij; E C8"(JRd ).

Proof. We will ignore domain questions, leaving these as an exercise for the reader. Compute d

2:>[lxl- 1pjlxl- 1,xj]

=

Ildlxl- 2

j=l

(d = space dimension = 3). Hence, using (4.1) again, d

ndlllxl- 1'1j;11 2 = -2 L 1m(lxl- 1pj Ixl- 1'1j;, Xj'lj;) j=l

and therefore, using

we obtain

d

ned -

2)lllxl- '1j;11 = -21m L(Pj'lj;, 1:12 'Ij;). 1

2

J-1

Now the Cauchy-Schwarz inequality implies d

1t;(Pj'lj;, ,:12 '1j;)1 : : :

('Ij;, IpI2'1j;)1/211Ixl-1'1j;11

(prove this!), which together with the previous equality gives

4.3 Application: Stability of Hydrogen

11ld -

33

2111Ixl- 17jJ11 2:::; 2(7jJ, IpI27jJ)1/211Ixl-17jJ11.

Squaring this, and observing that 112(7jJ, -d7jJ) yields (for d ;::: 3)

Illxl- 17jJ11 2 =

(7jJ, Ixl- 27jJ) and (7jJ, IpI27jJ)

=

which, for d = 3, implies the desired result. 0

4.3 Application: Stability of Hydrogen In classical mechanics, the hydrogen atom is unstable: as the electron orbits the nucleus (proton), it radiates away energy and falls onto the nucleus. The demonstration that this is not so in quantum mechanics was one of the first triumphs of the theory. The statement that the hydrogen atom is stable is expressed mathematically by the property that the Hamiltonian, H (and therefore the energy), is bounded from below. The hydrogen atom is described by the Schrodinger operator

112 e2 H=--d--

2m

Ixl

on L2(~3) (m and -e are the mass and charge of the electron respectively). The refined uncertainty principle gives

The right hand side reaches its minimum at

Ixl- 1= 4me 2/112

and so

2me 4 H- > - 11 -2- ' Thus, the energy of the hydrogen atom is bounded from below, and the electron does not collapse onto the nucleus.

5 Spectral Theory

Our next task is to classify the orbits (i.e. solutions) of the Schrodinger equation

in fJ'l/J = H'l/J fJt

with given initial condition

'l/Jlt=o = 'l/Jo according to their behaviour in space-time. Naturally, we want to distinguish between states which are localized for all time, and those whose essential support moves off to infinity. Such a classification is made with the help of a very important notion - the spectrum of an operator. We begin by describing the general theory, and then we proceed to applications.

5.1 The Spectrum of an Operator Definition 5.1 The spectrum of an operator A on a Hilbert space H is the subset of C given by cr(A) :=

p. E C I

A - A is not invertible (has no bounded inverse)}

(here and below, A - A denotes A - AI). The complement of the spectrum of A in C is called the resolvent set of A: p(A) := C\cr(A). For A E p(A), the operator (A - A)-I, called the resolvent of A, is well-defined. The usual reasons that A - A is not invertible are

= 0 has a non-zero solution, 'l/J E H. Then A is called an eigenvalue of A, and 'l/J a corresponding eigenvector. 2. (A - A)'l/J = 0 "almost" has a non-zero solution. More precisely, we say {'l/Jn} cHis a Weyl sequence for A and A if a) II'l/Jn II = 1 for all n b) II(A - A)'l/Jnll---> 0 as n ---> 00 c) 'l/Jn ---> 0 weakly as n ---> 00 (this means (cp, 'l/Jn) ---> 0 for all cp E H). This is what we mean by (A - A)'l/J = 0 "almost" having a non-zero solution. 1. (A - A)'l/J

S. J. Gustafson et al., Mathematical Concepts of Quantum Mechanics © Springer-Verlag Berlin Heidelberg 2003

36

5 Spectral Theory

Definition 5.2 The point spectrum of an operator A is iTp(A) = {A

Eel

A is an isolated eigenvalue of A with finite multiplicity}

(isolated meaning some neighbourhood of A is disjoint from the rest of iT(A)). Here the multiplicity of an eigenvalue A is the dimension of the subspace

Null(A - A)

:=

{v

E 1{

I

(A - A)V

=

O}.

Problem 5.3 1. Show Null(A - A) is a vector space. 2. Show that if A = A*, eigenvectors of A corresponding to different eigenvalues are orthogonal. Definition 5.4 The continuous spectrum of an operator A is iTc(A)

= {A

I there is a Weyl sequence for A and A}.

Remark 5.5 Some authors use the terms "discrete spectrum" and "essential spectrum" rather than (respectively) "point spectrum" and "continuous spectrum'. The following result says that when A is self-adjoint, the disjoint sets iTp(A) and iTc(A) comprise the whole spectrum (see, eg., [HS] for a proof):

Theorem 5.6 (Weyl) If A = A*, then the spectrum of A is the union of the point spectrum of A and the continuous spectrum of A:

Problem 5.7 Show that if U : 1{ ----7 1{ is unitary, then iT(U* AU) = iT(A), iTp(U* AU) = iTp(A), and iTc(U* AU) = iTc(A). Example 5.8 This example determines the spectrum of the operators Xj and Pj on L2(~d). We show that 1. iT(Pj)=iTc(Pj)=~ 2. iT(Xj) = iTc(Xj) = ~ Proof of the second fact: assume, for simplicity, d = 1. For any A E ~, we find a Weyl sequence for x and A. This sequence is such that its square approximates the delta-function 6>. (x) = 6(A - x) which solves the equation

(x-A)6>. = 0 exactly. Such a sequence is sketched in Fig. 5.1.

5.1 The Spectrum of an Operator

37

Fig.5.l. Weyl sequence for x, A. How do we construct such a sequence 'l/Jn? Let ¢ be a fixed non-negative function supported on [-1,1], and such that

We compress this function, increasing its height, and shift the result to A:

Then

and

as n ----- 00. Thus A E 17(x), at least. Now we show that 'l/Jn ----- 0 weakly. Indeed, for any f E L2(JR d),

which ----- 0 as n ----- 00 by a well-known result of analysis. Thus, A E 17c (x). It is easy to convince yourself that x has no eigenvalues. Now we prove the first fact. We seek a Weyl sequence, {'l/Jn}, for p and A. Using properties of the Fourier transform, we have

Take for 'l/J~ the Weyl sequence constructed above:

38

for

5 Spectral Theory

¢ supported on

[-1,1], and

J 1¢1 2 = 1. So we have II'l/Jnll = II'!f7nll = 1 and

and so 'l/Jn is a Weyl sequence for p and A. Thus o'(p) us see how 'l/Jn looks. We have

= O'c(p) = R Now let

Suppose, for example, that ¢ == 1 for Ixi ::; 1/2. Then 'l/Jn looks like a plane wave (with amplitude n- 1 / 2 and wave vector A), cut off near 00 by ¢(x/n) (I'l/Jnl is sketched in Fig. 5.2).

-nl2

nl2

Fig.5.2. Weyl sequence for p, A. We remark that the fact O'(pj) = O'c(pj) = JR also follows directly from the fact O'(xj) = O'c(xj) = JR, together with Problem 5.7 and properties of the Fourier transform. The following exercise asks for the spectrum of two of our other favourite operators. Problem 5.9 Prove

1. For V : JRd ---+ C continuous, the spectrum of the corresponding multiplication operator on L 2(JRd) is O'(V) = range(V). 2. On L2(JR d), 0'(-..:1) = O'c(-..:1) = [0,(0). We observe that self-adjoint operators have real spectrum. Theorem 5.10 If A

= A*,

then O'(A) C R

Proof This follows immediately from Theorem 2.21. D

Proposition 5.11 Let A be a self-adjoint operator on 'H. If A is an accumulation point of O'(A), then A E O'c(A).

5.2 Functions of Operators and the Spectral Mapping Theorem

39

Proof. Suppose Aj E O"(A), Aj ---t A as j ---t 00. Then we can find '¢j E O"(A) with lI'¢jll = 1 satisfying II(A - Aj)'¢jll :::; l/j (by the Weyl criterion, Theorem 5.6). Since

as j ---t 00, we see A E O"(A) (either '¢j ---t 0 weakly, in which case A E O"c(A), or else one can show that A is an eigenvalue of A). Since A is an accumulation point of O"(A), it is not isolated, and so A tj. O"p(A). Hence A E O"c(A). D If the operator A is a Schrodinger operator on L2 (Jl~d), we can strengthen Theorem 5.6 by showing that it is enough to consider Weyl sequences whose support moves off to infinity.

Definition 5.12 Let A be an operator on L 2(JR.d). A sequence {'¢n} C L 2(JR.d) is called a spreading sequence for A and A if 1. II'¢n II = 1 for all n 2. for any bounded set B C JR.d, sUPP('¢n) n B = 0 for n sufficiently large 3. II(A - A)'¢nll ---t 0 as n ---t 00.

Problem 5.13 Show that a spreading sequence for A, A is also a Weyl sequence for A, A. Theorem 5.14 If H = - ;~.::1 + V is a Schrodinger operator, with real potential V(x) which is continuous and bounded from below, then 0" c

(H)

=

{A I there is a spreading sequence for H and A}.

We skip the proof of this theorem (see [HS]).

5.2 Functions of Operators and the Spectral Mapping Theorem Our goal in this ~ection is to define functions f(A) of a self-adjoint operator A. We do this in the special case where A is bounded, and f is a function analytic in a neighbourhood of dA).

Problem 5.15 Let A be a bounded self-adjoint operator. Show O"(A) C [-IIAII, IIAII]· Hint: use that if Izl > IIAII, then II(A - z)ull ~ (lzl-IIAII)llull· Suppose f(A) is analytic in a complex disk ofradius R, {A Eel IAI < R}, with R > IIAII. SO f has a power series expansion, f(A) = E~=o anA n , which converges for IAI < R. Define the operator f(A) by the convergent series

L

(Xl

f(A)

:=

n=O

anAn.

(5.1)

40

5 Spectral Theory

We have already encountered an example of this definition: the exponential e A discussed in Section 2.2. As another example, consider the function j()...) = ()... - Z)-l, for Izl > IIAII, which is analytic is a disk of radius R, with IIAII < R < Izl, and has power-series expansion

j()...)

1

1

= --

zl-).../z

1

00

.

""().../z)J. z~

= --

The corresponding operator defined by (5.1) is the resolvent (5.2) and the series in (5.2) is called a Neumann series. Of course, the resolvent is defined for any z in the resolvent set p(A) = lC\lT(A). In fact, (A - Z)-l is an analytic (operator-valued) function of z E p(A). To see this, we start with the relation

(A - z)-l

=

(A - ZO)-l - (zo - z)(A - zo)-l(A - Z)-l

for z, Zo E p(A), which the reader is invited to verify (this relation is called the first resolvent identity). Thus

If Iz - zol < (II (A - zo)-lll)-l, the first inverse on the right hand side can be expanded in a Neumann series, yielding 00

(A - z)-l

=

L(z - zo)j (A - zO)-j-l. j=O

Thus (A - z)-l is analytic in a neighbourhood of any Zo E p(A). The following useful result relates eigenvalues and eigenfunctions of A to those of j(A). Proposition 5.16 (Spectral mapping theorem) Let A be a bounded operator, and j a function analytic on a disk of radius> ItAII. If A¢ = )...¢, then j(A)¢ = j()...)¢. Proof. If {aj} are the coefficients of the power series for j, then

o

00

00

j=o

j=o

5.3 Applications to Schrodinger Operators

41

We conclude with a brief discussion of alternatives to, and extensions of, the above definition. Suppose J is analytic in a complex neighbourhood of a(A). We can replace definition (5.1) by the contour integral (called a Riesz integra0 J(A) :=

~ 1 J(z)(A 27ft

Ir

z)-ldz

(5.3)

where r is a contour in C encircling a(A). The integral here can be understood in the following sense: for any 1/;, ¢ E 'H, 1. (1/;, J(A)¢) := -2 7ft

1 J(Z)(1/;, (A- z)-l¢)dz Ir

(knowledge of (1/;, J(A)¢) for all 1/;, ¢ determines the operator J(A) uniquely).

Problem 5.17 Show that the definition (5.3) agrees with (5.1) when J(A) is analytic on {IAI < R} with R > IIAII. Hint: by the Cauchy theorem, and analyticity of the resolvent, the contour r can be replaced by {Izl = R o}, IIAII < Ro < R. On this contour, (A-z)-l can be expressed as the Neumann series (5.2). A similar formula can be used for unbounded operators A to define certain functions J(A) (see [HS]). If A is an unbounded self-adjoint operator, and J is a continuous, bounded function, the bounded operator J(A) can still be defined. One example of this is the definition of e iA in Section 2.2. The definition of J(i'V) using the Fourier transform (Section 2.5.3) provides another example. The reader is referred to [RSI] for the general theory.

5.3 Applications to Schrodinger Operators In this section we address two central issues. Firstly, we want to know for which potentials, V, we can conclude that the Schrodinger operator H = - 2'::..1 + V is self-adjoint. As we saw in Chapter 2, only when H is selfadjoint do we know that the quantum dynamics exist. Secondly, we wish to describe the spectrum, a(H), of H. As we shall see in the next section, knowledge of a(H) gives us important information about the nature of the solutions of Schrodinger's equation (and hence about the evolution of physical states). Our first result covers confining potentials - that is, potentials which increase to infinity with x.

Theorem 5.18 Let V(x) be a continuous function on]Rd satisfying V(x) 2: 0, and V(x) ---> 00 as Ixl ---> 00. Then 1. H

= -..1 + V is self-adjoint on L 2(JR3)

2. a(H) consists of isolated eigenvalues

{An}~=l

with An

---> 00

as n

---> 00.

42

5 Spectral Theory

Remark 5.19 Here and elsewhere, the precise meaning of the statement "the operator H is self-adjoint on L2(JR d)" is as follows: there is a domain D(H), with Co(JR d) c D(H) C L2(JRd), for which H is self-adjoint, and H (with domain D(H)) is the unique self-adjoint extension of -.:1+ V(x), which is originally defined on Co(JRd). The exact form of D(H) depends on V. If V is bounded (a case we consider shortly), then D(H) = D(.:1) = H 2(JRd). Proof The proof of self-adjointness is fairly technical, and can be found in [HS], for example. To prove the second part, suppose ..\ is in the continuous spectrum of H, and let {'l/Jn} be a corresponding spreading sequence. Then as n -7 00,

o ~ ('l/Jn, (H -

..\)'l/Jn)

=

= ~

('l/Jn, -.:1'I/Jn)

+ ('l/Jn, V'l/Jn)

J1V''l/JnI + J 2

inf

yE supp( 1/Jn)

VI'l/JnI 2

V(y) -..\

- ..\

-..\

-7

00

(because {'l/Jn} is spreading), which is a contradiction. Thus the continuous spectrum is empty (or consists of a point at (0). Further, there must be eigenvalues tending to +00. Indeed, let ..\l, ... ,..\n be the first n eigenvalues of the operator H (counting multiplicities), and let 'l/J1, ... ,'l/Jn be corresponding normalized eigenfunctions. In the proof of Theorem 5.28, we show that inf

{1/JE7t:k nD (H)

I 111/J11=1}

('I/J, H'I/J)

=

inf{u(H)\{..\l' ... ' ..\n}},

(5.4)

where Hn := span { 'l/J1, ... , 'l/Jn}. Since the right hand side is < inf uc(H) = 00, we conclude that (5.4) is the smallest eigenvalue of H which is ~ ..\n. Hence it is ..\n+1 (..\n+! = ..\n is allowed, but remember that the eigenfunctions for ..\n and ..\n+1 are orthogonal). By repeating this argument, and using the fact that the eigenvalues "\1, ..\2, ... have finite multiplicity, we see that H has an infinite number of eigenvalues. Since H has no continuous spectrum, these eigenvalues cannot accumulate at a finite number. Hence ..\j -7 00 as j -7 00.

o

Our next theorem covers the case when the potential tends to zero at infinity. Theorem 5.20 Let V : lR d Then

-7

lR be continuous, with V(x)

-70

as Ixl

-700.

1. H = -.:1 + V is self-adjoint on L 2(JRd) 2. uc(H) = [0,(0) (so H can have only negative isolated eigenvalues, possi-

bly accumulating at 0). Proof. For a proof that H is self-adjoint (with domain D(H) = D(.:1) = H2 (JR d)), we refer the reader to [HS], for example. We prove the second part.

We have, by the triangle inequality,

5.3 Applications to Schrodinger Operators

43

Suppose {'ljln} is a spreading sequence. Then the term IIV'IjIn I goes to zero as n ---+ 00 because V goes to zero at infinity and {'ljln} is spreading. So A is in the continuous spectrum of H iff A E O"c( -,1). We have (see Problem 5.9) O"c( -,1) = [0,00), and consequently O"c(H) = [0,00). D Next we consider a Schrodinger operator on a bounded domain, with Dirichlet boundary conditions. Theorem 5.21 Let A be a cube in ~d, and V a continuous function on A. Then the Schrodinger operator H = -,1+ V, acting on the space L2(A) with Dirichlet boundary conditions, has purely point spectrum, accumulating at

00. To be precise, the operator "H on L2(A) with Dirichlet boundary conditions" should be understood as the unique self-adjoint extension of H from Co(A). Proof Suppose A = [0, L]d. Consider the normalized eigenfunctions of the operator -,1 on L2(A) with Dirichlet boundary conditions:

(see Section 7.1), so that (5.5) Now we recall that the eigenfunctions ¢k, k 'IjI

=

L

E

L(Z+)d, form a basis for L2(A):

(¢k, 'IjI)¢k

kEicz+)d

for any 'IjI E L2(A) (this is a special case of a general phenomenon valid for self-adjoint operators). We show now that the operator H has no continuous spectrum. Assume on the contrary that A E O"c(H), and let Un be a corresponding Weyl sequence; i.e. Ilunll = 1, Un ---+ weakly, and II(H -A)unll ---+ 0. By the triangle inequality

°

I (H since

A)Un I 2: II (-,1 - A)unll -

IIVunll 2: II (-,1 -

A)Un II

Ilunll = 1. Writing Un = La~¢k k

where a~ = (¢k, un), and using (5.5), we compute

(-,1 - A)U n =

L(lkl2 k

A)a~¢k

-

sup IVI

(5.6)

44

5 Spectral Theory

and so by the Parseval relation (Proposition 1.3) II( -,1- .\)unI1 2 = ~)lkI2 - .\)2Ia~12.

(5.7)

k

The Parseval relation also gives 1

=

IIun l1 2 =

L

la~12.

(5.8)

k

Now choose K such that Ikl 2 -.\ ~ V2(sup IVI +1) for Ikl ~ K. Then by (5.7) and (5.8), 11(-L1-.\)un I1 2 ~ 2(suplVl+1)2

L

la~12 = 2(suplVl+1)2(1-

Ikl;:O:K Since

Un --'>

0 weakly, a~

= ((Pk, un)

L

la~12).

Ikl
--'>

0

as n --'> 00, for each k. Choose N sufficiently large that la~1 : : : (2K#)-1/2 for k with Ikl < K and n ~ N, where

Then for n ~ N, Llkl
Returning to (5.6), we conclude that for n ~ N, II(H - .\)unll ~ 1, which contradicts the property II(H - .\)unll --'> O. Hence no finite .\ can be a point of the continuous spectrum of H. Proceeding as in the proof of Theorem 5.18, one can show that H has an infinite number of eigenvalues which accumulate at 00. D We conclude this section with a characterization of the continuous spectrum of a Schrodinger operator in a manner similar to the characterization of the point spectrum as a set of eigenvalues.

Theorem 5.22 (Schnol-Simon) Let H be a Schrodinger operator with a bounded potential. Then

CT(H) = closure {A I (H - .\)'lj; = 0 for 'lj; polynomially bounded }. So we see that the continuous spectrum also arises from solutions of the eigenvalue equation, but that these solutions do not live in the space L 2(JR3).

5.3 Applications to Schrodinger Operators

45

Proof. We prove only that the right hand side C a(H), and refer the reader to [RSIVl for a complete proof. Let 'l/J be a polynomially bounded solution of (H - )..)'l/J = O. Let Cr be the box of side-length 2r centred at the origin. Let jr be a smooth function with support contained in C r + 1 , with jr == 1 on Cr, o ::; jr ::; 1, and with sUPr,x,I"'I:S218;;jr(x)1 < 00. Our candidate for a Weyl sequence is

jr'l/J Wr := IlJr'l/Jll·

Note that Ilwrll any R,

=

1. If'l/J

r

J1xl
00.

tf- L2,

we must have Iljr'l/Jll --+

00

as r --+

00.

So for

r

Iw r l2 < _1_2 1'l/J12 --+ 0 - Iljr'l/Jll J1xl
We show that

(H -

)")W r

--+

o.

Let F(r) = fer 1'l/J12, which is monotonically increasing in r. We claim there is a subsequence {rn} such that

F(rn F(r n

+ 2)

-=-'----:---+1.

If not, then there is a

-

1)

> 1 and ro > 0 such that F(r + 3)

~

aF(r)

for all r ~ ro. Thus F(ro + 3k) ~ akF(ro) and so F(r) ~ (const)b r with b = a 1 / 3 > 1. But the assumption that 'l/J is polynomially bounded implies that F(r) ::; (const)r N for some N, a contradiction. Now,

(H - )..)jr'l/J = jr(H - )..)'l/J + [-Ll,jrl'l/J· Since (H - )..)'l/J = 0 and [Ll,jrl = (Lljr)

+ 2Vjr . V,

we have

(H - )..)jr'l/J = (-Lljr)'l/J - 2Vjr· V'l/J. Since 18"'jrl is uniformly bounded,

So

and so

II( H-)")

II(H -

II
),,)wrn II

--+

o. Thus {w rn }

is a Weyl sequence for Hand )... D

46

5 Spectral Theory

5.4 Spectrum and Evolution In this section we explain how the spectrum of a Schrodinger operator gives us useful information about the solutions of the Schrodinger equation. We begin with some simple notions. Definition 5.23 A subspace W c 1{ of a Hilbert space 1{ is invariant under an operator A if Aw E W whenever w E W. Problem 5.24 Show that 1. The span of the eigenfunctions of A is invariant under A 2. For A symmetric, if W is invariant under A, then so is W~ :=

{'4" E 1{ 1('4", w)

= 0 I;f wE

W}

If we restrict a self-adjoint operator A to the invariant subspace orthogonal to the span of all its eigenfunctions, the Weyl theorem (plus some abstract spectral theory) implies that

AI{span of eigenfunctions of

A}l-

has a purely continuous spectrum. N ow we will see how the spectrum of a Schrodinger operator H, acting on L2(JR. d ), gives us a space-time characterization of the quantum mechanical evolution, '4" = e- iHt / fi '4"o. We assume all functions below are normalized. Suppose first that '4"0 E { span of eigenfunctions of H }. Then for any E > 0, there is an R such that infl t Ixl::::R To see this, note that if H'4"o

1

Ixl2:R

1'4"12 ~

1- E.

(5.9)

= )..'4"0, then e- i';;t '4"0 = e- i~t '4"0, and so

1'4"12 =

1

Ixl2:R

1'4"01 2 ----> 0

as R ----> 00. Such a '4" is called a bound state, as it remains essentially localized in space for all time. A proof of (5.9) in the general case is given at the end of this section. On the other hand, if

'4"0

E

{span of eigenfunctions of H} ~

then for all R,

(5.10)

5.4 Spectrum and Evolution

as t

f (t)

----7 ----7

47

Strictly speaking, this convergence is in the sense of ergodic mean: 0 in ergodic mean as t ----7 00 means

00.

T1 Jro

T

f(t)dt

----7

0

as T ----7 00. This result is called the Ruelle theorem (see, eg, [CFKS, HS] for a proof). Such a state, 'l/J, is called a scattering state, as it eventually leaves any fixed ball in space. We conclude that the classification of the spectrum into point and continuous parts corresponds to a classification of the dynamics into localized (bound) states and locally-decaying (scattering) states. Finally, we prove equation (5.9) in the general case. If

'l/Jo

E {

span of eigenfunctions of H},

then 'l/Jo can be written as 'l/Jo = I: j aj'l/Jj where aj E "j'l/Jj. We will assume that the above sum has only a finite number of terms, say 'l/Jo = I:f=l aj'l/Jj. Otherwise an additional continuity argument is required below. The solution 'l/J = e-iHt/n'l/Jo can be written as

'l/J =

L e-iAjt/naj'l/Jj N

j=l

and therefore

Applying the Cauchy-Schwarz inequality to the sum on the right hand side, we obtain

(5.11) The first factor on the right is just

(5.12) To estimate the second factor on the right hand side, for any c > 0, we choose R such that for all j, k,

48

5 Spectral Theory

(5.13) Combining equations (5.11)-(5.13) yields

and so (5.9) follows.

5.5 Variational Characterization of Eigenvalues We consider, for the moment, a self-adjoint operator H, acting on a Hilbert space H. By applying variational techniques to the "energy" functional ('l/J, H'l/J), we will derive an important characterization of eigenvalues of H. We begin with some useful characterizations of operators in terms of their spectra. Theorem 5.25 Let H be a self-adjoint operator with u(H) H 2': a (i.e. (u, Hu) 2': allul1 2 for all u E D(H)).

c [a, 00).

Then

°

Proof. Without loss of generality, we can assume a = (otherwise we can consider H - al instead of H). First we suppose H is bounded (we will pass to the unbounded case later). If b > IIHII, then the operator (H + b)-l is positive, as follows from the Neumann series

(H

+ b)-l =

b- 1 (1

+ b- 1 H)-l

'2) _b00

=

b- 1

1 H)n

j=O

(see (5.2)). For all >. > 0, the operator (H + >.)-1 is bounded, self-adjoint, and differentiable in >. (in fact it is analytic - see Section 5.2). Compute

a

Hence for any

°<

a>. (H + >.)-1 c

< b,

=

-(H + >.)-2 < 0,

5.5 Variational Characterization of Eigenvalues

49

and therefore (H +C)-I > (H +b)-I > O. Now any U E D(H) can be written in the form u = (H + C)-Iv for some v E 1i (show this). Since

(u, (H + c)u)

=

(v, (H

+ C)-IV) > 0

for any c > 0, we conclude that (u, Hu) 2: 0 for all u E D(H), as claimed. In order to pass to unbounded operators, we proceed as follows. Let c > 0 and A := (H + c)-I, a bounded operator since -c rf'. (J(H). For any'\ =I- 0, we have (5.14)

Hence for ,\ > 0, the operator A + ,\ is invertible, and so (J(A) C [0,00) (in fact, one can see from (5.14) that A + ,\ is also invertible if ,\ < -cI, and so (J(A) C [0, c- I]). By the proof above, A := (H + C)-I 2: O. Repeating the argument at the end of this proof, we find that H + c 2: O. Since the latter is true for any c > 0, we conclude that H 2: o. D. Theorem 5.26 Let S('lj;) := ('lj;, H'lj;) for 'lj; E D(H) with 11'lj;11 = 1. Then inf (J(H) = inf S. Moreover, ,\ := inf (J(H) is an eigenvalue of H iff there is a minimizer for S('lj;) among'lj; E D(H) with the constraint II 'lj; II = 1.

Proof In the proof of Theorem 2.21, take z E lR satisfying z < inf S to conclude that z rf'. (J(H). Therefore inf S ::::: inf (J(H). Now let ,\ := inf (J(H). By Theorem 5.25, ('lj;, H'lj;) 2: '\11'lj;112 for any'lj; E D(H). Hence inf S 2: ,\ = inf (J(H), and therefore inf S = inf CJ(H) as required. Now if ,\ = inf (J(H) is an eigenvalue of H, with normalized eigenvector 'lj;o, then S( 'lj;o) = ('lj;o, H'lj;o) = ,\ = inf S, and therefore 'lj;o is a minimizer of S. On the other hand, if'lj;o is a minimizer of S, among 'lj; E D(H), 11'lj;11 = 1, then it satisfies the Euler-Lagrange equation (see Section 12.3)

S'('lj;o)

=

2'\'lj;o

for some ,\. Since S'('lj;) = 2H'lj; (see Section 12.2), this means that 'lj;o is an eigenvector of H with eigenvalue ,\. Moreover,

Since S('lj;o) = inf S = inf (J(H), we conclude that ,\ = inf (J(H) is an eigenvalue of H (with eigenvector 'lj;o). 0 This result leads us to the Ritz variational principle: for any'lj; E D(H),

('lj;,H'lj;) 2:'\ = inf(J(H) and equality holds iff H'lj; = '\'lj;.

50

5 Spectral Theory In particular, if we can find a'ljJ (called a test function) with 11'ljJ11

= 1 and

('ljJ, H'ljJ) < inf O"c(H), then we know that H has at least one eigenvalue below its continuous spectrum. Indeed, by Theorem 5.26, inf O"(H)

= inf S < ('ljJ, H'ljJ) < inf O"c(H),

so A = inf O"(H) must be an eigenvalue of H. Example 5.27 As an example, consider the Schrodinger operator describing the Hydrogen atom n,2 e2 H=--.d-2m Ixl acting on the Hilbert space L 2(JR3) (see Section 4.3). Take the (normalized) test function 'ljJ(x) = Vft3/7fe-ll-lxl for some ft > to be specified later and compute (passing to spherical coordinates)

°

To compute the above integrals, note

and

1

1

d e-arrdr = - e-ardr. da 0 Since fooo e-ardr = a-I, we find fooo e- ar r 2 dr = 2a- 3 , and fooo e-arrdr = a- 2. Substituting these expressions with a = 2ft into the formula for ('ljJ, H'ljJ), we obtain 00

00

o

!i2

('ljJ, H'ljJ) = 2mft2 - e2ft. The right hand side has a minimum at ft

=

me 2/!i2, which is equal to

Since O"c(H) = [0, (0) (according to Theorem 5.20, which can be extended to cover singular potentials like the Coulomb potential - see the remark in Section 5.6) we conclude that H has negative eigenvalues, and the lowest negative eigenvalue, AI, satisfies the estimate

5.5 Variational Characterization of Eigenvalues

51

me4 21t

Al < ---. 2

-

This should be compared with the lower bound

found in Section 4.3. The variational principle above can be extended to higher eigenvalues. Theorem 5.28 (Min-max principle) The operator H has at least n eigenvalues (counting multiplicities) less than inf uc(H) iff An < inf uc(H), where the number An is given by

An =

inf max ('IjJ, H'IjJ). I dimX=n} {.pEX 111.p11=1}

{XCD(H)

(5.15)

In this case, the n-th eigenvalue (labeled in non-decreasing order) is exactly

An. Sketch of proof. We prove only the "if" part of the theorem. The easier "only if" part is left as an exercise. We proceed by induction. For n = 1, the statement coincides with that of Theorem 5.26. Now assume that the "if" statement holds for n ::; m - 1, and we will prove it for n = m. By the induction assumption, the operator H has at least m - 1 eigenvalues, AI, ... ,Am-l (counting multiplicities), all < infuc(H). We show that H has at least m eigenvalues. Let Vm - l denote the subspace spanned by the (nc:irmalized) eigenvectors 'ljJ1, ... , 'ljJm-l, corresponding to AI, ... , Am-I' Then 1. the subspace Vm -

l , and its orthogonal complement, V~_l' are invariant under the operator H, and 2. the spectrum of the restriction, Hlv-L ,of H to the invariant subspace =-1

V~_l is U(H)\{Al,"" Am-I}.

Problem 5.29 Prove statements 1 and 2. Now apply Theorem 5.26 to the operator Hlv-L

=-1

inf

{.pEV';;_l nD (H)

111.p11=1}

('IjJ, H'IjJ)

=

to obtain

inf{ u(H)\ {AI, ... ,Am-I}}.

(5.16)

On the other hand, let X be any n-dimensional subspace of D(H). There exists ¢ E X such that ¢ ~ Vm - l , and II¢II = 1. We have,

(¢, H ¢)

~

inf

{.pEv';;_lnD(H)

and therefore Am defined by (5.15) obeys

111.p11=1}

('IjJ, H'IjJ)

52

5 Spectral Theory Am~

Since Am

inf

{,pEv';;-_lnD(H)

11;111I>11=1}

(1/;,H1/;).

(5.17)

< inf aAH) by assumption, and due to (5.16), we have (5.18)

is the m-th eigenvalue of H. Moreover, Equations (5.16)-(5.18) imply that

Am

~

A;".

Now we show that Am ::; A;". Let 1/;m be a normalized eigenvector corresponding to A;", and let Vm = span{ 1/;1, ... ,1/;m}. Then

Hence Am = A;". Thus we have shown that H has at least m eigenvalues (counting multiplicities) < inf CYc(H) , and these eigenvalues are given by (5.15). 0 This theorem implies that if we find n independent (normalized) vectors (PI, ... ,
for each j, then H has at least n eigenvalues less than inf CYc(H), and these eigenvalues satisfy the bound

This result will be used in Section 5.6. There is another formulation of the min-max (more precisely sup-inf) principle, in which Equation 5.15 is replaced by the equation

An =

sup inf (1/;, H1/;). I dimX=n} {,pEX-L 1111I>11=1}

{XCD(H)

A proof of this theorem is similar to the proof above. The following useful statement is a simple consequence of the min-max principle. Suppose A and B are self-adjoint operators with A ::; B. Denote the j-th eigenvalue of A below its continuous spectrum (if it exists) by Aj(A) (and similarly for B). Suppose also that the eigenvectors of B corresponding to the eigenvalues A1(B), ... , Aj(B) lie in D(A). Then Aj(A) ::; Aj(B). To see this, let Vj denote the span of the first j eigenvectors of B, and observe that max{,pEVj 1111I>11=1} (1/;, B1/;) = Aj(B). Since Vj c D(A), (5.15) gives A (A) < J

-

< -

max (.1, A·I,) 1111I>1I=1} ,/" '/'

{1I>EVn

max (.1, B·I,) = A (B). 1111I>11=1} ,/" '/' J

{1I>EVn

5.6 Number of Bound States

53

5.6 Number of Bound States Let H

V(x)

= - :;: ~ + V (x) ---+ 00

as

Ixl

---+ 00.

be a Schrodinger operator acting on L2 (JR3). Assume In this section, we address the questions

- Does H have any bound states? - If so, how many bound states does it have? We begin with a very simple example. Suppose that V(x) ~ O. Clearly 0, and so H certainly has no negative eigenvalues. On the other hand, <Jc(H) = [0(0) by Theorem 5.20. So H has no isolated eigenvalues. The next example is physically clear, but mathematically more subtle: H

~

If

X· \7V(x) ::::: 0, i.e. V(x) is repulsive, then H has no eigenvalues.

Let us assume here that V is twice differentiable, and that V(x), together with X· \7V(x), vanish as Ixl ---+ 00. The proof of the above statement is based on the following virial relation: define the self-adjoint operator 1

A := "2(x, p

+ p. x).

This operator is the generator of a one-parameter group of unitary transformations called dilations:

for () E R Let us show formally (i.e. ignoring domain issues) that if'¢ is an eigenfunction of H, then (,¢, i[H, A]'¢) = O. (5.19) Letting A be the eigenvalue corresponding to ,¢, and using (4.1), we have

(,¢, i[H, A]'¢) = (,¢, i[H - A, A]'¢) = -2Im((H - A),¢, A'¢). Since (H - A)'¢ = 0, (5.19) follows. On the other hand, a simple computation (left as an exercise) yields i[H, A]

11,2

= --<1- X· \7V(x). m

(5.20)

Therefore i[H, A] ~ - ~ ~ by the repulsivity condition on V(x), which implies

(,¢, i[H, A]'¢)

~ nm2 11 \7,¢112 > 0,

a contradiction to (5.19). Thus the operator H, with V(x) satisfying x . \7V(x) ::::: 0 has no eigenvalues.

54

5 Spectral Theory

Problem 5.30 Show (5.20).

We turn our attention now from a situation where there are no bound states, to one where there are many. In particular, we will prove that the Schrodinger operator li 2 q H=--Ll-(5.21 ) 2m Ixl with q > 0, has an infinite number of bound states. The potential V(x) = -qllxl is called the (attractive) Coulomb potential. For appropriate q, the operator H describes either the the Hydrogen atom, or a one-electron ion. In Section 7.7, we will go further, solving the eigenvalue problem for H exactly, finding explicit expressions for the eigenfunctions and (infinitely many) eigenvalues. Nevertheless, it is useful to be able to prove the existence of infinitely many bound states using an argument that does not rely on the explicit solvability of the eigenvalue problem. We would first like to apply Theorem 5.20 to locate the continuous spectrum. However, the fact that the Coulomb potential is singular at the origin is a possible obstacle. In fact, Theorem 5.20 can be extended to cover this case (see, eg, [CFKS]), and we may conclude that H is self-adjoint, with continuous spectrum equal to the half-line [0,00). To prove that the operator (5.21) has an infinite number of negative eigenvalues, we will construct an infinite sequence of normalized, mutually orthogonal test functions, un(x), such that (5.22) The "min-max principle" (described in Section 5.5) then implies that H has an infinite number of eigenvalues. We begin by choosing a single function, u(x), which is smooth, and which satisfies

Ilull = 1

and

supp(u)

c {x

E R3 11

< Ixl < 2 }.

Then we set un(x) := n- 3 / 2 u(n- 1 x) for n = 1,2,4,8, .... Problem 5.31 Show that (u m , un)

n.

= 6mn , and that (u m , HUn) = 0 if m =j:.

Given the results of the exercise, it remains to show (5.22) for n sufficiently large. Indeed, changing variables to y = n-1x, we compute

5.6 Number of Bound States

55

for n sufficiently large, as the second term - the potential term - prevails for large n. Thus we have proved that the operator (5.21) has an infinite number of negative eigenvalues. With this example in mind, we address the question of whether the Schrodinger operator fi2

H=-2m Ll + V (x) has a finite (including possibly 0) or infinite number of negative eigenvalues. Assume V (x) behaves at infinity as

V(x) = clxl- a

for

Ixl

sufficiently large

(5.23)

for some constant c. We test the operator H on the functions un(x) constructed above. It is left as an exercise to show that, as above, (u m , HUn) = 0 for m =f. n, and if if

0: 0:

< 2 and c < 0 >2

for n sufficiently large. Problem 5.32 Prove these last two statements. Invoking the min-max principle again, we see that if 0: < 2 and c < 0, the operator H has an infinite number of bound states. It is shown below that H has only a finite number of bound states if 0: > 2. The borderline for the question of the number of eigenvalues is given by the inequality 1 -Ll ~ 41xl 2 (the Uncertainty Principle - see Section 4.2). This inequality shows that the kinetic term - ::: Ll dominates at 00 if 0: > 2, or if 0: = 2 and ~c > - ~. Otherwise, the potential term favouring eigenvalues wins out. This simple intuition notwithstanding, there is presently no physically motivated proof of the finiteness of the point spectrum for 0: > 2. The proof presented below uses mathematical ingenuity rather than physical intuition. We now prove finiteness of the number of eigenvalues for 0: > 2. To simplify the argument slightly, we assume the potential V(x) is non-positive and denote U (x) := - V (x) ~ O. Let >.. < 0 be an eigenvalue of H with eigenfunction ¢. The eigenvalue equation (H - >..)¢ = 0 can be re-written as fi2

(--Ll- >..)¢ 2m Since >..

=

U¢.

< 0 is in the resolvent set of the operator - ::: Ll, we can invert

(- ~ Ll - >.) to obtain

56

5 Spectral Theory

the homogeneous Lippmann-Schwinger equation. By introducing the new function v := U 1 / 2 ¢>, this equation can be further re-written to read

v

=

K(>..)v

where the operator K(>..) is defined as

We can summarize the above derivation as follows:

>.. < 0 EV H and so

f--+

1 EV K(>..)

#{A < 0 I >.. EV H} = #{A < 0

11 EV K(>..)}.

(5.24)

The next step is to prove that

#{A < 0

11 EV K(>")} = #{v > 1 I v EV K(O)}.

(5.25)

To prove (5.25), we begin by showing that

:>.. K(>..) > 0 and

K(>..)

->

0 as

V

>..:s

>.. ->

(5.26)

0

(5.27)

-00.

Writing

(¢>, K(>")¢» = (U1/2¢>, (- : : Ll- >..)-lU 1/ 2¢» and differentiating with respect to

:>.. (¢>, K(>")¢»

>.., we obtain

=

(U1/2¢>, (- : : Ll - >..)-2U 1/ 2¢»

=

II( -~Ll- >..)-lU 1/ 2¢>112 > 0 2m

which proves (5.26). To establish (5.27), we need to derive the integral kernel of the operator K(>"). Using the fact that the operator (- ;~ Ll - >..)-1 has integral kernel l>2 ( __ '&_Ll_ >..)-l(x,y) 2m

=

m 2nn2 1x

- yl

/2"'1>-11

e- Y ~

x-y

1

(see Equation (2.22) of Section 2.5.3), we find that the integral kernel for

K(>..) is

5.6 Number of Bound States

57

(see the mathematical supplement, Section 5.7). Using the equation

IIKII:s;

(

r

J'iiI!.3 X 'iil!.3

1/2

IK(x, yWdxdy )

(again, see Section 5.7), we obtain

Since exp( -J2rr;Y·llx - yl) ~ 0 as .A ~ -00, Equation (5.27) follows. Now we show that the relations (5.26) and (5.27) imply (5.25). By (5.27), for all .A sufficiently negative, all of the eigenvalues of K(.A) are less than 1. By (5.26), the eigenvalues, vm(.A), of K(.A) increase monotonically with .A. Hence if vm(.A m ) = 1 for some .Am < 0, then vm(O) > vm(.A m ) = 1. Similarly, if vm(O) > 1, then there is a .Am < 0 such that vm(.A m ) = 1. In other words, there is a one-to-one correspondence between the eigenvalues vm(O) of K(O) which are greater than 1, and the points .Am at which some eigenvalue vm(.A) crosses 1 (see Fig. 5.3).

Fig. 5.3. Eigenvalues of K(.A). Thus (5.25) follows. The relations (5.24) and (5.25) imply

#{.A < 0

I

.A EV H} = #{v > 1

I v EV K(O)}.

(5.28)

The quantity on the left hand side of (5.28) is what we would like to estimate, while the quantity on the right hand side is what we can estimate. Indeed, we have

58

5 Spectral Theory

L

#{v> 1 I v EV K(O)} =

1:::;

: :;

v;'

)

v m > 1,v",EV K(O)

v m>l,vm EV K(O) 1/2

L

(

1/2

L

(

v;'

)

=

(tr(K(0)2)) 1/2

vmEVK(O)

(5.29) (see Section 9.14 for the definition of tr, the trace). On the other hand (see Section 5.7),

tr(K(0)2) = / IK(O)(x, y)1 2dxdy = (2:n2) 2 /

~~x2~j;) dxdy

Collecting equations (5.28)- (5.30) and recalling that V(x) obtain

#{,\ < 0 I ,\ EV H} :::;

~ 27rn2

-U(x), we

( / JV(x)V(y)1 dXdY) 1/2

Ix -

Yl2

Under our assumption (5.23) on the potential V(x), with a

/

=

(5.30)

JV(x)V(y)1 d d X Y < Ix-y 12

> 2,

00,

so that the number of negative eigenvalues of the operator H is finite. This is the fact we set out to prove. The argument used above (Equation (5.28) in particular) is called the Birman-Schwinger principle, and the operator K('\) is the Birman-Schwinger operator.

5.7 Mathematical Supplement: Integral Operators Let K be an integral operator on L 2 (lR,d):

(K¢)(x):= where K : lR,d x lR,d include

---+

r K(x, y)¢(y)dy

JlRd

C is the integral kernel of the operator K. Examples

1. K = g( -ih,\!) for which the kernel is

K(x, y) = (27rn)-d/2g(x - y) (see Section 2.5.3).

(5.31 )

5.7 Mathematical Supplement: Integral Operators

2. K

=

59

V (multiplication operator) for which the kernel is

K(x, y) = V(x)J(x - y). The following statement identifies the kernel of the composition of integral operators. The proof is left for the reader. Proposition 5.33 If K1 and K2 are integral operators (with kernels K1 and K 2), then the integral kernel of K := K1K2 is

Problem 5.34 Prove this.

Recall (see Problem 2.18) that if the kernel K(x, y) satisfies K(y, x) = K(x, y), then the corresponding integral operator K is symmetric. Proposition 5.35 Let K be an integral operator with kernel, K(x, y), satisfying K(x,y) = K(y,x), and which lies in L2 of the product space: K (x, y) E L2 (JRd XJRd). Then K is a bounded, self-adjoint operator on L2 (JRd) , and (5.32) Proof To show that the operator K is bounded, we estimate by the CauchySchwarz inequality,

IJ K(x, Y)U(Y)dyl ::; (J IK(x, YWdY) (J lu(y)12dY) 1/2 1/2

This implies

IIKuI12::;

J

IK(x, y)1 2dxdy

J

lu(y)1 2dy

which in turn yields (5.32). Since K is bounded and symmetric, it is selfadjoint by Lemma 2.19. 0

6 Scattering States

In this chapter we study scattering states in a little more detail. As we saw in Section 5.4, scattering states are solutions of the time-dependent Schrodinger equation

in a7/J at

= H 7/J

(6.1)

with initial condition orthogonal to all eigenfunctions of H: (6.2) where Hb := span{ eigenfunctions of H} is the subspace of bound states of H. We will have to make a more precise assumption on the potential V(x) entering the Schrodinger operator H = - ; : L1 + V (x): we assume, for simplicity, that la~V(x)1 ::::: 0(1 + Ixl)-JL-1a l (6.3) for lal ::::: 2 and for some J.L > 0 (and 0 a constant). The notation needs a little explanation: a is a multi-index a = (aI, a2, a3) with each aj a non-negative integer, lal = L~=l aj, and 3

aax = II aaj

Xj'

j=l

The question we want to address is what is the asymptotic behaviour of the solution 7/J = e- iHt / li 7/Jo of (6.1)-(6.2) as t --+ 00. First observe that (5.10) shows that 7/J moves away from any bounded region of space as t --+ 00. Hence, since V(x) --+ 0 as (6.4) Ixl --+ 00, we expect that the influence ofthe potential V(x) diminishes as t --+ 00. Thus the following question arises: does the evolution 7/J approach a free evolution, say cP = e-iHot/licpo, where Ho = - ;: L1, for some CPo E L2(JR.3), as t --+ 007 Put differently, given 7/Jo E is there CPo E L 2(JR.3) such that

Ht,

(6.5)

S. J. Gustafson et al., Mathematical Concepts of Quantum Mechanics © Springer-Verlag Berlin Heidelberg 2003

62

6 Scattering States

as t --+ oo? This (or a modification of this question discussed below) is the problem of asymptotic completeness. It is the central problem of mathematical scattering theory. It conjectures that all the possible free motions, together with the bound state motions, form a complete set of possibilities for the asymptotic behaviour of solutions of time-dependent Schrodinger equations. There are two principal cases depending on the decay rate, /-L, of the potential V(x) (more precisely, the decay rate, /-L + 1, of the force -\i'V(x)), and it is only in the first of these cases that the asymptotic completeness property formulated above holds. We now discuss these cases in turn.

6.1 Short-range Interactions: J.L

>

1

In this case, asymptotic completeness can be proved under condition (6.3). We can reformulate the asymptotic completeness property by defining the wave operator, n+: n+ ¢:= lim eiHt/lie-iHot/Ii¢. t-+oo

(6.6)

Below we will show that under the condition (6.3) with /-L > 1, the limit exists for any ¢ E L 2(JEt3). Further, the operator n+ is an isometry: (6.7) Indeed, (6.6) implies

Iln+ ¢II = t-+oo lim IleiHt/lie-iHot/Ii¢11 = II¢II since the operators eiHt / 1i and e-iHot/1i are isometries. The existence of the wave operator n+ means that given a free evolution e-iHot/li¢o, there is a full evolution e-iHt/Ii'I/Jo such that (6.5) holds. To see this, note that since eiHt / 1i is an isometry, (6.5) can be re-written as

as t

--+ 00,

which is equivalent to the relation 'l/Jo

=

n+¢o.

(6.8)

Thus the existence of the wave operator n+ is equivalent to the existence of scattering states - i.e. states e-iHt/Ii'I/Jo for which (6.5) holds for some ¢o. We have Ran(n+) c Indeed, for any ¢o E L 2(JEt3) and 9 E Jib such that Hg = )..g, we have

Hr.

(g, n+¢o)

= lim (g, eiHt/lie-iHot/li¢o) = lim (e-iHt/lig, e-iHot/li¢o) t~oo

t---+oo

(6.9)

6.1 Short-range Interactions:

J-L

>1

63

There is a general theorem implying that the right hand side here is zero, but we will prove it directly. Recalling expression (2.15) for the action of the free evolution operator e-iHot/li, we find

and therefore, if ¢o and 9 are integrable functions, i.e.,

r l¢o(x)ldx <

Jffi.3

r Ig(x)ldx <

and

00

Jffi.3

00,

we have

and thus the right hand side of (6.9) vanishes. For general ¢o and 9 the result is obtained by continuity. This argument can also be extended to cover the case where 9 is a linear combination of eigenfunctions. The property of asymptotic completeness states that or i.e., that the scattering states and bound states span the entire state space L2(JR3). We can similarly define the wave operator n- describing the asymptotic behaviour as t -+ -00:

n- ¢:=

lim

eiHt/lie-iHot/Ii¢.

t-+-oo

This operator maps free states e-iHot/li¢o into states e-iHt/Ii'l/Jo which approach these free states as t -+ -00. One can also define the scattering operator

s:= n+*n-

which maps asymptotic states at t =

e-iHJlfi

L

-00

into asymptotic states at t =

00:

64

6 Scattering States Fig.6.1. S: ¢_

--+

¢+

By changing variables, we can obtain the following intertwining relations

and Differentiating these relations at t = 0, we obtain

(6.10) and

HoS

=

SHoo

The property of asymptotic completeness implies that the Hamiltonian H restricted to the invariant subspace is unitarily equivalent to the free Hamiltonian Ho: H = n±Hon±* on

Ht

Ht.

Let n± (x, y) be the integral kernels of the operators n±, and let '¢± (x, k) denote their Fourier transforms with respect to the second variable, y. Equation (6.10) implies that

H'¢±(x, k) =

Ikl 2 '¢±(x, k). 2m

(6.11)

In other words, '¢± (x, k) are generalized eigenfunctions of the operator H, with eigenvalue Ik1 2 /2m. These generalized eigenfunctions are of the form

as Ixl of H.

--+ 00

(see, eg., [Va, RSIII]). They are called scattering eigenfunctions

Problem 6.1 Prove relation (6.11).

6.2 Long-range Interactions: JL

<1

In this case, the relation (6.5) must be modified to read

IleiHt/Ii,¢o -

e-iS(t)¢oll

--+

0

--+ 00, where for ~ < f-L :::; 1, the operator-family S(t) is defined to be S(t) = S(p, t), where the function S(k, t) satisfies the equation

as t

6.3 Existence of Wave Operators

:tS(k, t)

= Ho(k) + V

65

(~ kt) .

Here H 0 (k) = 2~ Ik 12 is the classical free Hamiltonian function. The operator S(p, t) is defined according to the rules described in Sections 2.5.3 and 3.5. Note that in this equation, the coordinate x in the potential Vex) is replaced by its free, classical expression, !nkt. With this modification, the asymptotic completeness property can again be established under assumption (6.3) with ~ < J.L ::; 1. For 0 < J.L ::; ~, the expression for the operator-family Set) is more complicated (see [DC]). As in the short-range case, one can introduce modified wave operators and investigate their properties (see the references given in Chapter 17).

6.3 Existence of Wave Operators Finally, as promised, we prove the existence of the wave operators Q±. To simplify the proof, we impose a somewhat stronger condition on the potential: Vex) E £2(JR3). Below, we also make use of the space

Theorem 6.2 If V E £2(JR3), then the wave operators Q±, introduced formally above, exist.

Proof Denote Qt := eiHte-iHot (we drop the h in what follows, just to simplify the notation). Since IIQtl1 ::; canst, uniformly in t (in fact, IIQtl1 = 1), it suffices to prove the existence of the limit (6.6) (and similarly for t -+ -(0) on functions ¢> E £2(JR3) n £1 (JR3) (the existence of the limit for ¢> E £2 will then follow by approximating ¢> by elements of the dense subspace £2 n £1). For t 2: t', we write the vector-function Qt¢> - Qt' ¢> as the integral of its derivative: Qt¢> _ Qt' ¢> = .!!:....Qs¢>ds. ds Using the relation d~eiHs = iH e iHs , and similarly for e-iHos (see Chapter 2), we find

rt

it'

as H - Ho

= V. Since IleiHsl1 = 1, and using the inequality I

J ¢>(s)dsll

::;

J II ¢>( s) II ds (which can be proved by writing the integral as a limit of Riemann

sums and using the triangle inequality - see [Fo]), we have

(6.12)

66

6 Scattering States

We estimate the integrand as follows:

IlVe-iHos¢112 =

r lV(x) (e-iHOS¢) (x)1 2dx (e-iHOS¢) (X'W r lV(x)1 2dx, E1R JlR

JlR 3

:::: sup I x/

yielding

IlVe-iHos¢11 ::::

3

3

11V11£2 sup l(e-iHos¢)(x')I· x'

Finally, recalling the bound (see (2.16))

from Section 2.5.3, we obtain

and so

1

00

IlVe-iHos¢llds::::

(const)IIVIIL211¢11£1·

This shows that the right hand side in (6.12) vanishes as t', t ---+ 00. In other words, for any sequence tj ---+ 00, {ntj ¢} is a Cauchy sequence, and so {nt¢} converges as t ---+ 00. Convergence for t ---+ -00 is proved in the same way. 0

7 Special Cases

In this chapter we will solve the Schri:idinger eigenvalue equation in a few special cases (i.e., for a few particular potentials) which not only illustrate some of the general arguments presented above, but in fact form a basis for our intuition about quantum behaviour.

7.1 The Infinite Well Let W be the box [0, Lj3 C ~3. We take V(x) = { 0 x

E W ()()X~W

as our potential. This means we take '¢ == 0 outside W, and that we impose Dirichlet boundary conditions

(7.1)

'¢i&w = 0

on the wave function inside W. It is a simple matter to solve the eigenvalue equation

(7.2) in W with the boundary condition (7.1), using the method of separation of variables. Doing so, we obtain eigenvalues (energy levels) >02

En

=

2

3

7r """" 2 2mL2 ~nj It

)=1

with corresponding eigenfunctions (bound states) 3

'¢n(x) = IIsin(7rn j Xj)

.

)=1

L

for each integer triple n = (nl' n2, n3), nj 2:: 1. We see that the eigenvalue En occurs with degeneracy equal to #{(ml, m2, m3)i L m; = L nj}. We remark that the ground-state (lowest) energy

E(1,I,I) =

~~Z~ is non-degenerate.

S. J. Gustafson et al., Mathematical Concepts of Quantum Mechanics © Springer-Verlag Berlin Heidelberg 2003

68

7 Special Cases

Problem 7.1 Determine the spectrum of the operator x and the eigenvalues of the operator p on L2(W) with zero boundary conditions.

7.2 The Torus Now we consider a particle on a torus T = ]R3 I'll}. This corresponds to taking V == 0 in the cube W, but this time imposing periodic boundary conditions. That is, we solve the eigenvalue equation (7.2) with boundary conditions 1jJ(x)IXj=o fJ1jJ I fJxk IXj=o

= 1jJ(X)lxj=L

= fJ1jJ I fJxk IXj=L

for all j, k. This leads to (separation of variables again) eigenfunctions 1jJn(x)

= II 3

j=l

j Xj)} { S . (2nn lll-

cos

(2n!;:jXj) L

with eigenvalues En

=

-----v L n; m 2n 2 li 2

3

1

for nj 2: O. The ground state energy, E(o,o,o) = 0 is non-degenerate, with eigenfunction 1jJ(O,O,O) == 1. The spacing between energy levels is greater than for the infinite well, and the degeneracy is higher. Problem 7.2 Determine the spectrum of x, and the eigenvalues of p, on L2(W) with periodic boundary conditions. Challenge: show that p has no

continuous spectrum (hint - use the Fourier transform for periodic functions).

7.3 A Potential Step Here we examine the one-dimensional potential V(x)

=

{~

x>O x:S;O

with Vo > O. Consider the "eigenvalue" problem _!f.1jJIf + V1jJ 2m

= E1jJ.

(7.3)

We put the term eigenvalue in quotation marks because we will allow solutions, 1jJ, which are not L2-functions. Solving this eigenvalue problem separately in the two different regions gives us a general solution of the form

7.4 The Square Well

'Ij;

= {

69

(n2k5 =E) x 0

There are no bound states (L2 solutions), but we can say something about the scattering states. Suppose 0 < E < Vo, and take kl = iK where K = v(2m/!i2)(Vo - E) > O. Then for a bounded solution, we require D = O. Imposing the condition that 'Ij; be continuously differentiable at 0, that is

leads to the equations

iko(A - B) = -KG.

A+B=G,

After some manipulation, we find that

B A

ko - iK ko + iK'

or making the dependence on the energy E explicit,

B A Similarly, if E

v'E - iJVo v'E + iJVo -

E E'

> Vo, we obtain B A

and

R(E) :=

v'E-JE-Vo v'E+JE-Vo'

1~ 12 = ~2 [2E -

Vo - 2VEVE - Vo]2

(R(E) is called the reflection coefficient). In particular, if R(E)

~ 1-

EvoVo

«

1, then

4J E - Vo Vo

and almost all of the wave is reflected. This is in spite of the fact that the energy of the particle lies above the barrier. In classical mechanics, the particle would pass over the barrier.

7.4 The Square Well Now we consider a potential well of finite depth Vo, and width a (see Fig. 7.1).

70

7 Special Cases V(x)

-aJ2

aJ2 x

Fig. 7.1. The finite well. The determination of the point and continuous spectra is straightforward, and is left as an exercise. Problem 7.3 Show

1. CTc(H) 2.

= [0,00)

CTp(H) c (- Vo, 0)

3. the equations for the eigenvalues E, for - Vo ::; E ::; 0, are ktan(ak/2) = K

kcot(ak/2) = -K

where

We study the scattering states - i.e. solutions of (7.3) for E > 0 which converge to plane waves as x --+ ±oo - in more detail. Consider the situation where a plane wave 'l/Jinc(X) = Aeikx is incoming from the left. Then on the left of the well, 'l/J is a superposition of incoming and reflected plane waves:

'l/J = Aeikx + AB(E)e- ikx

x < -a/2

while on the right of the well, 'l/J is an outgoing (transmitted) plane wave:

'l/J = AC(E)eik(x-a)

x > a/2

Problem 7.4 Show that

C(E)

=

_

[cos(ka) - i

p + k2 2kk

where and

_

sin(ka)r 1

k =

v

2mE 11,2 •

7.5 The Harmonic Oscillator

71

This implies

(T(E) is the transmission coefficient) which is sketched in Fig. 7.2.

0+----------------------------Fig. 7.2. Transmission coefficient. We see that at the energies satisfying sin(ka) n 2 rr2fi2

E

= - Vo + 2ma 2 > 0,

= 0,

i.e.

n = 1,2, ...

T(E) has maxima (T(E) = 1) which are called resonances. The corresponding values of E are the resonance energies. We remark that for large n these are approximately equal to the energy levels of the infinite well of the same width.

7.5 The Harmonic Oscillator The harmonic oscillator Hamiltonian is H

fi2

1

2m

2

2

2

= --.::1+ -mw Ixl .

By Theorem 5.18, a(H) consists of isolated eigenvalues, increasing to infinity. We will solve the eigenvalue problem explicitly for this operator. For simplicity, we consider the one-dimensional case. First, to remove all the constants we re-scale: x f--+ AX so that p f--+ where p = -ifi(djdx). Setting

A=

a

tP

gives H

f--+

fiwHnew where

72

7 Special Cases

with pnew = -i(d/dx). The number). gives the length scale in the original problem (the scale on which the eigenfunctions of the Hamiltonian Hare concentrated) . We derive a representation of the operator H which facilitates its spectral analysis. It also prepares us for a similar technique we will encounter in the more complex situation of quantum field theory (see Chapter 14). We introduce the creation and annihilation operators

The commutation relation [a, a*] is easily verified. The Hamiltonian a* as follows:

=1

Hnew

H new

(7.4)

can be re-written in terms of a and 1 2

= a*a + -.

We say that this expression is in nO'T'mal form because a* appears to the left of a. The particle number operator, N:= a*a satisfies the relations Na=a(N-1) Na*

= a*(N + 1)

(7.5) (7.6)

(use (7.4) to check these). Clearly,

Theorem 7.5 We have 1. N

~

0

2. IJ(N) = Z+ (the non-negative integers), each eigenvalue having multiplicity 1. Proof.

1. This is easy, because a* is the adjoint of a, so

for any 'Ij;.

7.5 The Harmonic Oscillator

73

2. Therefore, N'IjJ = 0 iff a'IjJ = O. Note that the function

'ljJo(x) := ce- x2 / 2 (c a constant) is the unique family of solutions of a'IjJ

=

1

..j2(x + d/dx)'IjJ

= 0,

and hence of N'IjJ = O. Thus 'ljJo (normalized by setting c := (27r)-1/4) is the ground state. The commutation relation (7.6) implies

Na*'ljJo

= a*'ljJo

and in general Thus

¢n := (a*)n'IjJo is an eigenfunction of N with eigenvalue n.

Problem 7.6 Show that II¢n11 2 = n!. Hint: write II¢n11 2 = ('ljJo, an (a*)n'IjJo). then push the a's through the a*'s (including the necessary commutators) until they hit 'ljJo and annihilate it. So

'ljJn:=

~(a*)n'IjJo

vn!

is a normalized eigenfunction of N with eigenvalue n. We now show that these are the only eigenfunctions. It follows from the commutation relations that if 'IjJ is any eigenfunction of N with eigenvalue A > 0, then

(7.7) If we choose m so that A - m But this implies

< 0 we contradict N

~

0 unless am'IjJ

= O.

aj'IjJ = c'IjJo for some integer j (c a constant), so by (7.7), A = j. Applying (a*)j to this equation, and using the commutation relations, we can show 'IjJ = c'IjJj (c another constant), so we are done. D

Corollary 7.7

cr(Hnew) = {n + 1/21 n = 0,1,2, ... } with eigenfunctions 'ljJn

= (l/Vnl)(a*)n'IjJo.

Finally, then, the spectrum of the original harmonic oscillator Hamiltonian is cr(H) = {1lw(n + 1/2) 1n = 0,1,2, ... } with eigenfunctions obtained by rescaling the 'ljJn's.

74

7 Special Cases

7.6 A Particle on a Sphere We consider a particle moving on the unit sphere §2 = {x E ~3 I Ixl = I} in ~3. Its Hamiltonian is -;~Lln, the Laplace-Beltrami operator on §2, given in spherical coordinates (B, rjJ), by

The eigenfunctions of Ll n are the well-known spherical harmonics, (7.8) where l = 0,1, ... ; k E {-l, -l + 1, ... ,l - 1, l}; Czk is a constant; and the Legendre function FZk can be written as (7.9) The spherical harmonics satisfy

-LlnY/ = l(l Problem 7.8 bit tedious).

+ l)Y/.

(7.10)

Check (7.10) using (7.8) and (7.9) (unfortunately, this is a

yt,

It turns out that the spherical harmonics, comprise an orthonormal basis of the Hilbert space L2(§2; dD) of L2 functions on the sphere, with the measure dD = sin2(B)dBdrjJ (see, eg, [LL]). We make a few remarks about the connection to angular momentum. The quantum-mechanical angular momentum L = (L1' L 2, L 3 ) is the self-adjoint (vector-valued) operator L =x xp where p = -ifi,\! as usual. We define also the squared magnitude of the angular momentum, L2 = Li+L§+L§. The following facts are easily checked: 1. L 2Yzk = fi 2l(l + l)Yi k 2. L 3Yzk = fikYik.

Thus we see that the spherical harmonics are simultaneous eigenfunctions of the angular momentum operators L3 and L2.

7.7 The Hydrogen Atom A hydrogen atom consists of a proton and an electron, interacting via a Coulomb force law. Let us make the simplifying assumption that the nucleus

7.7 The Hydrogen Atom

75

(the proton) is infinitely heavy, and so does not move. Placing the nucleus at the origin, we have the electron moving under the influence of the external potential V(x) = -e 2 flxl, where e is the charge of the proton, and -e that of the electron. The appropriate Schrodinger operator is therefore

lt2 H = --.1 - e 2 flxl 2m acting on the Hilbert space L 2(JR3). In Chapter 8 we will see how to reduce the problem of the more realistic hydrogen atom - when the nucleus has a finite mass (a two-body problem) - to the problem studied here. As usual, we want to study the spectrum of H. The first step is to invoke Theorem 5.20. As remarked in Section 5.6, Theorem 5.20 can be extended to cover the Coulomb potential V(x) = _e 2 flxl, which is singular at the origin (see, eg, [CFKS]). We may conclude that H is self-adjoint, with continuous spectrum equal to the half-line [0,(0). Our goal, then, is to find the bound-states (eigenfunctions) and bound-state energies (eigenvalues). It is a remarkable fact that we can find these explicitly. Indeed, aside from the infinite well, the only multi-dimensional potentials for which the Schrodinger eigenvalue problem can be solved explicitly are the harmonic oscillator and the Coulomb potential. Because the Coulomb potential is radially-symmetric (depends only on r = lxI), it is natural to work in spherical coordinates (r, (), ¢), where Xl

= rsin(()) cos(¢),

a :::; () < 7r, a :::; ¢ < 27r.

X2

= rsin(()) sin(¢),

X3

= rcos(()),

In spherical coordinates, the Laplacian becomes

where .1r is the "radial Laplacian" given by

.1r

[)2

=~ ur

2 [)

+--;:)

r

Ur

(.1 r depends only on the radial variable), and .1 n is the Laplace-Beltrami operator on §2, introduced in Section 7.6. To solve the eigenvalue problem, we seek eigenfunctions of H in the separated-variables form 1jJ(r, (), ¢)

= R(r)Y/((), ¢)

where Yzk is a spherical harmonic. Plugging this into the eigenvalue equation H 1jJ = E1jJ, we obtain

(7.11 )

76

7 Special Cases

The solutions of the ODE (7.11) are well-studied (see, eg, [LL]). Without going into details, we remark that one can show (by power-series methods) that (7.11) has square-integrable solutions only for

n:=

e Fm nV2if E {l+1,l+2, ... }. 2

The corresponding eigenfunctions,

are of the form

Rnl

where p = 2::;,~2 r, and Fnl is a polynomial. In full, then, the solutions of the eigenvalue problem H'Ij; = E'Ij; are

where

l=0,1,2, ... ;

kE{-l,~l+l,

... ,l};

nE{l+1,l+2, ... };

and the eigenfunctions are (7.12) So we see that the Hydrogen atom has an infinite number of bound states below the continuous spectrum (which starts at zero), which accumulate at zero (this result was obtained in Section 5.6 by a general technique, without solving the eigenvalue problem). The ground state energy, attained when l = k = 0, n = 1, is El = -me 4 /21l? An easy count finds the degeneracy of the energy level En to be n-l

2)2l + 1) = n 2 . l=O

Finally, we note that the expression (7.12) is in agreement with the empirical formula ("Balmer series") 1

1

l1E = R(- - -).

n}

n;

Here 1 ::; nf < ni are integers labeling the final and initial states ofthe atom in a radiation process, R is a constant, and l1E is the difference of the two energy levels. This formula predates quantum mechanics, and was based on measurements of absorption and emission spectra.

7.8 A Particle in an External EM Field

77

7.8 A Particle in an External EM Field Now we extend our quantization procedure to the case of a charged particle in an external electro-magnetic field. Of course, if the external field is purely electric, then it is a potential field, and fits within the framework we have considered already (as we saw in Section 7.7). Suppose, then, that a magnetic field B, and an electric field, E, are present (and are time-independent: B, E : ]R3 f----> ]R3). We know from the theory of electro-magnetism (Maxwell's equations) that these fields can be expressed in terms of the vector potential, A ]R3 f----> ]R3, and the scalar potential, ifJ : ]R3 f----> ]R via

E=-V'ifJ

B

=

curlA

(we are using units in which the speed of light, c, is equal to one). According to our general quantization procedure, we write the classical Hamiltonian function for a particle of charge e subject to the fields E and B, 1 2m

hex, k) = -(k - eA(x))2

+ eifJ(x)

and then replace the classical canonical variables x and k with the quantum canonical operators x and p. The resulting Schrodinger operator is 1 2m

H(A, ifJ) = -(p - eA?

+ eifJ

acting on L 2 (]R3). We remark that the self-adjointness of H(A, ifJ) can be established by using Kato's inequality (see [CFKS]). An important feature of the operator H(A, ifJ) is its gauge invariance. We recall that in the theory of electro-magnetism, the vector potential A is not uniquely determined by the magnetic field B. In fact, if we add the gradient of any function X to A (a gauge transformation), we obtain the same magnetic field B:

curl(A + V'X) = curlA = B.

Gauge invariance of the quantum Hamiltonian H(A, ifJ) is reflected in the relation H(A + V'X,ifJ) = eieX/IiH(A,ifJ)e-iex/li. (7.13)

Problem 7.9 Check that equation (7.13) holds. Thus if A and A differ by a gradient vector-field, then the operators H(A, ifJ) and H(A, ifJ) are unitarily equivalent via the unitary map

on L 2 (]R3). Thus the two Hamiltonians are physically equivalent. Of course, this is to be expected as A and A correspond to the same magnetic field.

78

7 Special Cases

One can impose restrictions (called gauge conditions) on the vector potential A in order to remove some, or all, of the freedom involved in the choice of A. A common choice is divA = 0, known as the Coulomb gauge. By an appropriate gauge transformation, the Coulomb gauge can always be achieved. We now consider an important special case - a constant magnetic field with no electric field present. A possible choice for A is 1

2B x x.

A(x) =

(7.14)

Another possibility, supposing B to be directed along the (0,0, b)) is A(x) = b( -X2, 0, 0).

X3

axis (B

=

(7.15)

Problem 7.10 Check that both (7.14) and (7.15) yield the magnetic field B, and that the two are gauge-equivalent. Using the second choice for A, the appropriate Schrodinger operator is 1

H(A) = -[(PI 2m

2 + ebx 2) 2 +2 P2 + P3l·

To analyze H(A), we apply the Fourier transform to only the first and third variables (XI,3 f---+ k l ,3). This results in the unitarily equivalent operator 1 2 mw 2 H = 2mP2 + -2-(X2

1

2

1

2

+ eb kl ) + 2m k3

where w = eb/m and kl' k3 act as multiplication operators. We remark that acts as a harmonic oscillator in the variable X2, and as a multiplication operator in kl and k 3 • In the following problem you are asked to determine the spectrum of this operator.

if

Problem 7.11 1. Show that the energy levels of

iI

(called Landau levels) are given by 1

a2 2m

(n+-)11w+2

where n = 0,1,2, ... and a eigenfunctions are

E

JR. Show that the corresponding generalized

where ¢n is the nth eigenfunction of the harmonic oscillator. 2. Analyze the same problem in two dimensions, with the magnetic field perpendicular to the plane.

8 Many-particle Systems

In this chapter, we extend the concepts developed in the previous chapters to many-particle systems. Specifically, we consider a physical system consisting of N particles of masses m1, ... , mN which interact pairwise via the potentials Vij(Xi - Xj), where Xj is the position of the j-th particle. Examples of such systems include atoms or molecules - i.e., systems consisting of electrons and nuclei interacting via Coulomb forces.

8.1 Quantization of a Many-particle System According to our general framework, we begin with the classical Hamiltonian formulation of an N-body system. The appropriate phase-space is lR~N x lR~N where x = (X1, ... ,XN) are the particle coordinates, and k = (k1, ... ,kN) are the particle momenta. The Hamiltonian function is 1

N

h(x, k) = "

- kJ2

~2m·

+ V(x)

J

j=1

where V is the total potential of the system, given in this case by

(8.1) The quantization proceeds in the standard way by replacing lR~N x lR~N with L2(lR~N), x with the multiplication operator x, k with p = -ifi'Y x, and h(x, k) with the operator HN = h(x,p). Explicitly, the Schrodinger operator HN is HN

=

N

1 2m PJ

j=1

J

L

+ V(x)

(8.2)

acting on L 2 (lR 3N ). Here Pj := -ihV xj • Example 8.1 Consider a molecule with N electrons of mass m and charge -e, and M nuclei of masses mj and charges Zje, j = 1, ... , M. In this case, the Schrodinger operator, Hmoz, is

S. J. Gustafson et al., Mathematical Concepts of Quantum Mechanics © Springer-Verlag Berlin Heidelberg 2003

80

8 Many-particle Systems (8.3)

acting on £2(JR3(N+Ml). Here x = (Xl, ... , XN) are the electron coordinates, Y = (YI, ... YM) are the nucleus coordinates, Pj = -ifi\l Xj is the momentum of the j-th electron, qj = -ifi\lYj is the momentum of the j-th nucleus, and

is the sum of Coulomb interaction potentials between the electrons (the first term on the r.h.s.), between the electrons and the nuclei (the second term), and between the nuclei (the third term). For a neutral molecule, we have M

LZj

=

N.

j=l

If M = 1, the resulting system is called an atom, or Z-atom (Z = ZI). Since nuclei are much heavier than electrons, in the leading approximation one can suppose that the nuclei are frozen at their positions. One then considers, instead of (8.3), the Schr6dinger operator 1

HBO

= 2m

N

LP7 + V(x, y) l

on £2(JR3N), the positions Y E JR3M of the nuclei appearing as parameters. This is called the Born-Oppenheimer approximation. It plays a fundamental role in quantum chemistry, where most computations are done with the operator HBO. The eigenvalues of the operator HBO are functions of the coordinates, y, of the nuclei. Minimizing the lowest eigenvalue - the ground state energy - with respect to Y gives the equilibrium positions of the nuclei, i.e. the shape of the molecule. One can show that if the pair potentials Vij satisfy the condition

(Le. each Vij can be represented as the sum of an £2 function and an £= function) then the operator HN is self-adjoint (see [CFKS, HuS]). Observe that the Coulomb potentials Vij(Y) = eiejlyl-I satisfy this condition:

lyl- I = lyl-Ie- 1yl

+ lyl-I(1- e- 1yl ),

for example, with lyl-Ie- 1yl E £2(JR3) and lyl-I(1 - e- 1yl ) E £=(JR3).

8.1 Quantization of a Many-particle System

81

However, the spectral analysis of the operator HN for N ~ 3 is much more delicate than that of the one-body Hamiltonians we have considered so far. (We will show shortly that for N = 2, the operator HN is reduced to a one-body Hamiltonian.) We are faced with the following issues: - separation of the center-of-mass motion - complicated behaviour ofthe potential V(x) at infinity in the configuration space ]R3N - identical particle symmetries. In the subsequent sections, we will deal with the first two issues in some detail. In the remainder of this section, we comment on all three of them. In what follows, we shall assume "Vij(x) -. 0 as Ixi -. 00, though for many considerations this condition is not required. Separation of the centre-oJ-mass motion. The Schrodinger operator (8.2) commutes with the operator of total translation of the system

and one can show that, as a result, its spectrum is purely continuous. So in order to obtain interesting spectral information about our system, we have to remove this translational invariance ("break" it). One way of doing this is by fixing the centre of mass of the system at, say, the origin: N

Lmjxj =

o.

j=l

We will describe a general mathematical procedure for fixing the centre of mass below, but first will show how to do it in the case of two particles (N = 2). In this case, we change the particle variables as follows: Xl,

X2

Y

f----*

=

Xl -

X2,

Z

=

mlXl

+ m2x2

(8.4)

Here y is the coordinate of the relative position of the two particles, and z is the coordinate of their centre of mass. Using this change of variables in the two-particle Schrodinger operator H2

acting on

L2(]R6),

=

1

2

-2-PI ml

1

2

+ -2m2 -P2 + V(XI

- X2)

we arrive easily at the operator H2

1

2

1

2

= 2p,P + 2M P + V(y)

where P = -ifiVy, P = -ifiV z, p, = ;:~m~2 (the reduced mass), and M = ml + m2 (the total mass). In fact, it can be shown that H2 and H2 are

82

8 Many-particle Systems

unitarily equivalent, with the equivalence given by a unitary realization of the change of coordinates (8.4). The point now is that one can separate variables in the operator H2 . In formal language, this means that H2 can be written in the form

H2 = H on L 2(JR6)

= L2(JR~) @ L2(JR~)

@

1

+ 1 @ HCM

where 1

H = _p2 + V(y) 2p 1

HCM = 2M P

2

acts on L2(JR~) (see Section 8.8 for a description of the tensor product, @). Clearly Hand HCM are the Schrodinger operators of the relative motion of the particles, and of their centre of mass motion, respectively. It is equally clear that of interest for us is H, and not HCM. Note that H has the form of a one-particle Schrodinger operator with external potential V(y). All the analysis we developed for such operators is applicable now to H.

Behaviour of V(x) at infinity. The second issue mentioned above arises from the geometry of the potential (8.1). The point is that V(x) does not vanish as x ----* 00 in certain directions, namely in those directions where Xi = Xj for at least one pair i =1= j (we assume here that Vij(O) =1= 0). This property is responsible for most of the peculiarities of many-body behaviour. In particular, the spectral analysis of Chapter 5 does not work in the manybody case, and must be modified in significant ways by taking into account the geometry of many-body systems. Identical particles. Many-particle systems display a remarkable new feature of quantum physics. Unlike in classical physics, identical particles (i.e., particles with the same masses and charges, or, more generally, which interact in the same way) are indistinguishable in quantum physics. This means that all probability distributions which can be extracted from an N-particle wave function 'I/J( Xl, ... , XN ), should be symmetric with respect to permutations of the coordinates of identical particles. This is equivalent to the property that 'I/J( Xl, ... , XN) is invariant under such a permutation modulo a change of sign. Assume for simplicity that all N particles are identical. Then the invariance property of 'I/J(XI, ... ,XN) formulated above means that 'I/J(Xl, ... ,XN) belongs to a representation of the symmetric group SN (the group of permutations of N indices) corresponding to a Young tableau with at most two columns (see, eg., [LL] for an explanation of Young tableaux). The shape of the Young tableau is determined by the spin of the particles involved. Onecolumn Young tableaux - i.e. purely symmetric 'I/J(XI, ... , XN) - correspond to

8.2 Separation of the Centre-of-mass Motion

83

particles with integer spins, or Bosons, while two-column tableaus correspond to particles with half-integer spins, or Fermions. This relation between the symmetry properties of wave functions and the spin of particles, is known as the relation between spin and statistics. We will not go into this topic here, and refer the interested reader to any of the standard books on quantum mechanics given in the references.

8.2 Separation of the Centre-of-mass Motion This section is devoted to a description of a general method for separating the centre-of-mass motion of a many-body system. After applying this method, one is left with a many-body system whose centre-of-mass is fixed at the origin. We begin by equipping the N-body configuration space ]R3N with the inner-product N

(x, Y) := L

(8.5)

mixi . Yi

i=l

where ml, ... , mN are the masses of the N particles, and Xi . Yi is the usual dot-product in ]R3. Next, we introduce the orthogonal subspaces

X:= {x

I

E ]R3N

N

Lmixi

= O}

(8.6)

1::/ i,j}

(8.7)

i=l

and Xl. := {x E ]R3N

I

Xi = Xj

of ]R3N. Recall that orthogonality (X ..l Xl.) means (x, y) = 0 for any x EX, Y E Xl.. To see that X ..l Xl., suppose x E X, and Y E Xl.. Since Yj = Yl for all j, we have n

(x,

y) = L

n

miXi . Yi

y=l

= Yl . L

mixi

=0

y=l

by the definition of X. We recall here the definition of the direct sum of subspaces. Definition 8.2 If Vl , V2 are orthogonal subspaces of a vector space, V, with an inner product, then Vi EB V2 denotes the subspace

Problem 8.3 Show that ]R3N

= X EB Xl..

(8.8)

84

8 Many-particle Systems

X is the configuration space of internal motion of the N-particle system, and X..L is the configuration space of the centre-of-mass motion of this system. The relation (8.8) implies that (8.9) (see Section 8.8 for an explanation of the tensor product, 18l). Let Ll denote the Laplacian on ]R3N in the metric determined by (8.5), i.e.

1

N

Ll = '" -Ll x J·. Lm. j=l

J

Under the decomposition (8.9), the Laplacian decomposes as (8.10) where Ix and IX-L are the identity operators on L2(X) and L2(X..L) respectively (see again Section 8.8). Problem 8.4 Show that Llx-L 1 M

"N uj=l mjxj.

= ~LlXCM where M = 2:f=l mj and XCM =

Let 7rx be the orthogonal projection operator from (7rXX)i

= Xi

1 -

]R3N

to X. Explicitly,

N

N

2: j =l mj

L mjxj.

(8.11 )

j=l

Problem 8.5 Show that (8.11) is the orthogonal projection operator from ]R3N to X. Find the orthogonal projection operator, 7rX-L, from ]R3N to X..L.

Equation (8.11) implies that (7rXX)i - (7rxx)j

=

Xi - Xj.

Hence the many-body potential (8.1) satisfies V(X)

=

V(7rxx).

(8.12)

Equations (8.10) and (8.12) imply that the operator HN given in (8.2) can be decomposed as (8.13) where H

!i 2

= -2Llx + V(x)

is the Hamiltonian of the internal motion of the system, and

8.3 Break-ups

85

is the Hamiltonian of the motion of its centre-of-mass. Equation (8.13) is the centre-of-mass separation formula, and H is called the Hamiltonian in the centre-of-mass frame. It is a self-adjoint operator under the assumptions on the potentials mentioned above. It is the main object of study in many-body theory.

8.3 Break-ups Here we describe the kinematics of the break-up of an N-body system into non-interacting systems. First we introduce the notion of a cluster decomposition a={C1 ... C s } for some s ::; N. The C j are non-empty, disjoint subsets of the set {1, ... ,N}, whose union yields the whole set: s

UC

j

= {l, ... ,N}.

j=l

The subsets C j are called clusters. An example of a cluster decomposition for N = 3 is a = {(12), (3)}. The number of clusters, s, in the decomposition a will be denoted by #(a). There is only one cluster decomposition with #(a) = 1, and one with #(a) = N. In the first case, the decomposition consists of a single cluster f!

= {(I. .. N)},

and in the second case the clusters are single particles:

a = {(l) ... (N)}. In the first case the system is not broken up at all, while in the second case it is broken into the smallest possible fragments. To each cluster decomposition, a, we associate the intercluster potential Ia(x):=

L

Vij(Xi - Xj),

(ij)\la

where the notation (ij) r:t. a signifies that the indices i and j belong to different clusters in the decomposition a. Similarly, we associate to a the intracluster potential Va(X):=

L

(ij)Ca

Vij(Xi - Xj)

86

8 Many-particle Systems

where (ij) c a signifies that i and j belong to the same cluster in a. Thus Ia(x) (resp. Va(x)) is the sum of the potentials between particles from different (resp. the same) clusters of a. The Hamiltonian of a decoupled system (in the total centre-of-mass frame) corresponding to a cluster decomposition, a, is

acting on L2(X). For a = {C1 , ... , Cs}, the Hamiltonian Ha describes s noninteracting sub-systems C 1 , ... , Cs. For s = 1, Ha = H, and for s = N, Ha = - ~2 Ll x . The operators Ha are also self-adjoint. If s > 1, then the system commutes with relative translations of the clusters C 1 ... Cs: for hEX satisfying

hi = h j

if

(ij) c a.

'Ij;(X1

+ h1, ... ,XN + hN)

Here

Th : 'Ij;(Xl, ... ,XN)

---+

for h = (h 1, ... , hN). As a result, one can show that the spectrum of Ha is purely continuous. This is due to the fact that the clusters in a move freely. One can separate the centre-of-mass motions of the clusters C 1 ... C s in a and establish a decomposition for Ha similar to that for HN (equation (8.13)); we will do this later.

8.4 The HVZ Theorem In this section we formulate and prove the key theorem in the mathematical theory of N-body systems - the HVZ theorem. The letters here are the initials ofW. Hunziker, C. van Winter, and G.M. Zhislin. This theorem identifies the location of the continuous spectrum of the many-body Hamiltonian, H. Theorem 8.6 (HVZ Theorem) We have

uc(H) = [17, (0), where

17:= min E a , #(a»l

with

Note that 17 is the minimal energy needed to break the system into independent parts.

8.4 The HVZ Theorem

87

Proof. We begin by showing that a(Ha) C a(H) for #(a) > 1. Suppose a(Ha). Then for any c > 0, there is 'lfJ E L2(X) with 11'lfJ11 = 1, such that II(Ha - .\)'lfJ11 < c. Let hEX satisfy hi = h j if (ij) C a, and hi =I- h j otherwise. For s > 0, let Tsh be the operator of coordinate translation by sh. Note that Tsh is an isometry. As remarked in the previous section, Tsh commutes with H a , and so .\ E

On the other hand, HTsh'lfJ ~ HaTsh'lfJ as s ~ 00, because the translation Tsh separates the clusters in a as s ~ 00. So for s sufficiently large, II (H '\)Tsh'lfJll < c. Since c > 0 is arbitrary, and IITsh'lfJll = 11'lfJ11 = 1, we see .\ E a(H). So we have shown a(Ha) C a(H). As we remarked earlier, Ha has purely continuous spectrum. In other words, a(Ha) = [Ea, (0). Thus we have shown [17, (0) C a(H). It remains to prove that a c (H) C [17, (0). To do this, we introduce a "partition of unity", i.e., a family {ja} of smooth functions on X, indexed by all cluster decompositions, a, with #(a) > 1, such that

L

j~(x) == 1.

(8.14)

#(a»l

We can use {ja} to decompose H into pieces which are localized in the supports of the ja, plus an error term: (8.15) Indeed, summing the identity

over a, and using (8.14), we obtain

Computing [ja, [ja, H]] = -11,21\7 jal 2 finishes the proof of of (8.15). Now we construct an appropriate partition of unity, namely one satisfying (8.16) for some c

> o. Indeed, the sets Sa

:=

{x E X I Ixl

=

1; IXj - xkl

>0

\:j

(jk)

ct a}

88

8 Many-particle Systems

form an open cover of the unit sphere, S, of X (i.e., Sa are open sets and S c Ua Sa). For each a, let Xa be a smooth function supported in Sa, and equal to 1 in a slightly smaller set (such that these smaller sets still cover S). Then the functions ja := Xa/(I: X;)1/2 form a partition of unity on S, with supp(ja) C Sa. In fact, (8.16) holds (with Ixi = 1), because supp(ja) is compact. We extend ja(x) to all of X by setting ja(x) := ja(x/lxl) for Ixl > 1, and for Ixl < 1, choosing any smooth extension of ja(x) which preserves (8.15). Thus the partition {ja} satisfies

for

ja(.)..x) = ja(x)

Ixl:::: 1,'>" :::: 1

(8.17)

as well as (8.16). By (8.17),

lV'ja(x) 12 By (8.16),

----*

0

Ixl ----* 00.

as

ja(H - Ha)ja = ja1aja

----*

0

Ixl ----* 00.

as

Returning to (8.15), we conclude that

H =

L

jaHaja

+K

#(a»l

where K is multiplication operator vanishing at infinity. An argument similar to the proof of the second part Theorem 5.20 shows that

Since H :::: (inf O"(H))l for any self-adjoint operator, H, we see

('I/J,jaHaja'I/J) = (ja'I/J, Hja'I/J) :::: 17allja'I/J112 ::::

17all'I/J112

for any 'I/J E X (note 17a :S 0). Thus jaHaja :::: 17a, and therefore I:a jaHaja :::: 17, yielding O"c(I:ajaHaja) C [17,(0), and consequently O"c(H) C [17,(0). This completes the proof of the HVZ theorem. D

8.5 Intra- vs. Inter-cluster Motion As was mentioned in Section 8.3, the Hamiltonians Ha describing the system broken up into non-interacting clusters have purely continuous spectra. This is due to the fact that the clusters in the decomposition move freely. To understand the finer structure of many-body spectra, we have to separate the centre-of-mass motion of the clusters, as we did with the centre-of-mass

8.5 Intra- vs. Inter-cluster Motion

89

motion of the entire system. Proceeding as in Section 8.2, we define the subspaces X a := {x E X I mjxj = 0 V i}

L

JEGi

and

Xa

:=

{x

E

X I Xi

=

Xj if (ij) c a}.

Problem 8.7 Show that these subspaces are mutually orthogonal (xa -L Xa) and span X: (8.18)

xa is the subspace of internal motion of the particles within the clusters of the decomposition a, and Xa is the subspace of the centre-of-mass motion of the clusters of a. As before, (8.18) leads to the decomposition L2(X) = L2(xa) ® L2(Xa) of L2(X) and the related decomposition of the Laplacian on X: ..dx = ..dxa ® lXa

+ lxa

® ..dxa

where ..dxa and ..dxa are the Laplacians on the spaces xa and Xa (or L2(xa) and L2(Xa)) in the metric (8.5). Again, if 7rXa is the orthogonal projection from X to X a , then

Va(X)

=

Va(7rxa X)

and consequently we have the decomposition

Ha = H a ® lXa where

fj,2

+ lxa

H a = -2..dxa

® T a,

+ Va(x)

is the Hamiltonian of the internal motion of the particles in the clusters of a, and Ta = - ~2 ..dxa is the Hamiltonian of the centre-of-mass motion of the clusters in a. Applying the HVZ theorem inductively, we arrive at the following representation of the continuous spectrum of H:

(Jc(H) =

U [A, (0),

)..Er(H)

where the discrete set T(H), called the threshold set of H, is defined as

T(H):=

U

(Jp(Ha) U{O},

#(a»l

the union of the point spectra of the break-up Hamiltonians and zero. The points of T(H) are called the thresholds. Thus one can think of the continuous spectrum of a many-body Hamiltonian H (in the centre-of-mass frame) as a union of branches starting at its thresholds and extending to infinity.

90

8 Many-particle Systems

8.6 Existence of Bound States for Atoms and Molecules Bound states of an atom or molecule described by a many-body Hamiltonian, H, correspond to the point spectrum, rYp(H). Here we turn our attention to the study of rYp(H), addressing the question of how many bound states H produces. For N = 2, the results in Section 5.6, show that the point spectrum of H is finite if the potential V(x) is "short-range", whereas a "long-range" attractive potential produces an infinite number of bound states. The borderline between short- and long-range potentials is marked by the asymptotic behaviour V(x) rv Ixl- 2 as Ixl ~ 00 (which is different than the borderline asymptotic behaviour of Ixl- 1 which we encountered in scattering theory in Chapter 6). For N > 2, however, the question of whether rYp(H) is finite or infinite cannot be answered solely in terms of the asymptotic fall-off of the intercluster potentials, fa (x); the nature of the threshold E at the bottom of the continuous spectrum plays a decisive role. Here we restrict our attention to the case where E is a two-cluster threshold. This means that for energy E and slightly above, the system can only disintegrate into two bound clusters, C 1 and C 2 . (In the case of an atom, one of the clusters contains a single electron, and the other contains the remaining electrons and the nucleus.) This situation can be represented by a product wave function (8.19) where ¢ is the eigenfunction of Ha with eigenvalue E, Ha¢ = E¢, and Xa and x a denote the components of x along the subspaces Xa and xa respectively. Here u is chosen so that (u, TaU) is arbitrarily small. The condition that E is a two-cluster threshold means that E is a discrete eigenvalue of Ha, and as a consequence it can be shown that ¢(xa) decays exponentially as Ixal ~ 00: (8.20) Ct > O. Using states of the form (8.19) as trial states to make ('IjJ, H'IjJ) < E, we can show that rYp(H) is infinite if the intercluster potential fa(x) has a long-range attractive part. For simplicity, we write this out in the case of Coulomb potentials:

for some

with 2: iE C 1 ,kEC2 eiek < 0, assuming that the clusters have opposite total charges. Using the exponential bound (8.20), it follows that

Observe that fa(xa) = -qllxal with q < O. Since H and Ha¢ = E¢, the last inequality implies

= Ha 01a + 1 a 0Ta + fa

8.7 Scattering States

(7f;, (H - E)7f;! :::: / U, (Ta - -qII \ Xa As in Section 5.6, we let

U

+ (COnst)IXal-2) U)

E CO'(lR 3 ) satisfy

SUpp(U) C {xa

I 1<

£2(Xa)

91

.

Ilull = 1 and

IXal < 2}.

Then the functions n = 1,2,4,8, ... are orthonormal, and have disjoint supports. Thus the corresponding trial states 7f;n(x) = ¢(xa)un(xa) satisfY (7f;n, H7f;m! = 0 for n =I- m, and q

(7f;n, (H - E)7f;n! :::: n

1

+ (const)2 <0 n

if n is sufficiently large. We can now apply the min-max principle (see Section 5.5) to conclude that H possesses infinitely many discrete eigenvalues below the threshold E. The same argument applies for attractive pair potentials with asymptotic behaviour like lxi-I-' with 0 < f.L < 2, and also if additional short-range potentials are present in Ia(x). This result shows that neutral atoms and positive ions always have infinite point spectrum. The accumulation of eigenvalues at E can be studied by similar arguments. Consider trial wave functions, 7f;nm, constructed as above, with unm(xa) a hydrogen atom eigenfunction of energy -n- 2 (in suitable units) (see Section 7.7). Then one can show

for some a > 3, where En = E - n- 2 . This implies that H has groups of eigenvalues close to En compared to the spacing En+1 - En as n -+ 00 (Rydberg states).

8.7 Scattering States Unlike in the one-body case, the many-body evolution 7f; = e- iHt / n7f;o behaves asymptotically as a superposition of several (possibly infinitely many) free evolutions, corresponding to different scenarios of the scattering problem. These scenarios, called scattering channels, can be described as follows. For a given scattering channel, the system is broken into non-interacting clusters C 1 , ... ,Cs corresponding to some cluster decomposition a, and its motion in each cluster is restricted to a cluster bound state. (If a given decomposition contains a cluster for which the cluster Hamiltonian has no bound state, then this cluster decomposition does not participate in the formation of scattering

92

8 Many-particle Systems

channels.) In other words, a scattering channel is specified by a pair (a, m), where a is a cluster decomposition such that the operator Ha has some point spectrum, and m labels the the eigenfunction 'lj;a,m of H a (we suppose that for fixed a, the 'lj;a,m are chosen orthonormal). The evolution in the channel (a, m) is determined by the pair

(1i a,m, Ha,m) where 1i a,m := 'lj;a,m ® L2(Xa) is the channel Hilbert space, and Ha ,m := Ha Ir1i a,rn = Ea,m

+ Ta

where Ea,m is the eigenvalue of Ha corresponding to the eigenfunction 'lj;a,m. Thus the channel evolution is given by

f E L2(Xa). As in the one-body case (see Chapter 6), the existence and asymptotic form of scattering states for t ---> 00 depends crucially of the rate at which the potentials tend to zero for large separations. We suppose that the intercluster potentials satisfy aCa )Ia(x) ::::: (const)lxl;;:-JL-la l (8.21 ) for

as Ixl a --->

00,

for

lal : : : 2. Here

Ixl a is the intercluster distance, i.e.

As in the one-body case, p, = 1 marks the borderline between short-range and long-range systems, for which the scattering theory is quite different.

Short-range systems: p, > 1. In this case, for a given scattering channel, (a, m), the wave operator Q+

Ca,m)

=

s-lim

t---too

eiHtjfie-iHa,=tjfi

can be shown to exist on 1ia ,m. The proof is similar to that for the one-body case (see Chapter 6). This wave operator maps free channel evolutions to full evolutions: setting of, '= Q+ ® f) , 'P . Ca,m) (of,a,m 'P we have

e-iHtjfi'lj;

--->

e-iHa,=tjfi('lj;a,m ® f)

as t ---> 00. The wave operator Qd,m is an isometry from 1ia ,m to 1i. Moreover, the ranges 1id,m := Ran(Qd,m) satisfy Hd,m -.l 1ib,n

if

a

=1=

b

or

m

=1=

n

8.8 Mathematical Supplement: Tensor Products

93

which follows from the fact that lim (e-iHa,,,,t/Ii('l/Ja,m ® j), e-iHb,nt/Ii('l/Jb,n ® g))

=

0

t-->oo

if a =f. b or m subspace

=f. n. Therefore, the outgoing scattering states form a closed a,m

Under the condition (8.21) with f..t > 1, it has been proven that the property of asymptotic completeness holds - i.e., that H+ = L2(X) (see Chapter 17 for references). In other words, as t ---; 00, every state approaches a superposition of channel evolutions and bound states (the bound states are the channel with #(a) = 1). Long-range systems: f..t ::::: 1. As in the one-body case, it is necessary to modify the form of the channel evolutions in the long-range case. The evolution e-iHa,,,,t/1i with Ha,m = >..a,m + Ta is replaced by e -iHa,,,,t/li-iCl'.a,t (Pa) where Pa = -iff'V X a. Here aa,t(Pa) is an adiabatic phase, arising from the fact that classically, the clusters are located at Xa = Pat(1 + O(rf1)) as t ---; 00. The modification aa,t(Pa), whose precise form we will not give here, is similar to that for the one-body case (see Chapter 6). We refer the interested reader to the references listed in Chapter 17 for further details. We remark only that with this modification in place, the existence of the (modified) wave operators, and asymptotic c9mpleteness, have been proved for f..t not too small (see Chapter 17 for references).

8.8 Mathematical Supplement: Tensor Products We collect here a few facts about tensor products of Hilbert spaces, and tensor products of operators and their spectra (see [RSI] for details and proofs). Let HI and H2 be two separable Hilbert spaces. The tensor product of HI and H2 is a Hilbert space HI ® H2 constructed as follows. To 'l/Jl E HI and 'l/J2 E H2, we associate a map

'l/Jl ® 'l/J2 : HI x H2 ---; C (ft, h)

f-)

(ft, 'l/Jl)'Hl (12, 'l/J2)'H2

which is conjugate linear in each component ('l/Jl ® 'l/J2 (aft, h) = ti'I/Jl ® 'l/J2(ft, h), 'l/Jl ® 'l/J2(ft + gl, h) = 'l/Jl ® 'l/J2(ft, h) + 'l/Jl ® 'l/J2(gl, h), and the same for the second component). On the vector space, V, of all finite linear combinations of such conjugate bilinear maps, we define an inner-product by setting

94

8 Many-particle Systems (8.22)

and extending by linearity (it is straightforward to check that this is welldefined). Then HI Q9 H2 is defined to be the completion of V in the innerproduct determined by (8.22). A simple example, which appears in Section 8.1, is

for positive integers m, n. This Hilbert space isomorphism is determined by the map f Q9 g f-+ f(x)g(y) (see, eg., [RSI] for details). Given bounded operators A and B acting on HI and H 2 , the operator A Q9 B, which acts on HI Q9 H2, is defined by setting

extending by linearity to all finite linear combinations of elements of this form, and then by density of these finite linear combinations, to HI Q9 H2. This produces a well-defined operator. This construction can be extended to unbounded self-adjoint operators A and B, yielding a self-adjoint operator A Q9 B (see, eg., [RSI]). Of particular interest for us are operators of the form A Q9 1 + 1 Q9 B, acting on H = HI Q9 H2, where A and B are operators acting on HI and H2 respectively. This is an abstract version of the "separation of variables" situation of differential equations. It is intuitively clear that we should be able to reconstruct characteristics of such operators from the corresponding characteristics of the operators A and B. As an example, we have the following important (and simple) description of the spectrum of A Q9 1 + 1 Q9 B under certain conditions on A and B, and, in particular, for A and B self-adjoint:

O"(A Q9 1 + 1 Q9 B) = O"(A) + O"(B) O"p(A Q9 1 + 1 Q9 B) C O"p(A) + O"p(B) C { ev's of A Q9 1 + 1 Q9 B} O"c(A Q91 + 1 Q9 B) = O"c(A) + O"c(B) U [O"c(A) + O"p(B)] U [O"p(A) + O"c(B)]. Rather than prove any such statements (an involved task, requiring further assumptions), let us just do a simple, suggestive computation. Suppose A7J!1 = AI7J!1 and B7J!2 = A27J!2. Then note that

(A Q9 1 + 1 Q9 B)7J!1 Q9 7J!2 = A7J!1 Q9 7J!2 + 7J!1 Q9 B7J!2 = AI7J!1 Q97J!2 +7J!1 Q9 (A27J!2) = (AI + A2)7J!1 Q9 7J!2, which shows, in particular, that O"p(A)

=

AI(7J!1 Q97J!2) +A2(7J!1 Q97J!2)

+ O"p(B) c { ev's of A Q9 1 + 1 Q9 B}.

9 Density Matrices

9.1 Introduction In this chapter we extend the basic notions of quantum mechanics to the situation of open systems and positive temperatures. This topic is called quantum statistical mechanics. Since the notion of temperature pertains to systems with an infinite number of degrees of freedom - infinite systems which are in states of thermal equilibrium, we should explain what the first sentence really means. By a quantum system at a positive temperature, we mean a quantum system coupled to an infinite system which is initially in a state of thermal equilibrium at a given temperature. In what follows, we develop a mathematical framework with which to handle such situations. We will see that we will have to replace the notion of wave function (i.e. a square integrable function of the particle coordinates - an element of L2 (JR3)) with the notion of density matrix, a positive, trace class operator on the state space L 2(JR3). The notions of ground state and ground state energy go over to the notions of Gibbs state and (Helmholtz) free energy. We consider some elementary examples and formulate some key problems. The notions of trace and trace class operators will be defined in the "mathematical supplement" at the end of this chapter.

9.2 States and Dynamics Consider a physical system described by a quantum Hamiltonian H acting on a Hilbert space 1{ (say, H = -L1 + V(x) on L2(A), where A c JR3). Let {7/Jj} be an orthonormal basis in H. Then any state 7/J E 1{ can be expanded as 7/J = 'Laj7/Jj. Given an arbitrary observable A (say, position or a characteristic function of position), its average in the state 7/J is given by

(9.1) m,n

Now suppose that we know only that the system is in the state 7/Jn with a probability Pn for each n. We thus have much less information than before. Now the average, (A), of an observable A, is given by the expression

S. J. Gustafson et al., Mathematical Concepts of Quantum Mechanics © Springer-Verlag Berlin Heidelberg 2003

96

9 Density Matrices (9.2) n

This corresponds to the situation when the parameters an in (9.1) are independent random variables with zero mean, and variance E(lan 2 ) = Pn. Observe that (9.2) can be written as I

(A)

=

tr(Ap) ,

(9.3)

where p = 2:;n PnP,pn' Here P,p stands for the rank-one orthogonal projection onto the vector 'l/J (P,pf = ('l/J, j)'l/J, or P,p = I'l/J)('l/JI in Dirac's notation). p is a trace class, positive (since Pn 2: 0 and P,pn 2: 0) operator, with trace 1: tr p = 2:;n Pn = 1. We extrapolate from this the assumption that generalized states are given by positive, trace class operators p on 1i, normalized such that trp = 1. Such operators are called density matrices. If the vectors 'l/Jn in the expression p = 2:;PnP,pn evolve according to the Schrodinger equation, ifiWf: = H'l/J, then the equation governing the state p is: (9.4) One takes this equation to be the basic dynamical equation of statistical mechanics. It is called the Landau-von Neumann equation, or the quantum Liouville equation. Problem 9.1 Derive equation (9.4) for p

= 2:;PnP,pn

with 'l/Jn as above.

If the spectrum of an operator H consists of isolated eigenvalues (of finite multiplicity) converging to 00, then an operator f(H) is trace class for any function f vanishing at (X) sufficiently fast (see Section 9.14). Thus, for any such positive function, f(H) is a density matrix (up to normalization). The operator f(H) commutes with H ([f(H), H] = 0), and therefore p := f(H) is a time-independent solution of equation (9.4). It is a stationary solution. We summarize our conclusions. In the situation when only partial information about a quantum system is available - namely, we know only that the system occupies certain states with certain probabilities - it is natural to describe states of such a system by positive trace-class operators p 2: 0 (normalized by trp = 1), called density matrices, with the equation of motion given by (9.4), and averages of observables computed according to the prescription (9.3). If the operator H has purely point spectrum, then Equation (9.4) has an infinite-dimensional family of time-independent solutions stationary states - of the form f (H), where f is a positive function, decaying sufficiently fast at infinity.

9.3 Open Systems There are two situations which necessitate passing from wave functions to density matrices. In one case, one deals with a system of infinitely many de-

9.3 Open Systems

97

grees of freedom. We consider this situation in the next section. In the other case, the system of interest is coupled to (or is a subsystem of) another, usually much larger, system, which we call the environment. Such a system is called an open system. Even if the total system is described by a wave function, say 'ljJ(x, y), where x and yare the coordinates of the system of interest and of the environment, respectively, the system itself cannot generally be described by a wave function. More specifically, given 'ljJ(x, y), there is, in general, no wave function cp(x) for our system such that, for any observable A = Ax associated with the system,

('ljJ, A'ljJ) = (cp, Acp) . However, there is a density matrix, p = P1j;, for our system, so that

('ljJ, A'ljJ)

=

tr(Ap)

(9.5)

for any system observable A. Indeed, the operator p is defined by its integral kernel (9.6) p(x, x') = 'ljJ(x,y)'ljJ(x',y)dy .

J

Problem 9.2 Check that (9.5) holds for any operator A acting on the variable x, provided p is given by (9.6).

One way to show (9.5) is as follows. Define the rank-one projection operator Then (show this!) (9.7)

for any operator (observable) A. Now, if A is a system observable, i.e., an operator which acts only on the variables x, then taking the trace first with respect to the environment variables, and then over the system variables (see Section 9.14) one obtains

tr(AP1j;)

=

trsysttrenvir(AP1j;)

= tr syst [Atr en vir (P1j;)]. Denote p := tr env ir(P1j;). This is a positive, trace class operator, acting on the system's wave functions, normalized by

The integral kernel of p is precisely (9.6). Thus p is a density matrix, and we have shown

tr(AP1j;) = trsyst(Ap)

and, due to (9.7), also (9.5).

98

9 Density Matrices

The states (density matrices) which are rank-one operators - i.e. which are of the form p = P'lj; for some normalized wave function 'Ij; - are called pure states. They are equivalent to wave functions in the sense that they are in one-to-one correspondence with wave functions, up to a phase, and produce exactly the same expectations for arbitrary observables as the corresponding wave functions.

9.4 The Thermodynamic Limit The aim of this section is to provide an informal discussion of the notion of thermodynamic limit. Consider the concrete physical situation of a system of N particles (say, a gas) in a volume A c IR3 described by the Schrodinger operator

acting on L2(AN) with Dirichlet or periodic boundary conditions, where Vi and Wij are bounded (see Chapter 8). Our goal is to understand the main features of the corresponding dynamics (9.4), which are independent of N and A for Nand IAllarge and N IIAI fixed. First note that if A is bounded, then !(HN,A) is a trace class operator for any! which decays at 00 sufficiently fast (show this! - see Section 9.14). Hence if! is also positive, then !(HN,A) is a stationary state (up to normalization). A basic question is whether or not these stationary states are stable (in an appropriate sense) as solutions of the Landau-von Neumann equation (9.4). In general, such states are stable in the Lyapunov sense. Roughly speaking, this means that solutions of (9.4) which are close to the given stationary state at time t = 0, remain close to it for later times. A stationary state is said to be asymptotically stable if solutions of (9.4) which start close to the given stationary state, converge to it as t ----; 00. The stationary states described above do not enjoy this stronger property of asymptotic stability. However, as a few examples accessible to analysis show, for N very large there is a finite-dimensional sub-family of states which appear to be asymptotically stable on extremely large time intervals (of length eN where e is a large number), while convergence to them is very fast. Clearly these exceptional states playa key role in the description of large particle systems. To isolate such states, and to prove their long-time asymptotic stability, one passes to the idealized situation of an infinite number of particles in infinite volume, but with lim(NIIAI) finite and positive. This is the thermodynamic limit.

9.6 The T

--->

0 Limit

99

9.5 Equilibrium States We are interested in stationary states which are asymptotically stable on long time intervals when the number of particles is very large. It is conjectured that (in the absence of transport) such states coincide with the states of thermal equilibrium ~ the (thermal) equilibrium states. The latter can be isolated as follows. Assume we have only one conserved quantity ~ the energy. Then, following the second law of thermodynamics, we can characterize the equilibrium states in a finite volume as follows: p is equilibrium iff p maximizes S(p), provided E(p) is fixed (E(p)

= E,

say) .

(9.8)

Here S(p) := -tr(plnp) is the Gibbs entropy, and E(p) := tr(Hp) is the internal energy. The criterion above is called the principle of maximum entropy. This principle can be extended in an appropriate form to infinite systems. Variational problem (9.8) can be easily solved (see Chapter 12) to give the following one-parameter family of positive operators PT = e~H/T /Z(T) , where Z(T) := tre~H/T ,

as equilibrium states (for a definition of the operator e~H/T, see Section 2.2). These states are called the Gibbs states and T, which is the inverse of the Lagrange multiplier, is called the temperature. The quantity Z(T) = tre~H/T (or Z(f3) = tre~{3H for f3 = l/T) is called the partition function (of the system described by the Hamiltonian H, at temperature T). It is conjectured that in the absence of conserved quantities other than the energy, all equilibrium states of infinite systems can be obtained as (weak) limits of Gibbs states: that is, the expected value of any observable A (Le. a bounded, self-adjoint operator on our Hilbert space) in an equilibrium state WT (at temperature T) is given by wT(A) = lim tr(ApT A) Aiffi,3

'

where PT,A is the Gibbs state at temperature T for the Hamiltonian H restricted to a bounded region, A, of physical space (see [BR]).

9.6 The T

~

0 Limit

The Lagrange multiplier theorem of variational calculus implies that an equilibrium state minimizes the Helmholtz free energy FT(p) := E(p) - TS(p)

where T, or f3 = T~\ is the Lagrange multiplier to be found from the relation tr(HpT) = E.

100

9 Density Matrices

Problem 9.3 Show this.

By a straightforward computation, the equilibrium free energy, F(T) .FT(PT), is given by F(T) = -TlnZ(T) . The next result connects Gibbs states and the free energy, to ground states and the ground state energy. Let 7/Jo be the (unique) ground state of the Hamiltonian H, and Eo the corresponding ground state energy. Let P'IjJ denote the rank-one projection onto the vector 7/J. We have Theorem 9.4 (Feynman-Kac Theorem) As T

PT

---+

F(T)

and

P'ljJo

---+

---+

0,

Eo .

Proof Let Eo < El ~ E2 ~ ... be the eigenvalues of H (our standing assumption is that H has purely discrete spectrum, running off to 00), and let 7/Jo, 7/Jl,'" be corresponding orthonormal eigenstates. Then by completeness of the eigenstates, and the spectral mapping theorem (see Section 9.14),

where Pn = e- En / T /Z(T). We can rewrite Pn as Pn

=

/2:= 00

e-(En-Eo)/T

e-(En,-Eo)/T .

n'=O

We see that Pn

~

1 and as T

0

---+

n=O n?l

It follows easily that

F(T)

= -TIn

we see that F(T)

IlpT -

P'ljJo II

(~e-En/T)

---+

Eo as T

---+

---+

0 as T

---+

= Eo - TIn

O. Furthermore, since

(1 + ~

e-(En-Eo)/T ) ,

O. 0

9.7 Example: a System of Harmonic Oscillators As an example, we compute the free energy and its thermodynamic limit for a system of N non-interacting one-dimensional harmonic oscillators. The Hamiltonian of such a system is

9.7 Example: a System of Harmonic Oscillators

acting on

N

® L2(JR 1 )

~

101

L2(JR N ) (elements of many-body theory are described

1

in Chapter 8, and the symbol Q9 is explained in Section 8.8). Such oscillators describe quantized normal modes of lattice vibrations in solids (quantized elasticity waves, or phonons), or a quantized electromagnetic field in a cavity (photons). The operator HN has purely point spectrum. By separation of variables (see Section 7.5 and Section 8.8), the eigenvalues of HN are given by for

nj

=

0,1,2, ....

Consequently the partition function, ZN((3) = tre- f3HN , is 00

j=l

=

n=O

II (1 - eN

f3Wj

)-l

j=l

and the free energy, F N ((3)

=-

FN ((3)

~ In Z N ((3), is

1 N

=

!3 I)n(1 - e- f3Wj ). j=l

In order to pass to the thermodynamic limit N -7 00, we have to know how the frequencies Wj = wj(N) behave as N -7 00. In the case of phonons in a one-dimensional crystal with N atoms spaced at a distance a from each other, one has 2nc j Wj = ---:;: N (9.9) where c is the speed of sound. Then, using

N

I:

j=l

f(wj) ~ 2~C J~7rc/a f(w)m.u for

V = N a large, we obtain an expression for the free energy per unit volume in the thermodynamic limit:

o as N

-7

00.

102

9 Density Matrices

9.8 A Particle Coupled to a Reservoir Typically, one is interested in describing a small system interacting weakly with a large system (a small part of a gas interacting with the rest of the gas, for example, or an atom interacting with radiation). Often the large system is (or, for the given purpose, can be) described by a rather simple Hamiltonian. We call the large system an environment or (thermal) reservoir. The environment (the electromagnetic field, say) can be assumed to have an infinite number of degrees of freedom, and to be in a state of thermal equilibrium. We describe here such a model for the case of a single particle (in 3 dimensions, and in a potential V(x)) coupled to an environment. As a reservoir, we take a photon or phonon gas or, in general, a free, massless quasi-particle gas. In an approximation by a finite number of degrees of freedom, such a gas can be described by a collection of independent, one-dimensional harmonic oscillators:

acting on

N

® L2(JR)

~

L 2(JRN). The simplest coupling of a particle to the

I

reservoir is given by the multiplication operator

IN

=

g(x)· x,

where g is a map from JR3 to JRN, and x = (Xl' ... ' XN) E JRN. Denoting the particle Hamiltonian by

("8" standing for "system") acting on JR3, we arrive at the following expression for the Hamiltonian of the total system - the particle and reservoir coupled together:

HN := H S

(9

1R

+ 1S

(9

HfJ

+ )"IN

acting on L2(JR3) (9 L 2(JRN). Eventually, when we study the states and dynamics ofthe Hamiltonian HN, we will want to send N to 00, while assuming a particular dependence of Wj on j (eg. (9.9)).

9.9 Quantum Systems We will distinguish three types of systems described by 8chrodinger operators H

=

li,2

--,1

2m

+ V(x)

:

9.11 Hilbert Space Approach

confined systems

V(x)

---+ 00

asymptotically free systems unstable systems

f--7

V (x)

as

Ixl

V(x)

---+ - 00

as x

103

---+ 00

---+

---+ 00

canst as

Ixl

---+ 00

in some direction

9.10 Problems One of the central questions of Statistical Mechanics is how the behaviour of a given system is affected by its interaction with an infinite environment which is in a state of equilibrium. There are three basic problems in this context: I. (Return to equilibrium). Show that a confined system is driven to a state of total equilibrium with the environment. II. (Thermal evaporation). Show that an asymptotically free system ends up eventually in a state of unbounded motion (diffusion);

III. (Metastable states). Find the probability of decay of unstable systems. These problems are extensively discussed in the physics literature. As far as rigorous results are concerned, at present there are partial answers to questions I, II, and III (I: [JP1, BFS, DJ, Mer]; II: [FMS, FM]; III: [SV]). This is a quickly developing area of (non-equilibrium) statistical mechanics, and we refer the interested reader to the original papers for more details. Topic III is discussed in Chapter 13.

9.11 Hilbert Space Approach Statistical dynamics can be put into a Hilbert space framework as follows. Consider the space 'HHS of Hilbert-Schmidt operators acting on the Hilbert space 'H. These are the bounded operators, K, such that K* K is trace-class (see Section 9.14). There is an inner-product on 'HHS, defined by

(F, K) := tr(F* K). Problem 9.5 Show that (9.10) defines an inner-product.

(9.10)

104

9 Density Matrices

This inner-product makes HHS into a Hilbert space (see [BR, RSI]). On the space HHS, we define an operator L via 1

LK= -n[H,Kl, where H is the Schrodinger operator of interest. The operator L is symmetric. Indeed,

n(F, LK) = tr(F* [H, K]).

Using the cyclicity of the trace, the right hand side can be written as

tr(F* HK - F* KH) = tr(F* HK - HF* K) = tr([F*, HlK) = tr([H, Fl* K) = n(LF, K) and so (F, LK) = (LF, K) as claimed. In fact, for self-adjoint Schrodinger operators, H, of interest, L is also self-adjoint. Now consider the Heisenberg equation i ak

at

= Lk

(9.11)

where k = k(t) E HHS. Since k(t) is a family of Hilbert-Schmidt operators, the operators p(t) = k*(t)k(t) are trace-class, positive operators. Because k(t) satisfies (9.11), the operators p(t) obey the equation

.ap

~ at

1

= Lp= -n[H,pl·

(9.12)

If p is normalized - i.e., trp = 1 - then p is a density matrix satisfying the Landau-von Neumann equation (9.12). The stationary solutions to (9.11) are just eigenvectors of the operator L with eigenvalue zero. To conclude, we have shown that instead of density matrices, we can consider Hilbert-Schmidt operators, which belong to a Hilbert space, and dynamical equations which are of the same form as for density matrices. Moreover, these equations can be written in the Schrodinger-type form (9.11), with self-adjoint operator L, sometimes called the Liouville operator. Some key problems about the dynamics (see Section 9.10) can be reduced to spectral problems for the operator L. For more details, see [BR, Ha, HKT, JP1, BFSl.

9.12 Appendix: the Ideal Bose Gas Consider a system of identical, non-interacting particles subject to Bose statistics, in a box A C lRd . We assume periodic boundary conditions. If the number of particles in the box A is n, then their Hamiltonian is

9.12 Appendix: the Ideal Bose Gas

HAn ,

1

n

=

105

~ - - L1x ~ 2m ' i=l

acting on the space ®~ L2(A) := L;YID (An) with periodic boundary conditions. Here ® is the symmetric tensor product, and L§YID (An) is the L2 space of functions symmetric with respect to permutations of variables belonging to different factors in the product An. Computing the partition function

(9.13) is not a simple matter (try it!). Instead, we compute the generating function 00

00

n=O

n=O

(9.14) where z = e{3/-L. In principle, the original partition function, ZA,n(J3), can be recovered by computing the n-th derivative, ar; ZA({3, /1)lz=o, of ZA({3, /1) with respect to z at z = 0, if this derivative exists. There is a more natural way (from the point of view of statistical mechanics) to recover Z A,n ({3) from ZA({3, /1) which we discuss below in this section. Meanwhile, we compute ZA({3, /1), which is not difficult. Using separation of variables, and the result of Section 7.2 for a single particle in a box with periodic boundary conditions, we conclude that the spectrum of the operator HA,n is (9.15) where Ck = 2~ Ik1 2 . The eigenvalues have multiplicity 1 since the eigenfunctions lie in the subspace L;ym of L2(JRnd) (they are symmetric with respect to permutations of the variables Xi E JRd). Problem 9.6 Show (9.15).

The expression on the right hand side can be rewritten as

CJ(HA,n)

= {

L kE

2;; Zd

Cknk

1

nk

= 0,1, ... , , L

nk

= n}.

Using equation (9.28), which expresses the trace of an operator in terms of its eigenvalues, we obtain the following expression for Z A ({3, /1): 00

ZA({3,/1) = L

L

e-{3~f:knkzn

The latter expression can be transformed as

=

Lz~nke-{3~f:knk.

106

9 Density Matrices (9.16)

Note in passing that if for a self-adjoint operator A with a purely point spectrum P'j} accumulating at 1, we define detA:= I1~1 >'j whenever this is finite, then we can rewrite (9.16) as

where HI,A = - 2~ L1x is the one particle Hamiltonian acting on L2(A) with periodic boundary conditions. Next, we consider the quantity P((3, J-l) := J(3 In ZA((3, J-l), called the pressure. We have

(9.17)

Using that as V

----> 00,

f(k)

---->

J

f(k)ddk,

we find the following approximation for the sum in (9.17):

P((3'J-l)=-(27r~d(3 =-

J

1 (27r)d (3dt2

In(l-ze-(3ck)ddk

J

In (1 - z e- ck ) dd k

(9.18)

modulo terms vanishing as V ----> 00. To get the last integral, we changed variables k ----> k/V73. Now we show how to derive the partition function ZA,n((3) from the generating function ZA((3,J-l). The considerations below are heuristic, but can be made rigorous. Using expression (9.16) one can show that as a function of z = etJP" ZA((3, J-l) is analytic in the disk {Izl :::; c} for some c > O. Hence, due to (9.14), ZA,n can be computed by the Cauchy formula (9.19) By the definition of PA((3, J-l), we can write ZA((3, J-l) = eVtJPA «(3,p,). Writing also z-n = e- n1nz = e-Vvlnz, where v = n/V, (9.19) becomes

9.12 Appendix: the Ideal Bose Gas

107

Now we take V = IAI and n large, while v = njV remains fixed. Taking into account the fact that PA ((3, JL) has a limit as V ----+ 00, and applying (formally) the method of steepest descent to the integral above, we find (9.20) where lIn cl is uniformly bounded in V, and JL solves the stationary phase equation JL: (3z8 z PA((3,JL) = njV or equivalently (passing from z to JL)

Define PA,n((3) := /3~ lnZA,n((3). Then relation (9.20) can be rewritten as

That is, PA,n((3), as a function of n, is (in the leading order as V ----+ 00) the Legendre transform of PA((3, JL), considered as a function of JL. Similarly, we can pass from ZA((3, JL) to ZA,n((3) by taking the Legendre transform in the variable v = njV. Returning to the generating function Z A ((3, JL), we view it as a partition function with a variable number of particles, but with a fixed average number of particles. The latter is given by the formula fi

=

L 00

ntre-/3HA,ne/3/"n jZA((3, JL),

(9.21 )

n=O

and depends on the value of the parameter JL (this viewpoint will be substantiated at the end of this section). The formula (9.21) implies the relation

from which we obtain the expression 1

njV=V

(9.22)

To have n 2': 0, we should take 0 :::; z :::; 1. The terms in the sum on the right hand side are, as can be easily checked, the average numbers of particles having momenta k:

To show this, one uses

108

9 Density Matrices

where ZA({3, J-L) is considered as a functional of C = Ck. Equations (9.18) and (9.22) constitute the equation of state of the ideal Bose gas (parameterized by z). More precisely, solving Equation (9.22) for z as a function of the density p = n/V, and temperature T = 1/{3, and substituting the result into Equation (9.18), we find the pressure P as a function of p and T. However, if we seek the equation of state as a relation between P, E and V (which is, of course, equivalent to the expression involving P, p and T), the answer is much simpler. Indeed, using the definition of E

E

=

E({3, J-L, V)

:=

tr (HA P{3,I-',A)

and considering the generating function as a function, ZA({3, z), of {3 and z (and V) rather than of {3 and J-L (and V), and similarly for the pressure, we find a a (9.23) E=-a{3 InZA({3,z)=-a{3 (V{3P({3,z)). Taking into account (9.18) and setting d E =

~ 2

=

3 we find

PV.

(9.24)

This is the equation of state of the ideal Bose gas. Problem 9.7 It is an instructive exercise to re-derive the results of this section for the ideal Fermi gas.

Finally, we justify considering the generating function ZA({3,J-L) as a partition function for a system with a variable number of particles. We pass to a space describing a variable number of particles - the (bosonic) Fock space:

F = EB~=o ®~ L2(A), where the n = 0 term is defined to be C. On this space we define the Hamiltonian H A := EB~=o HA,n, where, by definition, the n ZA({3, J-L) can be written as

= 0 term is HA,O =

ZA({3,J-L)

1. The generating function

= tre-{3(HA-I-'NA) ,

where N A is the number of particles operator, defined as

9.13 Appendix: Bose-Einstein Condensation

109

(n is the operator of multiplication by n). The partition function ZA((3,p) is associated with the Gibbs state

for the new system. This state is obtained by maximizing the entropy, while leaving the average energy and average number of particles fixed (hence two Lagrange multipliers, (3 and p, appear). The state Pfl,i-' is called the grand canonical ensemble state, ZA((3, p) the grand canonical partition function, p the chemical potential, and z the fugacity.

9.13 Appendix: Bose-Einstein Condensation We consider the ideal Bose gas, and analyze formula (9.22) for the average number of particles. We would like to pass to the thermodynamic limit, V ----t 00. The point is that Equation (9.22) is the relation between the average number of particles n (or the average density P = n/V), the temperature T = 1/(3, and the chemical potential p (or fugacity z = e fl i-'). Recall also that 0 :::; z :::; 1. As long as P and (3 are such that z < 1, the right hand side of (9.22) converges to the integral

as V ----t 00. However, if the solution of Equation (9.22) for z yields, e.g., z = 1 - O(V- 1 ), then we have to consider the k = 0 term in the sum on the right hand side of (9.22) separately. In this case we rewrite (9.22) approximately as (from now on we set d = 3) (9.25) where we put

integration as k'

where ,\

=

a

no = =

l':z. Now using

Ck

= I;~, changing the variable of

k, and passing to spherical coordinates, we obtain

J27r (3/m (the thermal wave length), and

Thus Equation (9.25) can be rewritten as

llO

9 Density Matrices

no

P = V +,\

-3

(9.26)

g3/2(Z).

Recall that no = l':'z' and that this equation connects the density p, the thermal wave length ,\ (or temperature T = 27r / m ,\2), and the fugacity z (or chemical potential J.L = ~ In z), and is supposed to be valid in the entire range of values, 0 < z < 1, of z. Can z really become very close to 1 (within O(l/V)), or is the precaution we took in the derivation of this equation by isolating the term no/V spurious? To answer this question we have to know the behaviour of the function g3/2(Z) for z E (0,1). One can see immediately that

g3/2(0)

=0,

g~/2(Z)

> 0 and g~/2(1) =

00.

The function g3/2(Z) is sketched below.

g 1 (z)

g 3 (1)

"2

~

_______2_____ _

~-=~

____________

- L_ _~~

Z

1 Fig. 9.1. Sketch of g3/2(Z). We see now that if p,\3 < g3/2(1) (with g3/2(1)-p,\3 ;::: a positive number, independent of V), then the equation (9.27)

'iJ,

has a which is obtained from (9.26) by omitting the V-dependent term unique solution for z which is less than 1, and is independent of V. Consequently, taking into account the term no/V = v(Lz) would lead to an adjustment of this solution by a term of order O(l/V), which disappears in the thermodynamic limit V . . . . , 00. However, for p,\3 = g3/2(1), the solution of this equation is, obviously, z = 1, and for p,\3 > g3/2(1), the above equation has no solutions at all. Thus for p,\3 ;::: g3/2(1) we do have to keep the term no/V. Moreover, we have an estimate

(here z* is the solution to Equation (9.26)) which shows that in the case p,\3 > g3/2(1), a macroscopically significant (i.e. proportional to the volume

9.13 Appendix: Bose-Einstein Condensation

111

or the total number of particles) fraction of the particles is in the single, zero momentum - or condensed - state. This phenomenon is called Bose-Einstein condensation. The critical temperature, T e , at which this phenomenon takes place can be found by solving the equation

describing the borderline case for A, and remembering that T result we have 2/3 2 T-~ ( - - p - ) c m 93/2(1)

=

';~2' As a

From Equation (9.26) we can also find the fraction of particles, no/V, in the zero momentum (condensed) state as a function of temperature. This dependence is shown in the diagram below.

lr----

T Fig. 9.2. In this elementary situation, we have stumbled upon one of the central phenomena in macrosystems - the phenomenon of phase transition. Indeed, the states for which all the particles are in the single quantum state corresponding to zero momentum, and those for which the macroscopic fraction of the particles in the quantum state of zero momentum (and consequently in every single quantum state) is zero, can be considered two distinct pure phases of ideal Bose matter (gas). The first pure phase - called the condensate - occurs at T = 0, while the second pure phase takes place for T ~ T e . In the interval 0 < T < Tc of temperatures, the Bose matter is in a mixed state in which both phases coexist. Bose-Einstein condensation exhibits a typical property of phase transitions of the second kind: though all the thermodynamic functions and their first derivatives are continuous at the phase transition, some of the second derivatives are not. Typically one looks at the specific heat

c ._ v·-

aE(T,fJ, V) aT '

112

9 Density Matrices

the change of heat or energy per unit of temperature. Using Equations (9.18) and (9.23), one can show that while C v is continuous at T = Te, its derivative with respect to T is not. C v as a function of T is plotted below (see [Hu], Sect. 12.3]): Cv/n

~

1

.J. _ _ _ _ _ _

2

I

IT C

.I

~-==---~------~--------~T

::::. T 3/2

Fig. 9.3. Specific heat of the ideal Bose gas. One can show that the thermodynamic properties (eg., the equation of state - a relation between pressure, temperature and volume) of BoseEinstein condensation are the same as those of an ordinary gas - liquid condensation. The modern theory relates the phase transitions to superfiuid states in liquid helium (He 2 ) and to superconducting states in metals and alloys, to the phenomenon of Bose-Einstein condensation. In the mean field description of the phenomena of superfiuidity and superconductivity, the wave function of the condensate - the fraction of particles (or pairs of particles, in superconductivity) in the quantum zero momentum state - called the order parameter, is the main object of investigation. Of course, in both cases one deals with interacting particles, and one has to argue that Bose-Einstein condensation persists, at least for weakly interacting Bose matter. Problem 9.8 Extend the above analysis to an arbitrary dimension d.

9.14 Mathematical Supplement: the Trace, and Trace Class Operators This section gives a quick introduction to the notion of the trace of an operator, a generalization of the familiar trace of a matrix. More details and proofs can be found in [RSI], for example. Let p be a bounded operator on a (separable) Hilbert space, 1{. Since p* p 2:: 0, we can define the positive operator Ipi := vp*p (this operator can be defined by a power series - see [RSI]; see also Section 5.2). The operator p is said to be of trace class if

L(7/'j, Ipl7/'j! < 00 j

9.14 Mathematical Supplement: the Trace, and Trace Class Operators

113

for some orthonormal basis {'l/Jj} of H.. If p is a trace class operator, we define its trace to be trp = 2)'l/Jj, p'l/Jj) j

for some orthonormal basis {'l/Jj} of H.. This definition is independent of the choice of basis. Properties of trace class operators include . the trace class operators form a Banach space under the norm

IlpIII := tripi, and the trace is a linear functional on this space; that is, tr maps trace class operators to complex numbers, and it satisfies tr(aA+f3B) = atr A+ f3trB for any a, f3 E
tr(J(A))

= Lf(Ej )

(9.28)

j

if this sum converges absolutely.

Proof Let {'l/Jj} be an orthonormal basis of eigenvectors corresponding to the eigenvalues {Ej} of A. By the spectral mapping theorem (see Section 5.2), If(A)I'l/Jj = If(Ej)I'l/Jj, so L('l/Jj, If(A)I'l/Jj)

=

L

If(Ej)1

<

00

j

j

by assumption. Hence f(A) is trace class, and we may compute its trace as

j

o

j

114

9 Density Matrices

Example 9.11 Integral operators provide another useful example. Proposition 9.12 Let K be a continuous function on [a, bj2. Then the integral operator K, on £2([a, b]) defined by

K,f(x) = is trace class, with

trK, =

lb lb

K(x, y)f(y)dy

K(x, x)dx.

A bounded operator K, is called Hilbert-Schmidt if K,* K, is trace class. We have Proposition 9.13 An integral operator K, on £2(JRd) with kernel K £2(JRd X JRd) is Hilbert-Schmidt, and

trK,*K, =

r

IN.d xN.d

E

IK(x,y)1 2dxdy.

Proof Let {¢j} be an orthonormal basis in £2 (JRd). Then by the definition of the trace,

j

j

L 11K,¢jI12 = L ld 11d K(x,Y)¢j(y)dyI2dx = r L I r K(x, Y)¢j(y)dyI2dx. iN.d IN.d =

J

J

j

By the Parseval relation, this is

as required. D

10 The Feynman Path Integral

In this chapter, we derive a convenient representation for the integral kernel of the Schrodinger evolution operator, e- itH / n. This representation, the "Feynman path integral" , will provide us with a heuristic but effective tool for investigating the connection between quantum and classical mechanics. This investigation will be undertaken in the next section.

10.1 The Feynman Path Integral Consider a particle in IRd described by a self-adjoint Schrodinger operator

H

!i2 2m

= --,1

+ V(x).

Recall that the dynamics of such a particle is given by the Schrodinger equation . 8'IjJ z!iFt = H'IjJ. Recall also that the solution to this equation, with the initial condition

is given in terms of the evolution operator U(t):= e- iHt / n as 'IjJ

= U(t)'ljJo.

Our goal in this section is to understand the evolution operator U(t) = e- iHt / n by finding a convenient representation of its integral kernel. We denote the integral kernel of U (t) by Ut (y, x) (also called the propagator from x to y). A representation of the exponential of a sum of operators is provided by the Trotter product formula (Theorem 10.2) which is explained in Section 10.3 at the end of this chapter. The Trotter product formula says that

S. J. Gustafson et al., Mathematical Concepts of Quantum Mechanics © Springer-Verlag Berlin Heidelberg 2003

116

10 The Feynman Path Integral

where Kn := e2'::;':"Ll e- i;;,t . Let Kn(x, y) be the integral kernel of the operator Kn. Then by Proposition 5.33,

Ut(y, x)

=

lim

n-HXl

J... J

Kn(y, xn-d··· Kn(X2' xdKn(XI, X)dXn-1 ... dXI. (10.1)

Now (see Section 5.7)

since V, and hence e-iVt/nn, is a multiplication operator (check this). Using the expression (2.15), and plugging into (10.1) gives us

Ut(y, x) = lim

n->oo

where

JJ ( ...

eiSn / fi

2

·nt)

~ mn

-nd/2

dXI ... dXn-1

n-l

Sn

:=

I)mnlxk+l - xkl 2/2t - V(Xk+l)t/n) k=O

= x, Xn = y. Define the piecewise linear function ¢n such that ¢n(O) = x, ¢n(t/n) = Xl, ... , ¢n(t) = Y (see Fig. 10.1).

with Xo

J

y

x

tin

2t1n· ..

s

Fig. 10.1. Piecewise linear function. Then

Note that Sn is a Riemann sum for the classical action

S(¢, t)

=

lot {; 1¢(sW - V(¢(s))} ds

10.1 The Feynman Path Integral

117

of the path ¢n. So we have shown

(10.2) where P:'y,t is the (n - I)-dimensional space of paths ¢n with ¢n(O) = x, ¢n(t) = y, and which are linear on the intervals (kt/n, (k + I)t/n) for k = 0,1, ... , n - 1, and D¢n = e~::t)-nd/2d¢n(t/n) ... d¢n((n -I)t/n). Heuristically, as n -+ 00 ¢n approaches a general path, ¢, from x to y (in time t), and Sn -+ S(¢). Thus we write

L

Ut(y,x) =

eiS(,t)/nD¢.

(10.3)

x,y,t

Here Px,y,t is a space of paths from x to y, defined as

Px,y,t:= {¢: [O,tj-+

]R d1

l

t

11>12 <

00,

¢(O) = x,

¢(t) = y}.

This is the Feynman path integral. It is not really an integral, but a formal expression whose meaning is given by (10.2). Useful results are obtained nonrigorously by treating it formally as an integral. Answers we get this way are intelligent guesses which must be justified by rigorous tools. Note that P:'y,t is an (n - I)-dimensional sub-family of the infinitedimensional space Px,y,t. It satisfies P:'y,t C P;,~,t and lim n _ oo P:'y,t = Px,y,t in some sense. We call such subspaces finite dimensional approximations of Px,y,t. In (non-rigorous) computations, it is often useful to use finitedimensional approximations to the path space other than the polygonal one above. We can construct more general finite-dimensional approximations as follows. Fix a function ¢xy E Px,y,t. Then

Px,y,t

=

¢xy

+ PO,O,t.

Note Po,O,t is a Hilbert space. Choose an orthonormal basis define PJ:O,t:= span {~j}r and

P:'y,t

:=

¢xy

{~j}

in Po,O,t and

+ PJ:o,t·

Then P:'y,t is a finite dimensional approximation of Px,y,t. Typical choices of ¢xy and {~j} are

1. ¢xy is piecewise linear and {~j} are "splines". This gives the polygonal approximation introduced above.

118

10 The Feynman Path Integral

2. ¢xy is a classical path (a critical point of the action functional S(¢)) and {~j} are eigenfunctions of the Hessian of S at ¢xy (these notions are described in the following two chapters). In this case, if TJ E PI\O,t, then

we can represent it as

n

TJ =

L aj~j, j=1

and we have

DTJ

=

(2:tn)

-d/2

C:n Ii) Q n

daj.

It is reasonable to expect that if

lim

r

eiS(¢,t)/1i D¢

n----7oo}pn

x,y,t

exists, then it is independent of the finite-dimensional approximation, P;:'y,t, that we choose. Problem 10.1 1. Compute (using (10.3) and a finite-dimensional approximation of the path space) Ut for

a) V(x) = 0 (free particle) b) V (x) = m~2 x 2 (harmonic oscillator in dimension d = 1). 2. Derive a path integral representation for the integral kernel of e-j3H/Ii. 3. Use the previous result to find a path integral representation for Z((3) := tr e-j3H/1i (hint: you should arrive at the expression (11.10)).

10.2 Generalizations of the Path Integral Here we mention briefly two extensions of the Feynman path integral we have just introduced. 1. Phase-space path integral:

where

l/J7r

is the path measure, normalized as

(in QM, 1i 3p = d 3p/(21f)3/2). To derive this representation, we use the Trotter product formula, the expression e- iEH ~ 1 - iEH for E small,

10.3 Mathematical Supplement: the Trotter Product Formula

119

and the symbolic (pseudo differential) composition formula. Unlike the representation I eiSjnDcp, this formula holds also for more complicated H, which are not quadratic in p! 2. A particle in a vector potential A(x). In this case, the Hamiltonian is

H(x,p)

=

1 2m

-(p - eA(x))2 + V(x)

and the Lagrangian is

L(x,x) = ; x 2 - V(x)

+ ex· A(x).

The propagator still has the representation

but with

S(cp) =

rt L(cp, ¢)ds = iot (m ¢2 _ V(cp))ds + e iort A(cp) . ¢ds.

io

2

Since A(x) does not commute with 'V in general, care should be exercised in computing a finite-dimensional approximation: one should take

or and not

10.3 Mathematical Supplement: the Trotter Product Formula Let A, B, and A + B be self-adjoint operators on a Hilbert space 1i. If [A, B] i- 0, then ei(A+B) i- eiAe iB in general. But we do have the following.

Theorem 10.2 (Trotter product formula) Let either A and B be bounded, or A, B, and A + B be self-adjoint and bounded from below. Then for Re()..) :::; 0, eA(A+B) = s - lim (eA~eA*t n-+CXJ

120

10 The Feynman Path Integral

Remark 10.3 The convergence here is in the sense of the strong operator topology. For operators An and A on a Hilbert space H, such that D(An) = D(A), An ----'> A in the strong operator topology (written s-limn--->ooAn = A) iff IIAn¢ - A¢II ----'> 0 for all ¢ E D(A). For bounded operators, we can take norm convergence. In the formula above, we used a uniform decomposition of the interval [0,1]. The formula still holds for a non-uniform decomposition. Proof for A,B bounded: We can assume Tn = eA/ne B/ n. Now by "telescoping",

snn _ Tn = snn _ T nn sn-l n

.x =

1. Let Sn

= e(A+B)/n

and

+ Tnsn-l + ... _ Tnn n

n-l

= L,T~(Sn - Tn)S;;;-k-l k=O

so n-l

liS;;; - T:II ::::; L, IITnllkllSn - TnIIIISnlln-k-l k=O n-l

::::; L,(max(IITnll, IISnll))n-11ISn - Tnll k=O

::::; neiIAII+IIBIIIISn -Tnll· Using a power series expansion, we see IISn - Tnll = O(1/n 2 ) and so IIS~ ----'> 0 as n ----'> 00. D A proof for unbounded operators can be found in [RSI].

T: II

11 Quasi-classical Analysis

In this chapter we investigate the connection between quantum and classical mechanics. More precisely, taking advantage of the fact that the Planck constant provides us with a small parameter, we compute some key quantum quantities - such as quantum energy levels - in terms of relevant classical quantities. This is called quasi-classical (or semi-classical) analysis. To do this, we use the Feynman path integral representation of the evolution operator (propagator) e- iHt / n. This representation provides a non-rigorous but highly effective tool, as the path integral is expressed directly in terms of the key classical quantity - the classical action. The heuristic power of path integrals is that when treated as usual convergent integrals, they lead to meaningful and, as it turns out, correct answers. Thus to obtain a "quasi-classical approximation", we apply the method of stationary phase. Recall that the (ordinary) method of stationary phase expands the integral in question in terms of the values of the integrand at the critical points of the phase, divided by the square root of the determinant of the Hessian of the phase at those critical points. The difference here is that the phase - the classical action - is not a function of several variables, but rather a "functional", which (roughly speaking) is a function of an infinite number of variables, or a function on paths. Critical points of the classical action are the classical paths (solutions of Newton's equation) and the Hessians are differential operators. Thus we need some new pieces of mathematics: determinants of operators and elements of the calculus of variations. These are presented in supplementary Section 11.5 and Chapter 12 respectively. Below we consider a particle in ]Rd described by a Schrodinger operator fi2 H = - 2m .1x

+ V(x).

(11.1)

Let L be a length scale for the potential, and 9 its size. So roughly, 9 sUPx lV(x) I and L = g(supx IV'V(x)l)-l. Re-scaling the variable as x -+ x' x/L, we find H = gH' where

H'

=

= =

fi'2 - - . 1 ,+ V'(x') 2m' x

where V'(x') = g-lV(Lx') and fi'/v'riI! = fi/(Lvmg). Now the potential V'(x') is essentially of unit size and varies on a unit length scale. The pa-

S. J. Gustafson et al., Mathematical Concepts of Quantum Mechanics © Springer-Verlag Berlin Heidelberg 2003

122

11 Quasi-classical Analysis

rameter Ii/ is dimensionless. If h' / -.;rn; < < 1, we can consider it a small parameter. As an example suppose V(x) is of the order lOOme 4 //1,2, where me4//1,2 is twice the ionization energy of the ground state of the hydrogen atom, and varies on the scale of the Bohr radius (of the hydrogen atom) L = /1,2/(me 2). Then /1,'/-.;rn; = l/lO. In the expansions we carry out below, we always have in mind the operator H' and the dimensionless parameter n' with the primes omitted; that is, we think of (11.1) in dimensionless variables.

11.1 Quasi-classical Asymptotics of the Propagator The path integral (10.3) has the form of oscillatory integrals extensively studied in physics and mathematics. One uses the method of stationary phase in order to derive asymptotic expressions for such integrals. It is natural, then, to apply (formally) this method - with small parameter /1, - to the path integral, in order to derive a quasi-classical expression for the Schrodinger propagator e-itH/n(y, x). We do this below. But first, we quickly review the basics of the method of stationary phase (in the finite-dimensional setting, of course). The stationary phase method. We would like to determine the asymptotics of oscillatory integrals of the form

as /1, --+ 0 (here ¢ is a finite dimensional variable). The basic idea is that as /1, --+ 0, the integrand is highly oscillating and yields a small contribution except where 'VS(¢) = 0 (i.e., critical points). We study

1(/1,):= where!

E

r

JJRd

!(¢)eiS(e/»/nd¢

C[)(]Rd) and consider two cases:

1. supp(J) contains no critical points of S. Let 0 in ]Rd. Define

=

0e/> denote the gradient

where the inner-product here is just the dot product in ]Rd. Note that LeiS(e/»/n = eiS(e/»/n, so integrating by parts gives

for any m.

11.1 Quasi-classical Asymptotics of the Propagator

123

2. supp(J) contains only one critical point, ¢ of S, which is non-degenerate (det S"(¢) i:- 0). Expand S(¢) around ¢. Writing ¢ - ¢ = VOO we obtain S(¢)//i

= S(¢)//i + ~(a, S"(¢)a) + O(v1ia3 )

where S"(¢) is the matrix of second derivatives of S: [S"(¢)]jk 8 2S(¢)/8¢j8¢k. So J(/i) = /id/2 e iS(f,)/li

r

JJRd

f(¢

+ Voo)ei (a,S"(f,)a)/2 eiO(Vfia

3

)da.

N ow we use the formula

(11.2) (We can derive this expression by analytically continuing J e-a(a,S"a) from Re( a) > 0, though the integral is not absolutely convergent. Some care is needed in choosing the right branch of the square root function. An unambiguous expression for the right hand side is

where for a symmetric matrix A, sgn(A) denotes the number of positive eigenvalues minus the number of negative eigenvalues.) Noting that f( ¢+ voo) = f(¢) + O(VIi), we have the stationary phase expansion (for one critical point):

(11.3) We formally apply the method outlined above to the infinite-dimensional integral (10.3). To this end, we simply plug the path integral expression (10.3) into the stationary phase expansion formula (11.3). The result is e-iHt/li(y, x)

=

lx,y"

eiS(¢)/li D¢

L f, cp of

Mf,(det S,,(¢))-1/2 e iS(f,)/li(1

+ O(v1i))

(11.4)

S

where Mf, is a normalization constant, and the sum is taken over all critical points, ¢, of the action S(¢), going from x to y in time t. Critical points and Hessians of functionals are discussed in the mathematical supplement, Chapter 12. Note that the Hessian S"(¢) is a differential operator. The problems of how to define and compute determinants of Hessians are discussed in the mathematical supplement Section 11.5. We will determine M := Mf" assuming it is independent of ¢ and V. For V = 0, we know the kernel of the propagator explicitly (see (2.15)):

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11 Quasi-classical Analysis

e-iHot/li(y, x) = (27rint/m)-d/2eimlx-yI2/2Iit. So in particular, e-iHot/li(x, x) = (27rint/m)-d/2. Now the right-hand side of the expression (11.4) for e-itHo/li(x, y) is (to leading order in n)

M( det S~ (¢o) )-1/2 eiSo(¢o)/1i where the unique critical point is ¢o(s) = x + (y - x)s/t. Thus So(¢o) mly _x1 2 /2t, and S~(¢o) = -ma;, an operator acting on functions satisfying Dirichlet boundary conditions. Comparison thus gives us

and therefore

e-iHt/ li ( x) = '" (27ritn) -d/2 (det( -ma;)) 1/2 eiS(¢)/Ii(l y, _L m det S"(¢) ¢

cp

S

+

O(VYi))

(11.5)

as

n ---+ O. This is precisely the quasi-classical expression we were looking for.

We now give a "semi-rigorous" derivation of this expression. We assume for simplicity that S has only one critical point, ¢, going from x to y in time t. Let {~j}~l be an orthonormal basis of eigenfunctions of S"(¢) acting on £2([0, t]) with zero boundary conditions (the eigenfunctions of such an operator are complete - see the remark in Section 9.14). So S"(¢)~j = /-lj~j for eigenvalues /-lj. For the n-th order finite dimensional approximation to the space of paths in the path integral, we take the n-dimensional space of functions of the form n

¢(n) =

¢ + Laj~j j=l

with aj E R Expanding s(¢(n)) around

¢ gives

where ~:= ¢(n) -

¢=

n

Laj~j.

j=l We also have

n

D¢(n)

= On

II daj j=l

(On some constant). Now using the fact that n

n

(~,S"(¢)~) = L aiaj(~i,S"(¢)~j) = L/-lja;, i,j=l j=l

11.2 Quasi-classical Asymptotics of Green's Function

we have

(

i ¢(n):x-+y

eiS(¢(n))/n Dq)n) = eiS(¢)/n

J

eiI: I" j a;/2n(1

125

+ O(a 3 /1i))Cn dn a

(as in Section 11.1, the integrals here are not absolutely convergent). Setting bj := aj / VIi this becomes

lin/2CneiS(¢)/n

J

ei I: I" j b;/2 (1

+ O(b3 V1i) )dnb

which is (see (11.2))

C n (27rili)n/2( det(S" (¢) IFn ))-1/2e iS (¢)/n(1

+ O( VIi))

where Fn := {L:~ ajej} so that det(S"(¢)IFn ) = fG fJ,j. To avoid determining the constants Cn arising in the "measure" Dcp, we compare again with the free (V = 0) propagator. Taking a ratio gives us

which reproduces (11.5).

11.2 Quasi-classical Asymptotics of Green's Function Definition 11.1 Green's function GA(x, y, z) of an operator A is (A z) -1 (y, x), the integral kernel of the resolvent (A - z) -1 .

For A self-adjoint,

(A - z)-l =

i

Ii

roo e-iAt/n+izt/ndt

io

converges if Im(z) > O. Taking z = E f ----* 0, we define

+ if

(E real,

f

> 0 small), and letting

Note that the -iO prescription is essential only for E E O'c(A), while for E E lR\O'c(A), it gives the same result as the +iO prescription. Here we are interested in the second case, and so we drop the -iO from the notation. The above formula, together with our quasi-classical expression (11.5) for the propagator e- iHt / \ yields in the leading order as Ii ----* 0 (dropping the -iO from the notation)

126

11 Quasi-classical Analysis

(H - E)-l(y,X)

=

~

1

00

o

L ?>

cp

K?>ei(S(?»+Et)/lidt S

where the sum is taken over the critical paths, ¢, from x to y in time t, and

K?>:=

(

m ) d/2 (det( -mil;) ) detS"(¢)

1/2

27ritn

We would like to use the stationary phase approximation again, but this time in the variable t. Denote by [ = [(x, y, E) the critical points of the phase S(¢) + Et. They satisfy the equation

8S(¢)/8t = -E. The path WE := ¢It=t is a classical path at energy E (see Lemma 12.36 of Section 12.6). Introduce the notation So(x, y, t) := S(¢) for a classical path going from x to y in time t. Then the stationary phase formula gives (in the leading order as II, --* 0)

(H - E)-l(y,x) =

~ LDYE2eiWWE/Ii,

(11.6)

WE

where the sum is taken over classical paths going from x to y at energy E. Here we have used the notation DYE2 := KWE(27rin)1/2(82S0/8t2)-1/2It=t and we have defined WWE (x, y, E) := (So(x, y, t) + Et)lt=t (so WWE is the Legendre transform of So in the variable t). Lemma 11.2

D

WE

= (_1_8 2 S 0 )d-1 det [8 2W WE 8 2 WWE _ 8 2WWE 8 2 WWE ] 27rin 8t2

8x8y

8E2

8x8E 8y8E

.

(

11.7

)

Proof We just sketch the proof. The first step is to establish

-m8';) (m)

det( det(S"(w)) =

-T

-d

det

(8 S

2 0 )

8x8y

(11.8)

(we drop the subscript E for ease of notation). To see this, we use the fact that if for an operator A we denote by J A the d x d matrix solving AJ = 0 (the Jacobi equation) with J(O) = 0 and j(O) = 1, then (see (11.12)) det(-m8';)

det(S"(w))

det Lma; (t) det JSIf(W)(t)·

Next, we use 1 --JSIf(W)(t) = m

2S (8-8x8y -0)-1

11.2 Quasi-classical Asymptotics of Green's Function (Equation (12.6) of Chapter 12), and for the free classical path x)(s/t),

-~Lma2(t) = m

({;2S(1>O))-l 8x8y

s

=_

127

1>0 = x + (y-

(m)-11 t

to arrive at (11.8). We can then show that

equals the determinant on the right hand side in equation (11. 7) (see the appendix to this section for details). D We will show in Section 12.6 (see Lemma 12.37) that the function WWE (the action at energy E) satisfies the Hamilton-J aco bi equation

h(x, -

8~;E ) = E.

Differentiating this equation with respect to y gives

8h 8 2 WWE _ 0 8k 8x8y - , and we see that the matrix (8 2 W WE /8x8y) has a zero-eigenvalue. Thus its determinant is zero. So if d = 1, (11.7) yields D

__ 8 2 W WE 8 2 W WE 8x8E 8y8E WE -

(d = 1).

(11.9)

Formula (11.6), together with (11.7) or (11.9), is our desired quasi-classical expression for Green's function (H - E)-l(y, x). 11.2.1 Appendix

Proposition 11.3 At t

= t,

Proof We drop the subscripts from So and WWE to simplify the notation. Differentiating W = S + Etlt=f with respect to x, we obtain 8W 8x

= 8S + 8S 8t + E 8t 8x

which due to the relation 8S/8t

8t 8x

= - E gives

8x '

128

11 Quasi-classical Analysis

aw ax Similarly,

as ax'

aw aE =t. This last equation, together with as/at = - E yields aw ay

as ay

and

aaE2w = ~ = _ (a s)-1 aE at 2 2

2

Furthermore,

a2s axay

a2w a2w aE at axay axaE at ay a2s [ a2wa2w a2w a2w] = at 2 - axay aE2 + axaE aEay

--=--+-----

and the result follows. 0

11.3 Bohr-Sommerfeld Semi-classical Quantization In this section we derive a semi-classical expression for the eigenvalues (energy + V. We use the Green's levels) of the Schrodinger operator H = function expansion (11.6) from the last chapter. For simplicity, we will assume d = 1. Application of the expression (11.6) requires a study of the classical paths at fixed energy. Consider the trajectories from x to y at energy E. We can write them (using informal notation) as

-;:Ll

cPn = cPxy ± no: where 0: is a periodic trajectory (from y to y) of minimal period, at the energy E, while cPxy is one of the four "primitive" paths from x to y at energy E, sketched in Fig. 11.1. V( )

\

V(;) /

x/~~~

E~ E~_~ x

y

x

Fig. 11.1. Primitive paths at energy E.

y

11.3 Bohr-Sommerfeld Semi-classical Quantization

129

All these paths are treated in the same way, so we consider only one, say the shortest one. The space time picture of
x

y

<1>n

Fig. 11.2. Turning points of
where
=
and I

=

lot L(a, it)ds + Et.

But a is a critical path so its energy is conserved (see Lemma 12.35)

and so I

=

lot {mit

2 -

E}

+ Et =

1

k· dx,

where k(s) = mites) and dx = it(s)ds. Let w be a classical path at energy E, and let us determine Dw' We will show later (see (12.11) and Lemma 12.31) that

8Ww --z:;;: = and

8Ww 7iiJ =

880 8x It=t

= -k(O)lt=t =

=t=

V2m(E -

8~/ 8y It=t = k(t)lt=t = ±y 2m(E

Vex))

- V(y)).

Differentiating these relations with respect to E and using (11.9), we obtain Dw

=-

m2 k(x)k(y)

130

11 Quasi-classical Analysis

At a turning point xo, k(xo) = 0 and k changes sign (we think about k(x) as a multi-valued function, or a function on the Riemann surface of viz, so at a turning point Jk(x) crosses to a different sheet of the Riemann surface). Because k changes sign at each of the two turning points of the periodic trajectory, we conclude that

So our semi-classical expression (11.6) for Green's function GE(y, x) is

GE(y,x) =

~ N exp[i(Wq,/n+ n [~1 k· dx -

= NeiW",/fi

1-

1

eiU", k·dx/fi-Jr)

(N is a constant). We conclude that as 11, H has eigenvalues) when

11

k· dx

'if])]

=

--->

2'ifn(j

0, GE(y, x) has poles (and hence

+ 1/2) I

for an integer j. This is the Bohr-Sommerfeld semi-classical quantization condition (for d = 1). It is an expression for the quantum energy levels (the energy E appears in the left hand side through the periodic path 0: at energy E), which uses purely classical data!

Problem 11.4 Show that for the harmonic oscillator potential, the BohrSommerfeld condition gives all of the energy levels exactly.

11.4 Quasi-classical Asymptotics for the Ground State Energy Here we derive a quasi-classical expression for the ground state energy (the lowest eigenvalue) of the Schrodinger operator H = - ;: .1 + V when V (x) ---> 00 as Ixl ---> 00 (as fast as some power of lxi, say); i.e., V(x) is a confining potential. We first define a couple of quantities which are familiar from statistical mechanics (see Chapter 9; we repeat some of the definitions here).

Definition 11.5 The partition function, Z(f3), at inverse temperature (3 is Z((3):= tr e-{3H (the trace is well-defined since O"(H) fast).

= {En}o with En

---> 00

>0

sufficiently

11.4 Quasi-classical Asymptotics for the Ground State Energy

131

Definition 11.6 The free energy, F, is

F(j3) :=

-131 InZ(j3).

The free energy is a useful quantity for us here because of the following connection with the ground state energy of the Schrodinger operator H: lim F(j3) = Eo.

(3-+=

This is the Feynman-Kac theorem of Section 9.6. Our goal is to find the semi-classical asymptotics for Eo by deriving an asymptotic expression for Z(j3) from a path integral. As we have seen (Problem 10.1), the path integral expression for Z(j3) is

Z(T/n) = { }
e-Se(
(11.10)

T

where Se(¢) = Jt{~1¢12 + V(¢)} (note that this is not the usual actionthe potential enters with the opposite sign). Remark 11.7 The path integral appearing in (11.10) can be rigorously defined. We refer the reader to [Sim3, GJ, RSII] for details.

Mimicking the procedure we used for the Schrodinger propagator (i.e., the stationary phase method, which in the present context is called the Laplace method), we see that the quasi-classical expression for Z(T/n) is (11.11) minimal paths

w

(N a constant) where by a minimal path, we mean a path minimizing Se, and where Bw := det S~(wo) detS~(w)

with So(¢) = J;(m/2)1¢1 2 . A critical path for Se is a classical path for the inverted potential - V. We specialize to d = 1 for simplicity, and we assume V has only one minimum, at Xo. Then the minimal path is w(s) == Xo (a constant path), and S~(w) = -ma; + V"(xo). Because Xo minimizes V, V"(xo) > 0 and we write V"(xo) = mw 2 . Then using the method (11.12) of computing ratios of determinants (see Section 11.5 below), we easily obtain

Also, Se(w)

= TV(XO).

In this way, we arrive at the leading-order expression

132

11 Quasi-classical Analysis

F(T/n) ~ V(Xo) as n ---+ obtain

o. Letting f3 = T /n

---+ 00

+ nw/2 + O(l/T)

and using the Feynman-Kac formula, we

IEo ~ V(xo) + ~nw I

which is the desired asymptotic (as n ---+ 0) expression for the ground state energy. It is equal to the classical ground state energy, V(xo), plus the ground state of the harmonic oscillator with frequency JV"(xo)/m. This suggests that the low energy excitation spectrum of a particle in the potential V (x) is the low energy spectrum of this harmonic oscillator.

11.5 Mathematical Supplement: Operator Determinants In this section we discuss determinants of differential operators, which appear in the stationary phase expansion of the path integrals developed earlier in this chapter. For square matrices, the determinant function has the properties

1. A is invertible iff det A -=J- 0 2. A = A * =} det A E R. 3. det(AB) = det(A) det(B) 4. A> 0 =} det A = etr(inA) 5. det A = fL ev of A A We would like to define the determinant of a Schrodinger operator. Example 11.8 Let H = -.::1 + V on [0, L] with zero boundary conditions (assume V is bounded and continuous). Then using the fact that for nEZ, sin(7rnx/L) is an eigenfunction of -.::1 with eigenvalue (7rn/L)2, we obtain

II (H -

(7rn/ L )2) sin(7rnx/ L) II

=

IIV(x) sin(7rnx/ L) II

::;

(sup IVI) I sin(7rnx/ L) II,

and spectral theory tells us that H has an eigenvalue in the interval [(7rn/ L )2_ sup lVI, (7rn/ L)2 + sup IVI]. Since { J2/ L sin(7rnx/ L) I n = 1,2, ... } is an orthonormal basis in L2[0, L], we have

a(H) = {(7rn/L)2

+ 0(1) In E Z}.

fL ev H A = 00. For a positive matrix, A, we can define (A (s) := tr A = LA ev of A A

So trying to compute the determinant directly, we get

-8

Problem 11.9 Show in this case that det(A) = e-(~(O).

-8.

11.5 Mathematical Supplement: Operator Determinants

Now for H = -L1

133

+ Von [0, L]d with zero boundary conditions, A

ev of

H

exists for Re(s) > 1/2 (see Example 11.8 for d = 1). If (H has an analytic continuation into a neighbourhood of s = 0, then we define

So defined, det H enjoys Properties 1-4 above, but not Property 5. It turns out that for H = -L1 + V, (H does have an analytic continuation to a neighbourhood of 0, and this definition applies. It is difficult, however, to compute a determinant from this definition. In what follows, we describe some useful techniques for computation of determinants. U sing the formula

('ex)

1

A- s = r(s) Jo

tS-1e-tAdt

for each An E u(H) leads to

Now A is an eigenvalue of H iff e- tA is an eigenvalue of e- tH (this is an example of the spectral mapping theorem - see Section 5.2), and so e- tAn is the n-th eigenvalue of e- tH . Thus

This formula can be useful, as it may be easier to deal with tr( e- tH ) c tH (x, x)dx than tr(H- S).

J

Example 11.10 We consider H = -L1 in a box B periodic boundary conditions. In this case

[-L/2, L/2]d with

in B x B, if B is very large, and so tr e- tH

=

r

JBXB

e- tH (x, x)dx

~

r (21ft)-d/2 = (21ft)-d/2 vo l(B).

JB

But calculation of det H by this method is still a problem.

134

11 Quasi-classical Analysis

Remark 11.11 Often (and in all cases we consider here), we have to compute a ratio, ~:~ ~ , of determinants of two operators A and B, such that det A and det B must be defined through a regularization procedure such as the one described above, but the determinant of the ratio AB- 1 can be defined directly. Since ~:~~ = det(AB- 1 ), we can make sense of the ratio on the left hand side without going to regularization.

The most useful calculational technique for us is as follows. Let A and B be Schrodinger operators on £2([0, T]; JRd) with Dirichlet boundary conditions. Denote by JA the solution to AJA = 0 with JA(O) = 0, jA(O) = 1 (J a d x d matrix valued function on [0, TD. Then one can show (see, eg., [LS, KID detA det B

det JA(T) det JB(T)·

Remark 11.12 If A = SI/(¢;) for a critical path matrix along ¢; (see Section 12.5).

(11.12)

¢;,

then JA is the Jacobi

Problem 11.13 Let A(T) be the operator -a;+q(t) defined on £2([0, T]; JRd) with Dirichlet boundary conditions, and let JA be the corresponding Jacobi matrix. Show that the functions det A(T) and det J(T) have the same zeros of the same multiplicities (to is a zero of f (t) of multiplicity n if a k / at k f (to) = 0 for k = 0, ... , n - 1 and an /atnf(to) i= 0).

12 Mathematical Supplement: the Calculus of Variations

The calculus of variations, an extensive mathematical theory in its own right, plays a fundamental role throughout physics. This chapter contains an overview of some of the basic aspects of the variational calculus. This material in used in Chapters 11 and 13, in conjunction with the path integral introduced in the previous chapter, to obtain useful quantitative results about quantum systems.

12.1 Functionals The basic objects of study in the calculus of variations are functionals, which are just functions defined on Banach spaces (usually spaces of functions). If we specify a space, X, then functionals on X are just maps S : X --+ R In the calculus of variations, one often uses spaces other than L2(]Rd), and which are not necessarily Hilbert spaces. Among the most frequently encountered spaces are the LP -spaces (p ?: 1)

and the Sobolev spaces HS(]Rd), s = 1,2, ... , introduced in Section 1.5. Recall that the Sobolev spaces are Hilbert spaces. The space LP(]Rd), endowed with the norm

II¢IILP

:=

(ld 1¢(X)lPdX)

liP,

is a Banach space, but not (for p -1= 2) a Hilbert space. We will also encounter Sobolev spaces of vector-valued functions,

(s = 1,2, ... ) where f2 is an open domain in ]Rd, and aa was defined in Section 1.5. Spaces of continuously differentiable functions also arise frequently. For k E {O, 1,2, ... }, and an open set f2 C ]Rd, we define C k (f2; ]Rm) to be the set of all functions ¢ : f2 --+ ]Rm such that aa¢ is continuous in f2 for all letl :::; k, and for which the norm

S. J. Gustafson et al., Mathematical Concepts of Quantum Mechanics © Springer-Verlag Berlin Heidelberg 2003

136

12 Mathematical Supplement: the Calculus of Variations

11¢IICk(l?;IR=)

L

sup la"'¢(x)1 lal::;k xED

:=

is finite. Equipped with this norm, C k (Q; JRm) is a Banach space. Example 12.1 Here are some common examples of functionals, S, and spaces, X, on which they are defined.

= L2([a, bj; JR), f E X* = X is fixed (the meaning of the notation X* will be explained in the next section), and

1. X

S: ¢

lb

f-+

f¢.

Note that S(¢) is well-defined, by the Cauchy-Schwarz inequality. 2. Evaluation functional: X = C([a, b]), Xo E (a, b) fixed, and S: ¢

f-+

¢(xo)

Compare this with the first example by taking f(x) = 8(x - xo) (E X*; in this case X* i- X). 3. Let V : JRm ----+ JR be continuous. Set X = {¢ : JRd ----+ JRm I V(¢) E LI(JR d)}, and take S:¢f-+

4. Dirichlet functional: X

HI(JRd), and

=

S:

5. Classical action: fix a,b a, ¢(T) = b}, and define S:

r V(¢).

JIRd

E

¢

f-+

¢

f-+

~

2

r

JIRd

IV'¢12.

JR m , set X = {¢

E

C I ([o,T];JRm) I ¢(O)

loT {~ml¢12 - V(¢)}.

6. Classical action: Let L : JRm x JRm (a Lagrangian function), and

S: ¢

f-+

----+

JR be a twice differentiable function

loT L(¢, ¢)dt.

Here, X is as in Example 5. 7. Action of a classical field theory: fix X

f, g, E HI (JRd; JR m), set = {¢ E HI(JR d X [0, T]; JRm) I ¢(x,O) = f(x), ¢(x, T) = g(x)},

and define S:

where V : JRm

¢ ----+

f-+

loT ld {-~lat¢12 + ~IV'x¢12 + V(¢)}.

JR is a continuous function.

12.2 The First Variation and Critical Points

137

8. Quadratic form: let B be a a self-adjoint operator on a Hilbert space, X = D(B), and set

s: 9. Action of an EM field: X

S:A

f--4

cP

1

f--4

= {A

E

~ loT

l3

Hl(1R3 x [0, TJ; 1R3 )

{-IAI2 + 1\7

10. Lagrangian functional: Suppose £ : IR x IRd is a twice differentiable function, and set

S : cP

f--4

(12.1)

2(cp,Bcp).

--+

X

\7. A

= O}

and

AI2}.

IR (a Lagrangian density)

J

12(cp(x, t), \7cp(x, t))dxdt

(here X is a space of functions on space-time IR~ x IRt, the form of which depends on the form of 12). We will encounter many of these functionals in applications to quantum mechanics and quantum field theory.

12.2 The First Variation and Critical Points The notion of a critical point of a functional is a central one. It is a direct extension of the usual notion of a critical point of a function of finitely many variables (Le., a place where the derivative vanishes). The solutions of many physical equations are critical points of certain functionals (action or energy functionals) . In what follows, the spaces X on which our functionals are defined will generally be linear (i.e. vector) spaces or affine spaces. By an affine space, we mean a space of the form X = {CPo+CP cp E X o}, where CPo is a fixed element of X, and Xo is a vector space. We will encounter examples of functionals defined on non-linear spaces when we study constrained variational problems in Section 12.3. Let X be a Banach space, or an affine space based on a Banach space. 1

Definition 12.2 A path, CP>., in X is a differentiable function I ::1 A from an interval I C IR containing 0, into X.

f--4

CP>.

Definition 12.3 The tangent space, Tq,X, to X at cp E X is the space of all "velocity vectors" at cp:

Tq,X:=

{%>.I>.=o

1

CPA is a path in X,CPo = cp}.

138

12 Mathematical Supplement: the Calculus of Variations

Problem 12.4 Check that Tq,X is a vector subspace of X (or Xo in the affine case).

In fact, if X is a Banach space, then Tq,X = X (and if X is an affine space based on the Banach space X o , Tq,X = Xo). To see this, just note that for any ~ E X, ¢).. := ¢ +,\~ is a path in X, satisfying ¢o = ¢ and o¢)../o'\ = ~ (the reader is invited to check the corresponding statement for affine spaces). So the tangent space to a linear space is not very interesting. The notion of a tangent space is useful when working in non-linear spaces (we will see an example of this shortly).

Definition 12.5 A variation of ¢ E X along such that ¢o = ¢ and 0¢)../0'\1)..=0 =~.

~ E

Tq,X is a path, ¢).. in X,

Example 12.6 Define X xy = {¢ E C([a,b],X) I ¢(a) = x,¢(b) = y} (an affine space). Then Tq,Xxy = Xoo (see Fig. 12.1), and an example of a variation of ¢ E X xy in the direction ~ E Xoo is ¢).. = ¢ +,\~ E X xy . y

x

Fig. 12.1. Variations of a path. We wish to define a notion of differentiation of functionals which is a direct extension of usual differentiation of functions of finitely many variables. To do so, it is useful to introduce a new concept ~ that of the dual space.

Definition 12.7 1. A bounded linear functional on a Banach space, B, is a map l : B -----
for all

~,T/ E

B, and

0:,

+ (3T/)

= o:l(~)

+ (3l(T/)

(3 E
<

00

such that

:::: ClI~11

for all ~ E B. 2. The dual space of a Banach space, B, is the space, B*, of all bounded linear functionals on B. The dual space B* is also a Banach space, under the norm III liB' := sUP';EB,II';IIB=l Il(~)I· One often denotes the action of l E B* on ~ E B by l(~)

=

(l,~).

12.2 The First Variation and Critical Points

139

Definition 12.8 Let S : X -7 lR be a functional on a Banach space X, and let ¢ E X. We say that S is differentiable at ¢ if there is a linear functional, 8S( ¢) E X*, such that (12.2)

for any variation ¢A of ¢ along tional) derivative of S at ¢.

~ E

X. The functional 8S(¢) is the (varia-

Remark 12.9 1. The notion of differentiability introduced here is often called Gateaux differentiability. A stronger notion of differentiability, called Frechet differentiability, demands (for linear spaces) that S(¢+~)

= S(¢) + (8S(¢),~)

+o(II~llx)

as 11~llx -7 o. The reader can check that if S is continuously (Gateaux) differentiable at ¢o E X (i.e. 8S(¢) exists in a neighbourhood of ¢o, and is a continuous map from this neighbourhood into X*), then S is Fn§chet differentiable at ¢o. 2. We will sometimes use the notation S' (¢) for the variational derivative

8S(¢).

3. If X is a Hilbert space, then we can identify its dual, X*, with X itself, via the map X :3 ¢ f-+ l¢ E X* with l¢~ := (¢,~) for ~ E X (here the notation (-,.) indicates the Hilbert space inner-product). The fact that this map is an isomorphism between X and X* is known as the Riesz representation theorem. Thus when X is a Hilbert space, we can identify 8S(¢) with an element of X. Furthermore, the expression (12.2) is still valid if (.,.) denotes the Hilbert space inner-product. Example 12.10 We compute the derivatives of some of the functionals in Example 12.1. For simplicity, we suppose that whenever X is a space of functions, it is, in fact, an L2-space. Then the variational derivative can be found easily by integrating by parts (where necessary). 1. For the functional S(¢) (8S(¢),~)

=

J: f¢,

we compute

d = d)" S(¢>JIA=O =

=

lb f~

=

d

a

d

r

= d)" S(¢A)lhO = d)" JlRd =

8

f 8)" ¢AlhO

U,~)

and thus 8S(¢) = f. 3. For S(¢) = JlRd V(¢), we compute

(8S(¢),~)

lb

8 8)" V(¢A)IA=O

r V'V(¢)· ~ = (V'V(¢),~)

JlRd

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12 Mathematical Supplement: the Calculus of Variations

and so 8S(¢) = V'V(¢). 4. For S(¢) = IlR d 1V'¢12, we compute

!

r

d 1 8 2 (8S(¢),~) = d>" S(¢A)IA=O = "2 JlR d 8>..1V'¢AI IA=O

=

r V'¢.V'~= JlRr (-£1¢)~=(-£1¢,~)

JlR

d

d

where we integrated by parts (Gauss theorem), and used the fact that the functions decay at 00. Thus we see 8S(¢) = -£1¢. We leave the remaining examples as an exercise. Problem 12.11 Compute the variational derivatives for the remaining functionals in Example 12.1. You should find 5. 8S(¢) = -m¢ - V'V(¢) 6. 8S(¢) = -1t(8J,L) + 8¢L 7. 8S(¢) = D¢ + V'V(¢) where D := 8i: -£1 is the D'Alembertian operator 8. 8S(¢) = B¢ 9. 8S(A) = DA 10. 8S(¢) = -V'(8",¢L) + 8¢L

As in the finite-dimensional case, a critical point is a place where the derivative vanishes. Definition 12.12 An element ¢ E X is a critical point (CP) of a functional S: X ----+ lR if 8S(¢) = O. In fact, many physical equations are critical point equations for certain functionals. Example 12.13 Continuing with the same list of examples of functionals, we can write down some of the equations describing their critical points: 4. 5. 6. 7. 9. 10.

£1¢ = 0 (Laplace equation, ¢ a harmonic function) m¢ = -V'V(¢) (Newton's equation) 1t(8J,L) = (8¢L) (Euler-Lagrange equation) D¢ + V'V(¢) = 0 (nonlinear wave/Klein-Gordon equation) DA = 0 (wave equation) -V'(8",¢L) + 8¢L = 0 (classical field equation)

The following connection between critical points and minima (or maxima) is familiar from multi-variable calculus. Theorem 12.14 If ¢ locally minimizes or maximizes a functional S : X ----+ lR, then ¢ is a critical point of S. (We say ¢ is a local minimizer (resp. maximizer) of S if there is some 0 > 0 such that S(¢) :2': S(¢) (resp. S(¢) :::: S(¢)) for all ¢ with II¢ - ¢llx < 0.)

12.3 Constrained Variational Problems

141

Problem 12.15 Prove this (hint: it is similar to the finite-dimensional case).

As we have seen, the equation for a critical point of the functional

(see Example 12.1, no.5) is exactly Newton's equation of classical mechanics. This is a special case of the principle of minimal action: solutions of physical equations minimize (more precisely, make stationary) certain functionals, called action functionals. It is one of the basic principles of modern physics.

12.3 Constrained Variational Problems Let Sand C be continuously differentiable functionals on a Banach space X. We consider the problem of minimizing the functional S(¢), subject to the constraint C(¢) = O. That means we would like minimize S(¢) for ¢ in the (non-linear) space M:={¢EX I C(¢)=O}. We assume that C'(¢) =I 0 for ¢ E M (here, and below, C'(¢) and S'(¢) denote the variational derivatives of the respective functionals considered as functionals on all of X, rather than just M). Let us first observe the following:

T¢M = {.;

E

X

I

(C'(¢),';) = O}.

To see this, suppose ¢A is a variation of ¢ in M. Differentiating the relation C(¢A) = 0 with respect to >. at >. = 0 yields (C'(¢),';) = 0, where .; = ¢AIA=o. Thus T¢M c {.; E X I (C'(¢),';) = O}. Conversely, given .; E X such that (C'(¢),';) = 0, one can show (using the "implicit function theorem") that there is a path ¢ A E M satisfying ¢o = ¢ and ¢ AI A=O = .;. So'; E T¢M. Concerning the constrained variational problem, we have the following result:

tA

tA

Theorem 12.16 (Lagrange multipliers) Let Sand C be continuously differentiable functionals on a Banach (or affine) space X. Suppose ¢ locally minimizes S(¢) subject to the constraint C(¢) = 0 (i.e. ¢ locally minimizes S on the space M) and C'(¢) =I O. Then ¢ is a critical point of the functional S - >'C on the space X, for some>. E lR (called a Lagrange multiplier). In other words, ¢ satisfies the equations

S'(¢)

= >.C'(¢)

and

C(¢) =

o.

Proof The fact that ¢ minimizes S over M implies that ¢ is a critical point of S considered as a functional on M. This means that S'(¢) = 0 on T¢M. Since

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12 Mathematical Supplement: the Calculus of Variations

T(f>M = {~E X

1

(C'((fi),~)

= O}

we can rewrite the latter statement as (8'((fi) , ~) = 0 for all ~ E X such that (C'((fi),~) = O. Thus 8'((fi) is a multiple of C'(¢). 0

12.4 The Second Variation In multi-variable calculus, if one wishes to know if a critical point is actually a minimum (or maximum), one looks at the second derivative. For the same reason, we need to define the second derivative of a functional.

Definition 12.17 Let 'TJ, ~ E Tq,X. A variation of ¢ along 'TJ and ~ is a twoparameter family, ¢>',I" E X, such that ¢o,o = ¢, ¢>',I" 1>'=1"=0 = ~, and

tl" ¢>',I" 1>'=1"=0 = 'TJ.

t>.

Definition 12.18 Let 8 : X ----- JR be a functional. We say 8 is twice differentiable is there is a bounded linear map 8 2 S(¢) : Tq,X ----- (Tq,X)* (called the Hessian or second variation of 8 at ¢) such that

(12.3) for all

~,'TJ E

Tq,X and all variations ¢>',I" of ¢ along

~

and 'TJ.

Remark 12.19 1. The Hessian 8 2 S(¢) can also be defined as the second derivative of 8(¢), i.e., 8 2 8(¢) = 8· 88(¢). That is, we consider the map ¢ 1-7 88(¢) and define, for'TJ E Tq,X, 8 2 8(¢)'TJ := 88(¢>.), where ¢>. is a variation of ¢ along 'TJ. 2. We will often use the notation 8"(¢) to denote 8 2 8(¢).

t>.

Computations of the second derivatives of the functionals in our list of examples are left as an exercise (again, we suppose for simplicity that the action of the dual space is just given by integration).

Problem 12.20 Continuing with our list of examples of functionals, show that

= D2V(¢) (a matrix multiplication operator). = -,,1 (the Laplacian). 5. S"(¢) = -m81- D2V(¢) (a Schrodinger operator) acting on functions satisfying Dirichlet boundary conditions: ~(O) = ~(T) = O.

3. 8"(¢) 4. 8"(¢)

6.

S"(¢) = -d/dt(8JL)d/dt - (d/dta~¢L)

+ 8~L,

(12.4)

with Dirichlet boundary conditions. 7. 8" (¢ ) = 0 + V" (¢), with Dirichlet boundary conditions: ~ (x, 0) ~(x, T) = O.

=

12.5 Conjugate Points and Jacobi Fields

143

8. S"(¢) = B.

9. S"(¢)

= D (the D'Alembertian).

10.

The following criterion for a critical point to be a minimizer is similar to the finite-dimensional version, and the proof is left as an exercise. Theorem 12.21 Let ¢ be a critical point of a twice continuously differentiable functional S : X -+ lR. 1. If ¢ locally minimizes S, then S" (¢) 2 0 (meaning (S" ( ¢) ~,~) 2 0 for all ~ E

Tq,X).

2. If S"(¢) > c, for some constant c > 0 (i.e. (S"(¢)~,~) 2 cll~ll~ for all ~ E Tq,X), then ¢ is a local minimizer of S. Problem 12.22 Prove this.

Let us now pursue the question of whether or not a critical point of the classical action functional S(¢) =

lT

L(¢(s), ¢(s))ds

(which is a solution of the Euler-Lagrange equation - i.e., a classical path) minimizes the action. As we have seen, the Hessian S"(¢) is given by (12.4). We call a~L the generalized mass.

a;L

Theorem 12.23 Suppose a~2L > o. Suppose further that is a bounded function. Then there is a To > 0, such that S" (¢) > 0 for T ::; To.

Proof for L = !fj¢2 - V(¢). In this case S"(¢) = -md 2/ds 2 - V"(¢), acting on L2([0, T]) with Dirichlet boundary conditions. Since inf a( _d 2/ ds 2) = (7f/T)2, we have, by Theorem 5.25, -d2/ds 2 2 (7f/T)2. So S"(¢) 2 m(7f/T)2sup IV"I, which is positive for T sufficiently small. D Corollary 12.24 For T sufficiently small, a critical point of S (i.e., a classical path) locally minimizes the action, S.

12.5 Conjugate Points and Jacobi Fields In the remainder of this chapter, we study the classical action functional and its critical points (classical paths) in some detail. While such a study is of obvious importance in classical mechanics, it will also prove useful in the quasi-classical analysis of quantum systems that we undertake in Chapters 11 and 13.

144

12 Mathematical Supplement: the Calculus of Variations Thus we consider the action functional S(¢)

=

lt

L(¢(s), ¢(s))ds.

We have shown above that ift is sufficiently small, then S"(¢) > 0, provided (fP L/8¢2) > 0. So in this case, if ¢ is a critical path, it minimizes S(¢). On the other hand, Theorem 12.21 implies that if ¢ is a critical path such that S" (¢) has negative spectrum, then ¢ is not a minimizer. We will show later that eigenvalues of S"(¢) decrease monotonically as t increases. So the point to when the smallest eigenvalue of S" (¢) becomes zero, separates the t's for which ¢ is a minimizer, from those for which ¢ has lost this property. The points at which one of the eigenvalues of S"(¢) becomes zero playa special role in the analysis of classical paths. They are considered in this section. In this discussion we have used implicitly the fact that because S"(¢) is a Schrodinger operator defined on L2 ([0, t]) with Dirichlet (zero) boundary conditions, it has a purely point spectrum running off to +00. We denote this spectrum by {An(t)}]'" with .An(t) ----+ +00 as n ----+ 00. Note that if ¢ is a critical point of S on [0, tJ, then for T < t, ¢T := ¢I[O,T] is a ~ritical po~nt of Son [O,T]. Thus for T :::; t, {.An(T)} is the spectrum of S"(¢T) = S"(¢) on [0, T] with zero boundary conditions. We specialize now to the classical action functional S(¢)

=

lt{; 1¢1

2 -

V(¢)}

on the space X = {¢ E C 1 ([0,t];lRd) I ¢(O) = x, ¢(t) = y}, and continue to denote by ¢, a critical point ofthis functional (classical path). Theorem 12.25 The eigenvalues .An(T) are monotonically decreasing in T.

Sketch of proof Consider .AI(T), and let its normalized eigenfunction be 'l/J1. Define ¢1 to be 'l/J1 extended to [0, T+ E] by 0. So by the variational principle,

Further, equality here is impossible by uniqueness for the Cauchy problem for ordinary differential equations, which states the following: if a solution of a linear, homogeneous, second-order equation is zero at some point, and its derivative is also zero at that point, then the solution is everywhere zero. To extend the proof to higher eigenvalues, one can use the min-max principle. 0

°for some n is called a So c = ¢( To) is a conjugate point to x, then S" (¢) on [0, TO] has a ° eigenvalue. That is, there is some non-zero L2([0, TO]) with = = °such that Definition 12.26 A point ¢( TO) such that .An (TO) = conjugate point to ¢(O) = x along ¢. if

~ E

~(O)

~(TO)

12.5 Conjugate Points and Jacobi Fields

s" (¢)~ = O.

145

(12.5)

This is the Jacobi equation. A solution of this equation with called a Jacobi vector field.

~(O) =

0 will be

Definition 12.27 The index of S"(¢) is the number of negative eigenvalues it has (counting multiplicity) on L2([0, t]) with zero boundary conditions. We recall that for T small, S"(¢) has no zero eigenvalues on [0, T] (Theorem 12.23). Combining this fact with Theorem 12.25 gives the following result. Theorem 12.28 (Morse) The index of S"(¢) is equal to the number of points conjugate to ¢(O) along ¢, counting multiplicity (see Fig. 12.2).

Fig. 12.2. Index =

#

of conjugate points.

The picture that has emerged is as follows. For sufficiently small times, a classical path ¢(O) locally minimizes the action. As time increases, the path might lose this property. This happens if there is a point in the path conjugate to ¢(O). Example 12.29 An example of a conjugate point is a turning point in a one-dimensional potential (see Fig. 12.3, and remember that we are working with the functional of Example 12.1, no. 5). V(x) ',j

x

a Fig. 12.3. A turning point. The classical path ¢ starts at a, and turns back after hitting b at time

T.

Now

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12 Mathematical Supplement: the Calculus of Variations

SI/(¢) = -ma; - vl/(¢) and it is easy to check that SI/(¢)¢ = 0 (just differentiate Newton's equation). Since ¢(O) = ¢(T) = 0 (the velocity at a turning point is zero), b is conjugate to a. We return to the Jacobi equation (12.5), and consider its fundamental solution, J(s). J(s) is the d x d matrix satisfying

SI/(¢)J

=0

with the initial conditions

J(O) = 0

and

j(O) = 1.

J is called the Jacobi matrix. Proposition 12.30 The Jacobi matrix has the following properties 1. For any h E ]Rd, Jh is a Jacobi field. Conversely, any Jacobi field is of the form J h for some h E ]Rd.

2. ¢(TO) is a conjugate point to ¢(O) iff J(TO) has a zero-eigenvalue, i.e. det J(TO) = O. Proof. 1. The first part is obvious. To prove the second part let ~ be a Jacobi field, and let h = ~(O). Then ~ := Jh satisfies the same differential equation as ~ with the same initial conditions. Hence ~ = ~, and therefore ~ = Jh. 2. We have shown above that ¢( TO) is a conjugate point iff there is a Jacobi field ~ such that ~(TO) = O. By the previous statement, there is h -1= 0 such that ~ = Jh, which implies J(To)h = O. So J(TO) has a zero eigenvalue (with eigenvector h), and det J (TO) = O.

o

Now we give the defining geometric/dynamic interpretation of J. Consider a family of critical paths ¢v (s) starting at ¢( 0) with various initial velocities v E ]Rd. Denote v = ¢(O). Then

J(s) =

a¢;~s) Iv=v

is the Jacobi matrix (along ¢). Indeed, ¢v satisfies the equation as(¢v) = O. Differentiating this equation with respect to v, and using that SI/(¢) a¢aS(¢), we find

o=

a u¢S '" (¢v) = S1/ ( ¢v ) a¢v av av·

Thus, a¢v/avlv=v satisfies the Jacobi equation. Next,

12.6 The Action of the Critical Path

and

8 . 8v

8 8v

= -v =

-¢v(O)

147

1

which completes the proof.

12.6 The Action of the Critical Path In this section we consider the situation of Example 12.1 no. 6, and its special case, Example 12.1 no. 5. Thus we set

x

= {¢ E C 1 ([0, t];ll~m) I ¢(O) = x, ¢(O) = y},

and S(¢) =

lot L(¢(s), ¢(s))ds.

Suppose ¢ is a critical path for S with ¢(O) = x and ¢(t) = y. We will denote the action of ¢ by So(x, y, t) := S(¢) (the action from x to y in time t). The momentum at time s, is k(s) := (8L/8¢)(¢(s), ¢(s)). Lemma 12.31 We have

8So 8x

= -k(O)

Proof. Again, we specialize to L = integration by parts, we find 8S(¢)/8x

~~o

and

ml¢1 2/2 -

=

k(t).

V(¢). Using the chain rule and

lot {m¢. 8¢/8x - V'V(¢)· 8¢/8x}ds = lot {( -m¢ - V'V(¢)) . 8¢/8x} + m¢. 8¢/8xl~ =

which, si~ce ¢ is a critical point, (8¢/8x)(t) = 0, and (8¢/8x)(0) = 1, is just -m¢(O) = -k(O), as claimed. The corresponding statement for 8So/8y is proved in the same way, using (8¢!8y)(0) = 0, and (8¢/8y)(t) = 1. 0 This lemma implies 8k(0)/8y = -8 2 S0 (x, y, t)/8x8y. On the other hand, for L = ~1¢12_ V(¢), 8k(0)/8y = (8y/8k(0))-1 = mJ- 1(t) (as m(8y/8k(0)) is the deriv?-tive of the classical path ¢ at t with respect to the initial velocity k(O)/m = ¢(O)). This gives

8 2 S 0 (x, y, t) 8x8y which establishes the following result:

= _

m

J-1() t

(12.6)

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12 Mathematical Supplement: the Calculus of Variations

Proposition 12.32 If y is a conjugate point to x then det( 8 2 s;;~::,t)) =

00.

The following exercise illustrates this result for the example of the classical harmonic oscillator. Problem 12.33 Consider the one-dimensional harmonic oscillator, whose • 2 Lagrangian is L = !!}¢2 - m~ ¢2. Compute

So(x, y, t)

=

2 . ~ ) [(x 2 sm wt

+ y2) cos(wt) -

2xy]

and so compute

fPSo(x,y,t)/8x8y

=

-~.

smwt Note that this is infinite for t = mf / w for all integers n. Thus the points ¢(mf/w) are conjugate to ¢(O). Lemma 12.34 (Hamilton-Jacobi equation) The action So(x, y, t) satisfies the Hamilton-Jacobi equation

8So/8t = -h(y, 8So/8y)

(12.7)

where h is the classical Hamiltonian function associated with L. Proof. The integrands below depend on s (as well as the parameters x,y, and t), and ¢ denotes 8¢/8s. Since So = S(¢) = J~ L(¢, ¢)ds, we have

8S(¢)/8t = L(¢, ¢)Is=t

+ lot (8L/8¢. 8¢/8t + 8L/8¢. 8¢/8t)ds

= L(¢, ¢)Is=t + 8L/8¢. 8¢/8tl:~&

+ lot (8L/8¢ Since ¢(s) = y + ¢(t)(s - t)

+ O((s 8¢

at Is=t

Using this, the fact that

d/ds(8L/8¢» . 8¢/at.

t)2) (here we used ¢(t) = y), we have =

"-

-¢(t).

¢ is a critical point of S,

and 8¢/8tl s=0 = 0, we find

8S(¢)/8t = -((8L/8¢) . ¢ - L(¢, ¢»Is=t

= -h(¢,8L/8¢)ls=t. Since ¢Is=t follows. D

=

y and, by Lemma 12.31, (8L/8¢)ls=t

=

8So/8y, the result

12.6 The Action of the Critical Path

149

Lemma 12.35 (Conservation of energy) (8L. energy (¢):= -.' ¢ - L)

8¢

I

_ = const.

=

Proof We compute

Since for ¢ = ¢, the expression on the right hand side vanishes, ~~ . J; - LI is a constant (which we will denote by E). Now

hence E is the energy of ¢. 0 We want to pass from a time-dependent to a time-independent picture of classical motion. We perform a Legendre transform on the function So(x, y, t) to obtain the function W(x, y, E) via W(x, y, E)

=

(So(x, y, t)

+ Et)lt:8So /8t=-E'

(12.8)

We denote by [ = [(x, y, E), solutions of

8So/8tl t=t = -E.

(12.9)

There may be many such solutions, so in the notation W, we record the classical path ¢ we are concerned with (for which ¢(O) = x, ¢(f) = y). Note that from the energy conservation law

we have W(x, y, E)

=

i

t

8L

~

o 8¢

h

-'. ¢ds = _k . dx,

where f k· dx := f~ k(s) . ¢(s)ds and, recall, k(s) = 8L/8J;1=· Lemma 12.36 ¢It=t is a classical path at energy E. Proof By the Hamilton-Jacobi equation (12.7) and the conservation of energy (Lemma 12.35), ¢It=t is a classical path with energy -8So(x, y, t)/8tlt=t, which, by (12.9), is just E. 0

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12 Mathematical Supplement: the Calculus of Variations

Lemma 12.37 W¢ satisfies the Hamilton-Jacobi equation

(12.10)

h(x, -oW¢/ox) = E. Proof. Using (12.9) and Lemma 12.31, we compute oW¢ = oSo It=t + (oSo ax ax at

+ E)lt=t of = ax

So by conservation of energy, E

oSo It=t = -k(O). ax

(12.11)

= h(x,k(O» = h(x, -oW¢/ox). D

12.7 Appendix: Connection to Geodesics The next theorem gives a geometric reinterpretation of classical motion. We consider a classical particle in ]Rd with a potential V(x). Recall the notation f(x)+ := max(f(x) , 0). Theorem 12.38 (Jacobi theorem) The classical trajectory of a particle at an energy E is a geodesic in the Riemannian metric

(u, v)x = 2(E - V(x»+u· v (where u· v is the inner product in ]Rn) on the set {x classically allowed region).

E

]RnlV(x) ::; E} (the

Proof. By the conserv3;tion of energy (Lemma 12.35), a classical path ¢ has a fixed energy E = ml¢1 2/2 + V(¢). Hence ¢ is a critical point of the action S(¢) = J(ml¢1 2/2- V(¢»ds among paths in M := {¢ I ml¢1 2/2+ V(¢) == E}. Using the relation ml¢1 2/2 + V(¢) = E, we can write ml¢1 2/2 - V(¢) as

ml¢1 2/2 - V(¢) Hence

¢ is

=

ml¢1 2 = myh(E - V(¢»/ml¢l.

a critical point of the functional L(¢) :=

Jml¢IV

2(E - V(¢»/mds

on M. This functional gives the length of the path in the metric above. On the other hand, we can re-parameterize any path with V(¢) < E so that it satisfies ml¢1 2/2 + V(¢) = E. Indeed, replacing ¢(s) with ¢(A(S», we note 1¢(A)1 2 = ('\(s»21¢(A(S»12. We must solve ('\(S»2

which we can re-write as

= ~ E -.V(¢(A(S») m 1¢(A(S»12

12.7 Appendix: Connection to Geodesics

r

io

1¢('\(s'))I~(s')

JE - V(¢('\(s')))

ds'

=

151

(2s

V:;;:

which can be solved. Since the functional L( ¢) is invariant under reparameterizations (if'\ = a(s), a' > 0, then 1¢(s)lds = I¢('\)I~; gld,\ = 1¢('\)ld'\), if ¢ is a critical point of L(¢), then so are different reparameterizations of ¢, and in particular the one, ¢, with the energy E. This ¢ is also a critical point of L(¢) on M, and, by the above, a critical point of S(¢) at energy E. Thus classical paths are geodesics up to re-parameterization (so they coincide as curves). D Problem 12.39 Check that the Euler-Lagrange equation for critical points of L(¢) on M yields Newton's equation.

13 Resonances

The notion of a resonance is a key notion in quantum physics. It refers to metastable states - i.e., to states which behave like stationary states for long time intervals, but which eventually break up. In other words, these are states of the continuous spectrum (i.e. scattering states), which for a long time behave as if they were bound states. In fact, the notion of a bound state is an idealization: most of the states which are (taken to be) bound states in certain models, turn out to be resonance states in a more realistic description of the system. We sketch briefly the mathematical theory of resonance states. To characterize them in terms of spectral properties, we have to use more detailed information than just the spectrum, such as spectral densities, or else generalize the notion of the spectrum altogether. The resulting theory, which is elegant and powerful, is not considered here. Rather, we study the particular but central case of resonances due to tunneling, on which we illustrate some of the mathematics and physics involved.

13.1 Tunneling and Resonances Consider a particle in a potential V(x), of the form shown in Fig. 13.1; i.e., V(x) has a local minimum at some point xo, and V(xo) > limsupx---+(X) V(x), for x in some cone, say. If V(x) ----+ -00 as x ----+ 00 (in some cone of directions), then the corresponding Schrodinger operator, H, is not bounded from below. "would be

boun~d state" Vex)

tunnels under barrier ro

x

Fig. 13.1. Unstable potential.

S. J. Gustafson et al., Mathematical Concepts of Quantum Mechanics © Springer-Verlag Berlin Heidelberg 2003

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13 Resonances

If the barrier is very thick, then a particle initially located in the well spends lots of time there, and behaves as if it were a bound state. However, it eventually tunnels through the barrier (quantum tunneling) and escapes to infinity. Thus the state of the particle is a scattering state. It is intuitively reasonable that

1. the energy of the resonance ::::; the energy of a bound state in the well 2. the resonance lifetime is determined by the barrier thickness and height, and n. Since the resonances are very close to bound states if the barrier is large or n is small (there is no tunneling in classical mechanics), we try to mimic our treatment of the ground state (Section 11.4). But right away we run into a problem: if V(x) -1+ 00 as x ----+ 00 in some directions, then

Z(j3) = tr e-/3H =

00.

The paradigm for this problem is the divergence of the integral

for ,\ :::; O. However, we can define this integral by an analytic continuation. Z('\) is well-defined for Re(,\) > 0, and it can be continued analytically into ,\ E IR- as follows. Move ,\ from Re('\) > 0 into Re('\) :::; 0, at the same time deforming the contour of integration in such a way that Re('\a 2 ) > 0 (see Fig. 13.2).

Fig. 13.2. Contour deformation. Of course, in this particular case we know the result:

Z('\) =

( -2,\)-1/2 = -z. (21'\1)-1/2 7r

for ,\ < 0 (which is purely imaginary!).

7r

13.2 The Free Resonance Energy

155

There is a powerful method of rotating the contour which is applicable much beyond the simple integral we consider. It goes as follows. For 8 E lR, we change variables via a = e-ob. This gives Z(A) = e-O

1

00

e-)..e- 28 b2 /2db.

(13.1)

The integral here is convergent and analytic in 8 as long as (13.2) We continue it analytically in 8 and A, preserving this condition. In particular, for A E lR-, we should have 7r/4 < Im(8) < 37r/4. Now observe that the right hand side of (13.1) is independent of 8. Indeed, it is analytic in 8 as long as (13.2) holds, and is independent of Re(8) since the latter can be changed without changing the integral, by changing the variable of integration (b f---t Co' b, 8' E lR). Thus we have constructed an analytic continuation of Z(A) with Re(A) > 0 into a region with Re(A) < O. In fact, we have continued this function onto the second Riemann sheet! Finally, we define Z(A), for A < 0, by (13.1) with 8 obeying (13.2).

13.2 The Free Resonance Energy With a bit of wisdom gained, we return to the problem of defining the partition function Z({3) and free energy F({3) in the case when V(x) 00, as x ---+ 00, in some directions (or more precisely, sUPxEr V(x) < 00 for some cone r). We begin with a definition.

+

Definition 13.1 A family of operators, H(8), for 8 in a complex disc {181 ::::: E}, will be called a complex deformation of H if H (0) = H, H (8) is analytic in {I 8 I : : : E}, and H (8) is an analytic continuation of the family

H(8)

for 8

E

lR, where U(8), 8

E

= U(8)-1 HU(8)

(13.3)

lR, is a one-parameter unitary group.

If V (x) has certain analytic properties, then H (8) can be constructed by changing the variables in H as x f---t xe-o for 8 E lR, and then continuing the result analytically in 8. Let, for example, V(x) = -Cx3 as x ---+ +00. Then V(eOx) = -Ce3 0x 3 . Take 8 = -i7r/3. Then V(eOx) = Cx 3 is positive in the direction x ---+ +00. The group U(8) here is the group of dilations:

for 7/J E L2(lRn). Assume that we can construct a complex deformation, H(8), of H, such that

156

13 Resonances

Z((3) = tre-!3 H (O) <

(13.4)

00

for Im(()) > 0, or more generally for I()I :S f, Im(()) > O. (In fact, it is not a simple matter to define the exponential e-!3 H (O) rigorously - see [SV]. Below we will deal formally with e-!3 H (O), assuming it has all the properties which can be derived from the power series expression for the exponential.) Proposition 13.2 If tre-!3 H (O) is independent of ().

<

for ()

00

{I()I :S f},

Proof. e-!3 H (O) is analytic in

E

Dc

{I()I < f},

then tre-!3 H (O)

and satisfies

e-!3 H (O+s) = U(s)-le-!3 H (O)U(s) for s E R This last relation can be derived using the expression H (() + s) = U(s)-lH(())U(s) (which follows from (13.3)) and a power series expansion of the exponential (or Equation (5.3)). By cyclicity of the trace (tr(AB) = tr(BA)), tre-!3 H (O+s) = tre-!3 H (O). Hence tr e-!3 H (O) is independent of Re(()), and so is independent of (). D If there is a complex deformation, H(()), of H, such that (13.4) holds, we call Z((3) = tre-!3 H (O) an adiabatic partition function for H, and F((3) = -(1/ (3) In Z((3) the resonance free energy for H. We interpret

E((3)

:=

ReF((3)

as the resonance energy at the temperature 1/(3,

r((3)

:=

-ImF((3)

as the resonance decay probability per unit time (or resonance width) at the temperature 1/(3, and

T((3)

:=

1

r((3)

as the resonance lifetime at the temperature 1/(3. The resonance eigenvalue for zero temperature is given by Zr

Usually, IImZ((3) I «

=

Er -

irr := lim F((3). !3--+CXJ

IReZ((3)I. Hence, E((3) = ReF((3)

and

r((3) = -ImF((3) =

1

~

-731 ln (ReZ((3)) .ImZ ((3)

1 I mZ ((3)

73 Imln (1 + z ReZ((3)) ~ 73 ReZ((3) .

13.3 Instantons

157

In fact, one can show that for L1.1E < < t < < r-l, where iJ.E is the average gap between eigenvalues of H (8) (L1.1E gives a time scale for H),

+

e-iHt/n'l/Jo = e-izrt/n'l/Jo

small

if 'l/Jo lies near Er in the spectral decomposition of H (see [SV, FP]). Note that exhibits exponential decay at the (slow) rate picture of a resonance as a metastable state.

rr. This is consistent with our

Remark 13.3 The example given after Definition 13.1 does not lead to a unique self-adjoint Schrodinger operator H = - ;~ iJ. + V (x). Presumably, F((3) is independent of the self-adjoint extension chosen. For a large class of self-adjoint Schrodinger operators, Condition (13.4) does not hold, and the trace has to be regularized (see [SV]). In such a case, the potential can be modified at infinity in such a way that for the modified potential, Condition (13.4) holds. We expect that such a modification can be chosen so that it leads to a sufficiently small error in the tunneling probabilities. In any case, the results we discuss below (which are obtained by applying another non-rigorous technique - path integrals) coincide with those given by more involved rigorous analysis, wherever the latter is possible.

13.3 Instantons To compute Z((3) for the potential sketched in Fig. 13.1, we proceed as in the ground state problem; we represent Z((3) formally as a path integral, and then derive the formal quasiclassical expansion (see (11.11)):

w

(as usual, we ignore the factor (1 + O(v1i))). The sum is taken over critical points w of the "action" Se(CP) , of period T. N is a normalization factor independent of wand H, and

Bw

=

det S~(wo) detS~'(w)

where the operators S~'(w) and S/f(wo) are defined on L2([O, T]) with zero boundary conditions. Now w is a periodic classical path in imaginary time (or in the inverted potential - V (x)), with period T, which we take large (see Fig. 13.3).

158

13 Resonances

Vex)

"bounce"

static

Fig. 13.3. Paths in inverted potential. Two periodic solutions of arbitrarily large period are

(the subscript "s" for "static") and

("b" for "bounce"). The solution Wb is called an instanton or "bounce". Since Ws is a minimum of V, VI/(w s ) > 0, and so S~(Ws) =

-8; + rP

where [22 = VI/(O). We computed earlier

_

B Ws

-

[27 2[27 sinh([27) ~ eDT

for 7 large. Moreover, Se(w s ) = O. We will show later (Section 13.5) that (13.5) where

is the action of the "bounce", and (13.6) is the determinant of A restricted to the orthogonal complement of the null space, NullA, of A. Collecting these results, we have (for large 7)

E and

~

11,

--In(ReZ) 7

11,[2

~-

2

r ~ _~ ImZ ~ nS-1/2 T ReZ

b

(I

13.4 Positive Temperatures det.l S~/(Wb)l) det S:/(W s )

-1/2

159

e- Sb / Ii .

So the probability of decay of the state inside the well, per unit time, is

r

=

(const)e- Sb / Ii

where Sb = Se(Wb) is the action of the instanton (which equals the length of the minimal geodesic in the Agmon metric ds 2 = (V(x) - E)+dx2). This explains the sensitivity of the lifetimes of unstable nuclei to small variations of the parameters (for example, isotopes with different masses can have very different lifetimes). Finally, we note that

13.4 Positive Temperatures Here we consider quantum tunneling at positive temperatures (t = {3-1 > 0). We use the same approach as above, but let the parameter {3 be any positive number. Now we have to consider all three critical paths of period T = n{3 (see Fig. 13.4): WI = Ws == Xmin, W2, and W3 == Xmax , where W2 is a classical periodic trajectory in the potential - V(x) of period T = n{3.

V(x)

(03

\.

(01

\

x

Fig. 13.4. Paths of period

T.

Since V"(Xmin) > 0, WI is a minimal trajectory. As we will see, W2 is a saddle point of Morse index 1 (see Section 12.5). Finally, V"(x max ) < 0, and so W3 is also a saddle point. For k = 1,3, and for k = 2,3,

Se(wI) < Se(Wk).

The quasi-classical expression for the decay probability works out to be

160

13 Resonances

r

=

-_l_(lmBW2 e- S (W2)/n T

BWI

+ 1mBW e- S (w3)/n) 3'

Through which trajectory, W2 or W3, does the tunneling take place? W3 corresponds to a thermally driven escape (due to thermal fluctuations), and W2 corresponds to a quantum tunneling escape. If T is very small (large temperature), the transition occurs through W3, as only W3 can have arbitrarily small period. On the other hand, if T is very large (small temperature), W2 sits close to the bottom of the well, and one can show that Se(W2) < Se(W3). In this case, the transition occurs through W2. There is a critical value of T, Te ~ 27r I fl max where fl;'ax = -V" (x max ), at which a transition occurs; the transition is between the situations in which decay is due to tunneling, and in which it is due to thermal fluctuations. (Note that for T < T e , the decay rate differs from r by the factor !t'2;xT (see [Afj)). This transition can take place either continuously or discontinuously, depending on whether the energy of the periodic classical trajectory in the inverted potential - V (x) depends on its period continuously or discontinuously. In the first case, as temperature decreases below liTe (i.e. T increases above Te) the tunneling trajectory bifurcates from W3 and slips down the barrier (see Fig. 13.5). For T < Te tunneling takes place through W3.

<

't

00

't=

00

Fig. 13.5. Continuous transition. In the second case (see Fig. 13.6), there are no closed trajectories with period > T e , so the transition is discontinuous: decay jumps from W3 to a trajectory at the bottom of the barrier.

no trajectories for 't>'tc

't

= co

Fig. 13.6. Discontinuous transition. Thus for intermediate temperatures, the nature of decay depends radically on the geometry of the barrier.

13.5 Pre-exponential Factor for the Bounce

161

The results above support the following physical picture of the tunneling process. With the Boltzmann probability (canst)e- E/ T , the particle is at an energy level E. The probability of tunneling from an energy level E is (canst)e- SE / Ii where SE is the action of the minimal path at energy E. The probability of this process is (canst)e-E/T-SE/Ii. Thus the total probability of tunneling is (canst)

J

e-E/T-SE/Ii

~

(canst)e-Eo/T-SEo/1i

where Eo solves the stationary point equation fJ E fJE (T

SE

1

1 fJSE fJE

+ h) = T + Ii

= O.

Here -fJSE I fJE is the period of the trajectory under the barrier at the energy level -E, and SEo + nEolT is the action of a particle (in imaginary time) at energy Eo corresponding to the period niT = T.

13.5 Pre-exponential Factor for the Bounce The bounce solution, Wb, presents some subtleties. Since Wb breaks the translational symmetry of Se(¢), uh is a zero-mode of S~(Wb): S~/(Wb)uh

= o.

To establish this fact, simply differentiate the equation fJSe(Wb) respect to s and use the fact that S~' = fJ2Se (see Section 12.4). As a result, we have two problems:

o with

1. S~/(Wb) has a zero eigenvalue, so formally

[detS~/(wb)rl/2 =

(canst)

Je-(f"S~'(Wb)f,)j2IiD~

=

00

(13.7)

2. Wb has one zero (see Fig. 13.7), and so the Sturm-Liouville theory (from the study of ordinary differential equations) tells us that, in fact, S~(Wb) has exactly one negative eigenvalue.

cOb s

Fig. 13.7. This gives a second reason for the integral (13.7) to diverge.

162

13 Resonances

To illustrate these divergences, we change variables. Let {~d be an orthonormal basis of eigenfunctions of S~(Wb) with eigenvalues )..k, in increasing order. For ¢ near Wb, write ¢ = Wb + ~ with

Then

+L

00

Se(¢) ~ Se(Wb)

But

)..0

<

°and

)..1

)..k a %. o = 0, hence we have two divergent integrals:

for j = 0, 1. We already know that we can define the first integral by an analytic continuation to be

1 e->'oa~/2Iidao

=

00

-00

(2)..0)-1/2 trn

=

-i 12 )..0 7rn

1- 1/ 2

The second integral, correctly treated, is shown to contribute (see the following section) S;I/2TV27rn (13.8) where Sb is the action of the "bounce", Se(Wb). Hence

r

Jnear

e-Se(
where B 1/ 2 Wb

~

BWbe-Sb/1i

Wb

= _iTS- I/ 2 b

(I det~ S~'(Wb)l) det Sf{(wo)

(this is (13.5)) and det~ is defined in (13.6).

13.6 Contribution of the Zero-mode The virial theorem of classical mechanics gives

Define the normalized zero eigenfunction t

<,1

=

s-1/2.

b

Wb·

-1/2

13.7 Bohr-Sommerfeld Quantization for Resonances

163

Then

Hence

¢~

Wb(S

+ C1S;1/2) + L cn~n n#l

and therefore

II dCn = S;1/2ds II den. n

Integrating in

S

n#l

from 0 to t gives (13.8).

13.7 Bohr-Sommerfeld Quantization for Resonances The goal of this section is to derive a semi-classical formula for the resonance eigenvalues of a Schrodinger operator with a tunneling potential. We proceed by analogy with the treatment of a confining potential in Section 11.3 which led to the Bohr-Sommerfeld quantization rule. As in the rest of this chapter, we consider a tunneling potential of the form sketched in Fig. 13.8.

E

classically forbidden

x

Fig. 13.8. Resonance potential. The path-integral expression for Green's function of H is, as in Section 11.2, (13.9) We seek critical points (because, as always, we wish to apply the method of stationary phase) which are closed trajectories (x = y) at the fixed energy E. The trajectories in phase space are shown in Fig. 13.9.

164

13 Resonances p x

Fig. 13.9. Phase portrait. At energy E, phase space is partitioned into classically allowed, and classically forbidden regions. Classical trajectories at energy E are shown in Fig. 13.10. p

classically allowed

j

clastcany forbidden x

Fig. 13.10. Phase portrait at fixed energy. If we complexify the phase space

lRxlFb-+CxC

then the phase space at a fixed energy E becomes connected as shown in Fig. 13.11. Re(p)

x

Im(p)

Fig. 13.11. Complexified phase space at fixed energy. Thus, in addition to real paths, ¢(s), we consider complex paths of the form a(O") = 7/J( -iO"). Setting t = -iT, the action for such a path is

S(a, -iT) where

=

i

T

o

m·

( __ 7/J2

2

- V(7/J))( -i)dO"

=

iA(7/J, T),

13.7 Bohr-Sommerfeld Quantization for Resonances

A(1j1, T) = and so and

165

foT ; ~2 + V(1j1), oS( 0:, -iT) OT

. oA( 1j1, T) OT .

=Z

Thus the phase in (13.9) is

S(o:, -iT)

+ E( -iT) = i(A(1j1, T)

- ET).

Now, the real critical point (
01>S

=

so
WI

=

0,

=

OS(
-\7V(
S(
The complex critical point (
= 1j12 (-iu), iT)

satisfies

O1>S( 0:, -iT) = iO,pA( 1j1, T) = 0 and

OS(
so 1j12 has period T, and m~2 = \7V(1j12) (as in Fig. 13.8). Hence the phase is iW2 = i(A(1j12' T) - ET)laA(;,;;,T) =E We can characterize a general closed critical orbit by the list

meaning the real closed critical point is traversed once, the complex closed critical point is traversed ml times, the real critical point is followed again, then the complex critical point m2 times, etc. (we follow the real critical point several times in succession if some of the mi are zero). Applying the stationary phase method, we obtain the following contribution to the path integral (up to a constant, in the leading order as n --., 0):

~ m~m. e'(Hn)W,fn-(m, +··+m.)W,/n ~ ,'W,fn ~ (e,W,fn = eiWl/ 1i ~ 6

(eiWl/ 1i

n=O

= e iW1 / 1i

1 1 - eiWl/1i

fo

e-mw,/n) n

1 ) 1 - e-W2/1i

I-e

I

eiWl/Ii(l _ e- W2 / 1i )

1 - eiWl/ 1i - e- W 2/Ii'

W2/ 1t

n

166

13 Resonances

We want to identify values of E for which Green's function has a singularity, with resonance eigenvalues. Writing the lowest resonance eigenvalue as Eo - ii1E and expanding eiW,(E)/n to first order around Eo, and e- W2 (E)/n to zeroth order, gives the equation or

WI (Eo)

= 21flin,

n

= 0, ±1, ...

for Eo (i.e. Eo is the ground state energy, as before), and the expression

for i1E. The last two equations represent the Bohr-Sommerfeld quantization for resonances.

14 Introduction to Quantum Field Theory

The goal of quantum field theory (which we will often abbreviate as QFT) is to describe elementary particles and their interactions. QFT has deep connections with a variety of disciplines, including statistical mechanics and condensed matter physics, probability theory (Le. stochastic PDEs), and nonlinear PDEs.

14.1 The Place of QFT We have the following diagram displaying the place of QFT relative to classical mechanics (eM), which is a classical theory of point objects (particles), classical field theory (eFT), which is a classical theory of extended objects (fields), and quantum mechanics (QM), which is a quantum theory of point objects (here d is the number of degrees of freedom): quantization eM QM I I d-+oo I I d-+oo

1

1

eFT

QFT quantization

Among the most important eFTs are electro-magnetism (EM) (governed by Maxwell's equations) and gravity (governed by Einstein's equations). When one "quantizes" Maxwell's equations, one obtains the theory of quantum electrodynamics (QED). An appropriate quantization of the Einstein equations is unknown. In addition, there are many eFTs which are observed only in their quantized version. These include the theories given by the Klein-Gordon equation, and the Yang-Mills equations (gauge field theory). These are both variations on the EM theory. The former is obtained by replacing the vector field of EM by a complex scalar field, and the latter by replacing the abelian gauge group of EM by a non-abelian one.

S. J. Gustafson et al., Mathematical Concepts of Quantum Mechanics © Springer-Verlag Berlin Heidelberg 2003

168

14 Introduction to Quantum Field Theory

14.1.1 Physical Theories

A physical theory specifies what it means for a system to be in a particular "state" (the state space), how this state evolves in time (evolution law), and how this state corresponds to the outcomes of physical measurements on the system (observables). The table (3.4) displays the objects describing CM and QM. We will introduce QFT by quantizing CFT. We do this by analogy with the quantization of CM (i.e., the passage to QM - see Section 3.4). This analogy suggests we have to put CFT, which is originally given in terms of a PDE, into Hamiltonian form. This is done in two steps: introducing the action principle, and performing a Legendre transform. 14.1.2 The Principle of Minimal Action

The principle of minimal action (properly, of "stationary" action) states that evolution equations for physical states are Euler-Lagrange equations for a certain functional called the action (see Chapter 12 for examples of functionals and Euler-Lagrange equations). More precisely, one considers a space of functions, ¢, defined on spacetime, called the fields. The equation of motion for ¢ is given as as(¢) = 0, where S is an appropriate functional on the space of fields. In particular, we consider a class of examples from Chapter 12: S(¢) =

(T

Jo

r 12(¢(x,t), Vx¢(x,t),¢(x,t))ddxdt

J~d

(14.1)

for ¢ : lR~ x lR t --- lR. Here, £ : lR X lR d X lR --- lR is the Lagrangian density. Specializing further, we have the following important example of a classical field theory: Example 14.1 [Klein-Gordon field theory] The Klein-Gordon Lagrangian density is

for some continuous function F : lR --- lR. The corresponding Lagrangian functional, L(¢, ¢)

=

ld {~(¢2 -IV

x¢12) - F(¢)}

is defined on some subspace of HI (lR d ) x L2(lRd ) (to be more precise about the domain of L, we need to know more about the function F). We recall from Chapter 12 that the critical point equation for the functional S(¢) is O¢ + F'(¢)

the (nonlinear) Klein-Gordon equation.

= 0,

(14.2)

14.2 Klein-Gordon Theory as a Hamiltonian System

169

The set of functions r/J on which S is well-defined will, in general, be a space of paths r/J: [O,Tj--> X in some (infinite dimensional) space X. X is called the configuration space of the physical system, and its elements are called fields. In our case, X will always be a space of functions of several variables. In general, X could be a non-linear space (in which case the Lagrangian functional would be defined on (a subset of) TX, the tangent bundle of X), but for simplicity we will assume X is a Hilbert space.

14.2 Klein-Gordon Theory as a Hamiltonian System In this section we describe (infinite-dimensional) Hamiltonian systems on Hilbert spaces, and write the Klein-Gordon theory as a Hamiltonian system.

14.2.1 The Legendre Transform Definition 14.2 Let f be a twice differentiable function (or functional), on a reflexive Banach space, X, satisfying (p f > 0. Then the Legendre transform, of f, is a function, g, on X*, defined by

g(7r)

:=

SUPuEX((7r,U) - f(u))

(14.3)

= ((7r,u) - f(u))lu:8f(u)=n

(where the second expression follows from solving the variational problem in the first expression).

Problem 14.3 Show that 8 2 g > 0, and that (Legendre transform)2 = 1. Hint: use the fact that 8g(7r) = u(7r) whereu(7r) solves 8f(u(7r)) = 7r. (Strictly speaking 9 is twice differentiable only if 8 2f ::::: 01 for some 0 > 0. Moreover, we have 8 2 g(7r) = (8 2f(u(7r)))-l.) Example 14.4 The Legendre transform maps the given functions/functionals as follows (these are easily verified): 1. On X = JR.d, f(v) = m~12 f--> g(k) = ~ 2. On X = L 2 (JR.d) , f('Ij;) = ~ J 1'Ij;12 f--> g(7r) = ~ 3. Classical mechanics: ~v2

L(x, v) = -2- - V(x)

f--> (

J 17r1 2

Leg. trans. in v) H(x, k)

=

k2

2~

+ V(x)

4. Klein-Gordon classical field theory:

L(r/J, 'Ij;) = H(r/J,7r)

=

J{~(1'Ij;12 J{~(17r12 +

- 1V'r/J12) - F(r/J)}dnx 1V'r/J12) + F(r/J)}dnx

f--> (

Leg. trans. in 'Ij;)

170

14 Introduction to Quantum Field Theory

14.2.2 Hamiltonians

Suppose the dynamics of a system are determined by the action principle with a Lagrangian function/functional defined on a configuration space X x V (V a space of velocities): L:XxV----+R

Then a Hamiltonian junction/junctional for the system is the Legendre transform (in the tangent space variable v E V) of L H:X x V*----+R

(see items 3 and 4 in Example 14.4 above). 14.2.3 Poisson Brackets

Let Z be a Banach space, which we call the state space. For example, the classical mechanics state space for a single particle is Z = ]!t3 X ]!t3, and the Klein-Gordon (KG) state space is Z = H 1(]!td) X L 2(]!td). A Poisson bracket is a bilinear map {-, .}, from the space of differentiable functions (or functionals) on Z to itself, satisfying 1. {F,G} = -{G,F} (skew-symmetry) 2. {F, {G, H}} + {G, {H, F}} + {H, {F, G}} = 0 (the Jacobi identity) 3. {F,GH} = {F,G}H +G{F,H} (Leibniz rule)

for functions (functionals) F,G, and H. Remark 14.5 A space of such functions (or functionals), together with a Poisson bracket, has the structure of a Lie Algebra.

We also assume that the Poisson bracket is non-degenerate: {F, G} G =? F=O.

o 't:/

=

Example 14.6 If Z is a real inner-product space, the following construction produces a Poisson bracket: 1. Suppose there is a linear operator J on Z such that J* = -J (J is a

symplectic operator). Then {F,G}:= (8F,J8G) is a Poisson bracket (check this). 2. (eM) If Z = ]!t3 X ]!t3, then

1)

J = ( 0 -10

is a symplectic operator, yielding the Poisson bracket

(14.4)

14.2 Klein-Gordon Theory as a Hamiltonian System

3. (KG) If Z

=

H 1(JRd)

X

171

L 2(JRd) and J has the form (14.4), the resulting

Poisson bracket is (14.5) Definition 14.7 A Hamiltonian system is a Poisson space (a Banach space Z, with a Poisson bracket) together with a Hamiltonian defined on that space. Example 14.8 Our two main examples of Hamiltonian systems are 1. (eM) Z

=

JR3 X JR3

with bracket as above, and Hamiltonian

h(x, k)

2. (KG) Z = H1(JRd)

X

=

1

2 + V(x) -lkl 2m

L 2(JRd) with bracket as above and Hamiltonian

Remark 14.9 Our definition of a Hamiltonian system differs from the standard one in using the Poisson bracket instead of a symplectic form. The reason for using the Poisson bracket is its relation to the commutator. 14.2.4 Hamilton's Equations Suppose we have a Hamiltonian system on Z, a space of functions. The functional on Z which maps Z 3 tfJ f----7 tfJ(x) is called the evaluation functional (at x), which we denote (with some abuse of notation) as tfJ(x). Problem 14.10 Show that fJtfJ(y)

= Oy.

Definition 14.11 Hamilton's equations are

(14.6)

for all x. Example 14.12 If the Poisson structure is given by a symplectic operator J, then

{tfJ(x) , H}(tfJ) =

J

(fJtfJ(x)) (y)JfJH(tfJ) (y)dy =

J

ox(y)JfJH(tfJ)(y) = JfJH(tfJ)(x)

which leads to Hamilton's equation

(14.7)

We have the following basic result, whose proof is left as an exercise:

172

14 Introduction to Quantum Field Theory

Theorem 14.13

1. A path
for all functionals F. 2. If L(qJ,'I/J) and H(cp,1r) are related by a Legendre transform (in 'ljJ (-71r), then the Euler-Lagrange equations for

S(cp, ¢) =

lT

L(cp, ¢)dt

are equivalent to the Hamilton equation (14.6) with cp

= (cp, 1r) and

Problem 14.14 Prove this theorem. Hint: since H(cp,1r) is the Legendre transform of L(cp, ¢), 87rH(cp,1r) = ¢(1r) where ¢(1r) satisfies 8L(cp, ¢(1r)) =

1r. We verify this result in the Klein-Gordon case. With

and we recover the Klein-Gordon equation.

Problem 14.15 Show, formally, that with the Poisson bracket given in (14.5),

{1r(X), cp(y)} = o(x - y).

(14.8)

Remark 14.16 Equation (14.8) says that the evaluation functionals, 1r, and cp, are canonical coordinates. To explain the meaning of (14.8), we introduce, for 1 E CO'(lR d ), the functionals

cp(f): (cp,1r)

I--r

(f,cp)

and

1r(f) : (cp,1r)

I--r

(j,1r).

Then (14.8) means that {1r(f),cp(g)} = (j,g) for all 1,g E CO'(lR d ).

14.3 Maxwell's Equations as a Hamiltonian System

173

14.3 Maxwell's Equations as a Hamiltonian System Here we write Maxwell's equations in Hamiltonian form, as a prelude to quantizing the electromagnetic field. The Maxwell equations in vacuum are

V' x B = BE Bt V' x E

BB

(14.9) (14.10)

=--

at

for vector fields E : ~3+1 ---> ~3 (the electric field) and B : ~3+1 ---> ~3 (the magnetic field). The equations (14.10) imply the existence of potentials U : ~3+1 ---> ~ and A : ~3+1 ---> ~3 such that

B = V' x A,

E

=

-BAjBt - V'U.

There is a redundancy in the choice of A and U. Specifically, any gauge transformation Uf--->U-BxjBt A f---> A+ V'X, for X : ~3+1 f---> ~, results in new potentials A and U which yield the same fields E and B. By appropriate choice of X, we may take

U=o, which is called the Coulomb gauge. From now on, we work in this gauge. Thus, we have B=V'xA E = -BAjBt, and using the second equation of (14.9) results in

DA=O

V'. A

= O.

(14.11)

A vector field f : ~3 ---> ~3 is called transverse (or incompressible or divergence free) if V'. f = O. We recall from Chapter 12 that Equation (14.11) is the Euler-Lagrange equation for the action

S(A) =

~

2

r r {IAI2 - IV' lo lIR T

X

A12}

(14.12)

3

where the variation is among transverse vector fields. Problem 14.17 Show that the Hamiltonian corresponding to (14.12) is

where E is the dual field to A, and V' . E = O.

174

14 Introduction to Quantum Field Theory

The phase space for this Hamiltonian is

Z

= H 1 ,trans(JR3; JR3) E9 L 2 ,trans(JR3; JR3),

the direct sum of a Sobolev space of transverse vector fields and the L2 space of transverse vector fields. Let T be the projection operator of vector fields onto transverse vector fields: TF:= F - (.<1)-1\7(\7. F). Problem 14.18 Check that \7. (TF)

= O.

We define a Poisson bracket on Z by

{F, G} := (8A ,EF, Jr8 A ,EG) where

Jr:=

(~ -oT).

Problem 14.19 Check that Maxwell's equations are equivalent to the Hamilton equations ¢ = (A,E) ¢ = Jr8q,H(¢),

(cf. Problem 14.14). We make two final comments. 1. Note that

{Ei(X), Aj (y)}

=

Tij(X - y)

where Tij(X - y) is the matrix integral kernel of the operator T. We still refer to A and E as canonical variables. 2. The first Hamilton equation

A=

T8EH(A,E)

=

-TE =-E

shows that the conjugate field E is, in fact, the electric field (which justifies the notation).

14.4 Quantization of the Klein-Gordon and Maxwell Equations We are now ready to attempt to quantize the Klein-Gordon theory. We begin with Klein-Gordon theory in Hamiltonian form, and proceed, naively at first, by analogy with the passage from classical mechanics to quantum mechanics (which is summarized in the table (3.4)). We take, for now,

F(¢)

= ~m21¢12

with m > O. We have been dealing with real fields until now. Everything we have done so far, however, carries over to complex fields, and from now on our fields will be complex.

14.4 Quantization of the Klein-Gordon and Maxwell Equations

175

14.4.1 The Quantization Procedure

Under quantization, the classical phase space

with the Poisson bracket arising from the symplectic matrix J, becomes the quantum state space L2(Hl(JR.3), D¢) (i.e. L 2 (config. space) with respect to a "Lebesgue measure", D¢, on configuration space) together with the commutator .~J,.J of operators on the state space. The classical observables, real-valued functionals on the phase space, become quantum observables - self-adjoint operators on the state space L2(Hl(JR.3), D¢). In particular, the classical canonical variables

become the operators

¢OP(x) = multo by ¢(x)

and

on the state space L2(Hl(JR.3), D¢). The classical Klein-Gordon dynamics are generated by the Hamiltonian functional

H(¢, n) =

~

J

{lnl 2 + 1\7¢12

+ m 21¢12}

which, under quantization, becomes the Schrodinger operator

H = H(¢OP,n OP ) on the state space L2(Hl(JR.3), D¢). Before proceeding to discuss the problems with this naive approach, we remark that the state space L2(Hl(JR.3), D¢) is a space of functionals On an infinite-dimensional space of functions, and the quantum observables can be thought of as infinite-dimensional pseudodifferential operators. Right away, our approach has serious problems. Problem 1: There is, in fact, no Lebesgue (translation invariant, sigmafinite) measure On infinite-dimensional spaces. We are forced to resort to measures known to exist on such spaces: Gaussian measures. Correction 1: We replace D¢ with dp,c(¢), the Gaussian measure of mean 0 and covariance operator C. The operator C acts on L 2(JR.3). One way to describe dp,c is through finite-dimensional approximations of the function space. Suppose ... C

Fn

C

Fn+l

C ...

176

14 Introduction to Quantum Field Theory

is a sequence of finite-dimensional subspaces of Hl(JR 3), whose limit is

Hl(JR 3 ). Then

where Dcp is the usual Lebesgue measure on a finite-dimensional space, Cn is C restricted to Fn (i.e. C n = PnCPNIFn , where Pn is the orthogonal projection from H 1 (JR3) onto Fn ), and Mn is a constant chosen so that

We introduce the expected value of a functional F with respect to dpc:

E(F) :=

J

F(cp)dpc(cp).

The terminology "mean 0" and "covariance C" corresponds to the properties

E(cp(x)) = 0 and

E(cp(x)cp(y)) = C(x, y)

where C(x, y) is the integral kernel of C. For the Klein-Gordon theory, we take C= ~(_L1+m2)-1/2 2

(recall -L1 + > 0, so this makes sense). To get an idea of the origin of this choice, note that the classical Klein-Gordon Hamiltonian is m2

Integrating by parts, we obtain

J

J + =J + ~ J + I~C-lcpI2}.

{1V'CPI2 + m 21cp12} =

cp( -L1

I( -L1

and so

H(cp,7r) =

{7r 2

But we are not out of the woods yet. Problem 2: It turns out that

m 2)cp

m 2)1/2cpI2

(14.13)

14.4 Quantization of the Klein-Gordon and Maxwell Equations

177

To see that there is a problem, we compute formally E(j 1V'¢12) = j j 8(x - Y)V' x V'yE(¢(x)¢(y))d 3 xd3 y = j j 8(x - Y)V' x V' yC(x,y)d3 xd3 y. If C

= c( -iV' x), then C(x, y) = c(x - y) (see Section 2.5.3) and so

Thus the integral on the right hand side diverges, as we are integrating over infinite volume. Even if we had finite volume, however, this expression would still be infinite, since for C = (-..1 + m 2)-1/2 we have c(x)

=

(21f)-3/2

j e ik . x (lkI 2 + m 2 )-1/2d3 k

= (const)lxl- 2 + o(lxl- 2 ) as Ixl

--->

o.

Thus E(J 1V'¢12) = 00, indicating that f-tc(H I (JR3)) = o. This argument suggests that c( x) must be integrated at least twice in order to remove the singularity at x = o. We expect, then, that E(J 1V'-s¢1 2) < 00 for s > 1. One can show (see [GJ, Sim3, Sim5]) that f-tc(Q) = 1 where Q = {J I (1 + IxI 2)-t/2(1 - ..1)-s/2 f E L2} for sand t sufficiently large. Vectors in the space L2(Q),df-tc) are functionals F(¢) on Q such that

The most basic example of such a functional is as follows. Fix the map ¢EQ

f--+

f

E

H S • Then

¢(f)(= f(¢)) := j f¢

is a linear functional on Q which is in L2( Q, df-tc). (The integral on the r.h.s. is understood as J(l- ..1)s/2f· (1- ..1)-s/2¢.) We compute formally

E(I j f¢12)

=

j j f(x)f(y)E(¢(x)¢(y)) = (I, Ct) <

00.

More generally, let P(tl, ... , tn) be a polynomial in tl, ... , tn and fix iI,···, fn E

CD. Then the functional

F : ¢ E Q f--+ p(¢(iI),···, ¢(fn)) is an element of L2 (Q, df-tc). To summarize, we have made the following correction.

178

14 Introduction to Quantum Field Theory

Correction 2: replace L 2(H 1 (JR3), Drp) by L2(Q, dpc) where C m2)-1/2.

= (-,1+

Unfortunately, the first correction has generated a further problem: -iOq, is not symmetric on L2(Q, dpc). In fact, we have the following integration by parts formula

J

J =J

F( -iOq,G)dpc(rp) =

(-i8q,F)Gdpc(rp)

(-iOq,

+i

J

FG8q,dpc(rp)

+ iC-lrp)FGdpc(rp)

where we have used

To see this, think formally about the Gaussian measure as being

Thus we have

(-iOq,)*

=

-i8q, + iC-1rp

which leads us to the following correction. Correction 1a:

Note that n*

= n,

as desired.

Problem 14.20 Derive formally the commutation relations i

1i[n(x),rp(y)] = 8(x - y) i

1i[n(x),n(y)]

(14.14)

i

= 1i[rp(x), rp(y)] = o.

We have improved our situation somewhat, but a serious problem still remains: Problem 3: H(rp, n) = 00. Specifically, we will show the following a bit later: (14.15)

Note, however, that rp and n are operator-valued distributions (E Q), and there is no reason to expect this expression to be finite. Correction 3: We modify the quantization procedure as follows. We "Wick order" H

14.4 Quantization of the Klein-Gordon and Maxwell Equations

179

We will postpone the description of this modification for a little while. Let us now pause and summarize our corrected quantization procedure in the table below. KGCFT

KGQFT

state space

¢(x) = mult by ¢(x) 7r(x) = -i1l/)(x) + ¥-C- 1 ¢(x)

canonical variables

self-adjoint operators

: F(¢,7r): on L 2(Q,dp,c) dynamics

*

Hamiltonian H (¢cl , 7rcl) Schrodinger op. H =: H(¢, 7r) : Poisson brackets (commutator)

{7rcl (x), ¢cl (y)}

=

f[7r(x),¢(y)] = o(x - y)

o(x - y)

Before moving on, we make a remark about mathematical rigor. Strictly speaking, ¢ and 7r are operator-valued distributions. Though ¢(x), for example, is not well-defined, ¢(f) (for a test function f E COO) is well-defined. We think formally of ¢(f) as ¢(x)f(x)dx. In particular, the correct expression for the commutation relation (14.14) is

J

i

n[7r(f),¢(g)] TheoreIIl 14.21 For all f on L2(Q,dp,c).

E

=

-

(j,g).

COO, ¢(f) and 7r(f) are self-adjoint operators

We omit the proof (see [GJ, RSII]) 14.4.2 Creation and Annihilation Operators

From now on in this chapter, we set Planck's constant equal to one: 1i = l. Definition 14.22 The annihilation operator a(f), and the creation operator a* (f) are defined by

a(f) a*(f)

~¢( C- 1/ 2f) + i7r( C 1/ 2f)

(14.16)

= ~¢(C-1/2f) -i7r(C 1 / 2 f) . 2

(14.17)

=

2

14 Introduction to Quantum Field Theory

180

Problem 14.23 Establish the commutation relations

[a(f),a*(g)] = (/,g) [a(f), a(g)] = [a*(f), a*(g)] = 0 for j,g E

ego.

We now find the expression for the Hamiltonian operator

in terms of the creation and annihilation operators. Using the representation (14.13) for the Hamiltonian, and inserting (14.16)-(14.17), we obtain

where we have used the commutation relation for a and a*, and the selfadjointness of e- 1 / 2 . The first term on the right hand side is non-negative, and the second is infinite (recall e- 1 ox (x) = c(O)), which establishes, formally, Equation (14.15). We can make this argument rigorous as follows. First, we move to momentum space via the Fourier transform: a(x) 1--+ a(k), a*(x) 1--+ a*(k). We wish to show that I w(k)a(k)a*(k)dk = 00, where w(k) = (lkl 2 + m 2)1/2 (the dispersion law). Let ]R3 = U"'Ez3B", (a disjoint union), where B", is the unit box with center at a lattice point in Z3, a", = a, and w'" = minkEBa w(k). Then

IBa

so that

On the other hand,

14.4 Quantization of the Klein-Gordon and Maxwell Equations

181

14.4.3 Wick Ordering We now describe the process of Wick ordering or Wick quantization mentioned in an earlier section. Let A( ¢el , n el ) be a classical observable which is a power series in the classical fields ¢el and n el : A(¢el,n el )

=

L/

Amn(Xl, ... ,xm+n)¢el(xI) .. 'nel(xm+n)d(m+n)x.

m,n

We will use the following shorthand for such integrals: A(¢el,nel ) =

L/

Amn(¢el)m(nel)n.

(14.18)

m,n

1. We express A( ¢el, n el ) in terms of the functions a and a*, where

and to obtain

A(¢el, n el )

= B(a, a*).

2. In the expression for B, move all a*'s to the left of the a's, to obtain an expression of the form B(a,a*) =

L/

Bm.n(a*)ma n .

m,n

3. We quantize the observable A as follows: A(¢el,nel )

f--+

:

A(¢,n): :=

L/

Bm,n(a*)ma n .

(14.19)

m,n

This is the Wick ordered observable. As in (14.18), we have used a shorthand notation. For example, (14.19) stands for :A(¢,n):

=

L/

Bmn(Xl"",xm+n)a*(xl)···a(xm+n)d(n+m)x.

m,n

Here are some examples of Wick ordering: Example 14.24

182

14 Introduction to Quantum Field Theory

1.

+ a*)]2 : = : (C- 1/ 2a)2 + (C- 1/ 2a*)2 + C- 1/ 2a*C- 1/ 2a + C- 1/ 2aC- 1/ 2a* = (C- 1/ 2a)2 + (C- 1/ 2a*? + 2C- 1/ 2a*C- 1/ 2a.

: 1}: = : [C- 1/ 2(a

2. As before,

H

=

~

:

J

{7r 2 + IV' ¢1 2 + m 2¢2}

:=

~

J

a*C-1a.

(14.20)

Problem 14.25 (see [GJ]) Letc= U,C!) =E(¢U)2). Show 1. : ¢U)n := c n / 2Pn (C- 1/ 2¢U)) where Pn is the 2. : e¢(f) := e¢(f)-c/2.

nth

Hermite polynomial

Recall that C- 1 = 2J-Ll + m 2 . Passing to the Fourier transform, a(k) = (27r)-d/2 J e-ik,xa(x)dx, and using the Plancherel theorem, we obtain

H where w(k) = Jlkl2

=

J

w(k)a*(k)a(k)dk,

(14.21 )

+ m 2.

14.4.4 Quantizing Maxwell's Equations The procedure described above for the quantization of the Klein-Gordon theory carries over to the case of Maxwell's theory, whose Hamiltonian formulation was given in Section 14.3. There are two essential differences, which we list here (below we assume the speed of light is c = 1). 1. There is no mass (m = 0) in the EM case. That is, the covariance operator is C = ~ (_Ll)-1/2. 2. The quantized A(x) and E(x) are operator-valued transverse vector fields. That is, our quantum state space is L2 (Qtrans , d/-Lc). The Hamiltonian for the quantized EM theory is, of course, just

and the non-trivial commutation relation is

i[E(x), A(y)]

=

T(x - y)1

where T(x - y) is the integral kernel of the operator of projection onto the transverse vector fields (recall our convention n = 1). In this case, we obtain an expression for the Hamiltonian H, which is similar in form to (14.20):

H

=

14.5 Fock Space

J

183

w(k)a*(k) . a(k)dk,

but with w(k) = Ikl, and where a(k) and a*(k) are operator-valued transverse vector-fields. That is, a(k) = (al(k),a2(k),a3(k)) with k . a(k) = O. The commutation relations for ai (k), at (k) are

[ai(k), aj(k)] = (8ij -

~~~g )8(k -

k')

and

14.5 Fock Space We return now to the Klein-Gordon theory. Recall the definitions (14.16)(14.17) of the annihilation and creation operators, a(x) and a*(x) which act on L2(Q,dp,c). Note that a

= ~C-l/2¢ + iC 1 / 27[ = C 1 / 28¢. 2

Thus the only solution to the equation aD = 0 is D = const. We thus set D == 1, and call it the vacuum. Theorem 14.26 Any vector ¢ E L2(Q,dp,c) can be written as

(14.22) where 'f'n A,

E

L2sym (~nd) =

lOIn

'
sym L2(~d).

Here 0 sym denotes the symmetrized tensor product (see Section 8.8 for some information about tensor products). The space L;ym(~nd) consists of those functions in L2(~nd) which are symmetric with respect to permutations of the n variables Xj E ~d. For simplicity, we will denote the right hand side in (14.22) by

We prove the theorem under the additional condition that the span of vectors ofthe form TI~ ¢Uj)D, n ~ 1, is dense in L2. Proving this latter fact requires some general considerations from the theory of functional spaces. Proof.

1. We first remark that

184

14 Introduction to Quantum Field Theory

where

is the symmetrization of ¢. Here Sn is the group of permutations of the n variables, and for n E Sn, n(I, ... , n) = (n(I), ... , n(n)). 2. Now we use the fact that the span of vectors of the form I1~ ¢(Jj)D, n 2:: 1, is dense in L2. Note n

n

1

1

IT ¢(fJ)D = IT (a(C

1 / 2 fJ)

+ a*(C 1 / 2 fJ))D.

Using the commutation relations, we write I1(a+a*) in the normal form

IT

(a

+ a*)D =

1

But (a*)kaID

=

°

unless l

IT

L /

k+l::;n

Akl(a*)kal D.

= 0, so

¢(fJ)D =

1

L/

k::;n

AkO(a*)kD.

Thus vectors of the form

are dense in L2. 3. We have by straightforward computation

(/ ¢n(a*)n D , / Xm(a*)mD) =

{n!(¢~'Xn) ~::

(14.23)

Problem 14.27 Show (14.23). Thus

(14.24) is closed, contains a dense set, and hence is the whole L2 space. D

Definition 14.28 The (bosonic) Fock space is 00

:F:= E9[0~ym,lL2(lRd)l. n=O

We call

:Fn

:= 0~ym,lL2(lRd)

the n-particle sector. By convention, :Fo =
14.5 Fock Space

185

The previous theorem provides a unitary isomorphism

given by

Moreover it is easily checked that on F,

and a*(f) : ¢n E Fn

f--+

,,;:r;:+lf ®sym ¢n

E

Fn+ 1·

Proposition 14.29 Consider the Hamiltonian, H = J a*C-1a (see (14.20)), and particle number, N = J a*a, operators. In the Fock space representation,

where the subscript

Xj

in C Xj indicates that this operator acts on the variable

Xj.

Proof. This is simply a matter of using the commutation relations. We leave the details as an exercise. 0

Thus we have obtained a very simple realization of our state space L2( Q, dJ-Lc) which is independent of C, and in which the Klein-Gordon Hamiltonian acts as a direct sum of simple operators in a finite but increasing number of variables: n

H~EB~=oLJ-..::1xi +m 2 . 1

In particular, the spectrum of H is

rr(H) = {O} U {Un>l[nm,oo)} where the zero-eigenfunction is the vacuum, fl. Physically, this theory describes non-interacting particles (bosons) of mass m.

186

14 Introduction to Quantum Field Theory

Problem 14.30 Define the momentum operator as

P

=

P

=

J J:

a*(x)(-iV'x)a(x)dx.

Show that

7r(x)iV' x¢(x) : dx.

Find the spectrum of P. Show (formally) that the operators P, H, and N commute.

14.6 Generalized Free Theory Above we have considered a free quantum field theory with covariance operator C = ~(-.c1 + m 2)-1/2. In exactly the same way, we can construct a free quantum field theory based on an arbitrary positive operator C, acting on L2(JRd), which has a densely defined inverse, C- 1 = 2A. Such a theory is a quantized version of a classical Hamiltonian functional

(note that with A = v' -.c1 + m 2 , we recover the Klein-Gordon theory). We may now quantize as in the Klein-Gordon case, and construct Fock space analogously Defining creation and annihilation operators as in the Klein-Gordon case (eg. a(x) = ~A¢ + ~A-17r) leads to HA

=

J

a*(x)Aa(x)dx.

As with Klein-Gordon, the one-particle operator A determines all the properties of HA. In particular, n

HA ~ E9~=o

LA

xi •

1

Again, this describes a free (non-interacting) theory.

14.7 Interactions The quadratic Hamiltonians studied above correspond to non-interacting or "free" quantum field theories. In technical terms, the central goal of QFT is

14.7 Interactions

187

to understand how the interactions modify the picture constructed for the free theories. There are two ways of introducing interaction into a QFT. The first way is to add to the quadratic Hamiltonian in question a functional of ¢ of degree higher than two. Such a term represents a self-interaction of the field ¢. The second way is to consider an interaction of several fields, or an interaction of a field with a quantum-mechanical system. An interaction of this type will be considered in the next chapter. In this section we make a few remarks on the (very difficult) problem of constructing self-interacting quantum field theories. A more detailed treatment of this subject is beyond the scope of this book. It can be found in [GJ, Sim5]. We consider a Hamiltonian H=HKG+).W

where HKG is the Klein-Gordon Hamiltonian (with m > 0) treated in the previous section, ). > 0, and W

=

J:

F(¢):

where F is a polynomial which is bounded from below. The problem here is that if F is of order ~ 3, then W is not well-defined. The origin of the problem depends on the dimension. For d = 1, a spatial cut-off suffices in order to regularize W: the operator Wg :=

J

g(x) : F(¢(x)) : dx

is well-defined for g E L2 (JR.) real (see the discussion at the end of this section). One can then define the operator H by some limiting procedure, starting with Hg := HKG + W g, and taking g -+ 1. For d ~ 2 the situation is more complicated, and we have to institute, in addition, an ultraviolet (UV) cut-off. This amounts to smoothing out the field ¢(x). We describe here a convenient mathematical method for smoothing out a rough function ¢. Let 8 be a Coo function such that 8 ~ 0, f 8 = 1, and 8(0) = 1. Define Now we set ¢e := 8e * ¢. Then ¢e is smooth, and ¢e

-+

¢ as

t -+

O. Note that

So ¢e is essentially ¢ cut off at the momentum scale Ikl ::; l/t. Finally, we replace ¢(x) in the expression for W by ¢e(x) to obtain another treatable approximation to W:

188

14 Introduction to Quantum Field Theory

Note that if g(x) == 1, then Wg,e is not translation invariant. That is, if Pis the momentum operator (P = J a*(k)ka(k)dk), then [WE' P] = 0 iff Wn,E is independent of L Xi· Problem 14.31 Show

[WE,P] =

J: [t

-iV'j,Wn,E]¢@n: dnx.

1

Another way to do a UV cut-off is via a lattice approximation. In this case we replace ]Rn with the lattice (Ez)n, and let E -7 O. It turns out that removing the spatial cut-off (g -7 1) and the UV cut-off (E -7 0), however, is rather difficult, and not always possible. Finally, we explain why for the Klein-Gordon self-interacting theory with m > 1 and d = 1, the spatial cut-off suffices. Let F(¢) = Ln an¢n and consider the term ¢n. Using properties of the Fourier transform (see Section 2.5.2), we obtain

(n copies of ¢ appear in the convolution). Changing the variables of integration in the convolutions, we can rewrite the above as

(14.25) Now equations (14.16) and (14.17) imply

¢(x) where C

=

=

C 1 / 2 (a(x)

+ a*(x))

(-.:1 + m 2)-1/2. Taking the Fourier transform, we have ¢(k) = h(k)(a(k)

+ a*(k))

where h(k) = (lkl 2 + m 2)-1/2 and (abusing notation) a(k) stands for the Fourier transform of a(x). Substituting this expression for ¢(x) into the right hand side of (14.25) gives

:J

g¢n:=

L : J... Jg(t kj ) IT h(kj ) IT a

Uj

CTl, ... ,CTn

1

1

1

(kj )

IT

dkj : (14.26)

1

where (Jj E {-, +}, a_(k) = a(k), and a+(k) = a*(k). Since h(k) = (lkl 2 + m 2)-1/2 with m> 0, and d = 1, the coupling function in (14.26) is in L2:

14.8 Quadratic Approximation n

189

n

9(2:: k j ) II h(kj ) 1

E L 2 (lR. nd ).

1

Therefore the operator on the right hand side of (14.26) is well-defined for d

=

1.

14.8 Quadratic Approximation Presently, we understand quadratic Hamiltonians rather well. The first step in investigating more general, non-quadratic Hamiltonians, is finding the correct quadratic approximation. This is completely analogous to the situation in non-linear differential equations where the first step is practically always a linearization around a solution of interest. Indeed, quadratic Hamiltonians are in one-to-one correspondence with linear (Heisenberg) equations of motion. Consider a quantum field theory with Hamiltonian (14.27)

H=HKc+W

where W = >.-2

J:

F(>.¢) :

(we have presented W in a conveniently rescaled form). Splitting the term ~ J : IV'¢12 : from the operator HKc, and joining it with the operator W, we rewrite (14.27) as H

So V(¢) =

=

J~ :

~J:

7r 2 :

+ : V (¢)

:.

(14.28)

IV'¢12+>.-2F(>.¢) :dd x .

If F(¢) = o(¢2) and F(¢) ::::: 0, then the operator H KC is obviously a quadratic approximation of the operator H. This corresponds to linearization around a trivial solution ¢o == 0 in non-linear differential equations. However, there are many important cases where finding a correct quadratic approximation constitutes a considerable conceptual step. In these cases one deals with quantum fluctuations around nontrivial classical solutions, rather than trivial ones. These are exactly the cases we discuss briefly in this section. Before proceeding to this discussion, we give two simple, but rather important examples of non-trivial classical solutions ¢o. Both examples are in dimension one (d = 1), and for real fields. Let

where (in order to identify this with (14.27))

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14 Introduction to Quantum Field Theory

For our first example, we take

for some a > O. In this case, the functional V (¢) has a minimizer in a class of functions ¢ : lR ----+ lR, satisfying the boundary conditions

¢(x)

----+

a

±-

>.

as x ----+ ±oo, and with appropriate smoothness conditions. This minimizer can be found explicitly: a (14.29) ¢o(x) = tanh(ax)

:x

For our second example, take

F(¢) = a(cos(¢) - 1) for some a > O. In this case, V(¢) has a minimizer in a class of functions ¢ : lR ----+ [0,27r] with the boundary conditions ¢(x) ----+ 0 as x ----+ -00 and ¢(x) ----+ 27rj>. as x ----+ 00, and with appropriate smoothness conditions. Again, this minimizer can be found explicitly:

¢o(x)

=

:x4 tan-l (evla

(14.30)

X ).

Problem 14.32 Show that (14.29) and (14.30) are critical points of the corresponding functionals V (¢ ). The boundary conditions specified above are such that

¢o(x)

----+

null set of F(¢) as x

----+

±oo.

(14.31 )

This is needed if we want the functional V (¢) to be finite. There are three more boundary conditions satisfying (14.31) in each of the above cases. One of these boundary conditions has the minimizer ¢o( -x) while the other two have constant minimizers. Problem 14.33 Prove this last statement. The four boundary conditions split the functions of finite energy into four topologically inequivalent sectors: functions from one of these sectors cannot be continuously deformed into functions from another sector. Thus we have a topological conservation law - a classical solution starting in one of the sectors stays always in the same sector (an evolution is a continuous deformation, with time serving as a deformation parameter). The topological sectors can be characterized by the quantity

14.8 Quadratic Approximation

191

Q = ),(¢(+oo) - ¢(-oo)) which is a topological invariant. It is called the topological charge. The solutions ¢o (x) written above have topological charges 2a and 27r respectively, while solutions ¢o( -x) have charges -2a and -27r. The solutions ¢o(x) are called topological solitons and the solutions ¢o (-x) are topological antisolitons. Topological solitons exist also in higher dimensions (vortices, monopoles, etc.). We do not describe them here, and refer the interested reader to the books PT, CO, Raj. Now we return to our main subject, the quantum theory of non-trivial classical solutions. We consider a Hamiltonian of the form (14.27). We would like to understand the spectrum of H near its bottom. This means we want to study the spectrum of H near inf V (¢). We assume the functional V (¢) has a minimizer, ¢o. If ¢o is trivial (i.e. ¢o == 0) and F(¢) = O(¢2), then the correct quadratic approximation to H is just HKG (as discussed above). We are interested in situations where ¢o is non-trivial, such as is the case with topological solitons (and other interesting objects). In what follows, we give a formal but systematic treatment of this example. For simplicity, we consider the case of non translationally-invariant functionals V (¢) (e.g. V(¢) = f{(1/2)IV'¢1 2 + F(¢, x)}) and we will make a few remarks about the translationally- invariant case at the end. We represent the quantum field in the form

¢(x) = ¢o(x)

+ ~(x)

where ¢o(x) is the classical minimizer of V(¢) and ~(x) describes quantum fluctuations around the classical background ¢o(x). Our goal is to pass from the original quantum field ¢(x), to the new field ~(x). Note that on the scale of), (a small, dimensionless coupling constant), ¢o(x) = O(),-l) and ~(x) = 0(1). The first step is to expand V (¢) around ¢o to the third order:

V(¢) = V(¢o)

1

+ 2(~,HessV(¢o)~) +

O()'e),

since ¢o is a critical point of V. We compute formally how the change of variables from ¢ to ~ affects the Gaussian measure: d/-Lc(¢)

= Z- l e-(¢,C-'¢)/2D¢ =

Z- l e-(¢oH,C-'(¢oH»/2 D~

= e-(¢o,C-'¢o)/2-(¢o,C-'t;) d/-Lc(~). Define

192

14 Introduction to Quantum Field Theory

by

U : F(¢)

e-(e/>O,C- 1e/>o)/4-(e/>o,C- 1f;)/2 F(¢o

f---7

+ ~).

It is easily checked that U is unitary. Next, set

1ff

:=

U1fgu- 1

where as before, C

iC-1rl.

.",

1fe/> = -we/> + "2 Consequently,

fI

:=

UHU- 1 =

'P'

J:~(1ff)2 +

V(¢o

+~) :

acting on L2 (Q, d/-Lc(~)). Hence

fI = ~

J:

{(1ff)2 + ~A2~ + O(>,e)} : +V(¢o)

where

A2

:=

HessV(¢o)

=

-L1x

+ F"(>"¢o).

N ow we change the covariance

To this end we write

d/-LC = iPd/-LC A, where

iP =

ZCAZcle-(~,(C-1-CA1)~)/2.

We define another unitary operator

U1 : L 2 (Q,d/-Lc)

---+

L 2 (Q,d/-LCA)

by (U1F)(~) := iPl/2 F(~). Then

Define on L 2 (Q,d/-LCA)' Then

H new = Hquad where

+

J:

O(>..e) : +V(¢o)

(14.32)

14.8 Quadratic Approximation

Hquad =

~

J:('if~ + ~A2~)

: .

193 (14.33)

Passing now to the corresponding new creation and annihilation operators (see Section 14.6), we have Hquad

=

J

a*(x)Aa(x)dx.

(14.34)

Thus we have a new free theory with covariance ~A-l, and with one-particle operator A. A determines entirely the spectral properties of Hquad. Let us summarize the above analysis. We began with a Hamiltonian of the form (14.28), where the functional V(¢) has a minimizer ¢o, acting on the space L2( Q, d/Lc). We constructed a unitarily equivalent Hamiltonian (14.32) acting on the space L 2 (Q,d/LCA) where the covariance is CA = (2A)-1 with A = (HessV(¢0))1/2. Up to the additive constant V(¢o) (the classical energy of ¢o), the new Hamiltonian is a perturbation of the quadratic Hamiltonian (14.33) or (14.34) by a term which is of higher order - O(e) - in the new quantum field ~. Consequently, one expects that the low-energy spectrum of the original (and new) Hamiltonian is determined by the spectrum of this quadratic Hamiltonian (14.33) or (14.34), which, in turn, is determined by the spectrum of A, the square root of the Hessian of V (¢) at ¢o. Now we briefly discuss the translation-invariant case, continuing to work with real fields. First, we take another look at the minimizer ¢o. As a critical point of the potential functional V(¢), it is a stationary solution of the classical equations of motion for the classical Hamiltonian functional H(¢, 'if)

=

J~'if(X)2 +

V(¢)

(see Section 14.2.4). The corresponding Hamilton's equations are equivalent to the "Newton equation" (14.35) ¢ = -aV(¢), which is nothing but the familiar non-linear wave equation D¢ + '\F'('\¢)

=0

with 0 = al - ,1x the D'Alembert operator (this is the classical dynamics corresponding to the quantum dynamics given by the Schrodinger equation i8'IjJ / at = H'IjJ). This equation is invariant with respect to the relativistic group of motions - the Poincare group. In particular, because of translational invariance, for any y E ]Rd, ¢o(x - y) is also a stationary solution of the equation of motion (14.35). Thus we have a d-dimensional manifold

where Th is the "shift operator" mapping ¢(x) to ¢(x - h), each point of which is a stationary solution of (14.35).

194

14 Introduction to Quantum Field Theory

Now we expand the field ¢(x) "around" the manifold Me. In other words, we decompose any field ¢(x) as

¢(x) = ¢o(x - y)

+ ~(x -

y)

where ¢o(x - y) is the projection of ¢(x) onto the manifold Me, and ~(x - y) is the projection of ¢(x) in the orthogonal direction; i.e. ~y -.l T¢yMe where we have used the notation ~y(x) = ~(x - y), ¢y(x) = ¢(x - y), etc. Note that the tangent space, T¢yMc is spanned by the functions a¢o(x - y)/axj, j = 1, ... , d. Consequently, the above orthogonality condition becomes

J~(x)"\l¢o(x)dx

=

o.

(14.36)

Observe that the functions a¢o / ax j are zero-modes (eigenfunctions with eigenvalue zero) of the Hessian operator Hess V (¢o). Indeed, the functions ¢h(X) := ¢o(x - h) are all critical points of the functional V:

Differentiating this equation with respect to h j at h = 0, we find

Hess(V(¢o))a¢o/axj =

°

which establishes our assertion. The condition (14.36) states that the fluctuations are orthogonal to the zero-modes of HessV(¢o). Thus we pass from the field ¢(x) to the pair (y,~(x)) where y E]Rd and ~(x) satisfies (14.36). That is, y = y(¢) satisfies the equation

J

{¢(x

+ y) - ¢o(x)}"\l¢o(x)dx =

and ~(x)

°

= ¢(x + y(¢)) - ¢o(x).

Conversely, the field ¢(x) is reconstructed from y and equation

¢(x) = ¢o(x - y)

+ ~(x -

~(x)

according to the

y).

The transformation of the momentum field, 7r( x), can be found in the Lagrangian formalism. Then the transformations of both ¢(x) and 7r(x) yield a canonical transformation. The analysis presented above for the non translation-invariant case can be extended to the translation-invariant one. We do not pursue it here.

14.8 Quadratic Approximation

195

14.8.1 Further Discussion 1. Quantization: Physically, we can re-interpret what we have done above as quantization of the classical solution ¢o(x) in the first case, or the manifold of solutions ¢o(x - y) in the second. Indeed, we can arrive at the results above by first making the canonical transformation

¢(x)

f---+

~(x)

= ¢(x) - ¢o(x)

and

7r(x)

f---+

7r(x)

(14.37)

and then performing canonical quantization. In the translation invariant case, we have to quantize in the presence of a symmetry group. Have we encountered such a problem before? In fact we have, in quantizing the Maxwell equations. There we had to deal with gauge symmetries. As a result, we quantize the electromagnetic field in the direction transverse to the orbits of the gauge group. Passing to the field ~ in (14.37) can be considered as quantization in the direction transverse to the group of translations. 2. Relativistic invariance. The classical equation (14.35) is invariant with respect to the relativistic group of motions - the Poincare group. Therefore any stationary solution, ¢o, of this equation can be "boosted" by a Lorentz transformation into a traveling wave. For d = 1, the boost is given by

¢v(x,t)

= ¢o (x ~o).

(14.38)

Problem 14.34 Check by direct calculation that if d = 1, the function ¢v satisfies (14.35) provided ¢o is a critical point of V(¢). Thus in the translationally invariant case, a single stationary solution leads to an entire 2d-dimensional manifold of time-dependent solutions. These solutions are traveling waves, or solitons (in physics terminology). They are parameterized by their initial positions, Xo and their velocities, v. 3. Momentum conservation. The physical quantity which is conserved due to translation invariance is the field momentum. In the classical case, the latter is given by

P(¢,7r) =

J

7r(x)\l¢(x)dx.

The quantum field momentum, P, is obtained by quantizing P(¢,7r) in the standard way. Problem 14.35 Show that in the translation-invariant case, P(¢,7r), and its quantum counterpart, P, are conserved - i.e.

{P(¢, 7r), H(¢, 7r)}

=0

and

1

n[P,H] = O.

196

14 Introduction to Quantum Field Theory

Suppose d = 1. For soliton (14.38), the energy Esol = H(
The latter implies (14.40). Problem 14.36 Use the virial relation to prove the relativistic formula (14.39).

The quantum momentum P leads to the soliton center momentum p, conjugate to the soliton center coordinate.

15 Quantum Electrodynamics of Non-relativistic Particles: the Theory of Radiation

We conclude this book by outlining the theory of the phenomenon of emission and absorption of electromagnetic radiation by systems of non-relativistic particles such as atoms and molecules. Attempts to understand this phenomenon led, at the beginning of the twentieth century, to the birth of quantum physics. Only by treating the matter and the radiation as quantum mechanical can one give a consistent description of the phenomenon in question. Thus, our starting point should be a Schrodinger operator describing quantum particles interacting amongst themselves, and with quantum radiation. In mathematical terms, the question we address is how the bound state structure of the particle system is modified by the interaction with radiation. One expects that the ground state of the particle system survives, while the excited states turn into resonances. The real parts of the resonance eigenvalues - the resonance energies - produce the Lamb shift, first experimentally measured by Lamb and Retherford (Lamb was awarded the Nobel prize for this discovery). The imaginary parts of the resonance eigenvalues - the decay probabilities - are given by the Fermi Golden Rule (see, eg, [HuS]). This picture was established rigorously, under somewhat restrictive conditions, in [BFS1]- [BFS4], whose results we describe here. The method in these papers also provides an effective computational technique to any order in the electron charge, something the conventional perturbation theory fails to do.

15.1 The Hamiltonian In what follows, we work in physical units in which the Planck constant and speed of light are equal to 1: fi = 1, c = 1. In this case, the electron charge, -e, is a dimensionless constant equal to where a ~ 1~7 is the fine structure constant. Here we will consider a to be a small, dimensionless parameter. Recall from Section 14.4.4 that the quantized electromagnetic field is described by two canonical fields: the quantized vector potential (quantized connection) A'(x) and quantized electric field E(x), acting on Fock space :F:= 'Hj. They satisfy the standard commutation relations

-va,

i[E(x),A'(y)]

=

T(x - y) ·1,

S. J. Gustafson et al., Mathematical Concepts of Quantum Mechanics © Springer-Verlag Berlin Heidelberg 2003

198

15 Quantum Electrodynamics of Non-relativistic Particles

etc., where T(x - y) is the integral kernel of the projection operator from the vector fields on ]R3 to the divergence-free (transverse) vector fields. The quantum Hamiltonian governing the evolution is

which is just a quantization of the classical Hamiltonian functional. In terms of the creation and annihilation operators, these objects are written as follows: A'(x) = (eik,xa(k) + h.c.)g'(k)d3k,

J

where g'(k) = (27T)-3(2w(k))-1/2, w(k) mitian conjugate", and

= Ikl, and h.c. stands for the "her-

J

w(k)a*(k)a(k)d3k.

Hf =

The dynamics of a system of quantum matter (atom, molecule, etc., with fixed nuclei) is described by the Schrodinger operator N Hpart =

"~ 2m 1 Pj2

+ e2 v;,oul(;r., R)

j=l

acting on 1ipart = L2 (]R3N) (or a subspace of this space of a definite symmetry type). Here N is the number of electrons involved, and m, Xj and Pj = -iV Xj are the mass, position, and momentum, respectively, of the j-th electron. Furthermore, e is the absolute value of the electron charge, ;r. = (Xl, ... , XN), R = (R 1 , ... , R M ), and e2v;,oul(;r., R) is the total Coulomb potential for N electrons and M nuclei located at the positions Xl, ..• , X N , and at R l , ... , RM, respectively. Radiation interacts with matter via the minimal coupling (see Section 7.8). An equivalent way to arrive at it (which is couched in purely mathematical terms) is to remember that A'(x) is the quantum connection, so we replace the usual derivatives by covariant ones:

P

f--+

PA' := P - eA'(x) .

(15.1)

Thus to obtain the full, interacting Hamiltonian, we replace the particle momenta Pj in Hpart by the covariant momenta PA',j = Pj - eA'(xj), and add to the result the field Hamiltonian, H f' responsible for the dynamics of A' (x). However, the operator we obtain in this way is ill-defined: its domain of definition contains no non-zero vectors. The problem is that A'(x) is too rough an operator-valued function. To remedy it we institute an ultraviolet cut-off. Namely we replace A'(x) in (15.1) by an operator Ax(x):

15.1 The Hamiltonian

A'(x)

f---+

Ax(x)

:=

199

X * A'(x)

where X is a smoothed-out 8-function (recall that X denotes the inverse Fourier transform of a function X). In fact, the specific shape of X is not important for us. All we need to begin with will be encoded in the assumption (15.4) below. As a result, we arrive at the standard Hamiltonian of non-relativistic matter interacting with radiation:

acting on 1ipart l8l1if, where PAx,j = Pj - eAx(Xj), and Pj = -iVxj (note that the parameter e enters the definition of the operator PAx,j). One can show that H(e) is well-defined under some mild restrictions on X (see (15.4) below). Now we pass to dimensionless variables as follows 1

x----; - - x me 2

'

i.e., the electron coordinates are measured in units of the Bohr radius = m~2' Under this rescaling, the Hamiltonian H(e) is mapped into the Hamiltonian me 4 H(c), where

Tbohr

H(c) =

N 1

L

'2(iV Xj

+ cA(Xj))2 + V(;r) + Hf'

j=1

= e3 , A(x) = Ax(e 2x)

with X(k) replaced by x(me 4 k), and V(.JC) = 2 Vcoul(.JC, me E). Thus the energy is measured in units of me 4 = 2(Rydberg). The Hamiltonian H (c) is, of course, equivalent to our original Hamiltonian H(e). Understanding the spectral properties of H(c) is the object of this

where c

chapter. To simplify notation we only consider the case N 1 .

H(c) = '2(zV x + cA(x))2

= 1:

+ V(x) + Hf

,

(15.2)

where x E JR3. Now Hpart = -~Ll + V(x) acts on 1ipart = L2(JR 3 ), so that H(c) acts on L2(JR3 ) 18l:F. The operator A(x) has the same form as A'(x),

A(x) =

J

(eik.xa(k)

+ h.c.)g(k)d3 k,

(15.3)

but with the new coupling function g(k) = X(k)g'(k). We assume that 9 is real and satisfies

200

15 Quantum Electrodynamics of Non-relativistic Particles

J1~2

<

(15.4)

00.

Thus H(E) depends also on the (coupling) function (or form factor) g(k). At this point we forget about the origin of the potential V(x), but rather assume it to be a general real function satisfying standard assumptions. Such assumptions can be found in the literature cited at the end of this book. We assume for simplicity, either

(a)

lV(x) I ::; C(1

+ Ixj)-2-e

(b)

V(x):::: -C, and 3R,o: > 0, s.t. V(x)::::

for some e > 0, or

o:Ixl 2

for Ixl:::: R,

(15.5)

which is considerably more restrictive than necessary. One can show that under conditions (15.4) -(15.5), the operator H(E) is essentially self-adjoint on the domain D(H(O)). Remark 15.1 We have tacitly assumed that the ultraviolet cut-off is imposed at a scale 1 in our dimensionless units. A general cut-off at a scale K can be absorbed, after rescaling, into the parameter E, which would change from e 3 to e3 K.

15.2 Perturbation Set-up We first examine a system consisting of matter and radiation not coupled to each other. Such a system is described by the Hamiltonian

H(O) =

Hpart

® If

+ Ipart ® Hf,

(15.6)

which is obtained by setting the parameter E in (15.2) to zero. Using separation of variables (see Section 8.8), we obtain

O"(H(O))

= dHpart) + O"(Hf)

O"p(H(O)) = O"p(Hpart ) + O"p(Hf) O"cont

(H(O))

=

0"1

U

0"2

U

0"3

where 0"1 0"2 0"3

= O"p(Hpart) + O"cont(Hf)

= O"cont(Hpart ) + O"p(Hf) = O"cont(Hpart ) + O"cont(Hf)·

Recall the spectral structure of the Schrodinger operator H have

O"(Hpart)

=

{negative EV's, E j

}

U {

part .

Typically, we

continuum [0, (0) }

(15.7)

15.2 Perturbation Set-up

201

Here j = 0,1, ... and we assume Eo < El ::; .... The eigenfunction, 1fJbar t, corresponding to the smallest eigenvalue, Eo is called the ground state, while the eigenfunctions 1fJrrt for the higher eigenvalues E j with j ~ 1, are called the excited states. The generalized eigenfunctions of the continuous spectrum are identified with the scattering states.

a

t

\ t I

scattering states

bound states

Fig. 15.1. Spectrum of Hpart For the field, (Jcont(Hf) = [0, (0) and (Jp(Hf) = {O}. The eigenvalue 0 is non-degenerate, and corresponds to the vacuum vector: HfD = o. Thus we have

(J(H(O)) = {EV's E j } U{continuum[Ej,oo)} Ucontinuum[O,oo). (15.8) j?O

The spectrum of H(O) is pictured in Figure 15.2.

a

Eo E 1 ···

\ tI

bound states

Fig. 15.2. Spectrum of H(O). The eigenfunctions of H(O) corresponding to the eigenvalues E j , j = 0,1, ... , are (15.10) The generalized eigenfunctions corresponding to the branch [Ej, (0) of its continuous spectrum are

1fJr

n

art

®

II a*(ki)D

(15.11)

i=l

for various n ~ 1 and k 1 , ... , k n in ]R3 (the corresponding spectral points are Ej(k) = E j + L~=l w(ki )). The solutions of the time-dependent Schrodinger equation

202

15 Quantum Electrodynamics of Non-relativistic Particles i

~~ = H(O)1/J

(15.12)

with the initial conditions (15.10) and (15.11) are given by e- iEjt (1/Jrrt ®

and

D)

(1/Jyart ® II a*(ki)D) n

e-iEj(k)

,

i=l

respectively. The first of these states describes the particle system in the state 1/Jrrt with no photons around, while the second one corresponds to the system in the state 1/Jrrt and n photons with momenta k 1 , ... , k n . Both states are stationary in time. In the absence of coupling between matter and radiation, the system of matter and radiation placed in one of these states remains in the same state forever. Radiation is neither absorbed nor emitted by this system. The question we want to address is: how does this picture change as an interaction between the matter and radiation is switched on? Understanding how this picture changes is the main problem of the mathematical theory of radiation.

15.3 Results The rigorous answer to the question posed above is given in the theorem below. This theorem refers to the notion of resonance described in [BFSl, BFS2] (see also Section 13.2) and, for various statements, uses subsets of the following two conditions (in addition to conditions (15.4) and (15.5), which are assumed throughout the chapter without further mention, and a technical condition called positivity of the Fermi Golden Rule, which is satisfied except in a few "degenerate" cases): (A) ge(k) := e- 3e / 2 g(e-ek), as an L 2 (w- 1 d 3k)-function of k, has an analytic continuation in 8 from lR. into a complex neighbourhood of 8 = 0, satisfying the estimate Ige (k) I :::; CI k 1- 1 / 2 . (B) Ve(x) := V(ee x ), as a multiplication operator from CO"(lR.3 ) to 1ipart, has an analytic continuation in 8 from lR. into a complex neighbourhood of 8 = 0, satisfying either (i) Ve is bounded from D(L1x) to 1ipart , or (ii) ReVe(x) ~ -C and ReVe(x) ~ a'Ixl 2 for Ixl ~ R', for some a', R'. Theorem 15.2 Let c =1= 0 be sufficiently small. We assume (A) and (B) for statement (ii) below, and (15.5)(b), (A), and (B) for statement (iii). Then

(i) H(c) has a unique ground state, 1/Jc. This state converges to 1/Jgart ® D as c ----7 0, and is exponentially localized in the particle coordinates: i.e, I e'5Ix11/Jc I < 00 for some 8 > O.

15.3 Results

203

(ii) H(c:) has no other bound states. In particular, the excited states of Hpart (i.e. 7/Jrrt Q9 il, j ~ 1) are unstable. (iii) The excited states of Hpart turn into resonances of H(c:), c: -I=- 0 (see Fig. 15.3).

Fig. 15.3. Bifurcation of eigenvalues of H(O) (the second Riemann sheet). A complete proof of statements (i)-(iii) can be found in [BFS2, BFS3, BFS4]. We will describe the proof of (i) in Chapter 16. The proofs of (ii) and (iii) are similar. This theorem gives mathematical content to formal calculations performed in physics with the help of perturbation theory. The purpose of this theorem, as well as of the calculations mentioned, is to explain theoretically the following fundamental physical phenomena: (a) a system of matter, say an atom or a molecule, in its lowest energy state is stable and well localized in space, while (b) the same system placed in the vacuum in the excited state, e.g. in 7/Jjart Q9 il, j ~ 1, emits radiation and descends to its lowest energy state. The mathematical manifestation of the first statement is that H(c:) has a ground state, which is well localized in the particle coordinates, - statement (i) of the theorem - while the second statement, rendered in mathematical terms, says that the system in question has no stable states in a neighbourhood of the excited states of the particle system - statement (ii) of the theorem. Moreover, statement (iii) says that the excited states, 7/Jjart Q9 il, j ~ 1, of the particles in the vacuum lead to metastable states, i.e., solutions of the time-dependent Schrodinger equation = H(C:)7/J which are localized for long intervals of time, but eventually disintegrate. A metastable state is another term for a resonance. These metastable states are responsible for the phenomena of emission and absorption of radiation and their life-times tell us how long, on average, we have to wait until a particle system emits (or absorbs) radiation. A proof of Theorem 15.2 is outlined in Chapter 16, where the basic technique - the renormalization group - is introduced. We will close the present chapter with a brief discussion of this approach. For simplicity, consider just the ground state. Our goal is to understand the low energy spectrum of H (c:), for c: sufficiently small. Namely, we address the questions:

iWf

- Does H(c:) have a ground state? - What is the structure of the spectrum of H (c:) near its infimum?

204

15 Quantum Electrodynamics of Non-relativistic Particles Look first at the spectrum of H(E = 0), i.e., when the interaction is turned

off: rt(gion of mterest

(l) t Fig. 15.4. Region of interest w.r.t. specH(O). This picture suggests that only the ground state 'ljJgart of Hpart and the low energy states of H f are essential. The key idea here is to project out the inessential parts of the spectrum without distorting the essential ones. But how to do this? Let us try the first idea that comes to mind: (15.13) where Pp is the spectral projection for H(O) associated with the interval [Eo, Eo + pl. The operator Pp can be written explicitly as (15.14) Here pJ'art is the orthogonal projection onto 'ljJgart and E[o,pJ(Hf) is the spectral projection of Hf corresponding to the interval [0, p] (see Section 15.4.1 at the end of this chapter for the definitions). The latter "cuts-off" the energy states of H f with energy above p. The new operator PpH(E)Pp acts on the subspace L 2(JR.3) 09 :F which consists of states of the form

(see Section 15.4.2 below). This is exactly what we want. However, the low energy spectrum of the operator PpH (E )Pp is different from that of the operator H(E). So we have lost the spectral information we are after. In Section 16.1, we learn how to project operators to smaller subspaces without losing the spectral information of interest. But, as usual, there is a trade-off involved. While the map (15.13) acting on operators H is linear, the new map we introduce is not. This new map is called the decimation map (or Feshbach map). In order to analyze the spectrum of H(E), we change our viewpoint, and think of the operators HO(E, z) := H(E) - zl as points in a space of operators (a Banach space, in fact). On this space, we define the renormalization map

where Dp is the decimation map, and Sp is a simple rescaling map (see Sections 16.7 and 16.8). Dp maps operators which act on the subspace

15.4 Mathematical Supplements

205

RanE[o, 1] (Hf) consisting of states with photon energies S; 1, to operators which act on states with photon energies S; p. The rescaling Sp brings us back to the subspace RanE[o,l] (Hf). By design, the renormalization map Rp is isospectmZ in the sense that the operators K and Rp (K) have the same spectrum near 0, modulo rescaling. The effect of Rp when we apply it to Ho(c:, z), is to focus on a neighbourhood of the spectrum of Ho(c:, z) near O. The smaller p is, the smaller this neighbourhood is. We show that the renormalization map obeys a semi-group law (Rpl 0 Rp2 = R p1P2 )' Thus it gives rise to an isospectral flow on our Banach space of operators. We will see that under this flow, Ho(c:, z) approaches the operator wHf (for some w E C) as p ----+ O. In fact, CHf is a line of fixed points of the flow (Rp(wHf) = wHf). By studying the behaviour of the flow near this line of fixed points, we can relate the spectrum of Ho(c:, z) near 0 (and hence of H(c:) near z) to that of Hf, which we know well. This is the basic idea behind the proof of Theorem 15.2. This program, which is described somewhat vaguely here, will be studied in detail in Chapter 16.

15.4 Mathematical Supplements 15.4.1 Spectral Projections For a general definition of the spectral projections E L1 (A) of a self-adjoint operator A, see [RSIj. For the operator of interest above, such a projection can be constructed explicitly as follows. Recall (Section 14.5) that the operator Hf

=

J

w(k)a*(k)a(k)d 3 k

can be represented as the direct sum 00

00

on where Hf,o is the operator of multiplication by 0 on Fo := C, and for n

~

1,

n

Hf,n

=

Lw(ki ) i=l

are multiplication operators on Fn := ®~ L2(R.3 ). Then

where for n ~ 1, EL1(Hf,n) is the operator of multiplication by the function XL1(L~=l w(k i )), where XL1 is the characteristic function of the set L\ c R., acting on F n , and EL1(Hf,o) is the operator of multiplication by XL1(O) on Fo·

206

15 Quantum Electrodynamics of Non-relativistic Particles

15.4.2 Projecting-out Procedure

Let H be a Hilbert space, and P a projection on H (i.e. an operator satisfying p 2 = P). Let A be an operator acting on H. Then PAP is an operator acting on Ran(P): i.e. PAP: H ----t Ran(P) and consequently PAP: Ran(P) ----t Ran(P). The simplest example is when P is a rank-one projection: P = I¢)(¢I for some ¢ E H, II¢II = 1. Here we have used Dirac's bra-ket notation

I¢)(¢I : V;

f--4

¢(¢, V;).

Then PAP = (¢, A¢)P acts on Ran(P) = {z¢ I Z E C} as multiplication by the complex number (¢, A¢). This example can be easily generalized to a finite-rank projection, say P = I I¢i)(¢il where {¢j} is an orthonormal set in H. In this case the number (¢, A¢) is replaced by the matrix ((¢i, A¢j) )ij. For another typical example, take H = L2(JR) and take P to be multiplication of the Fourier transform by the characteristic function, XI, of an interval I c lR:

2::

where j(k) = (21f)-1/2 J~oo e- ikx f(x)dx is the Fourier transform of f. P is a projection since XJ = XI. For a given operator A on H, the operator PAP (assuming it is well-defined) maps functions whose Fourier transforms are supported on I into functions of the same type. Of interest to us is the following situation: the projection P acts on a space H = HI ® H2 as P = PI ® 1, where PI is a rank-one projection on HI, say PI = 1¢1)(¢II, with ¢I E HI· If A is an operator on H, then the operator PAP is of the form PAP = 1®A2 where A2 is an operator on H2 which can formally be written as A2 (¢I, A¢lhil·

16 Supplement: Renormalization Group

In this chapter we describe an operator version of the renormalization group method, due to [BFS1]-[BFS4] (see also [GaW, Weg, KM]). We demonstrate how this method works by applying it to the problem of radiation described in Chapter 15. In particular, we continue our study of the Hamiltonian H(E) which describes quantum particles coupled to the quantized EM field. We outline a proof of part (i) of Theorem 15.2, which states the existence of the ground state of the operator H(E) for sufficiently small 14 The problems of instability of the excited states and existence of the resonances - statements (ii) and (iii) of Theorem 15.2 - can be treated in the same way. Since we are dealing only with the ground state, we do not need the analyticity part of Condition (A) of Chapter 15. Moreover, to simplify the exposition, we strengthen Conditions (A) and (B) in other respects, and suppose - instead of Conditions (A) and (B) - that the following condition holds:

(A') The coupling function g(k) satisfies the estimate Ig(k)1 s:; CW(k)-V for some v

(16.1)

< 1/2.

Notice that estimate (16.1) with v = 1/2 suffices in order to treat the original Hamiltonian (15.2) (recall that the coupling function g(k) in (15.3) is given as g(k) = X(k)/(27r)3J2w(k)). On the other hand, if Condition (B) is satisfied, then the Hamiltonian H (E) can be transformed so that the new Hamiltonian has a coupling function satisfying (16.1) with v = -1/2 (the Pauli-Fierz transformation; see [BFS3]). In what follows, we will always assume Conditions (15.4), (15.5), and (16.1). We refer the reader interested in the analysis of the v = 1/2 case (without condition (B)) to [BFS3, BFS4].

16.1 The Decimation Map In this section we realize the first step in our analysis of the Hamiltonian H(E). Our goal is to pass from the family H(E) - z· 1 to a family HO(E, z)

S. J. Gustafson et al., Mathematical Concepts of Quantum Mechanics © Springer-Verlag Berlin Heidelberg 2003

208

16 Supplement: Renormalization Group

of operators which act on the space RanEp(Hf) C F, and whose spectra near 0 are of the same nature as the spectrum of H (c) near z. The family Ho(c, z) will turn out to be more accessible to spectral analysis. Passing from H(c) - z . 1 to Ho(c, z) will be referred to as elimination of the particle and high photon energy (actually photon energy ~ p) degrees of freedom. We begin by studying maps between sets of operators which are isospectml in the sense specified below. We shall apply the abstract theory with the operators H = H(c) - z· 1 and P = Ep(Hf). Let P and P be orthogonal projections (i.e. P,P are self-adjoint, p 2 = P, and p2 = P) on a separable Hilbert space X, satisfying P + P = 1. Denote by Cp the set of all closed operators, H, on X whose domains have dense (in Ran P) intersections with Ran P and which satisfy

IIRpl1 < 00, Here Hp

IIPHRpl1 <

00

and

IIRpHPl1 <

(16.2)

00 .

= PHP r RanP, etc. and Rp = PHi/Po We define the map Dp : C p

closed operators on RanP

----t {

}

by Dp(H)

=

P(H - HRpH)P

rRanP

.

(16.3)

We call D p the decimation map. The following (bounded) operators play an important role in our analysis: Q = Q(H) := P - RpH P

(16.4)

and Q#

=

Q#(H)

:=

P - PHRp

(16.5)

We have Q#(H) = Q(H*)*, but we do not use this property. The operators P, Q, and Q#, have the following properties: NullQ n NullH' and

= {O}

HQ = PH'

and and

NullP n NullH Q#H = H'p,

= {O}

(16.6) (16.7)

where H' = Dp(H). These relations are proved below. The main result of this section is the following Theorem 16.1 Assume (16.2) hold, i.e. H is in the domain of the map D p Then the operators Hand Dp(H) are isospectral at 0, in the sense that

(a) 0 E C5(H) {o}O E C5(Dp(H)), and (b) H'Ij; = 0 {o} Dp(H)¢ = 0 where 'Ij; and ¢ are related by ¢ = P'Ij; and 'Ij; = Q¢. Proof. We begin with a general statement.

.

16.1 The Decimation Map

209

Proposition 16.2 Assume conditions (16.2) are satisfied. Then (16.6)-(16.7) imply that 0 E a(H) =} 0 E IJ(H') (the part of property (a) which is crucial for us). Moreover, we have N ullP n N ullH = {O}.

Proof. Let 0 obtain

E

p(H'). Then we can solve the equation H' P = Q# H for P to P = H,-lQ#H.

The equation P

+P =

1 and the definition H p

(16.8)

= PH Pimply (16.9)

Substituting expression (16.8) for P into the r.h.s., we find

P = PHj,l(P _ PHPH,-lQ#)H . Adding this to Equation (16.8) multiplied from the left by P, and using P + P = 1, yields 1

= [pHj,l P - PHj,l PHPH,-lQ# + PH'-lQ#] H.

Since by our conditions P Hj, 1 PH P is bounded, the expression in the square brackets represents a bounded operator. Hence H has a bounded inverse. So o E p(H). This proves the first statement. The second statement follows from the relation 1

= QP+RpH,

(16.10)

which, in turn, is implied by Equation (16.9) and the relation P + P = l. Indeed, applying (16.10) to a vector ¢ E NullP n NullH, we obtain ¢ = QP¢+RpH¢=O. D Next we prove relations (16.6)-(16.7). The second relation in (16.6) is shown in Proposition 16.2, while the first one follows from the inequality

In the first equality, we used the fact that the projections P and Pare orthogonal. Now we prove relations (16.7). Using the definition of Q(H), we transform

HQ = HP - HPHj,lPHP = PHP+PHP- PHPHj,lPHP-PHPHj,lPHP =

PHP- PHPHj,lPHP

= PDp(H). Next, we have

(16.11)

210

16 Supplement: Renormalization Group

Q#H

=

PH - PHPHi/PH

= PH P + PH P - PH P Hpl PH P - PH P Hpl PH P

= PHP- PHPHplPHP = Dp(H)P. This completes the proof of (16.6)-(16.7). Now we proceed directly to the proof of Theorem 16.1. Statement (b) follows from relations (16.6)-(16.7). Proposition 16.2 implies that 0 E p(H) if 0 E p(H'), where H' := Dp(H), which is half of statement (a). Conversely, suppose 0 E p(H). The fact that 0 E p(H') follows from the relation (16.12) which we set out to prove now. We have by definition

H'PH-lp

= PHPH-lp- PHPH~lPHPH-Ip P = PH(l- P)H-lp - PHPHp I PH(l- P)H-lp =P.

Similarly one shows that P H- I PH' = P. Hence H' has the bounded inverse PH-Ip. So we have shown that 0 E p(H) {o}o E p(Dp(H)), which is equivalent to 0 E (J(H) {o}o E (J(Dp(H)). D Thus the decimation (or Feshbach) map, Dp, maps operators on X into operators on RanP in such a way that Hand Dp(H) are isospectral at O. To understand the spectrum of an operator H near a point zo, we apply Dp to the operators H - zl, with z near zoo Finally, we prove Proposition 16.3 (semigroup property) Assume the projections PI and P2 commute. Then Dpl 0 Dp2 = DplP2

Proof. Assume for simplicity that Hand Dp2(H) are invertible. Then the statement follows by applying equation (16.12) twice. D

16.2 Relative Bounds Before proceeding with the first application of the decimation map constructed above, we prove a key relative bound on the quantized vector potential A(x) used in this application. In this section II . IIF stands for the norm in the Fock space F.

16.3 Elimination of Particle and High Photon Energy Degrees of Freedom

211

Theorem 16.4 We have

IIA(x)v;IIF :::; 2 ( / I~

2) 1/2

( ) 1/2 IIHy2V;IIF + /191 2 11v;IIF.

(16.13)

A proof of this theorem follows readily from Lemma 16.5 below. Lemma 16.5 (relative bounds on a# )

(16.14) and (16.15) Proof. (Drop the subscript F.) By the Schwarz inequality

Ila(f)v;11 :::; /lflllaV;11 :::; (/ 1~2) 1/2 (/ w 1 aV;11 2) 1/2 Since

the first inequality follows. The second inequality follows from the first and the relation

D

16.3 Elimination of Particle and High Photon Energy Degrees of Freedom This is the first application of the decimation map. Our task is to map (particles + fields) into fields. We consider only the ground state. Excited states are treated similarly once H(s) is suitably prepared. Since we are looking at a vicinity of the ground state energy of H(s), the degrees of freedom connected to the excited particle states and to high photon energies should not be essential. So we eliminate them isospectrally using the decimation map. Recall that Ep(Hf) = EC-oo,p](Hf) denotes the spectral projection for art ® Ep(Hf), where, recall, the operator Hf onto energies:::; p. Let Pp =

pg

212

16 Supplement: Renormalization Group

npart LO

I proJectlOn . . onto t he groun d state "i,part =P >pr;art = h ort ogona '1-'0 0 f H part,

(16.16) art = l1jJgart)(1jJgart I in the Dirac notation (we have assumed tacitly i.e. that the operator Hpart has a unique ground state 1jJgart). We want to apply the map Dpp to the operators H(e) - z ·1 for z in a neighbourhood of the ground state energy Eo. Recall that the decimation map Dp is defined on operators H, s.t. D(H)n RanP is dense in RanP, provided 0 E p(Hp) and

pg

IIPHRpl1 <

and

00

IIRpHPl1 <

00 ,

where Rp = P(Hp)-l P. In what follows Eo and El are the ground state energy and the first excited state of Hparh and LJ.E = El - Eo. Let

set

[2 p : = {z Eel Rez::::: Eo

1

+ 4' p},

(16.17)

and let e < < P < 1. Theorem 16.6 Let lei « p. Then D~O) := Dpp is defined on operators of the form H(e) - z . 1, where z E [2p, and is isospectral (in the sense of Theorem 16.1).

Proof. The isospectrality follows from the first part of the theorem and Theorem 16.1. The first part of the theorem follows from the definition of the domain of the map Dp given above and Proposition 16.7 below. 0 Proposition 16.7 Assume

z

E

[2p,

lei «

p. Then (i) [2p C p(Hp)E)) and (ii) for (16.18)

Here, as usual, Pp := 1 - Pp.

Proof. In order to simplify the exposition somewhat in a couple of places below, we use also the condition Ig(k)1 2 d 3 k < 00. However, using more careful estimates this condition can be avoided. To simplify notation we assume z is real. Let p = -i'\1 x . Since div A(x) = 0, we obtain, by expanding (p - eA(x))2,

J

H(e) = H(O)

+ e' I(e),

where I(e) is the interaction energy

I(e) = -p. A(x)

1

+ 2eIA(x)12.

If we denote P = Pp, P = Pp and I = I(e) (so that Ip := PIP), then

16.3 Elimination of Particle and High Photon Energy Degrees of Freedom

Hp(c:) = Hp(O)

213

+ elp.

Since H p(O) - z > 0, we can define the invertible, positive operator (Hp(O)z) 1/2. We have the following identity

(Hp(C:) - z) = (Hp(O) - z)I/2[1 where

K

+ c:K](Hp(O) -

z)I/2,

(16.19)

= (Hp(O) - z)-1/2 Jp(Hp(O) _ Z)-1/2.

Using the definition of J, we find

IIKII ::::: IIp(Hp(O) - Z)-1/2111IA(x)P(Hp(0) - Z)-1/211 + ~11(Hp(O) - z)-IPA(x)IIIIA(x)P(Hp(O) - Z)-1/211 . The relative bound on A(x) proven above implies that

IIA(x)P(Hp(O) - z)-1/211 ::::: 2

(1 1~2) 1/2 1IH}/2(Hp (0) _ z)-1/2PII

+(

I

1g12 )

1/2

II(Hp(O) - z)-1/2PII . (16.20)

Our goal now is to estimate the first norm on the r.h.s. of (16.20). Recall rt 181 Ep(Hf) and therefore that P =

pr

(16.21 ) -part

part-

where Po = 1- Po v E D(Hp(O)),

and Ep(Hf) = 1- Ep(Hf). Hence we have for any

((Hp(O)-z)P)v = ((Hp(O)-z) (pg art l8l1f ))v+((Hp(O)-z)(lpartI8lEp(Hf)))v . Now using

H part ppart 0

> E 1ppart _ 0

,

Hpart ::::: Eo

and we estimate

(Hp(O) - z) Pv ::::: (El - z)(pgart 181 If)v + (Hf(pgart 181 If)

+ (Eo

- z+

~P)(lpart 181 Ep(Hf ))v

+ ~Hf(lpart 181 Ep(Hf)))v

.

Taking the smallest constants on the r.h.s. and remembering (16.21), we obtain

214

16 Supplement: Renormalization Group

(Hp(O) - z)Pv 2: (0 where 0 := min(E1 - z, Eo

+ ~p - z). u

=

1

+ 2Hf)Pv

,

Now let

(Hp(O) - z)1/2v .

Then the estimate above can be rewritten as

which gives, in particular, that (16.22) and (16.23) where we used that 0 2: ~p for zED n R These estimates, together with (16.20) imply that

IIA(x)P(Hp(O) - z)-1/211 :::; 4 (

JI~

2) 1/2

+2

(

J

Igl2

) 1/2

/ p1/2. (16.24)

Since IJ(Hp(O)) = [E,oo), where, E = min(E1' Eo + p), we have that IIp(Hp(O) - z)-1/211 :::; Cdist(z, [E, 00))-1/2. Thus IIKII :::; Clz - EI- 1 for Re z < E1 and therefore IIKII :::; Cp-1 , (16.25) if z E Dp. Pick now lei:::; 0/2C. Then IleK11 :::; 1/2 and therefore the right hand side of (16.19) is invertible. Hence

{z I Rez:::; Eo

1

+ 4P} c

provided lei « L1E. Moreover, since PH(e)P(Hp(e) - z)-1P estimate

p(Hp(e)),

=

IIPH(e)P(Hp(e) - z)-1 PII :::; (cliP. pll

ePIP(Hp(e) - z)-1P, we 2

+ e2

IIPA(x)ll)

x IIA(x)(Hp(e) - z)-1PII . Now equations (16.19), (16.23) and (16.24) imply

IIA(x)(Hp(e) - z)-1 PII :::; Clz - EI- 1 . Next, since p is bounded relative to H part , we have

16.3 Elimination of Particle and High Photon Energy Degrees of Freedom

215

IIP'pll:S C. Finally, bound (16.13) implies that

IIPA(x)11 :S C(pl/2 + 1)

.

Collecting the last four estimates and using the fact that Re z :S Eo arrive at (16.18). 0

+ p,

we

Remark 16.8 In the case of a non-self-adjoint operator H, to show invertibility of Hp(O) - z, one uses an auxiliary positive, invertible (Schrodinger) operatorTon RanF such that T- 1 / 2IpT- 1 / 2 is bounded, and T-l/2(Hp(0)z)T- 1 / 2 is invertible. This again would imply invertibility of the operator

Hp(c:) - z. Since we are interested in the part of the spectrum which lies in the set lR I Rez:S Eo + ~p}, we can study D~O)(H(c:) - z) instead of H(c:). We choose Po such that c: « Po < min(1, LlE). Observe that since the projection pgart has rank 1, the operator D~~)(H(c:) - z) is of the form

{z

E

where the operator Ho(c:,z), defined by this relation, acts on RanEpo(Hf). This operator can be explicitly written as

Ho(C:, z) == (?jJgart , D~~) (H(c:) - z . l)?jJgart)'H p art

(16.26)

Problem 16.9 Show this (hint: use (16.16)).

Thus we have passed from the operator H (c:) acting on Ji part 181 F to the operator Ho(C:,z) acting on the much smaller space RanEpo(Hf) C F, which is isospectral (in the sense of Theorem 16.1) to H(c:) - z . 1 at 0, provided z is in the set

[2po

:=

{z

Eel

Rez :S Eo

1

+ 4Po}.

We eliminated, in an isospectral way, the degrees of freedom corresponding to the particle and to the photon energies;::: Po; i.e., we projected out the part Ran (1 - pgart) of the particle space Ji part , and the part Ran( 1 - E Po (Hf)) of the Fock space F. The parameter Po is called the photon energy scale. Though the operator Ho(C:, z) looks rather complicated, we will show in Section 16.5 that the complicated part of it gives a very small contribution. Namely, we will show that the operator Ho(C:, z) is of the simple form

Ho(C:, z) = LloE + Hf

+

small terms ,

where LloE is a computable energy shift of the order O(c: 2 ). If we ignore the small terms, then in (-00, Eo + ~po)

216

16 Supplement: Renormalization Group

spec H (c:)

R:::

spec H f

+ Eo + LJ.oE.

Now note that the spectrum of the operator Hf+Eo+LJ.oE contains the eigenvalue Eo + LJ.oE, with eigenfunction il, and the continuum [Eo + LJ.oE, 00), with generalized eigenfunctions 1j;gart ® II a * (k j ) Q. Hence Q P po (H (c:) - z) il gives an approximate eigenfunction of H(c:) with approximate eigenvalue, Eo + LJ.oE and similarly for the continuum. In fact, we will find energies E(n) = Eo+O(c: 2 ) and numbers w(n) = 1+0(c: 2 ), such that for p = O(c:) «1 and any n 21,

H(c:) is isospectral to E(n)

+ w(n) Hf + O(c: 2 pn)

in the disk D(E(n), pn). This is called infrared asymptotic freedom.

16.4 Generalized Normal Form of Operators on Fock Space Though the operator Ho(C:, z) looks much more complicated than H(c:), in fact the complicated terms can be easily estimated and shown to give a relatively small contribution. These estimates are based on a presentation of the operator Ho(C:, z) in the generalized normal form. The operator Ho(C:, z) we are interested in has no Wick representation. However, it can be expressed in a closely related form which serves the same purpose as the Wick representation, but covers a much wider class of operators. We call this form, the generalized normal form. To describe this representation, we first introduce some abbreviations: n

(a#)n =

IT a#(k

j )

and

dnk = d3k 1 ... d3k n ,

j=1

where a# stands either for a* or for a and is the same throughout the product (we drop the parentheses on the left hand side in the case a# = a). The shorthand (16.27) j(a*rhrs(Hf)aS or sometimes j(a*rhrsaS stands for the operator

Here hrs(fl,k), k = (k 1 ... kr+s ), are measurable functions on [0,1] x ]R3(r+s) , called coupling functions. Their behaviour in k will be specified later. We say that an operator H is in generalized normal form if it can be written as

16.4 Generalized Normal Form of Operators on Fock Space

217

H = Ep(Hf) L j(a*Yh1's(Hf)aSEp(Hf) . 1',s2':O Operators of the form (16.27) will be called (rs)-monomials. So far these definitions are purely formal since we do not describe a class of coupling functions h1's for which the corresponding operators are, say, densely defined or bounded. This will be done later. For now we mention only that though H f can be expressed in the standard Wick form

Hf = j wa*a, the corresponding coupling function, W(k1)I5(k1 - k2 ), is more singular than we allow. But even if this coupling function were smooth, finding the Wick form of operators like Ep(Hf), or the h1's which in our case are composed of operators like E p (Hf), is not an easy matter. In what follows we manipulate the operators a*(k), a(k) and Hf as if they were independent. We use only the commutation relations between these operators: [a(k), Hf] = w(k)a(k) , etc. The point here is that though we do not take expectations of operators in the vacuum state (or equivalently, do not sandwich operators by projections onto the vacuum), we do sandwich them by projections, Ep(Hf) , onto the low lying spectral states of the free Hamiltonian, H f. As a result we show that terms containing at least one creation or annihilation operator give a relatively small contribution, and the greater the number of creation and annihilation operators, the smaller the contribution. To bring the operators we deal with into generalized normal form, we use the "pull-through" formulae (see [GJ, Fr2])

and

a(k)f(Hf) = f(Hf +w(k))a(k)

(16.28)

f(Hf)a*(k) = a*(k)f(Hf +w(k)),

(16.29)

valid for any piecewise continuous, bounded function,

f, on R

Problem 16.10 Using the commutation relations above, prove relations (16.28)(16.29) for f(H) = (Hf - z)-l, z E C rv JR+.

Using the Stone-Weierstrass theorem, one can extend (16.28)- (16.29) from functions of the form f(>-..) = (>-.. - z)-l, z E C\JR+, to the class of functions mentioned above. Alternatively, (16.28)- (16.29) follow from the relation N

f(Hf)

N

N

II a*(kj)D = f(Lw(k II a*(kj)D.

j=1

i ))

i=1

j=1

Problem 16.11 Prove this last relation, and derive (16.28)- (16.29) from it.

16 Supplement: Renormalization Group

218

16.5 The Hamiltonian Ho(e, z) Our next goal is to show that the operator Ho(c:, z), defined by the relation (16.26), for z E Dpo, can be represented in the generalized normal form

Ho(C:,z) = Ep(Hf)

L

r,s20

(16.30)

j(a*rho,rs(Hf)aSEp(Hf),

and to estimate its coupling functions ho,rs(J-l,!:£). Our results are summarized in the following Theorem 16.12 Let c: « Po :::; i1E (Po is the scale parameter entering the definition of Ho(c:, z)). The operator Ho(c:, z) has a generalized normal form, Equation (16.30), with coupling functions ho,rs which are analytic in zED, and satisfy the estimates

r+s lo;ho,rs(J-l,!:£) I :::; (const)8o

II (const· c:. POl. w(kj)-V) ,

(16.31 )

j=l

where the product is absent in the case 'r = S J-l E [0,1] for n = 0 and J-l E [0,1) for n = 1, and

=

O. Here n

0,1, with

IOfLho,oo(J-l) -11:::; (const)c: 2 jP6, for J-l

E

(16.32)

[0,1).

The proof of this theorem is simple but lengthy. It can be found in the Appendix to [BFS4]. Here we give just an idea of the proof. Using the definition of the decimation map V~~), we write the operator Ho(c:, z) in the form

Ho(c:, z) = Epo(Hf )[Hf + cI - c: 2 W]Epo(Hf) ,

where with the same definitions as in the proof of Proposition 16.7, W

= ('l/Jgart , IRpo(z)I'l/Jgart)?-lpa't ,

with Rp(z) = Pp(Hpp (c:) -Z)-IPp. Now using estimates obtained in the proof of Proposition 16.7, we expand the resolvent (Hp po (c:) - z)-l in a Neumann series to obtain 00

W = "'(ol,part IP(- c: R O,P-IP)n- 2 R O,P_Iol,part) ~ 'PO' 'Po ,

(16.33)

n=2

where we have used the notation P = PPQ ' P = Ppo , and Ro,p = (H(O)-

z)-lp.

Next we bring each term on the right hand side of (16.33) to generalized normal form. To do this we observe that the terms on the right hand side

16.5 The Hamiltonian Ho(c, z)

219

of (16.33) consist of sums of products of five operators: R o p, P, p, a(k) and a*(k). We do not touch the operators Ro,p, P and p, whil~ we move the operators a(k) to the extreme right and the operators a*(k) to the extreme left. In doing this we use the following rules: 1. a(k) is pulled through a*(k) according to the relation

a(k)a*(k') = a*(k')a(k)

+ o(k - k')

2. a(k) and a*(k) are pulled through Ro,p according to the relations a(k)RO,PRo,pa*(k)

=

RW(~) a(k) O,P

= a*(k)R~,~) ,

where R~,~) = Ro,p IHr-;Hf+w(k) , and similarly for pulling a(k) and a*(k) through other functions of Hf' such as P. The procedure above brings the operator HO(E, z) to generalized normal form (16.30), at least formally. It remains to estimate the coupling functions ho,rs entering (16.30). A direct estimate produces large combinatorial factors which we must avoid. To this end, one uses a special technique which amounts to a partial resummation of the series, in order to take advantage of cancellations. This can be found in the Appendix to [BFS4]. D To take advantage of Theorem 16.12 in order to estimate the operator HO(E, z), we need the following Proposition 16.13 (Key bound) Let H rs be an rs-monomial of the form (16.27). Then we have the following bound:

1/2

r

L

j=l

w(kj)~p,

r+s

L

j=r+l

w(kj)~p

sUPJt Ihrs(/-l,k.W dr+sk r+s TI w(kj )

(16.34)

j=l

Proof Denote Ep := Ep(Hf). Using the form (16.27) of H rs , taking the norm under the integral sign, and using the norm inequality for the product of operators, as well as IIA*II = IIAII, we have

(16.35) Here Ilhrsll is the operator norm of hrs(Hf,k.)· So Ilhrsll = sUPJt Ihrs(/-l,k.) I· Let f be a positive, continuous function on ]R3n. We will prove the estimate

220

16 Supplement: Renormalization Group

(16.36)

where Wj = w(kj ), which will imply (16.34). First, we prove the estimate for n = 1. By the pull-through formula (16.28), we have

+ w(k))a(k)Ep.

a(k)Ep = Ep(Hf

Since Hf 2': 0, this implies

Proceeding as in the proof of the relative bound on a(k) (see (16.14)), we obtain

J

flla(k)Epll ::::;

l~p flla(k)Epll

: :; J -

f 2) 1/2

(

w~p

: : ; (pi

w

w~p

IIH}/2 Epll

(16.37)

P)1/2 w

which implies (16.36) for n = 1. Now we prove (16.36) for arbitrary n 2': 1. First of all, applying the pull-through formula (16.28) n times, we find n

n

n

1

1

1

IT a(kj)Ep(Hf) = Ep(Hf + LW(kj )) IT a(kj)Ep. Hence (16.38) Secondly, applying the pull-through formula (16.28) n times again, we obtain

=

(Hf

n

n

1

1

+ Lw(kj ))-1/2IT a(kj )H}/2Ep.

This formula and inequalities (16.37) and (16.38) give

16.6 A Banach Space of Operators

221

Proceeding in the same fashion we arrive at (16.36). Applying the bound (16.36) with n = r to the integral filar EpllllhrslWk, we bound the r.h.s. of (16.35) by

Applying bound (16.36) with n = s to the outer integral gives (16.34). 0 Applying Proposition 16.13 to the operator

Ho(c,z) =

L

EpJHf)Ho,rs(c,z)Epo(Hf),

k+s2:0

and using the estimates (16.31) and

we obtain for r

+ s 2':

I,

These estimates, in turn, imply that the operator Ho(c, z) is of the form

3-2V)

Ho(c,z)=Eo·l+Hf+O (c 2 -

v

,

where Eo = ho,oo(O). Here we have used the fact that the worst terms in (16.30) are 1

ho,oo(Hf) - ho,oo(O) - Hf

=

J(h~,oo(SHf) -l)dsHf o

and

16.6 A Banach Space of Operators Instead of looking at the single Hamiltonian, Ho(c, z), we now consider an entire space of Hamiltonians of which Ho(c, z) is just one point (or a complex curve depending on the parameter z). Our aim is to establish an isospectral flow on this space which takes Ho(c, z) to a simple operator.

222

16 Supplement: Renormalization Group From now on we use the shorthand Ep := Ep(Hf) and Ep = Ep(Hf) :=

1- Ep(Hf).

Consider a positive function J on

]R3,

satisfying

Jp

~ <00.

w:'Ol

Define the Banach space, l3 J, consisting of formal expressions of the form E 1HE 1, with H = L H rs , where r+s~O

As before, the right hand side here is shorthand for the expression

JII r

j=l

r+s a*(kj)hrs(Hf, k1 ... kr+s )

II

a(kj)dr+sk.

j=r+1

The coupling functions hrs(/-L,k) (where, recall, k = (k 1, ... , kr+s)) are assumed to satisfy the estimates

Ilhrsll J

:=

sup[(J-1),ZI(r+s)lh rs ll <

(16.39)

00 .

,",k

Here we have used the notation J@n

= 1 for n = 0, and for n

~

1,

n

J@n

= II J(k j )

.

(16.40)

j=l

(16.41 ) Proposition 16.14 Let H rs be as above. Then

(16.42) where

)"(J,l):=(l

P)1/2

w:'Ol W

Thus E1H E1 = Lr,s E 1Hrs E 1 converges in norm, provided

1P -

w:'Ol W

<

1.

16.7 Rescaling

223

Proof. The proposition follows from Proposition 16.13, and from the relations Ilhrsll < IlhrsllJ . J0(r+s) (where IIhrsll is the operator norm of hrs(Hf,k)) and

J

(~2) On :::; (n!)-l A(J, l)n

.

n

L 1

Wj::;l

D Next we state without proof that the map

is one-to-one (see [BeFS, Thm. III. 3]) . Here the {h rs } satisfy (16.39). Hence the normed space B J is indeed a Banach space. In what follows, we do not display the cut-off functions El in formulae, writing H for operators El H E 1 . The latter are bounded operators on F as follows from Proposition 16.14. Moreover, we take J to be of the form where

~

>0

and

v

1 2

<-.

16.7 Rescaling We want to map our operators acting on the space EpF (where, recall, Ep = Ep(Hf)), into operators acting on a fixed, p-independent space. To this end, we rescale them as follows. Define a unitary group of operators, Ue, by

Ue

II a*(Ji)D = II a*(ueJi)D

(16.43)

where Ue is the rescaling transformation acting on L2(JR3), given by (16.44) In particular, UeD = D. Moreover, we have the relations (16.45) and (16.46) The last equation implies (16.47) Now we define the rescaling transformation on operators as

224

16 Supplement: Renormalization Group

e = -In p.

(16.48)

Note Sp(Hf) = Hf. By (16.47), if an operator H acts on RanEp, then the operator Sp(H) acts on RanE l , a p-independent space. Observe that we can define Sp directly as Sp = is(p), where the oneparameter group S(p) is defined by the properties 1. S(p)(A· B) = S(p)(A) . S(p)(B), 2. S(p)a#(f) = a#(u-lnpJ).

We can now discuss the notion of scaling dimension. Rescaling H rs J(a*thrsa s gives (16.49) where (16.50) If H rs E B J , J So

= ~w-v,

then h rs behaves for small Ikjl's like I1;~~ w(kj)-v.

Sp (H) rs

rv

P(3/2-v)(r+s)-lHrs

If r + s 2': 1, then IISp(Hrs)IIJ ---+ 0 as p terminology (for v < ~) is as follows:

r In the r

+ s 2': 1

+ s = 0 case, o. If hoo(p)

hoo(p) at p =

+-4

---+

.

0 for v < 1/2. The physics

irrelevant terms.

we have to specify the behaviour of the function pa for p rv 0, with a 2': 0, then

rv

Sp(Hoo)

rv

pa-l Hoo.

So as p ---+ 0, Sp(Hoo) blows up if a < 1, stays fixed if a = 1, and vanishes if a > 1. In physics terminology, Hoo is

relevant marginal irrelevant

if

a <1

if

a =1 a > 1.

if

We record also the action of the group Sp on the Banach space BJ (remember that J = ~w-V). The following equation follows from Equations (16.49)(16.50): (16.51 ) Applying these equalities to operators of the form H =

L

L

H rs and Hl =

r+s~O

H rs , we find

r+s~l

(16.52) for v :::;

1

2.

(16.53)

16.8 The Renormalization Map

225

16.8 The Renorrnalization Map We define the renormalization map as a composition of a decimation map and a rescaling map. Let Dp := D Ep ' where Ep = Ep(Hf), and Dp is the decimation map which is defined in (16.3). We define the renormalization map by (16.54) Without specifying a domain of definition of the map R p , expression (16.54) is purely formal. To describe such a domain we note first that the map Rp is defined on operators of the form w· Hf, w E C, acting on RanEI(Hf). Indeed, one can explicitly compute (16.55) In other words, the operators

W·

Hf, w E C are fixed points of Rp.

Problem 16.15 Show (16.55).

L
is sufficiently small, p < 1, and a is Now, one can show that if such that a ::::: min(~, ~), then the-map Rp is defined on a neighbourhood Ua:={HEBJIIIH-w.HfIIJ:::::lwl·a for somew =j=O}

of the fixed point set Mfp := C· Hf = {wHf I w E C}.

The proof of this fact is similar to the proof of Theorem 16.12 which we outlined above: expand Rp(H) in the perturbation W := H - H oo , reduce the resulting series to generalized normal form, and estimate the obtained coupling functions. We omit it here, referring the interested reader to [BFS4J. We mention only that what makes things work is the estimate (16.34). We explain briefly the key mechanism. Let E p = E p (Hf). Consider an operator H restricted to the subspace Ran Ep. We want to pass to a lower scale p' < p, i.e. to the subspace RanEp' C RanEp. The slice p' ::::: Hf ::::: p which is eliminated shows up as an effective interaction. For this procedure to work, and to estimate the effective interaction, we have to invert the operator (16.56) where E p' = 1 - E p ' is the spectral projection onto the subspace on which H f :2: p' (cf. Proposition 16.7). The leading part of this operator is bounded below as 1-2 Ep,HooEp' :2: 2P'Ep" while the perturbation HI := H - Hoo is bounded as

226

16 Supplement: Renormalization Group IIEpEpfHIEpfEpl1 :::; CE' pl/2

(the worst case, r+s = 1, of estimate (16.34)). Here E = IIHIIIJ (see (16.41). Hence operator (16.56) is invertible if p' » E • pl/2. In fact, a little trick (cf. (16.19) and estimates after this equation) allows one to improve this estimate to

.;-;; »

E'

Jp

or

~»

E.

Thus, if the ratio of scales satisfies this condition, then we can pass from the scale p to the scale p'! Next, we list some elementary properties of the map Rp which follow readily from the definitions 1. Rp is isospectral in the sense that p. Rp(H) and Hare isospectral at 0

in the sense of Theorem 16.1,

2. Rp is a semigroup: Rp2 0 Rpl = R P2P1 and Rl = 1, 3. Rp(wHf + zl) = wHf + ~zl Vw, z E C. In particular, CHf is a complex line of fixed points of Rp: Rp(wHf) = wHf Vw E C, and C· 1 is (a part

of) the unstable manifold. Problem 16.16 Prove the statements above.

We want to understand the dynamics of Rp as p ----? 0 in a vicinity of the fixed point manifold M fp, and to connect this dynamics with spectral properties of operators of interest. We do this in the next two sections, beginning with a study of the linearized dynamics in the next section.

16.9 Linearized Flow As is usual in the study of nonlinear dynamics, in order to understand the dynamics of the map Rp near its fixed points w . H f' w E C, we consider its linearization (variational derivative) aRp (w . H f) at those fixed points. Recall that the variational derivative of the map Rp at a point Ho is defined as

aRp(Ho)~

:=

a

Oc Rp(Ho

+ E~)lc=o

for any ~ E B J . Thus aRp(Ho) is a linear operator on B J . We show in the Appendix (Section 16.11) that the map Rp is continuously differentiable (in fact, it is analytic). A straightforward computation gives (16.57) To show this relation, we observe that due to the definition of Rp and the equation

16.9 Linearized Flow

227

pSp(Ep) = E1

it suffices to show that the map V p is differentiable, with derivative (16.58) Now the definition of the map V p, the relation Ep(w . Hf + ~)Ep = Ep~Ep (since Ep and Hf commute, and EpEp = 0), and arguments given in either the proof of Proposition 16.7 or Theorem 16.23 imply that Vp(W' Hf

+ e~) = Ep(w . Hf + e~ -

e2~Ep(wHf1)Ep~

+ O(e3 ))Ep

which implies (16.58), which in turn yields (16.57). We have shown that the operator 8Rp(w . Hf) is independent of wE C, and can be identified with the rescaling map Sp. Denote this operator as A := 8Rp(w . Hf) = Sp. The idea is to consider, near a fixed point W . Hf, the linearized dynamics (the dynamics of the linearized map A) instead of the full dynamics (the dynamics of the nonlinear map Rp). The linearized dynamics is characterized by the unstable, central and stable subspaces, Vu , Vc and v;, defined as follows: Vu , Vc and Vs span the entire Banach space BJ:

(16.59) they overlap only at {O}:

Vu n Vc = Vu n Vs = Vc n Vs = {O},

(16.60)

and they verify (16.61 ) and (16.62) where As, Au and Ac denote the restrictions of the operator A onto the subspaces Vs, Vu , and Vc, respectively (if V is not an invariant subspace of A then the restriction A I V of A onto V is defined on V by the expression PAP where P is the orthogonal projection onto V). Thus the operator A is a contraction on Vs, an expansion on Vu , and the identity on Vc. We introduce the subspace

T = {T(Hf)

IT: [0,1]

-+

C is C 1 with T(O) = T'(O) = O}.

Lemma 16.17 The stable, central and unstable subspaces, of the linearized renormalization group map at M fp are

v;, , Vc

(16.63) and Vu , (16.64)

and (16.65)

228

16 Supplement: Renormalization Group

Proof. The subspaces defined in (16.64)- (16.65) intersect only at {O} and span B J . The definition of the rescaling map Sp and equation (16.53) imply that Sp(v)

= p-1v

Sp(v)

=v

IISp(v)IIJ :::;

I::/v E Vu , I::/v E

v" ,

pl/2- v

l vI J

I::/v E

Vs ,

where Vu , v" , and Vs are defined in in (16.64)-(16.65). Since A = Sp, these relations imply (remember 1/ < 1/2) (16.62) and

where Au = A r Vu and As = A r Vs. We have used here that 1 2: ~ - 1/ for 1/ 2: -~. Hence the subs paces defined in (16.64)- (16.65) verify all of the conditions (16.59)- (16.62), and are therefore the unstable, central, and stable subspaces of A. 0 The results we have obtained allow for a complete analysis of the flow Rp near the fixed point manifold Mfp- This is the subject of the next section. Remark 16.18 The actual expressions for the unstable, central and stable subspaces Vs, v" and Vu depend on the parameter 1/ entering the definition of the Banach space BJ, J = ~w-v. We consider other ranges ofthis parameter: _ 1/ -

1

2'

Vu = C . 1 , Vc = C . Hf and

+{

L

Vs = { HrslHrs r+s2:2 ~
Vu =C·1+{

L

L

r+s=l E

H rs I Hrs E BJ }

BJ } +7

HrsIHrsEBJ},

r+s=l

and

Vs = {

L

HrslHrs

E

BJ } +7

r+s2:2

and so forth. When proceeding to 1/ 2: lone has to remember that the original operator H(c:) is not bounded from below if 9 behaves like w- v with 1/ 2: 1, as Ikl ---+ O.

16.9 Linearized Flow

229

Remark 16.19 Of course, the decomposition of the Banach space B J into the subspaces Vu , Vc and Vs is related to the spectral decomposition of the map A = Sp. Indeed, consider a generalized normal monomial H rs with a coupling function h rs which is homogeneous of degree -a in each variable k, and homogeneous of degree /3 in J.1,. Then, due to (16.50), we have

S p (Hrs ) -- p(3 j 2-a)(r+s)+,6-1 H rs

,

i.e. H rs is an eigenvector of Sp with eigenvalue p(3j2-a)(r+s)+,6-1. For H rs E BJ with J = t,w- v the worst case scenario is a = v and /3 = 0 for r + s :::: 1. In the case r = s = 0, the coupling function hoo is independent of the kj's and can be written as (16.66) with E = hoo(O), w = h~o(O) and h 1 (J.1,) = hoo(J.1,) - hoo(O) - h~o(O)J.1,. The first term on the r.h.s is of the type (a = 0, /3 = 0), the second, of the type (a = 0,/3 = 1), while the third is ofthe type (a = 0,/3 = 2) (provided hoo(J.1,) is twice differentiable). Thus the spectra of the restrictions of the operator A onto the invariant subspaces Vu , Vc and Vs, contain the sets {p-l} ,

{pm-I},

m :::: 1

and

{p(3 j 2-a)n-l

I Re a ::; v} ,n :::: 1 , (16.67)

respectively. Moreover, one can show that (16.67) are the entire spectra of the restrictions of A onto Vu , Ve , and Vs, and consequently, the total spectrum of A on the space BJ is the union of these sets. Thus the decomposition BJ = Vu EB Vc EB Vs is, in fact, the decomposition of BJ onto the spectral subspaces of the operator A corresponding to the pieces (16.67) of its spectrum. Remark 16.20 The stable subspaces for fixed points w . Hf' w E
Fig. 16.1. Stable, central and unstable subs paces

Remark 16.21 The case v = ~ is more subtle. The main problem here is that the central manifold is larger than the manifold of fixed points. One has

230

16 Supplement: Renormalization Group

to isolate the smallest possible central subspace - the central subspace for l/ = ~ mentioned above is too large; a big piece of it could, in fact, be taken into the stable subspace. To do this one has to use a mOre refined Banach space instead of B J. We do not pursue this case any further here, but refer the interested reader to [BFS4].

16.10 Central-stable Manifold for RG and Spectra of Hamiltonians We have realized the decomposition BJ = Vu + Ve + Vs of the space BJ into the invariant subspaces, Vu , Vc and Vs, for the linear map A. The dynamics of this map has distinctly different behaviour on each of these subspaces: An is expanding on Vu and contracting on Vs while it leaves the points of Vc fixed. In this section we extend this decomposition to the nonlinear dynamics of Rp by decomposing (nonlinearly) the space BJ into the manifolds M u , Me and Ms in such a way that (i) the spaces Vu , Vc and Vs are tangent spaces to these manifolds at Mfp, (ii) each manifold is invariant under the map R p , and (iii) the map Rp is expanding on Mu and contracting on Ms while it leaves the points of Me fixed. Formally, a local stable manifold can be defined as a manifold invariant under Rp such that Rp(H)

---+

Mfp as p

---+

o}.

(16.68)

In fact, as the definition of the map Rp implies (see Statement 3 in Section 16.8 above) \Iv E Vu and \Iv

E

Vc

and therefore Mu :2 Vu

and

Me:2

Vc .

This is a special feature of the problem at hand. What is shown in [BFS4] is that Mu

= Vu and Me = M fp = Ve

(16.69)

and that the stable manifold, M s , has co dimension 1. More precisely, Ms is a graph (16.70) Ms = {H E Un of a function e : (Ve

+ Vs) n Un

---+

Vu such that

16.10 Central-stable Manifold for RG and Spectra of Hamiltonians e(w·Hf)=O

'VwEC.

Here we have introduced the notation HT

:=

f4;;-,---

231 (16.71)

H - (H)n (so that (HT)n

=

0).

C.1 Mu =C.l'l+C.1

~H-l';1

Fig. 16.2. RG flow. Moreover, the following facts are proved in [BFS4]: dist(H, Ms) ::; rp'n

=}

H

E

D(R;) 'Vk ::; n,

dist(R~(H), Mfp) ::; ao'n

+ r,

(16.72) (16.73)

and (control of the projection of R~(H) on the unstable subspace Vu ): (16.74) where p' = p - POr(1) and 0' = p!-v + or(l). These facts describe the dynamics of the map Rp near the stable manifold Ms. Note that formula (16.70) implies that (16.75) (the left hand side is just the distance between the operators Hand e( HT) 1 + HT EMs). This finishes our analysis of the dynamics of the map Rp. But how is the information we obtained used to understand spectral properties of operators acting on Ran E1(Hf)? We explain this briefly. In what follows we ignore the difference between p and p' = p(l- POr(l)) and 0 = p:;,-v and 0' = p!-v + or(1) in (16.72) and (16.73), and for small p we define the map Rp by Rp := R~l where nand Pl are such that p = and Pl is sufficiently close to (but smaller than) 1. Given an operator H E UOil we show how to find its ground state energy (and its ground state) and the nearby spectrum. To simplify notation we assume (H)n = 0, and denote H(z) := H - z . 1 (to put it differently, we absorb (H)n into z). We know by (16.72) and (16.75) that the operator Rp(H(z)) is well defined if

pr

Iz + e(H)1 < rp.

(16.76)

232

16 Supplement: Renormalization Group

Again by (16.73), we know that there are w that

Rp(H(z)) = ((z)l

Using (Rp(H(z)))n

= w(H) and ((z) = ((z, H) such

+ w· H f + O(a: 2 p!-V).

= ((z) and (H(z))n = -z, we derive from (16.74) that (16.77)

On the other hand, by the isospectrality of the map Rp we have

for z such that [z + e(H)[ ::::; rp and modulo O(a: 2 p2(!-V»). Here, recall, (T# stands for (T, (Tp and (Te. Now remember (16.77), and H(z) = H - z· 1, and let Ao = -e(H) (so that ((z) :::::; _p-l(Z + Ao)). So the last relation implies z E (T#(H) f-4 z E (T#(Ao + p. w· H f ), which can be rewritten as

(T#(H) = (T#(Ao1

+ pw . Hf)

in the set {z E lR [ [z + e(H)[ ::::; rp} and modulo O(a: 2p!-V). If we take p ----+ 0 in the latter relation we obtain

AO is an eigenvalue of H (strictly speaking, in order to pass from ((z) to z, one needs to control the derivative (' (z), but a discussion of this point goes beyond the level of precision in the argument above). Of course, Ao is the lowest eigenvalue of the operator H. Thus we proved the existence of the ground state for the operator H and found the excitation spectrum near the ground state energy. Using a technique from Section 16.1, we can also construct the eigenfunction of the operator H corresponding to the eigenvalue Ao. The procedure above can be applied to a nonlinear eigenvalue problem when the family H(z) is not linear in z. One just has to replace z in (16.76) and (16.77) by (H(z))n. Now we apply this procedure to find the ground state and ground state energy, and the nearby spectrum, for the non-relativistic quantum electrodynamics Hamiltonian H(c:). Theorem 16.6 and the definition of the operator

H(O)(c:,z) := Spo(Ho(c:,z)), (recall Ho(C:, z) := Rpo (H(c:) - z· 1)) imply that

z

E

(T#(H(c:))

f-4

0 E (T#(H(O)(c:,z))

as long as z is in the set

[2po

:=

{z

E

lR [ z::::; Eo

1

+ 4Po }.

(16.78)

16.10 Central-stable Manifold for RG and Spectra of Hamiltonians Next, Theorem 16.12 above implies that for

0:

=

233

cmax(c, lEo - zl),

ll(O)(c,z) E [Ta.

Thus for 10 sufficiently small the theory developed above applies to the operators 1l(0) (10, z). In particular, we have - 0 is the ground state energy of 1l(0) (10, Eo(c)) , where Eo(c) solves the equation (ll(O)(c,z))n = e(ll(O)(c,zf), - 0 E O"#(ll(O)(c,z))

+-+

z E 0"# (Eo(c)l +w(c)llf)

for some w(c) E lR, modulo O(c 2p l/2-v). This together with (16.78) shows that 0"# (ll(c)) = 0"# (Eo(c)l + w(c)llf) for z s.t. Iz - Eo (c) I ::; p modulo O(c 2 p!-V) and in particular, Eo(c) is the smallest eigenvalue of ll(c).

Thus Eo (c) is the ground state energy of the operator II (c). Using the second of the expressions in Theorem 16.1 (more precisely, a similar expression) and the fact that the zero eigenfunction of llf (i.e. the ground state of llf) is the vacuum il, one can also construct the ground state of the operator ll(c). We will not do it here but refer the interested reader to [BFS4]. Remark 16.22 The discussion of the analytic properties of the map Rp in this section is rather informal. In a rigorous treatment, one needs the differentiability of the coupling functions hrs(JS., J-L) in J-L. This requires either a modification of the Banach space BJ if the original decimation map, D p , is used, or a modification of this map - the smooth decimation map introduced in [BCFS]. The latter arises when the projections P and P := 1 - Pare replaced by more general operators P and P forming a partition of unity

p2

+ p2 =

1.

In our case, P and P are chosen to be XH/p<5.1 and XH/p?:.l, where X>'<5.1 and X>.?:.l are smooth functions satisfying the relations

o ::;

_{I

X>'9, X>'?:.l ::; 1, and XA9 -

if A ::; ~ 0 if A> 1 (and the corresponding

relation for x>.>d. We do not explain here how this works, but refer the interested reader to [BeFS, BFS4] for details.

234

16 Supplement: Renormalization Group

16.11 Appendix In this appendix we show Theorem 16.23 The map Rp is analytic on Uc .

Proof. Let P := Ep and P = 1 - P. By the definition of R p, and equation (16.3), it suffices to show that R(H) := H f P(PHP)-1PHf is analytic as a map from Uc to the Banach space B4p -l/2J. The argument we use below is the same as the one used in showing that the operator PH P is invertible on RanP. Let H E Uc and assume 118HII ::; c, where 8H = H - H f , i.e. we take r = 1 here. Define the operator K = Hi1/2(8H)pHi1/2. Take c > 0 sufficiently small so that )..( J, 1) :=

(J ,,:)

1/2 ::;

i. Then equation (16.42) implies that

w::;1

IIKII ::; 4118HIIJ .

i,

Thus for 118HIIJ < the operator 1 PH P on RanP and we have R

+K

is invertible, and therefore so is

= H}/2 P[l + Kj-1 P H}/2

.

The operator [1 + Kj-1 can be expanded in a Neumann series. Now Proposition 15.1 of [BFS4j implies that

:E (-

K)n Hy2 converges in the Banach space B4p -l/2J. n=1 1/2 1/2 Since K = Hi P8H P Hi the claim follows. 0

Thus the series H}i2

1 7 Comments on Missing Topics, Literature,

and Further Reading

General references There is an extensive literature on quantum mechanics. Standard books include [Bay, LL, Me, Schi]. More advanced treatments can be found in [Sa, BJ, DR]. For rigorous treatments of quantum mechanics see [BS, T3]. Mathematical background is developed in the following texts: [Ar, BiS, CFKS, Fo, HS, LiL, RSI, RSII]. Further mathematical developments and open problems are reviewed in [Fe, HuS, Lil, Li5, Sim4]. Chapter 3 For a discussion of the relation between quantization and pseudo differential operators, see [Fe, BS, DS, Ho, Shu]. Chapter 4 For more discussion about the relation between the uncertainty principle and the stability of atoms, see [Lil]. Chapter 6 Chapter 6 can serve as an introduction to (one-body) scattering theory. References to books on scattering theory can be found at the end of this chapter. The wave operators for short-range interactions were introduced in [Mol, and for long-range interactions, in [Do, BuM]. Chapter 8 Chapter 8 is somewhat more advanced than the chapters preceeding it. Recommended reading for this chapter includes [CFKS, HuS, Ag]. Recently, mathematical many-body theory, especially scattering theory for many-particle systems, has undergone rapid and radical development. The latter is covered in [H uS, D J, G L] .

S. J. Gustafson et al., Mathematical Concepts of Quantum Mechanics © Springer-Verlag Berlin Heidelberg 2003

236

17 Comments on Missing Topics, Literature, and Further Reading

Certain aspects of the theory of Coulomb systems (atoms, molecules, and aggregates thereof) are reviewed in [Fe, Lil, Li3, Li4, Sigl, T3]. Chapter 9

In the chapter on density matrices, we touch upon the rich and fast-developing subject of non-equilibrium quantum statistics, or quantum mechanics at positive temperatures. For a review of this subject see [Sig3]. The original papers mentioned in the text are [BFS, DJ, JPl, JP2, JP3, Mer] More material on density matrices can be found in [T4]. Standard texts covering thermodynamics and statistical mechanics are [Fey, Hu, LL2, Ma2]. Rigorous treatments of equilibrium statistical mechanics can be found in [T4, Is, Ru, Sim6, BR, Ha]. Chapter 10

Standard references on path integrals are [FH, Kl, Schu, Ra]. Rigorous results can be found in [Sim3]. Important original papers are [La, Co, LS]. Chapter 11

An extensive rigorous treatment of quasiclassical asymptotics can be found in [Iv, Mas, MF, Ro]. Section 11.5

For a discussion of determinants, see [BuN, BR, Kl, LS, Ra, Schu, Schw] and references therein. Chapter 12

For elementary facts on the calculus of variations, one can consult [GF, BB]. Chapter 13

The mathematical theory of resonances started with the papers [AC, BC, SimI]. A classic, influential review of the theory of resonances is [Sim2]. Recent developments are reviewed in [HS]. Since the latter book appeared, there has been important progress on the time-dependent theory of resonances: see [BZ, CS, MS, SW]. Our approach to the theory of resonances is close to [SV], and its physical predecessors are [La, Co, Af, LOl, L02]. For complex classical trajectories, see, eg., [Bu, BuF, BuG, Vol, Vo2]. Our

17 Comments on Missing Topics, Literature, and Further Reading

237

treatment of positive temperatures is close to [L01, L02]. A more careful treatment involves the coupling of a quantum system to a thermal reservoir (see Sections 9.9-9.10). This, as well as many other relevant ideas can be found in [CL, L01, L02]. Metastable states at zero and positive temperatures are a subject of intensive study in condensed matter physics, quantum chemistry, nuclear physics, and cosmology. See [BFGLV, HTB, VS, SV, DHIKSZ] for reviews and further references. Chapter 14

Appropriate texts for further reading in quantum field theory are [Be, CDG, FS, Frl, Ge, GJ, Ja, LeB, Ra, Ry, Schw, Sim5, Zi]. Some of the recent developments are reviewed in [Fr2]. Chapter 15

The results on the theory of radiation, as well as the renormalization group approach, are taken from [BFS2, BFS3, BFS4, BCFS]. The existence of the ground state for the physical range of the parameters was proved in [GLL]. See reviews in [BFS1, HSp, Sig2]. Chapter 16

Texts on the standard renormalization group method include [LeB, Zi, FFS, Mal, BDH, FKT, Sal]. Missing topics

As was mentioned in the introduction, this book is not a standard text on Quantum Mechanics for - say - physics undergraduates. As a result, many standard (and sometimes key) topics are not included. The most important missing topics are -

Perturbation theory Spin Scattering theory The theory of atoms and molecules Perturbation theory for resonances and the Fermi Golden Rule Hartree-Fock and Thomas-Fermi approximations

Fortunately, these topics are well covered in the literature: see [Bay, LL, Me, Schi, T3], and, for perturbation theory, [Ba, Ka, Re, RSIV]. Moreover, [RSIV] gives an excellent exposition of the perturbation theory of resonances and mathematical questions of many-body theory. For the latter two topics, see also the review [HuS].

238

17 Comments on Missing Topics, Literature, and Further Reading

For a modern exposition of the Hartree-Fock and Thomas-Fermi theories see the seminal original papers [LiS1, LiS2], with further mathematical developments in [Lio], the book [T4], and the reviews [Fe, Lil, Li2]. The review [Lil] also describes the connection between Thomas-Fermi theory and the problem of stability of matter. The articles [LiSl, LiS2, Lil, Li2] mentioned above, can also be found in [Li4]. Scattering theory is presented in the textbooks mentioned above, as well as in [GW, Mel, Ne, Ta]. Rigorous results on scattering are reviewed in [HuS] (the many-body case), and in textbook form appear in [DG, RSIII, Ya].

References

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I. Affleck, Quantum Statistical Metastability, Phys. Rev. Lett. 46 no.6 (1981) 388-39l. S. Agmon, Lectures on Exponential Decay of Solutions of Second Order Elliptic Equations. Princeton University Press: Princeton, 1982. J. Aguilar, J.M. Combes, A class of analytic perturbations for one-body Schrodinger Hamiltonians. Comm. Math. Phys. 22 (1971) 269-279. V.I. Arnold, Mathematical Methods of Classical Mechanics. Springer, 1989. V. Bach, T. Chen, J. Frohlich, I.M. Sigal, Smooth Feshbach map and operator-theoretic renormalization group methods. Preprint, 2002. V. Bach, J. Frohlich, I.M. Sigal, Return to equilibrium. J. Math. Phys. 41 (2000) 3985-4060. V. Bach, J. Frohlich, I.M. Sigal, Mathematical theory of radiation. Found. Phys. 27 no.2 (1997) 227-237. V. Bach, J. Frohlich, I.M. Sigal. Adv. Math. 137 (1998) 205-298, and 137 (1998) 299-395. V. Bach, J. Frohlich, I.M. Sigal, Spectral analysis for systems of atoms and molecules coupled to the quantized radiation field. Comm. Math. Phys. 207 no.2 (1999) 249-290. V. Bach, J. Frohlich, I.M. Sigal, Application of invariant manifold theory to the problem of radiation (in preparation). E. Balslev, J.M. Combes, Spectral properties of many-body Schrodinger operators with dilation analytic interactions. Comm. Math. Phys. 22 (1971) 280-294. H. Baumgartel, Endlichdimensionale Analytische Storungstheorie. Akademie-Verlag, Berlin 1972. G. Baym, Lectures on Quantum Mechanics. Addison-Wesley (1990). G. Benfatto, G. Gallavotti, Renormalization Group. Princeton University Press: Princeton, 1995. F. A. Berezin, The Method of Second Quantization. Academic Press: New York,1966. F.A. Berezin, M.A. Shubin, The Schrodinger Equation. Kluwer: Dordrecht, 1991. H.A. Bethe, R. Jackiw, Intermediate Quantum Mechanics. Benjamin/Cummings: Menlo Park, 1986. M. Sh. Birman, M.Z. Solomyak, Spectral theory of self-adjoint operators in Hilbert spaces. Mathematics and its Applications (Soviet series). Reidel, 1987. J. Bjorken, S. Drell. Relativistic Quantum Fields. McGraw-Hill: New York, 1965.

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R. Israel, Convexity in the Theory of Lattice Gases Princeton University Press: Princeton, 1979. V. Ivrii, Microlocal Analysis and Precise Spectral Asymptotics. Springer, 1998. R. Jackiw, Quantum meaning of classical field theory, Rev Mod. Pys. 4g (1977),681-706. A. Jaffe, C. Taubes, Vortices and Monopoles. Birkhiiuser: Boston, 1980. V. Jaksic, C.-A. Pillet, On a model for quantum friction II. Fermi's golden rule and dynamics at positive temperatures. Comm. Math. Phys. 176 no.3 (1996) 619-644. V. Jaksic, C.-A. Pillet, On a model for quantum friction III. Ergodic properties of the spin-boson system. Comm. Math. Phys. 178 no.3 (1996) 627-65l. V. Jaksic, C.-A. Pillet, Mathematical theory of non-equilibrium quantum statistical mechanics. Preprint, 200l. T. Kato, Perturbation Theory for Linear Operators. Springer-Verlag: Berlin, 1976. S.K. Kehrein, A. Mielke, Theory of the Anderson impurity model: the Schrieffer-Wolff transformation reexamined. Ann. Phys. 252 # 1 (1996) 1-32. H. Kleinert, Path Integrals in Quantum Mechanics, Statistics, and Polymer Physics. World Scientific, 1995. L.D. Landau, E.M. Lifshitz, Theoretical Physics 3. Quantum Mechanics: Non-Relativistic Theory. Pergamon, 1977. L.D. Landau, E.M. Lifshitz, Theoretical Physics V. Statistical Physics. Addison-Wesley, 1969. J.S. Langer, Theory of the condensation point. Ann. Phys. 41 (1967), 108157. A.1. Larkin, Yu. N. Ovchinnikov, Damping of a superconducting current in tunnel junctions. Zh. Eksp. Theor. Fiz 85 (1983) 1510. A.1. Larkin, Yu. N. Ovchinnikov, Quantum mechanical tunneling with dissipation. The pre-exponential factor. Zh. Eksp. Theor. Fiz 86 (1984) 719. M. Le Bellac, Quantum and Statistical Field Theory. Oxford, 1991. S. Levit, U. Smilanski, A new approach to Gaussian path integrals and the evaluation of the semiclassical propagator. Ann. Phys. 103, no. 1 (1977) 198-207. E.H. Lieb, Stability of Matter. Rev. Mod. Phys. 48 (1976) 553-569. E.H. Lieb, Thomas-Fermi and related theories. Rev. Mod. Phys., 53 (1981). E.H. Lieb, Stability of Matter: From Atoms to Stars. Bull. Amer. Math. Soc. 22 (1990) 1-49. E.H. Lieb, Stability of Matter: From Atoms to Stars. Selecta Elliot H. Lieb. W. Thirring, ed. Springer-Verlag (New York), 1991. E.H. Lieb Inequalities, Selecta of E.H. Lieb. M. Loss and M.B. Ruskai, eds. E.H. Lieb, M. Loss, Analysis. American Mathematical Society, 200l. E.H. Lieb, B. Simon, Thomas-Fermi theory of atoms, molecules and solids. Adv Math. 23 no.1 (1977) 22-116. E.H. Lieb, B. Simon. The Hartree-Fock theory for Coulomb systems Comm. Math. Phys. 53 no.3 (1977) 185-194. P.L. Lions, Hartree-Fock and related equations. Nonlinear partial differential equations and their applications. College de France Seminar, Vol. IX. Pitman Res. Notes Math. Ser. 181 (1988) 304-333.

[Iv] [Ja] [JT] [JP1]

[JP2] [JP3] [Ka] [KM]

[KI]

[LL] [LL2] [La]

[L01] [L02] [LeB] [LS]

[Lil] [Li2] [Li3] [Li4] [Li5] [LiL] [LiS1] [LiS2]

[Lio]

References

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S.K. Ma, Modern Theory of Critical Phenomena. Benjamin: New York, 1976. [Ma2] S.K. Ma, Statistical Mechanics. World Scientific, 1985. [Mas] V.P. Maslov, Theorie des Perturbations et Methods Asymtotiques. Dunod, 1972. V.P. Maslov, M.V. Fedoryuk Semiclassical approximation in quantum me[MF] chanics. Contemporary mathematics, 5. Reidel: Boston, 1981. R. Melrose, Geometric Scattering Theory. Cambridge, 1995. [Mel] [Mer] M. Merkli, Positive commutators in non-equilibrium quantum statistical mechanics: return to equilibrium. Comm. Math. Phys. 223 no. 2 (2001) 327-362. M. Merkli, I.M. Sigal, A time-dependent theory of quantum resonances. [MS] Comm. Math. Phys. 201 (1999) 549-576. A. Messiah, Quantum Mechanics. Interscience, 1961. [Me] C. M!2iller, General properties of the characteristic matrix in the theory [M!2i] of elementary particles I. Danske. Vid. Selske. Mat.-Fys. Medd. 23 (1945) 1-48. R.G. Newton, Scattering Theory of Waves and Particles. McGraw-Hill: [Ne] New York, 1966. R. Rajaraman, Solitons and Instantons. North Holland, 1982. [Ra] [RSI] M. Reed, B. Simon, Methods of Modern Mathematical Physics, Vol I. Functional Analysis. Academic Press, 1972. [RSII] M. Reed, B. Simon, Methods of Modern Mathematical Physics, Vol. II. Fourier Analysis and Self-Adjointness. Academic Press, 1972. [RSIII] M. Reed and B. Simon, Methods of Modern Mathematical Physics III. Scattering Theory. Academic Press, 1979. [RSIV] M. Reed, B. Simon, Methods of Modern Mathematical Physics IV: Analysis of Operators. Academic Press, 1978. F. Rellich, Perturbation Theory of Eigenvalue Problems. Gordon and [Re] Breach: New York, 1969. D. Robert, Autour de l'Approximation Semiclassique. Birkhauser, Basel, [Ro] 1983. G. Rapstorff, Path integral approach to quantum physics: an introduction. [Ra] Springer, 1994. D. Ruelle, Statistical Mechanics. Benjamin: New York, 1969. [Ru] L.H. Ryder. Quantum Field Theory. Cambridge, 1996. [Ry] J.J. Sakurai, Advanced Quantum Mechanics. Addison-Wesley: Reading, [Sa] 1987. M. Salmhofer, Renormalization Group. Springer-Verlag, 2000. [Sal] [Schul L.S. Schulman, Techniques and Applications of Path Integration. John Wiley & Sons., 1981. [Schw] A. Schwarz, Quantum Field Theory and Topology. Springer-Verlag, 1993. [Schi] L.I. Schiff, Quantum Mechanics. McGraw-Hill: New York, 1955. [Shu] M. Shubin, Pseudodifferential Operators and Spectral Theory. Springer, 1987. [Sig1] I.M. Sigal, Lectures on Large Coulomb Systems, CRM Proceedings and Lecture Notes Vol. 8, 1995 73-107. [Sig2] I.M. Sigal, Renormalization group approach in spectral analysis and the problem of radiation, CMS Notes 33 no.3 (2001) 3-9. [Mal]

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LM. Sigal, Quantum mechanics at positive temperatures. Preprint, 2002. LM. Sigal, B. Vasilijevic, Mathematical theory of quantum tunneling at positive temperature. Ann. Inst. H. Poincare (2002) (to appear). B. Simon, The theory of resonances for dilation analytic potentials and the foundations of time-dependent perturbation theory. Ann. Math. 97 (1973) 247-274. B. Simon, Resonances and complex scaling: a rigorous overview. Int. J. Quant. Chern. 14 (1978) 529-542. B. Simon, Functional Integration and Quantum Physics. Academic Press, 1979. B. Simon, Fifteen problems in mathematical physics. Perspectives in Mathematics, 423-454. Birkhiiuser, 1984. B. Simon, The P(¢h Euclidean (Quantum) Field Theory. Princeton University Press: Princeton, 1984. B. Simon, Statistical Mechanics of Lattice Gases, Vol. 1 Princeton Series in Physics, Princeton University Press, 1993. Va. Sinai, Theory of Phase Transitions: Rigorous Results. Pergamon: Oxford, 1982. A. Soffer, M. Weinstein, Time-dependent resonance theory, Geom. Fun. Anal. 8 (1998) 1086-1128. J. R. Taylor, Scattering Theory: The Quantum Theory on Nonrelativistic Collisions. John Wiley & Sons: New York, 1972. W. Thirring, A Course in Mathematical Physics, Vol. 3: Quantum Mechanics of Atoms and Molecules. Springer, 1980. W. Thirring, A Course in Mathematical Physics, Vol. 4. Quantum Mechanics of Large Systems. Springer-Verlag, 1983. A. Vilenkin, E.P.S. Shellard, Cosmic Strings and other Topological Defects. Cambridge, 1994. A. Voros, Spectre de l'equation de Schrodinger et method BKW. Publications Mathematiques d'Orsay 81 (1982). A. Voros, The return of the quartic oscillator: the complex WKB method. Ann. Inst. H. Poincare Sect. A 39 (1983) no. 3, 211-338. F. Wegner, Ann. Physik (Leipzig) 3 (1994) 555-559. S. Weinberg, The quantum theory of fields. Vols. 1-3. Cambridge University Press: Cambridge, 1996. U. Weiss, Quantum Dissipative Systems. World Scientific, 1993. A.S. Wightman, Introduction to R. Israel, Convexity in the Theory of Lattice Gases Princeton University Press: Princeton, 1979. D. Yafaev, Scattering Theory: Some Old and New Problems. Springer Lecture Notes in Mathematics No. 1937, Springer-Verlag, 2000. J. Zinn-Justin, Quantum Field Theory and Critical Phenomena. Oxford, 1996.

[SimI]

[Sim2] [Sim3] [Sim4] [Sim5] [Sim6] [Sin] [SW] [Ta] [T3] [T4] [VS]

[Vol] [Vo2] [Weg] [We] [Wei] [Wi]

[Val [Zi]

Index

c k (n),138

o

C (D),8 C (lR d ),8 HS(R d), 137 H S (D),137

o

£l(R d), 23, 67 £2(R d), 4, 7

U(Rd), 137 n-particle sector, 184 action, 5, 118, 138, 146, 149, 168 adiabatic partition function, 156 adjoint, 19 affine space, 139 analytic continuation, 154, 162 angular momentum, 28, 76 annihilation operator, 74, 179, 183, 186, 198 asymptotic completeness, 64, 65, 95 asymptotically free systems, 105 asymptotically stable, 100 average value, 97 Balmer series, 78 Banach space, 9, 137 basis, 8 Birman-Schwinger - operator, 60 - principle, 60 Bohr-Sommerfeld quantization, 132, 163, 166 Born-Oppenheimer approximation, 82 Bose-Einstein condensation, 113 boson, 85, 106, 185 bound state, 48, 54, 63, 92 bounded operator, 9 canonical

- commutation relations, 30 - coordinates, 172 - operators, 30 - variables, 30, 175 Cauchy problem, 13 Cauchy-Schwarz inequality, 8 causality, 5 central subspace, 227 centre-of-mass motion, 83-85 channel evolution, 94 chemical potential, 111 classical - action, 145 - field theory, 138 - path, 130, 145 cluster decomposition, 87, 93 commutator, 11, 27, 30, 175 complex deformation, 155, 156 condensate, 113 configuration space, 169 confined systems, 105 confining potential, 43 conjugate point, 146, 147 conservation of energy, 29, 150 conservation of probability, 13 constraint, 143 continuous spectrum, 38 convolution, 23, 187 correspondence principle, 5 Coulomb gauge, 80, 173 Coulomb potential, 55, 77, 92 covariance operator, 175, 182, 186, 192 covariant derivative, 198 creation operator, 74, 179, 183, 186, 198 critical point, 124, 125, 139, 142 critical temperature, 113

246

Index

D'Alembertian, 145 decimation map, 204, 210 delta function, 24, 38 dense set, 8 density matrix, 97-99 determinant - of an operator, 133, 135 dilation, 54, 155 Dirac delta function, 24 direct sum, 85 Dirichlet boundary conditions, 69 Dirichlet functional, 138 dispersion , 33 - law, 180 distribution, 24 domain of an operator, 8 dual space, 140 eigenvalue, 37 electric field, 79, 173 electromagnetic field, 139 entropy, 101 environment, 99, 104 equation of state, 110 equilibrium - state, 101 - thermal, 97 ergodic mean convergence, 48 Euler-Lagrange equation, 142, 168 evaluation functional, 171 evolution operator, 17, 117 excited state, 201 existence of dynamics, 14 expected value, 176 exponential decay, 92 - of an operator, 14 Fermi golden rule, 197 fermion, 85 Feshbach map, 204, 210 Feynman path integral, 119 Feynman-Kac theorem, 102, 133 field, 168 fine structure constant, 197 first resolvent identity, 42 fixed point, 205, 225 Fock space, 110, 184, 197 Fourier inversion formula, 23

Fourier transform, 22 Fnlchet derivative, 141 free - evolution, 63, 65 - Hamiltonian, 18 - propagator, 18, 125 free energy, 97, 101, 103, 133, 155 - Helmholtz, 101 free propagator, 18 fugacity, 111 functional , 137 - linear, 140 Gateaux derivative, 141 gauge - invariance, 79 - transformation, 79, 173 Gaussian, 18 Gaussian measure, 175, 191 generalized eigenfunction, 66 generalized normal form, 216 generator of dilations, 54 Gibbs entropy, 101 Gibbs state, 97, 101 grand canonical ensemble, 111 Green's function, 127, 129, 163 ground state, 97, 201 ground state energy, 132, 134 group property, 17 Hamilton's equations, 29, 30, 171 Hamilton-Jacobi equation, 5, 129, 150, 151 Hamiltonian - function, 5, 30, 150 - functional, 170 - operator, 28, 30, 185 - system, 171 harmonic function, 142 harmonic oscillator, 73, 102, 104, 134, 150 Heisenberg - equation, 29, 30 - representation, 29 - uncertainty principle, 33 Hessian, 120 - of a functional, 144 Hilbert space, 7, 137 Hilbert-Schmidt operator, 116

Index HVZ theorem, 88, 91 hydrogen atom, 35, 55, 76 ideal Bose gas, 106, 110 identical particle symmetry, 84 index, 147 infinite systems, 97 initial condition, 13, 37 initial value problem, 13 inner product, 6, 7 instanton, 158, 159 integrable function, 22 integral kernel, 9, 58, 60, 66, 99, 117, 176 integral operator, 9, 60, 116 intercluster distance, 94 intercluster potential, 87, 94 internal energy, 101 internal motion, 86 intertwining relations, 66 intracluster potential, 87 invariant subspace, 47 inverse - of an operator, 10 irrelevant term, 224 isometry, 15, 16, 64 isospectral map, 208, 226 Jacobi equation, 128, 147 identity, 170 matrix, 136, 148 vector field, 147 Josephson junction, 159 Kato's inequality, 79 kernel - of an integral operator, 9 Klein-Gordon equation, 142, 168, 171 Lagrange multiplier, 143 Lagrangian, 121, 139 - density, 168 - functional, 168 Lamb shift, 197 Landau levels, 80 Landau-von Neumann equation, 98, 100 Laplace method, 133

247

Laplace-Beltrami operator, 76, 77 Laplacian, 6, 9, 24, 144 lattice approximation, 188 Lebesgue measure, 175 Legendre function, 76 Legendre transform, 128, 151, 169 Lie algebra, 170 linear functional, 140 linear operator, 8 linearization, 226 Lippmann-Schwinger equation, 57 long-range interaction, 66 - potential, 92, 95 magnetic field, 79, 173 marginal term, 224 Maxwell's equations, 173, 182 mean value, 27 metastable state, 105, 153, 157 min-max principle, 52, 56, 93 minimal coupling, 198 minimizer of a functional, 142, 145 momentum operator, 8, 24, 28 monopole, 191 Morse theorem, 147 multi-index, 7, 63 multiplication operator, 8 multiplicity of an eigenvalue, 38 Neumann series, 42, 218 Newton's equation, 29, 142 non-equilibrium statistical mechanics, 97 norm, 7 normal form, 74 observable, 27, 28, 97, 175, 181 one-parameter unitary group, 18, 155 open system, 97, 99 operator ,8 angular momentum, 28 bounded, 9 evolution, 17, 117 Hamiltonian, 28, 30, 185 Hilbert-Schmidt, 116 integral, 9, 60, 116 Laplacian, 9, 24 linear, 8

248

Index

momentum, 8, 24, 28 multiplication, 8 particle number, 185 position, 27 positive, 21, 98 Schrodinger, 6, 9, 13, 54, 117, 144, 175 self-adjoint, 20 symmetric, 20 trace class, 98, 114 unitary, 16 operator-valued distributions, 179 order parameter, 114 orthogonal projection, 86, 98, 208 orthonormal set, 8 Parseval relation, 8 particle number operator, 74, 110, 185 partition function, 101, 103, 107, 132, 155 partition of unity, 89 path integral, 119, 125, 133, 157, 163 periodic boundary conditions, 70 permutation, 184 phase space, 30 phase transition, 113 phonon, 103, 104 photon, 103, 104, 202 Plancherel theorem, 23 Planck's constant, 6, 179 plane wave, 24, 40 Poincare group, 193, 195 point spectrum, 38, 92 Poisson bracket, 30,170,171,174,175 polarization identity, 14 position operator, 27 positive operator, 21, 98 positive temperatures, 97 potential, 6 pressure, 108 principle - of maximum entropy, 101 - of minimal action, 143, 168 probability distribution, 27 projection, 206 orthogonal, 86, 98 - rank-one, 98, 99, 215 - spectral, 204, 205 propagator, 17, 117

- free, 18 pseudodifferential operators, 31, 175 pure state, 100 quadratic form, 139 quantization, 29, 175, 195 quantum statistical mechanics, 97 quasi-classical analysis, 129, 134, 157 range - of an operator, 21 rank-one projection, 98, 99 reduced mass, 83 reflection coefficient, 71 relative motion, 84 relevant term, 224 renormalization group, 207 renormalization map, 204, 225 repulsive potential, 54 reservoir, 104 resolvent - of an operator, 37, 42 - set, 37, 57 resonance, 73, 153, 166, 197, 203 decay probability, 156 eigenvalue, 156 energy, 73, 154, 156 free energy, 156 lifetime, 154, 156 width, 156 return to equilibrium, 105 Riesz integral, 43 Ruelle theorem, 48 Rydberg states, 93 scalar potential, 79 scaling dimension, 224 scattering channel,93 eigenfunction, 66 operator, 65 state, 48, 63, 64, 153, 201 Schrodinger eigenvalue equation, 69 equation, 5, 6, 13, 37, 63, 117 operator, 6, 9, 13, 54, 117, 144, 175 representation, 29 second law of thermodynamics, 101 second variation of a functional, 144

Index self-adjoint operator, 20 semi-classical analysis, 129, 134, 157 semi-group, 205, 210, 226 separation of variables, 96, 200 short-range - interaction, 64 - potential, 92, 94 Sobolev space, 7, 137 soliton, 195 specific heat, 113 spectral projection, 204, 205 spectrum - continuous, 38 - of an operator, 37, 185 - point, 38, 92 spherical coordinates, 77 spherical harmonics, 76 spin, 84 spreading sequence, 41 stable - asymptotically, 100 - in Lyapunov sense, 100 - manifold, 230 - subspace, 227 state space, 4, 30, 170, 175 state vector, 4 stationary phase method, 125, 128, 133, 163 stationary state, 98, 100 statistical mechanics, 97 strong operator topology, 122 superposition principle, 5 symmetric operator, 20 symmetrization, 184 symplectic - form, 171 - matrix, 175 operator, 170 tangent space, 139 temperature, 101 tensor product - of Hilbert spaces, 95 - of operators, 96 thermal equilibrium, 97, 101 thermal evaporation, 105 thermodynamic limit, 100, 111

threshold, 91, 92 - set, 91 - two-cluster, 92 topological (anti-) solitons, 191 - charge, 191 - conservation law, 190 - sector, 190 trace - of an operator, 59, 115 trace class operator, 98, 114 transmission coefficient, 73 transverse vector-field, 173 triangle inequality, 7, 44 Trotter product formula, 117, 121 tunneling, 154, 159, 163 two-cluster threshold, 92 ultraviolet cut-off, 187, 198 uncertainty principle, 33, 56 unitary - operator, 16 unstable manifold, 230 subspace, 227 systems, 105 vacuum, 183, 201 variational derivative, 141, 226 vector potential, 79, 121, 197 virial - relation, 54, 196 - theorem, 162 vortex, 191 wave equation, 142, 173 - nonlinear, 142 wave function, 4, 27, 97 wave operator, 64, 65, 67, 94 weak convergence, 37 Weyl - criterion, 38 - sequence, 37 Wick - ordering, 178, 181 - quantization, 181

249

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