A D V A N C E S IN IMAGING A N D ELECTRON PHYSICS
VOLUME 137
D O G M A OF THE C O N T I N U U M AND THE CALCULUS OF FINITE D I F F E R E N C E S IN Q U A N T U M PHYSICS
EDITOR-IN-CHIEF
PETER W. HAWKES CEMES-CNRS Toulouse, France
ASSOCIATE EDITOR
BENJAMIN K A Z A N Palo Alto, California
HONORARY ASSOCIATE EDITOR TOM MULVEY
Advances in
Imaging and Electron Physics Dogma of the Continuum and the Calculus of Finite Differences in Quantum Physics HENNING F. HARMUTH Retired, The Catholic University of America Washington, DC, USA
BEATE MEFFERT Humboldt- Universitgit Berlin, Germany
V O L U M E 137
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CONTENTS
PREFACE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
FUTURE CONTRIBUTIONS
ix
. . . . . . . . . . . . . . . . . . . . . . . . .
DEDICATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xi .
xvii
FOREWORD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xix
LIST OF FREQUENTLY USED SYMBOLS . . . . . . . . . . . . . . . . . . . .
xxiii
1
Introduction
1.1
Modified Maxwell Equations and Basic Relations . . . . . . . . . . .
1.2
Basic Concepts of the Calculus of Finite Differences . . . . . . . . . .
8
1.3
Lagrange F u n c t i o n and Modified Maxwell Equations . . . . . . . . .
18
1.4
D o g m a of the C o n t i n u u m in Physics
26
1.5
Concept of Space Based on Finite Differences
2
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
30
Modified Klein-Gordon Equation
2.1
Differential E q u a t i o n with Magnetic Current Density
. . . . . . . . .
2.2
Modified K l e i n - G o r d o n Difference E q u a t i o n . . . . . . . . . . . . .
52
2.3
Solution of the Difference E q u a t i o n of Wx0 . . . . . . . . . . . . . .
61
2.4
Time-Dependent Solution of Wx0(ff, 0) . . . . . . . . . . . . . . . .
70
2.5
H a m i l t o n Function and Quantization . . . . . . . . . . . . . . . .
78
2.6
Plots for First-Order A p p r o x i m a t i o n
86
. . . . . . . . . . . . . . . .
44
Equations are numbered consecutively within each of Sections 1.1 to 6.10. Reference to an equation in a different section is made by writing the number of the section in front of the number of the equation, e.g., Eq. (1.1-45) for Eq. (45) in Section 1.1. Illustrations are numbered consecutively within each section, with the number of the section given first, e.g., Figure 1.2-1. References are listed by the name of the author(s), the year of publication, and a lowercase Latin letter if more than one reference by the same author(s) is listed for that year.
vi
CONTENTS
3
Inhomogeneous Difference Equation
3.1 3.2
Inhomogeneous Term of Eq. (2.2-31) . . . . . . . . . . . . . . . . Evaluation of Eq. (3.1-32) . . . . . . . . . . . . . . . . . . . . .
92 102
3.3 3.4 3.5
Quantization of the Solution for Ax >> h / m o c . . . . . . . . . . . . Evaluation of the Energy Uc . . . . . . . . . . . . . . . . . . . . Plots for Second-Order Approximation . . . . . . . . . . . . . . .
115
4 4.1 4.2 4.3 4.4
Klein-Gordon Difference Equation for Small Distances
Evaluation of the Difference Equation for Ax << h / m o c . . . . . . . . Evaluation of Eq. (4.1-4) . . . . . . . . . . . . . . . . . . . . . . Quantization of the Solution for Ax << h / m o c . . . . . . . . . . . . Evaluation of the Energy Uc for Small Distances . . . . . . . . . . .
5 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 5.10 5.11
143 147 155 163
Difference Equation in Spherical Coordinates
Charged Particle in an Electromagnetic (EM) Field . . . . . . . . Relativistic Mass Variation . . . . . . . . . . . . . . . . . . . Quantization with Differential Operators . . . . . . . . . . . . . Difference Operators . . . . . . . . . . . . . . . . . . . . . Spatial Difference Equation . . . . . . . . . . . . . . . . . . Quantization of the Solution q~r0(P, ~9) . . . . . . . . . . . . . . Convergence for Small Values of Ar . . . . . . . . . . . . . . . Plots for Sections 5.5 and 5.7 . . . . . . . . . . . . . . . . . . Origin of the Coulomb Field . . . . . . . . . . . . . . . . . . Unbounded Bosons in a Coulomb Field . . . . . . . . . . . . . Antiparticles . . . . . . . . . . . . . . . . . . . . . . . . .
6 6.1 6.2 6.3 6.4 6.5 6.6 6.7
122 135
. . . . . . . . . . .
. . . . . . . . . .
183 189 198 203 211 222 226 230 243 244 257
. . . . . . .
. . . . . . .
262 266 268 278 284 288 293
Appendix
Difference Operators of Higher Order . . . . . . . . . . . Extension of Section 3.1 for Ax << Ac . . . . . . . . . . . Solution of Inhomogeneous Difference Equations . . . . . . Calculations for Section 4.2 . . . . . . . . . . . . . . . Formulas for Spherical Coordinates . . . . . . . . . . . . Difference Equations Solved by Polynomials . . . . . . . . Discrete Spherical Harmonics . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . .
CONTENTS
vii
6.8
Solution of the Differential Equation (5.5-12) . . . . . . . . . . . . .
298
6.9
C o n f o r m a l M a p p i n g for Section 5.7 . . . . . . . . . . . . . . . . .
300
C o n f o r m a l M a p p i n g for Section 5.10
. . . . . . . . . . . . . . . .
311
. . . . . . . . . . . . . . . . . . . . . . .
314
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
318
6.10
References and Bibliography Index
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PREFACE
This is H. F. Harmuth's eighth long contribution to these Advances and it adds a new chapter to his many studies of Maxwell's equations and his more recent preoccupations with finite difference equations instead of differential equations, in which he is joined by B. Meffert. A first examination of these questions formed volume 129 of these Advances and here, the work on quantum mechanics is pursued more deeply. The Klein-Gordon equation is at the heart of this volume but chapters are also devoted to the many difficult and little-studied problems that arise when discreteness is imposed and finite difference equations must be solved. I am delighted to include this work in these Advances and hope, by doing so, to provoke much discussion among the theoreticians of quantum mechanics. Peter Hawkes
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FUTURE CONTRIBUTIONS
G. Abbate New developments in liquid-crystal-based photonic devices
S. Ando Gradient operators and edge and corner detection A. Asif Applications of noncausal Gauss-Markov random processes in multidimensional image processing C. Beeli Structure and microscopy of quasicrystals M. Bianchini, F. Scarselli, and L. Sarti Recursive neural networks and object detection in images G. Borgefors Distance transforms A. Bottino Retrieval of shape from silhouette A. Buchau Boundary element or integral equation methods for static and time-dependent problems B. Buchberger Gr6bner bases J. Caulfield Optics and information sciences C. Cervellera and M. Muselli The discrepancy-based approach to neural network learning
T. Cremer Neutron microscopy H. Delingette Surface reconstruction based on simplex meshes A. R. F aruqi Direct detection devices for electron microscopy
R. G. Forbes Liquid metal ion sources
xii
FUTURE CONTRIBUTIONS
J. Y.-I. Forrest Grey systems and grey information
E. F6rster and F. N. Chukhovsky X-ray optics A. Fox The critical-voltage effect L. Godo & V. Torra Aggregation operators A. Giilzh~iuser Recent advances in electron holography with point sources K. Hayashi X-ray holography M. I. Herrera The development of electron microscopy in Spain D. Hitz Recent progress on HF ECR ion sources D. P. Huijsmans and N. Sebe Ranking metrics and evaluation measures K. Ishizuka Contrast transfer and crystal images J. Isenberg Imaging IR-techniques for the characterization of solar cells K. Jensen Field-emission source mechanisms
L. Kipp Photon sieves G. K6gel Positron microscopy T. Kohashi Spin-polarized scanning electron microscopy
W. Krakow Sideband imaging
FUTURE CONTRIBUTIONS R. Leitgeb Fourier domain and time domain optical coherence tomography B. Lencovfi Modern developments in electron optical calculations R. Lenz (vol. 138) Aspects of colour image processing W. Lodwick Interval analysis and fuzzy possibility theory R. Lukac Weighted directional filters and colour imaging L. Macaire, N. Vandenbroucke, and J.-G. Postaire Color spaces and segmentation M. Matsuya Calculation of aberration coefficients using Lie algebra S. McVitie Microscopy of magnetic specimens L. Mugnier, A. Blanc, and J. Idier Phase diversity
K. Nagayama (vol. 138) Electron phase microscopy M. A. O'Keefe Electron image simulation J. Orloff and X. Liu (vol. 138) Optics of a gas field-ionization source D. Oulton and H. Owens Colorimetric imaging N. Papamarkos and A. Kesidis The inverse Hough transform K. S. Pedersen, A. Lee, and M. Nielsen The scale-space properties of natural images E. Rau Energy analysers for electron microscopes
xiii
xiv
FUTURE CONTRIBUTIONS
H. Rauch The wave-particle dualism E. Recami Superluminal solutions to wave equations
J. Rehficek, Z. Hradil, J. Pefina, S. Pascazio, P. Facchi, and M. Zawisky Neutron imaging and sensing of physical fields G. Ritter Lattice-based artifical neural networks J.-F. Rivest Complex morphology
G. Schmahl X-ray microscopy G. Sch6nhense, C. M. Schneider, and S. A. Nepijko Time-resolved photoemission electron microscopy
F. Shih General sweep mathematical morphology R. Shimizu, T. Ikuta, and Y. Takai Defocus image modulation processing in real time S. Shirai CRT gun design methods N. Silvis-Cividjian and C. W. Hagen Electron-beam-induced deposition T. Soma Focus-deflection systems and their applications Q. F. Sugon Geometrical optics in terms of Clifford algebra
W. Szmaja Recent developments in the imaging of magnetic domains I. Talmon Study of complex fluids by transmission electron microscopy I. J. Taneja (vol. 138) Divergence measures and their applications
FUTURE CONTRIBUTIONS
xv
M. E. Testorf and M. Fiddy Imaging from scattered electromagnetic fields, investigations into an unsolved problem M. Tonouchi Terahertz radiation imaging
N. M. Towghi Ip norm optimal filters Y. Uchikawa Electron gun optics
K. Vaeth and G. Rajeswaran Organic light-emitting arrays J. Vald~s (vol. 138) Units and measures, the future of the SI D. Walsh (vol. 138) The importance-sampling Hough transform G. G. Walter Recent studies on prolate spheroidal wave functions B. Yazici Stochastic deconvolution over groups
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To the memory of Max Planck (1858-1947) Founder of quantum physics and distinguished participant of the Morgenthau Plan, 1945-1948.
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FOREWORD
The ancient Greeks did not distinguish between mathematics as a science of the thinkable and physics as a science of the observable. This cast long shadows over the development of physics. As a first example we cite the dogma of the circle. Ptolemy expressed it as follows: ... we believe it is the necessary purpose and aim of the mathematician to show forth all the appearances of the heavens as products of regular and circular motion. (Ptolemy, 1952, Almagest, Book III, 1; p. 83, {}2) It is generally assumed that Kepler ended the dogma of the circle, but this is true only for astronomy. The superposition of deferents and epicycles of Ptolemy and Copernicus developed into the Fourier series in complex notation. We meet the old circle under the new name exponential function e i~ in the complex plane. Another circle in disguise is the character group {e iyx} of the topologic group of real numbers. The word the in the title provides the connection with Greek thinking. Mathematics justifies only the name a character group . . . . A second long shadow was cast by Euclid's geometry. Navigators had been using spherical trigonometry since about 1500 to chart their course across the oceans. But the greatest mathematicians struggled three centuries later with the question of whether Euclid's geometry was the only possible one. Here we are concerned with a third long shadow, the dogma of the continuum of physical space and time. It can be traced back to the Eleatic school of the Greeks in southern Italy. Zeno of Elea (c. 490-c. 430 BCE) advanced the paradox of the race between Achilles and the turtle as well as that of the arrow that does not fly, to refute the continuum or the infinite divisibility of distances in space and time. Zeno's paradoxes were in turn refuted by Aristotle in his Physica. Aristotle's arguments in favor of a mathematical continuum for the physical space and time were so convincing that they were questioned rarely since. The physics of space and time became a branch of mathematics. Newton demonstrated the perception of physics as a branch of mathematics when he wrote Absolute, true and mathematical time, of itself, and from its nature, flows equably without connection with anything external, .... (Newton, 1971, p. 6) Newton and Leibniz carried the concept of infinite divisibility from the denumerable infinite of the Greeks to the nondenumerable infinite of differential calculus. The development of non-Euclidean geometries and the experimental verification of the acoustic Doppler effect changed our thinking about time and space to the concepts used in the special and the general theory of relativity,
xix
xx
FOREWORD
and beyond. A quotation of Einstein from his later years shows this development: But to connect every instant of time with a number, by the use of a clock, to regard time as an one-dimensional continuum, is already an invention. So also are the concepts of Euclidean and non-Euclidean geometry, and our space understood as a three-dimensional continuum. (Einstein and Infeld, 1938, p. 311) The straightforward proof of a continuum of physical space and time would be the observation of events at two spatial points x and x § dx or two times t and t § dt. What is physically possible are observations at x and x § Ax or t and t + At, where Ax and At may be very small but must be finite. Any finite interval Ax, At can be divided into nondenumerably many subintervals dx, dt, which means we are a long way from a mathematical continuum. If we want to use finite differences Ax, At instead of differentials dx, dt we must use the calculus of finite differences instead of the differential calculus. This is a true generalization since no fixed values for Ax, At are specified at the beginning of the calculation. When solving for the eigenfunctions of a difference equation in relativistic quantum physics we typically get well-behaved functions if the spatial resolution Ax is large enough, but sequences of random numbers for too small values of Ax. This is how the calculation represents the Compton effect. The theory goes beyond Heisenberg's uncertainty relation since it puts a lower limit on Ax rather than on the product A x A p . Consider elementary particles within the framework of differential calculus. We must match the physical situation to the mathematical method and we do so by defining elementary particles to be "point-like" to avoid giving them any spatial features. Using the calculus of finite differences we must demand only that an elementary particle is smaller than an arbitrarily small but finite distance Ax to avoid any observable spatial feature. The difference theory clearly offers the better choice. A particle with mass m0 can become an antiparticle with mass -m0 without a quantum jump in the difference theory. Under certain conditions a finite spatial resolution Ax permits such a transition without violating any physical laws. Generally, the theorem of H61der stated in 1887 that the gamma function can be defined by a simple difference equation F(x + 1) -- xF(x) but by no algebraic differential equation. This implies that differential and difference equations define different classes of functions. The calculus of finite differences predates the differential calculus since differentials are obtained as limits of finite differences. The success of differential calculus in science and engineering stimulated its enormous development. There were no comparable applications for the calculus of finite differences before its usefulness for relativistic quantum physics was discovered and there was thus
FOREWORD
xxi
little development. Our Bibliography lists only 10 mathematical books on the calculus of finite differences published in the twentieth century ~. We want to thank Humboldt-Universit~it of Berlin for help with computer and library services. Henning F. Harmuth
1Our search was limited to books in English, French, German, Russian, and Spanish. We would be grateful for information about books in Chinese or Japanese.
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List of Frequently Used Symbols
Ae Am B
b~,b*~
b;,b+ r D~i (0)
b i(o)
D~i(O) D
d(~c) din, d~i (0) E,E E
e F (0 ge gm
GI(~, O) G2(~, O) Gc~ (0) G,~ (0) H,H
H~ (0) Hs~ (0) 7{ h h=h/27t IT mo N Pl (~, O) Pc
As/m Vs/m Vs/m 2 m/s As/m 2 V/m VAs As A/m 2 V/m 2 A/m -
electric vector potential magnetic vector potential magnetic flux density Eq. (2.5-28) Eq. (2.5-29) 299 792 458; velocity of light (definition) Eqs. (6.3-17)-(6.3-21) Eqs. (6.3-25)-(6.3-28) Eqs. (6.4-6)-(6.4-9) electric flux density Eq. (2.5-18) Eqs. (6.3-47)-(6.3-50) electric field strength energy electric charge Eqs. (2.3-7), (2.3-15) electric current density magnetic current density Eq. (3.2-4) Eq. (4.2-3) Eq. (3.2-32) Eq. (3.2-33) magnetic field strength Eq. (6.3-29) Eq. (6.3-1)
-
Hamilton function 6.626 075 5 x 10 -34, Planck's constant 1.054 572 7 x 10 -34 Eq. (2.4-29) rest mass T/At, Eq. (2.2-6) Eq. (4.2-7)
-
Eq. ( 4 . 4 - 1 0 )
Js Js kg
-
pN (~, o)
-
Eq. (4.2-5)
Q
-
Eq. (1.1-45)
-
AP, E q . ( 5 . 5 - 2 )
-
Eqs. (3.2-20), (3.2-54)
s~ (o)
(Continued)
xxiii
xxiv
F R E Q U E N T L Y USED SYMBOLS s
V/Am
T
s
(0) t
s
At U
VAs
v
m/s
Z - la/C
V/A
s
Z
Ze2/2h7.297 535 • 10-3, Eq. (1.1-45) ZecAe/moc 2, Eq. (1.1-45)
o~ ~e
6 c = 1/Zc A
As/Vm
AI Ar
A 0 b K0
21 22 23 2r
D
m
2 #Pe Pm
Z/c
Vs/Am As/m 3 Vs/m 3
Pr
P~ ff
magnetic conductivity arbitrarily large but finite time interval Eqs. (3.2-20), (3.2-55) time variable arbitrarily small but finite time interval Eq. (2.5-2) Eqs. (3.2-8), (3.2-10)-(3.2-13) velocity Eqs. (3.2-20), (3.2-21) 376.730 314; wave impedance of empty space 1, 2,...; charge number
A/Vm
Eq. (5.5-2) Eqs. (2.4-11), (2.4-13), (2.4-14) 4rtZ~, Eq. (5.5-2) Eq. (5.5-2) 1/#c2; permittivity symbol for difference quotient: ~IF/Ax, Eq. (1.2-1) left difference quotient, Eq. (1.2-5) right difference quotient, Eq. (1.2-4) symbol for finite difference: x + Ax x/cAt, normalized distance; Eqs. (2.2-6), (2.3-1) t/At, normalized time; Eq. (2.2-6) Eq. (2.3-18) Eq. (2.4-32) ecAtAmox/h; Eqs. (2.3-2), (3.2-36) Eqs. (2.3-2), (3.2-36) Ckeo/cAmox; Eqs. (2.3-2), (3.2-37) h/moc = 8.89 x 10-15 for ~+ and ~-, Eq. (1.1-45) Eq. (5.5-2) 4~z x 10-7; permeability electric charge density magnetic charge density ar, Eq. (5.5-1) constant, Eq. (2.3-25) electric conductivity, Eq. (1.1-7)
FREQUENTLY
De 4'm ~o, q~l tI/x0 , tI'/x 1
tPxOxj kI'/x 1xj
V A
USED SYMBOLS
electric scalar potential magnetic scalar potential Eq. (2.3-38) Eq. (2.1-38) Eq. (2.1-8) Eq. (2.1-8) Eq. (2.1-25), (2.2-26), (2.1-37) Eq. (2.1-37)
xxv
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ADVANCES IN IMAGINGAND ELECTRONPHYSICS,VOL. 137
1 Introduction
1.1 M O D I F I E D M A X W E L L EQUATIONS AND BASIC RELATIONS
During the last century an enormous number of books and journal articles has been published on solutions of Maxwell's equations. Practically all of them were about sinusoidal waves. These solutions were outside the conservation law of energy, since periodic sinusoidal waves, infinitely extended, must have infinite energy unless their power is zero. We have found only two papers that used Gaussian pulses rather than periodic sinusoidal functions to keep the energy finite (King and Harrison 1968, King 1993). The Gaussian pulse still starts at minus infinity, which prevents its use as a signal. A signal has to be zero before a certain finite time. This need was recognized by mathematicians who coined the term causal functions for functions that are zero before a finite time. Since signals--like any producible or observable electromagnetic wave--must have a finite energy, they are mathematically represented by quadratically integrable causal functions or signal solutions. The lack of signal solutions of Maxwell's equations must always have been a problem for its serious students. The best equations of the theory of electromagnetism did generally not provide the only solutions that could be produced or observed experimentally. Stratton (1941) tried to overcome this problem but his mathematical derivations were not well received and we do not find them in the textbooks published later. From 1986 on it was recognized that Maxwell's equations could generally not have solutions that satisfied the causality law, which explained the absence of signal solutions (Harmuth 1986a,b,c; Hillion 1991, 1992a,b; 1993). The addition of a magnetic (dipole) current density term corrected this shortcoming 1 (Harmuth 1986a,b,c; Anastasovski et al. 2001). Rotating magnetic dipoles produce magnetic dipole currents just as rotating electric dipoleswe.g, in a material like barium-titanate--produce electric dipole currents. Maxwell's equations modified by a magnetic dipole current density can be written in the following form with international units in a coordinate system at rest: 1publications of Lehnert (1995, 1996) as well as Lehnert a n d Roy (1998) call for an "extended electromagnetic theory" that keeps Maxwell's gm -- 0 in the following Eq.(2) but replaces Pe [As/m 3] by a/c [(A/m2)(s/m)] in Eq.(3). For an overview see Barrett (1993).
ISSN 1076-5670/05 DOI: 10.1016/S1076-5670(05)37001-7
Copyright 2005, ElsevierInc. All rights reserved.
2
1 INTRODUCTION
0D
curl H = - - ~ + g,
(1)
0B - curl E = --~ + gm
(2)
div D = pe div B = 0
(3) or div B = Pm
(4)
We use here an old-fashioned notation but the problem of Maxwell's equations with the causality law was found with this notation. It may well be that the physical meaning of equations is easier to grasp with this old notation. The operators V and [--1 will be used when mathematical compactness is more important than physical meaning. The symbols E and H stand for the electric and magnetic field strength, D and B for the electric and magnetic flux density, ge and gm for the electric and magnetic current density, Pe and tim for the electric and a possible magnetic charge density. The existence of magnetic charges is not accepted as confirmed experimentally, but there are serious theoretical arguments for their existence. The magnetic dipole current density g m does not depend on the existence of magnetic charges. The existence of magnetic dipoles is not disputed and rotating dipoles create dipole currents. The electric current density ge has always stood for electric monopole current densities requiring charges, but also for electric dipole and higher order multipole current densities whose total charge is zero. Without dipole currents no electric current could flow through a capacitor whose dielectric is an insulator for electric monopole currents. The use of the term signal solution should not mislead one to believe that the modified Maxwell equations are only of interest in low-energy effects typically associated with information transmission. The electromagnetic pulse of an atomic bomb explosion or the electromagnetic radiation produced by a supernova explosion fit the definition of a signal solution too. The electric and magnetic field strengths produced in these examples are generally not defined by Maxwell's equation but they are defined by the modified Maxwell equations. Equations (1) to (4) are augmented by constitutive equations that connect D with E, B with H, ge with E, and gm with H. In the simplest case this connection is provided by scalar constants called permittivity e, permeability #, electric conductivity (7, and magnetic conductivity s. The electric and magnetic conductivities may be monopole current conductivities as well as dipole or higher order multipole current conductivities:
D=eE B=#H ge = erE
(5) (6) (7)
gm
(8)
--" s H
1.1 MODIFIED MAXWELL EQUATIONS AND BASIC RELATIONS
3
In more complicated cases e, #, a, and s may vary with location, time, and direction, which requires time-variable tensors for their representation. In still more complicated cases Eqs.(5) to (8) may be replaced by partial differential equations. For sinusoidal time variation of E, H, D, B, ge, and gm one may use functions of frequency e(w), #(w), a(w), and s(w) but this takes one beyond the conservation law of energy and the causality law. The widespread use of e(w), #(w), and or(w) demonstrates our ability to derive useful results from wrong theories. A number of basic relations derived from the modified Maxwell equations will be needed. They are listed here without derivation. References for their derivation are given. The electric and magnetic field strength in Maxwell's equations are related to a vector potential Am and a scalar potential r
0Am
E =
grad Ce
Ot c
H = ~ curlAm
(9) (10)
For the modified Maxwell equations we have to add a vector potential Ae and a scalar potential era. Equations (9) and (10) are replaced by the following relations2:
E = - Z c curl Ae c
H = ~ curl Am
0Am
grad Ce
Ot
0Ae Ot
grad Cm
(11) (12)
The vector potentials are not completely specified since E q s . ( l l ) and (12) define only curl Ae and curl Am. Two additional conditions can be chosen that we call the e x t e n d e d L o r e n t z convention:
1 0q~e divAm + c20q $ - - 0
(13)
1 0r div Ae 4- c-~ 0--'~ -- 0
(14)
The potentials of E q . ( l l ) and (12) then satisfy the following inhomogeneous partial differential equations: 2Harmuth and Husain 1994, Sec. 1.7; Harmuth et. al. 2001, Sec. 1.6.
4
1 INTRODUCTION
1 02Ae c 2 0t2 -- [:]Ae -
V2Ae V2Am
1 0 2 A m _ [-]Am
C2 Ot 2
-
=
1 zcgm Z
---ge
c
1 02r _ [-1r = - Z c p . c2 0 t 2
V 2r
1 02 ~)m : [-](~m =
V2r
C2 a t 2
"~_C - - Z pm
(15) (16)
(17) (18)
Particular solutions of these partial differential equations may be represented by integrals taken over the whole space. We note that the magnetic charge density Pm may be always zero, which implies Cm ---0:
Ae(x,y,z,t)= 4~zcl / / / gm(~,~,r Am(x,y,z,t) = ~cZ ///" ge(~,rl,~,t- r/c) d~
(20)
Zc / / / p~(~,~?,~,r t - r/c) d~ d~ d~
(2~)
~ ~ Zc ///" pm(~,U,~,t-r/C)cl~d~d -r"
(22)
Ce(X,y,z,t) = ~
r
(19)
Here r is the distance between the coordinates ~, U, ~ of the current and charge densities and the coordinates x, y, z of the potentials:
= [(~ - ~)' + (y - ~)' + (z - c) '] ' j '
(23)
If there is no magnetic charge Pm, the scalar potential Cm drops out; if in addition there are no magnetic dipole current densities gm, the vector potential Ae drops out too. Equations (11) and (12) are then reduced to the conventional Eqs.(9) and (10). The field strength H(~, 9) does not have defined values but E(~,0) has. This implies that Am in Eq.(10) is undefined but OAm/Ot in Eq.(9) must be defined to yield defined values for E(~, 0). Hence, Eqs.(9) and (10) contain a contradiction and cannot be used 3. We note that only Eqs.(1)-(4) are needed to derive Eqs.(11)-(22), the constitutive equations (5)-(8) are not used. The Lagrange function and the Hamilton function shall be needed for a particle with mass m, charge e, and velocity v in an electromagnetic field. From the Lorentz equation of motion 3For a more detailed discussion of this contradiction see Harmuth et. al. 2001, Sec. 3.1.
1.1 MODIFIED MAXWELL EQUATIONS AND BASIC RELATIONS
0 ( m v ) = eE q
Ot
Ze
c
v x H
5
(24)
one can derive for v << c from the original Maxwell equations the Lagrange f u n c t i o n 4 f~M:
1
~M -- -~m( j:2 +
y2
+
~2
) -Jr-e.(--r -Jr-nmxSC + nmy~l + nmz2;)
(25)
The modified Maxwell equations yield a Lagrange function represented by a matrix 5 L that is derived in some detail in Section 1.3 and is shown in Eq.(1.324). A more compact representation is by means of unit vectors e~, ev, ez:
-- ~-~M -[- ~ c -- (f'aM -[- L~x)e~
+ (f-~M AwLcy)ey + (f-aM -~- Lcz)ez
(26)
The term LM is the same as in Eq.(25) while Lc~ is defined by
Zef[
LCX ---
Z~J:(yAez - ~Ao,) + --
---j
~
~
- c 2 ( OA ~z Oy The terms LCy and Lr
OCm- - y OCm + A o z 9 -
~
-5;
Ao~
OA ey ( ~I~y 0 ~0 )]dx Oz ) + + )(~lAez-zA~y ]
are obtained from Lr
x ~ y - - - , z ~ x and x ~ z ~ y ~
(27)
by the cyclical replacements
x.
The derivatives 2, y, i in Eq.(27) can and should be replaced by the components of the moment p:
p = pxez + pyey -b Pzez Ze OL~ O(Lm + Lc~) _ 'm2 + eAm~ + - - ( A e z ~ ] - Aey:Z) P~ = 02 02 c Ze OLv -- m{l +eAmy + ~ ( A e x z Aezx) PY= 09 c Ze OLz P z - 0:~ = m i + eAmz + - -c( A ~ y 2 - Ae~])
(28)
_
w
(29)
(30) (31)
This is a major effort and we rewrite only the first component of Lcx, denoted f~cxl, in this form 6' 4The subscript M refers to 'Maxwell'. 5The subscript c refers to 'correction'. 6For the other components see Harmuth et. al. 2001, Sec. 3.2.
6
1 INTRODUCTION
__
ze(
Lcxl = Z e (A~zy - Aeyz,)'Jc = m2c2
c
-f- - -
Aex(p - eAm)y - A~y(p - eAm)z
Aez[Ae x (p - eAm)]y - Aey[Ae • (p - eAm)]z
'lTtC
x
(p -- eAm)x q-
~c
AexAe" (P - eArn)
-t---[Ae x (p-eAm)]x
?T/,C
i
1+
--
?T/,C
A
(32)
The Hamilton function Jt: derived from the Lagrange function/5 of Eq.(26) can be represented as a vector too. If either the energy m c 2 is large compared with the energy due to the potential A~ or the magnitude of the potential Am is large compared with the magnitude of Ar we obtain the following simple equations:
JC = ~ z e x + ~ y e y + 9fzez
(33)
1 )2 9s = 9--~m(p- eArn -~-"eCe -- fOcx
(34)
1
~c~ = ~-~ (p - CAm) 2 + r162 --
~r
(35)
9s = ~---(P -- eArn) 2 -4- eCe -- s
(36)
Zm-
The terms (1/2m)(p - cAm) 2 + eCe equal the conventional one derived from Maxwell's original equations. If the simplifying assumptions made for the derivation of Eqs.(34)-(36) are not satisfied one obtains the following exact but much more complicated Hamilton function:
1[ )2 ( Z e ) 2 { = ~m ( p - e A r n -i- - - m c 2[Ae.(p-eAm)]2+
[nex(p-eAm)]2
}
-4-(-~c)4Ae2[Ae- (P - eAm)] 2] [l-~-(Z---~)2Ae2]-2-~-eCe-~'~c (37) Terms multiplied by ( Z e c A e / m c 2 ) 2 or ( Z e c A e / m C 2 ) 4 have been added to the simplified terms of Eqs.(34)-(36). Dropping the simplifying restriction v << c we obtain more complicated expressions. In particular, the Hamilton function can be written with the
1.1 MODIFIED MAXWELL EQUATIONS AND BASIC RELATIONS
7
help of series expansions only 7. The relativistic generalization of the Lagrange function of Eq.(26) is:
v2/c2) 1/2 + e ( - r
/5 = - m 0 c 2 ( 1 -
+ Am" v) + f~c
(38)
In analogy to Eqs.(34)-(36) we first write an a p p r o x i m a t i o n for the three c o m p o n e n t s of the H a m i l t o n function t h a t holds if the energy due to the potential Ae is small compared with the energy m o c 2 / ( 1 - v2/c2) 1/2 and the m a g n i t u d e of A~ is small compared with the m a g n i t u d e of Am:
9{.~ = c[(p - eArn) 2 +
m2c 2] 1/2 -~- e r
~ v = c[(p - eArn) 2 +
e __ f--'cx
(39)
m20c2] 1/2 +
e C e -- f-~cy
(40)
9[z = c[(p - eArn) 2 + mo2C2] 1/2 +
e r e __ ' ~ c z
(41)
If we leave out the correcting terms Lcx, key, Lcz we have the conventional relativistic Hamilton function for a charged particle in an electromagnetic field, written with three c o m p o n e n t s rather than one. We call these equations the zero order a p p r o x i m a t i o n in ae = ae(r, t) = ZecA./moc 2. Let us note t h a t a , is a dimension-free normalization of the m a g n i t u d e of the potential A~(r, t). A first-order a p p r o x i m a t i o n in a , is provided by the following equations:
[}{~x -- c[(p -- eArn) 2 +
m2c2]1/2(1 -k oteQ) --}-eCe - f~cx
(42)
9~y = c[(p - cAm) 2 +
m2c2]1/2(1 + aeQ) +
(43)
eCe -- f~cy
9-(,z=c[(p-eAm)2 +m2c2]l/2(l +oleQ)--beCe-~cz 1
(44)
(p -- eAm)2[Ae 9(p - eAm)] 2
Q : m20C2 [1 + (p ae = ae(r, t) =
-
eAm)2/m2c2] 3/2 A 2 (p
eArn) 2
ZecA~(r, t) = 2 ze--~2 h Ae(r, t) moc 2 2h moc e
h
Ae(r, t)
?7t0C
e
= 2a--
-
= 2a
AcAe(r,t) e
ae = 2.210 • 105Ae(r, t) for electron, ae = 1.204 • 102Ae(r, t ) f o r p r o t o n
Ze 2 a
=
2h
7.297535 • 10 -3 fine structure constant -
Ac = '
h moc
(45)
T h e fine s t r u c t u r e constant a is a universal constant of q u a n t u m physics. The factor (h/c)Ae/moe normalizes the m a g n i t u d e A~ of the potential A~ by the 7Harmuth et al. 2001, Sec. 3.3.
8
1 INTRODUCTION
mass and charge of a particle interacting with the field; the factor h/c makes ae dimension-free. The correcting terms Lc~, Lcv, Lcz are defined by Eq.(27) and the text following it. The first term Lcxl of Lcx is shown by Eq.(32). Using first-order approximation in ae it becomes:
~cxl = ~Ze ( A e z ~ ]
c
- Aey~,)'Jc =
2C~C em0
Aez (P
-- e A m ) y --
[1 + (p
X
-
Aey ( p
(p -- e t m ) x
[1 + (p
-
-- e A m ) z
eAm)2/rn2oc2]l/2
+ O(a~)
(46)
etm)2/m2c2] 1/2
There was no need so far to distinguish between commuting and noncommuting factors, but our equations make it evident that this problem will have to be addressed eventually. A rule is needed that makes the replacement of two commuting factors by noncommuting factors unique. We use the rule that a commuting product ab is replaced by
1 (ab + ba) ab ---, -~
(47)
if a and b are not commuting. This choice is recommended by the simplicity of Eq.(47). 1.2 BAsic
C O N C E P T S OF THE CALCULUS OF F I N I T E D I F F E R E N C E S
We need the equivalent of differential quotients for the calculus of finite differences. For the first-order difference quotient we generally use the symmetric quotient
dA(O.__~)--, AA(O) = A(O + AO) - A ( O - AO) dO AO 2A0
(1)
The symbol z3 is used for difference operators while A is used for a finite difference. It is usual to simplify writing by the substitution e' = e / / x e ,
,,xe' = , x e / A e
= 1
(2)
and then drop the prime. Equation (1) becomes with this convention
dA(O) AA(O) 1 [A(0 + 1) - A(0 - 1)] dO --~ AO ='2
(3)
According to Fig.l.2-1 one obtains in this way a first-order difference quotient for 0 = 1, 2, ... but not for 0 = 0. To overcome this limitation one may define a right first-order difference quotient
1.2 BASIC CONCEPTS OF THE CALCULUS OF FINITE DIFFERTENCES
0
9
1
2 3 0 ----~ FIG.1.2-1. The average slope of the symmetric first-order difference quotient is defined for 0 = 1, 2, ... but not for 0 = 0.
/•(0+1) ,~
I
A(0-1)
,6 "4
h k'o'~
I
I
-- 0-1
0
I
0+1
0---, FIG.1.2-2. Symmetric difference quotient of first-order [A(O + 1) - A(O - 1)]/2, right difference quotient A(O + 1 ) - A(0), and left difference quotient A(O) - A ( O - 1).
z~rA(0)
= A(O + 1) - A(O)
(4)
t h a t works for 0 = 0. T h e r e is also a l e ~ first-order difference q u o t i e n t
,~lA(O)
Ao
= A ( O ) - A ( O - 1)
(5)
T h e left first-order difference q u o t i e n t is not likely e n c o u n t e r e d if 0 is interp r e t e d as time. If 0 is a spatial variable one needs the right first-order difference q u o t i e n t for a b o u n d a r y condition on the left side a n d the left one for a b o u n d a r y c o n d i t i o n on the right side. F i g u r e 1.2-2 shows the t h r e e difference q u o t i e n t s a n d explains the use of 'left' and 'right'. T h e differential q u o t i e n t dA(O)/dO can be derived from either one of the three difference q u o t i e n t s AA(O)/AO:
10
1 INTRODUCTION
dA(O) dO
. lira 1 [A(O+ AO) - A ( O - AO)] /',O--.dO lim
1 [A(O + A0) - A(0)]
A O---,d O " ~
lira ~--~ 1 [A(0)- A ( O - A0)] ~O--.dO
(6)
The second-order difference quotient is practically always used in the symmetric form:
A2A(O) zi2A(0)
d2A(O)
(z~O) 2
dO 2
A(O + AO) - 2A(O) + A ( O - AO)
z~O2
(AO) 2
=A(O+I)-2A(O)+A(O-1)
forA0=l
(7)
The second-order difference quotient does not formally follow from using twice the first-order symmetric difference quotient. However, one may use first the left and then the right first-order difference quotient or vice versa: /~r ( / ~ l A ( 0 ) ) /~r [A(O)-A(O-1)]=[A(O+I)-A(O)]-[A(O)-A(O-1)] 7he Ao = = A(O + 1) - 2A(0) + A ( O - 1)
Al
Xe
(/~rA(0))
7~0
/~1 [A(O+I)-A(O)] = [A(O+I)-A(O)]-[A(O)-A(O-1)]
= -~
= A(O + 1) - 2A(0) + A ( O - 1)
(8)
Mathematicians are usually satisfied with the right difference quotient. But its use in physics introduces an asymmetry that is strictly due to mathematics and that may lead to divergencies that are avoided by the symmetric difference quotient 1. Hence, we use whenever possible Eqs.(1), (3), and (7) for the first- and second-order difference quotient. As a further example of the importance of symmetry consider the difference quotient of the product u(O)v(O) in various notations:
~[~(O)~(0)]
AO
1
= 2zx0 [~(o + ~ o ) ~ ( o + AO) - ~(0 - ZxO)~(O - ZX0)] = 2AO {~(0 + ~XO)[v(O + ~XO) - v(O- AO)]
+ ~(O - AO)[~(O + AO) - ~ ( O - ~XO)]}
= ,~(0 + AO)A~,(O) A~(0) ,50 + v(O- AO) AO 1Harmuth 1989, Sec. 8.2.
(9)
1 . 2 BASIC C O N C E P T S OF T H E CALCULUS OF F I N I T E D I F F E R T E N C E S
11
Since u(O) and v(O) are not treated equally on the right side of Eq.(9) even though they are treated equally on the left side, we rewrite the second line of Eq.(9):
A~(o)~,(o) 1 = 2zxo {v(o + A0)[~(0 + ZX0)- ~(0 - zX0)] Ao + ~(0 -/',0) [v(0 + zX0) - v(0 - zX0)]} Av(O) A~(o) = ~ ( o - Ao) Ao
+ ~(o + zxo) ,4O _
(10)
The sum of Eqs.(9) and (10) divided by 2 is symmetric in u(0) and v(0): 1 /~(o)~(o) (o_. / ~___~) = -- [~(o + Ao) + ~ ( o - f0)] Ao 2 Ao
+ ~1 [v(O+ A0)
+ v ( 0 - A0)]
A~(o) .....
AO
(11)
According to Eq.(7) we have the relations
~(0 + A0) + ~(0 -/xo) = 2~(o) + /~2~(o) /~o~ (f0)~ ~(o + Ao)+ ~(o- Ao) = 2~(o)+ ...../~0~" (/xo)~ and Eq. (11) becomes:
/~(o)~(o) = (~(0)~ (zx~ z l v (z~~z~02( ~ Ao
A0 (
+ = ~(o)Av(o)
A~(o)
(A0)~ A~(0)) Zl~(0) ~,(o)-~
2
Ao + ~(o) Ao + O(fO)~
Ao2
Ao (12)
The symmetrizing of Eq.(9) yields the exact equivalent of the expression for d[u(O)v(O)]/dO of the differential calculus. If one does not like to ignore the terms O(A0) 2 one can use Eq.(11) instead of Eq.(12). We do not know whether the use of Eq.(11) leads to significantly different results than the use of Eq.(12). We encounter here the first time the lack of development of the calculus of finite differences and we choose to use the simpler Eq.(12).
12
1 INTRODUCTION
Differential calculus permits us to define integration by means of the differential equation
du(x) =~(x) dx
(13)
u(x) = f f du(x) dx dx = / qp(x)dx
(14)
and its formal solution
A very similar process leads in the calculus of finite differences from the difference quotient of first order to summation rather than integration. In order to use the results of NSrlund and Milne-Thomson 2 we follow closely their derivation. This forces us to use the notation
~ ( ~ ) = ~(~ + ~) - ~(~) = ~(x)
(15)
co
To connect Eq.(15) with our notation in Eq.(1) we choose first co = 2Ax and then co = 2' 1 2Ax [u(x + 2Ax) - u(x)] = ~o(x)
(16)
W i t h the substitution x = x I - Ax we get:
~(x' + zxx) - ~ ( ~ ' 2Ax l [ u ( x , + 1) 2
Ax) = ~ p ( x ' - Ax),
u(x'
1)]
~o(x'
1),
x ' = x + Ax x'
x + 1 Ax '
(17) 1
(18)
Consider now a function f(x)
f ( x ) = Co - co[qo(x) + qo(x + co) + ~(x + 2co) + . . . ] (19)
= c0-~~~(~+~) s-'-0
and the shifted function f ( x + w)" c~
f (x -t- co) -- Co - co ~
qo(x -t- co -t- sco)
s'--O
2NSrlund 1924, Ch. 3; Milne-Thomson 1951, Ch. VIII.
(20)
1.2
BASIC
CONCEPTS
OF
THE
CALCULUS
OF
FINITE
13
DIFFERTENCES
A formal solution of Eq.(15) is obtained by the substitution of f(x) for u(x). For a reason to be seen presently we may replace the constant Co by a definite integral oo
Co = f to(v)dv
(21)
0
HauptlSsung or principal solution of Eq.(15), which is also called of the function to(x), may then be written in the following form:
The
oo
the sum
oo
= / v(.)d. d 0
+
(22)
s=O
The integral is used instead of the constant Co because a divergency of this integral may compensate a divergency of the sum, which a constant Co in Eq.(19) could not do. The principal solution of Eq.(15) is thus obtained by summing the function to(x). NSrlund introduced the following notation for this summation: x
oo
(23) c
0
s=O
The function F(x l w) is said to be obtained by summing to(x) from c to x. The integral in Eq.(23) represents an 'integration' or 'summing' constant like the c in an integral fc f(x')dx' if it and the sum converge. In this case one may write Co for the integral rather than evaluate Eq.(21). NSrlund generalized the definition of F(x Iw) beyond what is shown in Eq.(22) to functions that can be made summable by means of an exponential function:
F(x,w)=.---.(olfimto(v)e-'~(~')dv-w E to(x+sw)e-'~('+'')) O0
0
O0
(24)
s=0
This limit process makes some functions to(x) summable. It is needed to obtain the sum of the constant a in Table 1.2-1 (Milne-Thomson 1951, p. 203). We write F(x[w) for the symmetric difference quotient on the left side of Eq.(17), substituting first w = 2Ax, x = x I - Ax and then w = 2, x = x / - 1:
14
1 INTRODUCTION
for c O = 2 A x , X ! --
AX
x=x
t-Ax
oo
oo
P
+ 2sAx) M,]
J
c
0
s=0
X
f'~
G(x)
p
oO
OO
= F(~ 12A~)= , ~ ( ~ ) a ~ = / ~ ( ~ ) ~ - 2 z x ~ ~(x+ 2~zx~) X,..A
J
C
c
s--O
A{M u = A u for w = 2 A x
(25)
For the choice co = 2, x = x' - 1 we get the following result that will be used from here on unless Eq.(25) is specifically indicated"
for c 0 = 2 , X ! --
x=x'-I
1
OO
OO P
F(x' -
112) =
,~ ~(~)a, = /
~(~)d.- 2 E ~(~'- 1 + 2s)
J
c
0
s=0
X
P
OO
OO
G(x) J c
C
A u=Au co
s--O
forw=2
(26)
Let us derive the summation directly in detail from Eq.(17) by the substitutions
f(x'-Ax)
= Co-2Ax[cp(x'--Ax)+~o(x'-Ax+2Ax)+cp(x'-Ax+4Ax)+
...]
oo
= Co - 2Ax E
~o(x' - A x + 2sAx)
S'--0
f(x' +Ax)
--- C o - 2 A x [ c p ( x ' + A x ) - f - ~ ( x ' - 4 - A x + 2 A x ) + q p ( x '
+Ax+4Ax)+
oo
= Co - 2Ax E
~o(x' + A x + 2sAx)
8--0
Substitution into Eq.(17) yields: 1 2)Xx [f(x' + Ax) - f ( z ' -
Az)] = ~ ( z ' -
Ax)
Hence, we may define the principal solution in analogy to Eq.(22)'
...]
1 . 2 BASIC C O N C E P T S OF THE CALCULUS OF FINITE D I F F E R T E N C E S
/.oo
15
oo
F(x'12Ax ) = ] r
- 2Ax E
d
0
~o(z' + 2sAx)
s-----0
/.oo
oo
y(~' - A~ 12A~) = ] ~(~)d~ - 2 ~ z ~ J 0
~(~' - A~ + 2~A~)
s--0
oo
oo
P
F(x ,2A~) = ] ~(~)d~ - 2A~ ~ J
0
~(~ + 2~A~),
~ = ~' - A x
s=O
This is again Eq.(25) for w = 2Ax. Let ~ ( x ) in Eq.(15) be the exponential function e - ~ and let w as well as x be real and positive. We obtain:
x
oo
C
--
e -c-
We
oo
8--0
C
-x
(27)
1 - e -~'
Consider now the following difference equation to see what the s y m m e t r i c f e r e n c e quotient and the choice w = 2 do to E q . ( 2 7 ) "
1 [v(x' + 1) - v ( x ' A ~ ( ~ ) = -~ -- l[u(x' 2
+ 1)-
u(x'-
u(x') -- ev(x')
1)] = e - ~ ' ---- ~p(x'),
1)] = e -(x'-l)
This equation corresponds to Eq.(13). x'=x+l"
=
dif-
1)
(iO(X ! --
(2s)
We get with the help of Eq.(26) and
~(x') = ~-1~(~,) x !
oo
'
c
= e -1
e -c-
l_e_-------~
oo
s---O
c
e-e-
1
e-e-
1
C
e -x'
(29)
16
1 INTRODUCTION
T h e integral f exp(-x')dx' yields C - e x p ( - x ' ) . E q u a t i o n (29) differs by a factor 2 / ( e - e -1) - 0.85092 from this result. It has been shown a t h a t Eq.(27) not only holds for the exponential function qo(x) = e -x in Eq.(15) but generally for the exponential function e ~ in the complex plane as long as the condition ]w] < 2~/]~] is satisfied: x
e" ~ A v =
e"~
~
"
'T
1
C
Iw[ < -27r -
we'Y~
I'YI
-e.~'
X !
8
e~(~+ 11Au =
2e~X'
e~(c+l) 7
e - - 7 _ e-7 ~
I'Y! <
(30)
a
One m a y use this relationship to solve Eq.(15) for ~p(x) = sin 7x:
_1 [u(x
1•
.
.
+ w) - u(x)] = sin -)Ix = ~ (e '~x - e -*'y~)
co X
u(x) = ~1 8 (e ''r'' " - e -''r') Av C.
1 ( =
2~
e i'Yc i7
cos 7c
we i'~x
e-i~c
1 - ei~ "~ +
-i 7
w cos-y(x - w/2) 2 sin('),w/2) '
we-i'~x I + 1 - e -~"~
Iwl
2~ < I'yl
(31)
We need the corresponding result for Eq.(28). I n t e r m e d i a t e steps are given to help with verification'
1(.,
/~v(x') = -~1[v(x' + 1) - v(x' - 1)] = sin 7x' = ~-~ e wx - e -i'yx 1
~[V 1 (X' "F 1) -- V I(x'
1 ei.~x, ,
-- 1)] "-- ~"
1
tt I(x')
---- 2ie-i'Yvl
,)
(32)
(x')
9 ,
(33)
~[Ul(X' Jr- 1) -- ?.tl(X' -- 1)] = e '~(~ -1) = ~ ( x ' - - 1) 1
1
-i•x'
~[v2(x' + 1 ) -
v2(x'- 1)] = - ~ e
1 ~[u2(x' + 1) -
u 2 ( x ' - 1)] = e -i~'(x'-l)
3NSrlund
1 9 2 4 , p. 81; M i l n e - T h o m s o n
, :
1 9 5 1 , p. 231.
u2(x') = - 2 i e i ' r v 2 ( x ~ ( X ' - - 1)
') (34)
1.2 BASIC CONCEPTS OF THE CALCULUS OF FINITE DIFFERTENCES
17
T A B L E 1.2-1 SUMS v ( x ' ) = v(x) OF CERTAIN FUNCTIONS ~p(x') = ~ ( x ) ACCORDING TO EQ.(26) FOR w = 2. THE INTEORALS OF ~ ( x ) ARE SHOWN FOR COMPARISON.
x
~(z)
v(x) = 8
./~(x)dx
~(u + l)Au
c
a e -~
a(x - c -
ax + C
e - ~ + e -(c+1) sh 1 e'rX
e'Y~
1)
- e -~ + C
e-)'(c+l)
e'r x
+~
sh 3" 3' 3" complex, [3"1 < 7r
1)] I"rl
+ C
3"
sin 3"x
cos 3"x + cos[~(~ + sin 3" 3"
cos 3"x
sin 3"x sin[3"(c + 1)] , sin 3' + 3" ca 3"x sh3"
ch ~(c + 1), i~[ < 3"
3"______xx ch + C
sh3"x
s h ' r ( c + 1), I'r[ < 3"
3"___x_x sh + C
ch 3"x
sh'rx sh 3"
e~X sin 3"x
A2 +.),2( A ~ 1 7 6 1 7 6 1 7 6
I~1
_ cosf__x + C 3"
<
+sin3"x + C
<
3"
3" 3" e Ax
e )~X
Ao=
3"0 =
Co =
A2 +
2(A2+3"2)shAcos7 cos 23" - ch 2A 2 (A2 + 3"2) ch A sin 3" cos 23" - ch 2A e)~(c+l)
A2 + 3"2{Asin[3'(c + I)] -3" cos[3"(c +
1)1}
e Ax
e ~ cos 3"x
e Ax
A2 + 3"2 (A~ cos 3"x + 3"0 sin 3"x) + C1 Ao and 3"0 are shown above
c1 =
.),2 (A sin 3"x - 3" cos 3"x) + C
eX(C+l) ~ + ~ {~ cosb(~ + 1)] - 3 ' sin[3"(c -t- 1)]}
A2 + 3' 2 (A cos 3"x + 3" sin 3"x) + C
1g
1 INTRODUCTION
1
1
v ( ~ ' ) = ~x(~') + ~2(~') = g ~ ' ~
(~') - ~ - ' ~ 2 ( ~ ' )
x /
x /
x /
= S sin 3`(u + l)Au = lei'Y Sei'yvAu-- 2ie c
12
1 /'
ei7(c+1)
= -~ ~ --
i~/
cos[3,(c + 1)] = 3'
i~ Se-i"/UAt,, c
2e i7(x'+l) e -in(c+1) 2e-i7(x' + 1) ) + 1 -- e -2i~ -- 1-- e2i7 + --i 7 cos 3`x' , 171 < 7r sin 7
(35)
The integral f sin 7x' dx' yields -(cos 7 x ' ) / 7 + C. Equation (35) has the factor 1/sin "7 rather than 1/7. Table 1.2-1 shows a collection of sums. Except for the last two they are all due to NSrlund. There is a long way to go to turn Table 1.2-1 into something comparable to our available tables of integrals. The integral f qo(x)dx in the first row of Table 1.2-1 is a simplified form of f f ~(u)du. A corresponding simplification will sometimes be used here for the summation in the first row of Table 1.2-1: x
u(x):
8
x
qp(u+l)Au=
e
8qO(x+l)Ax=
8qo(x+l)Ax
(36)
e
Summation and differentiation are inverse operations in the sense that integration and differentiation are inverse operations4: x
x
c
c
For the symmetric first-order difference quotient of Eq.(1) and the secondorder difference quotient of Eq.(7) we must adopt the convention: 0 8 z~2A(u) Au = /~A(O) z~u2 z~0 + C
(38)
e
1.3 LAGRANGE FUNCTION AND MODIFIED MAXWELL EQUATIONS We have previously derived a Lagrange function for the modified Maxwell equations 1. We give here a more detailed and hopefully more understandable derivation. The starting point is the Lorentz equation in MKSA notation: 4NSrlund 1924, Ch. 3, w3/22; Milne-Thomson 1951, Sec. 8.1. 1Harmuth et al. 2001, Sec. 3.2.
1.3
LAGRANGE FUNCTION AND MODIFIED MAXWELL EQUATIONS Ze
d (my) = eE + ~ v
dt
19
• H
c
(1)
The field strengths E and H of the modified Maxwell equations are written in the potential form. We use the old-fashioned notation curl and grad instead of V• and V since it seems to make the physical processes more lucid:
0Am
Ot
E = - g a c u r l Ae c H = ~ curl Am
0Ae
Ot
grad r
(2)
grad r
(3)
The substitution of E and H into Eq.(1) yields
d (my)
dt
e(
0Am
grad q~e - 0----~ + v • curl Am
Ze( c
) 0Ae
vxgradr
--~+
c2
curlAe
)
(4)
The first line is the conventional equation of motion of an electrically charged particle in an electromagnetic field. The second line contains additional terms due to the modification of Maxwell's equations. Two of them with the potential Ae apply to the undisputed magnetic dipole currents produced by the rotation of magnetic dipoles as well as to hypothetical magnetic monopole currents produced by so far not reliably observed magnetic charges. The term with the potential r is strictly hypothetical until the observations of magnetic charges are widely accepted. Equation (4) is to be rewritten with the help of a Lagrange function L, spatial variables xj and velocity variables Oxj/Ot - ~cj. The Lagrange function must satisfy the Euler equation
d(O,L) d--t ~
OL - Oxj'
j=l'
2, 3
(5)
which derives the equation of motion from the principle of least action by Maupertuis and Hamilton. In order to transform Eq.(4) we use a relation found in the literature (Morse and Feshbach 1953, part I, p. 295) dAm
dt
0Am -- ~ -[- v - ( Y A m )
Ot
which assumes for Cartesian coordinates the previously used form
(6)
20
1 INTRODUCTION dAm dt
0Am =
Ot
0Am +
0Am
Ox ic +
0Am
Oy ~] +
(7)
Oz :k
In analogy to Eq.(6) we get the further equation d(v • Ae) dt
0(v • Ae) =
Ot
+ v . [V(v • A~)]
0v 0A~ = 0---t-x A~ + v • - - ~ + v . [V(v • Ae)]
(8)
For Cartesian coordinates we again get the previously used form
d(v•
d-7
0Ae
5-i-
(
0
0
0)(v•
)
(9)
Substitution of Eqs.(6) and (8)into Eq.(4) brings: Ze d
d (my + eAm) + -(v • Ae) d-'t c d-t = e l - g r a d Cm + v" (VAIn) + v • curl Am] + --c
- v x grad ~m nt- "~- x Ae -Jr-v . [V (v x Ae)] - (I2 curl Ae
(10)
For Cartesian coordinates we had obtained: Ze d
d (mv + eArn) + - (v x Ae) d'-~ c d-t ( 0Am 0Am 0Am ) = e - g r a d C e + Ox :i: + Oy ~1+ Oz ~ + v x c u r l A m -~---c - v •
~xx+Y~y -~- ~z - c 2 curl Ae]
(11)
We may break Eq.(10) into two equations. The first equals the one obtained from the original Maxwell equations and the second contains the terms due to the modification of Maxwell's equations: d ( m y + eAm) dt
e l - g r a d r + v (VAm) q- v • curl Am]
(12)
1.3 LAGRANGE FUNCTION AND MODIFIED MAXWELL EQUATIONS
Ze d ( v x A , ) = ~Ze (_ v x g r a d Cm+-E7 o~Ovx A e + v - [ V ( v • c dt c
21
2 curiA ~) (13)
The use of these equations is complicated by the dyadics VAm in Eq.(12) and V(v x A~) in Eq.(13). They are avoided for Cartesian coordinates, a fact which greatly favors them. Instead of Eqs.(12) and (13) we use the two corresponding equations derived from Eq. (11):
d (my 4. eArn)
e(
dt
gradr oqAm
i)Am
Ohm
4- Ox dc 4- Oy 94- Oz ~"4. v • curl Am
)
(14)
Ze d (v • Ae) = ~Zec [[ - v x grad (~m 4. "Ov~ X Ae c dt +
5:-a-- + YT- +/~7-
-
curl Ae
(15)
With the help of the expressions
v-
x e x 4. [ley + Zez
Am "-Amze~ 4- Amyey 4- Am~e. grad Ce
0r e + 0r ez
/)r e~ + =-h7 -5gy
((~nmz
(~Amy~
curlAm--~ ~
-57
G?ex+\"
(Oamx
OAmz.~ ((~Amy Ox /l ey + \ ~x
OAm%~
~y 7] ez (16)
we rewrite Eq.(14) in component form: d
d
~-7(rnv 4. eArn) --~-7[(rnd: + eAmx)ex 4. (mfl 4. eAmy)ey 4. (m:~ 4. eAmz)ez] -
0 e~xx(-r
+ xAmx + 9Amy + ~Am~)ex
0 + e~--(-r Y
+ ~Amx + 9Amy + ~Am~)ey
(;9 4. e-a--(--r + dgAmx 4- ~]A my 4. z A mz ) ez oz- -
(17)
22
1 INTRODUCTION
We note that in lines 2, 3 and 4 the factors of the unit vectors ex, ey, e~ have the same form except for the differential operators O/Ox, O/Oy, O/Oz. These factors plus the differential operators match the right side of Eq.(5) for every j. In order to get the left side of Eq.(7) correctly we must add a term 89 4- 92 4- 22) to the common factors of lines 2, 3 and 4. We obtain the Lagrange function f~M of the conventional Maxwell equations:
1
2
~/2
~2) 4. e ( - r
4- "x,Amx 4- ~]Amy 4- 7,Amz)
-~ !my2 4. e v . Am - e(~e
(18)
2
One may readily verify that L = f~M in Eq.(5) satisfies this equation for every xj or every unit vector e~, ey, e~ in Eq.(17). We turn to Eq.(15) and rewrite it in component form too. The following terms are required:
A e - A e x e x + Aeyey + A~ze~ Cm :
v•
Ov
0-~ • A ~ =
~Y-~z
-~y ] ex+ (2:~ff-ff -~ - x--~-z ] ey4. ~, -~y - y - ~ ) ez
(ijA~z - iiA~)e~ + (iiAex - iiA~)ey + (itAly - ~lA~x)e~
v • A e = (gAez curlAm ~ curl Ae
zAey)ex + (~Aez - xAez)ey + (J:Aey - ~]Ae~)ez (19)
for m ~ e
We substitute Eqs.(19) into Eq.(15):
Ze Ze d c dtd (v • Ae) _ --~-~[(~]Aez-zAey)ex+(z, Aex-xAez)ey+(xAey-~]Aex)]ez
_ z~ ~
-9-~z + ~ +
~+y~+
~
(gA~-~Ae~)
c2 ( OA~z Oy \
+ --~ [X-~z -- 2 ; ~
+
OAey O z ) ] e~
-'~ + -~x + Y ~ + ~Z (2;Aex - xAez)
Oz
Ox
ey
Ze [. OCm . (~r ( (~ 0 0 O) + --~ LY--~-z - ~~oy + ~ + ~ + 9 ~ + ~-5; (~Ae~ - 9A~) -c2(
0''Aey'0x
OAe~)lezoy (20)
1.3 LAGRANGE FUNCTION AND MODIFIED MAXWELL EQUATIONS
23
The factors of e~, ey, ez in lines 2- 7 of this equation are not equal as the corresponding factors in Eq.(17) are. To make progress we use intuition and integrate the factor of e , over x:
__ : [:~-~y 'r - fl--~-z Or + (0'~ + x"~x 0 + Y.~y0 + 2-~z 0) (flA,, - z.Aey)
Lc, - Z._.s
_ C2 ( O A e z
Oy
__
OAey
O z ) ] dx
Z e f [,~ Odt)m OdDm [ -~y - 9 ~O z
-- Z-'-~ex ( ~]A e - z A ey ) + ~c +
( 0--~ + y-~y . 0 + 2 ~0 )
( (~]dez - z,dey) - c 20Aezoy
OAeYoz
) ]dx
(21)
The derivative of L~x with respect to x yields the factor of ex in the first line of Eq.(20):
(22)
Z e OLc~, _- ~Ze (flAe. - 2,Aey)
c Oic OL/O2j on
c
This expression fits the left side of Eq.(5), while the derivative OLc.~/Ox of Eq.(21) fits the right side OL/Oxj. Hence, Eq.(21) yields the previously published I x-component Lcx of the Lagrange function L~, where the subscript c stands for 'correction'. The cyclical replacements x ~ y ~ z ~ x in Eq.(21) yield the components L~v and Lc, of the correcting Lagrange function/5c. It can be written as a matrix:
s
=
0
Lay
0
0
0
L~z
(23)
The sum of the Lagrange function LM of Eq.(18) and/Le yields the complete Lagrange function/L. Using the unit matrix we can write this sum as the sum of two matrices:
-- ~-~M q- ~ c
"-
(1. ~mv
~- e v . A m - eCe
(100) 0
1
0
0
0
1
+
0
0)
0
Lcy
0
0
0
Lcz
(24)
If one wants these matrices displayed in the vector component form of Eqs.(17) or (20) one can multiply them with the matrix
24
1 INTRODUCTION
(
ex 0 0 )
exyz --
0
ey
0
0
0
e~
(25)
We observe that the term gm was added in Eq.(1.1-2) to overcome the problem of Maxwell's equations with causality. That this term led to the use of a matrix for the Lagrange function was an unintended consequence. The matrix notation is spread to anything that uses electromagnetic theory, e.g., the Klein-Gordon equation. We recognize here how impossible it is to foresee where a new theory will take us. The Lagrange function LM in Eq.(18) was derived for Cartesian coordinates but it could be written in the second line of Eq.(18) free of any reference to coordinates. We may check this for spherical coordinates. The three components of the velocity v of the point re~ + r~geo + r sin ~)~oe~o are wanted for the directions er, eo,
e~:
v=
Or Or O(rO) O0 O(r sin 0 ~p)0~ Or ot er q O~ ot e~ + 0~0 -~e~
-- re~ + rOeo + (rsin 0)~be~ = xler + x2eo + x3e~o
(9.6)
(27)
Am -- Am~e~ + Amoeo + Am~oe~ 1 2 + r2~2 + (r sin v9)2~21 ~M = ~m[~ + e[--r
+ rArer + r~)Amo + (r sin v~)~Am~]
(2s)
OLM i)LM __ mrS) + eAmo 0r = m r + eAm~, O(rO) OLM
O(r sin 0 ~) 0LM 0 Or = e ~ r ( - r 0L M
0
O ( r O ) -- e O(r?~)
0L M
= e
0
= m r sin ~) ~ + eAm~
+ §
(--r
"~-§ (-r
+ rOAmo + r sin v9~bAm~)
+ r(~Amo + r sin va~bAm~)
(29)
(30)
(31)
+ rArer + rOAmo + r sin ~ ~bAm~) (32)
One recognizes the equivalence of Eqs.(29)-(32) and Eq.(17) if one chooses the variables r, r0, (rsin vg)~o rather than r, ~, ~o for the spherical coordinates.
1.3 LAGRANGE FUNCTION AND MODIFIED MAXWELL EQUATIONS
25
The Lagrange function suggests using the coordinates r, tO, (rsinO)qo. Both notations work. There is an evident advantage in physics of having the same dimension of length for all coordinates as in Cartesian coordinates. No such advantage exists in pure mathematics since it does not use physical dimensions. More effort is required to write the correcting Lagrange function Toc of Eqs.(23) and (24) for spherical coordinates since the components L~x, L~y, L~z apply only to Cartesian coordinates. We can obtain the correcting terms Lr Lc~, Lc~o for spherical coordinates from Eq.(13). The calculation is shown in Section 6.5. Here we list only Lr Lr Lay, which have to be substituted for Lcx, Lcy, Lcz in Eqs.(23) and (24). The vectors ex, ey, ez in Eq.(25) must be replaced by er, eo, e~,:
L
c,r
Ze ~(r~A~ ~ - r sin ~ qbAeo) + -Ze / [ sin ~b -
=
c
c
+ r~Ae~o -- r sin d qbAee +
(~ ~
O(~m O (~(~m O0
sin 0 0 ~
0 )(r~Ae~~
+ q3~
OAe~)]dr
c 2 ( 0 ( A ~ , s i n v Q) r sin ~ 0~
ge
Ze f [ ?~ + ~ c J[rsin0
,
L ~ = ~ r ~ [ r sin ~ CAe~ - § c
+ r sin ~ ~5Ae~ - i~Ae~ +
(0
§ ~ r + qo
r
Lr
0(~m 0~
(+Ao
-
- -r
Or
sinv9 099
Ze Z e f [ 0r sin ~ qb(rAe# - r~Aer) + - r~ c c J[ Or +
r sin ~ r
(33)
0(~m
Or
(r sin ~ qbA~ - §
= ~r +
0q0
r0r
r O~ OAe
rdO
(34)
+ rAeo - rOAer h]
Or
For an explanation of the Lagrange function of Eq.(24) consider a spinning bullet shot upward. The kinetic energy is transformed into potential energy on the way up and back into kinetic energy on the way down. Such transformable energies are represented by the Lagrange function LM of Eq.(18). The rotational energy of the spinning bullet will not be transformed into either kinetic or potential energy. A more complicated bullet that extends helicopter blades when its kinetic energy approaches zero could transform the rotational energy into potential energy--but only in the atmosphere, not above it. Which energies can be transformed into each other depends on the particular physical
26
1 INTRODUCTION
circumstances. An electrically charged particle traveling through an electromagnetic field permits transfer between the kinetic energy 89 2 and the energies e v . Am or eCe according to Eq.(18). The transformation between the 1 e n e r g y ~mv 2 and the energies (Ze/c)vxvyAez or ( Z e / c ) f Vz(OCm/Oy)dx may be more complicated but it is not prohibited. 1 . 4 DOGMA OF THE CONTINUUM IN PHYSICS
The concept of a continuous space and time or a space-time continuum seems to have originated at the Eleatic school of the Greeks in southern Italy. Zeno of Elea (c. 490-c. 430 BCE) advanced the paradoxes of the race between Achilles and the turtle as well as of the arrow that cannot fly to show that infinite divisibility of space and time was not possible. Aristotle (384-322 BCE) made it clear that the concept of the continuum was discussed before he wrote his Physica: Now a motion is thought to be one of those things which are continuous, and it is in the continuous that the infinite first appears; and for this reason, it often happens that those who define the continuous use the formula of the infinite, that is, they say that continuous is that which is infinitely divisible. [Apostle 1969, Book III (F), 1, w2] Aristotle refuted the paradox of the race between Achilles and the turtle by pointing out that the sum of infinitely many distances Ax + A x / 2 + A x / 4 + . . . is not infinite. He argued the concept of the continuum for space, time, and motion so successfully, that it has been challenged very rarely ever since 1. Isaac Newton (1642-1727) took this concept apparently so much for granted that he did not mention where it came from, even though he was very meticulous in listing and elaborating his assumptions. The Greeks thought of denumerable in/~nite when they talked about infinite divisibility. Gottfried Wilhelm Leibniz (1646-1716) and Newton generalized or abstracted this concept to nondenumerable in/inite when they developed the differential calculus and arrived at space and time differentials dx and dr. The concept of a continuum is clearly thinkable, which is all t h a t is required in mathematics. When we mention space and time we are no longer in mathematics but in physics and we require that a concept like space-time continuum must be observable 2. If we could observe something at the location x and something else at x + dx or at the time t and t + dt we could prove the existence of a space and time continuum by observation. One would not expect that any one proposing such a proof would be taken seriously. A spatial 1Book V(E) 3, w5 defines continuous; Book VI(Z) 1, 2 elaborates the concept of the continuum further; Zeno of Elea is refuted in Book VI(Z) 2 and 9 (Apostle 1969; Aristotle 1930). 2We note that pure mathematics has no space or time variables. Instead it has complex variables, real variables, integer variables, random variables, rational variables, etc. No physical units like meter or second are attached to variables in pure mathematics.
1.4 DOGMA OF THE CONTINUUM IN PHYSICS
27
resolution dx or a time resolution dt is no more observable t h a n the number of angels that can dance on the tip of a pin. How did we get into the situation that some of the most basic concepts of physics are inherently not observable? We treated physics as a branch of mathematics! This was quite acceptable at the time of the ancient Greeks. Today we recognize mathematics as a science of the thinkable and physics as a science of the observable. They cannot be more for each other than a tool or a source of inspiration. To see how concepts of pure mathematics were transformed into concepts of physics consider the definition of the continuum with the help of the real numbers on the numbers axis. A one-dimensional continuum has the same topology as the real numbers. It is usual to map any real number x by means of the exponential function e i y z onto the unit circle in the complex plane. The resulting map { e iyx } is called the character group of the topologic group of real numbers. W h y do we map the real numbers on a circle and not on an ellipse or a polygon? The definition of a character group does not require the circle, it only permits it (van der Waerden 1966, Part 1, w54). A shift along the numbers axis becomes a rotation along the unit circle. Only the circle remains unchanged under infinitesimal as well as finite rotations. If we do away with the unobservable concept of infinity and use an arbitrarily large but finite (time) interval T, and if we further do away with the unobservable infinitesimal and replace it with an arbitrarily small but finite (time) interval At we obtain a finite number N = T / A t of (time) intervals. Instead of a circle we would use a regular polygon with N corners. It is not yet evident why the concepts of time and space should be connected either to the circle or a regular polygon. We can only assume that the use of the circle is due to the dogma of the circle, which has influenced European thinking at least since Plato (c. 428-c. 348 BCE). It is usually connected with astronomy but its influence goes much further. Claudius Ptolemy (c. 90-c. 168) stated it as follows: ... we believe it is the necessary purpose and aim of the mathematician to show forth all the appearances of the heavens as products of regular and circular motions. (Ptolemy 1952, Alrnagest, Book III, 1, p. 83, second paragraph) If one believes that strongly in the circle one should have no trouble believing in the continuum derived with the help of the dogma of the circle. It is worth remembering that Nicholas Copernicus (1473-1543) could not overcome the dogma of the circle. We celebrate Johannes Kepler (1571-1630) for breaking the hold of the circle on our thinking with his book Astronomia Nova in 1609. Actually, this hold was never completely overcome. The deferents and epicycles of Ptolemy and Copernicus evolved into the Fourier series in complex notation. The representation of a function by a Fourier series expansion is as pervasive today as the description of the orbits of celestial objects by a superposition of circles before Kepler. It prevented electrical communications for
28
1 INTRODUCTION
almost a century to advance from a steady-state theory to a transient theory, even though in retrospect the use of a steady-state theory for the transmission of information is hard to believe. This short summary of the dogma of the circle should suffice to show that deriving the dogma of the continuum from the dogma of the circle worked in its time but is no longer a good idea. What we can observe are distances between objects measured with some form of a ruler. In other words, we observe relative locations or points in a man-made coordinate system. Furthermore, we can observe changes relative to a standard change. This standard change was once our own aging process due to its unique importance for us, refined by the orbiting of the Earth around the Sun and the rotation of the Earth around its own axis. Progress in technology replaced this original standard of change by man-made standards, e.g., by the pendulum clock, the quartz clock, and the cesium clock. The words space, time, and space-time permit us to talk efficiently with few words about points in a coordinate system, relative changes, and points in moving coordinate systems, all of which are observable as long as we stay clear of the concept of the continuum (Harmuth 1989, Chapters 2-4). But these words took on a life of their own and led to the continuum of space, time, and space-time. Since these continuums cannot be observed they cannot be described in terms of physics. Mathematics was used to overcome the difficulty. This created the impression that physics is a branch of mathematics. Let us illustrate this development with the help of a reference: From the essence of space remains in the hands of the mathematician, using such abstraction, only one truth: that it is a three-dimensional continuum 3. (Weyl 1921, 1968, vol. II, p. 213) Einstein (1879-1955) is remembered for his curved space-time. But in his later years he had a change of mind and called it an invention: The psychological subjective feeling of time enables us to order our impressions, to state that one event precedes another. But to connect every instant of time with a number, by the use of a clock, to regard time as an one-dimensional continuum, is already an invention. So also are the concepts of Euclidean and non-Euclidean geometry, and our space understood as a three-dimensional continuum. (Einstein and Infeld 1938, p. 311) A collection of quotations about the concepts of space and time from the days of Aristotle to 1976 is presented by Harmuth (1989, Sec. 1.4). For a simple example of what finite differences Ax, At rather than differentials dx, dt can do for us consider the cases of infinite energy that currently are resolved by renormalization. It has been pointed out before that some of these cases are caused by a problem of Maxwell's equations with the causality law (Harmuth et al. 2001, Sections 4.4, 4.5). The modified Maxwell equations with a term for magnetic current densities provide one solution to the problem 3Vom Wesen des Raumes bleibt dem Mathematiker bei solcher Abstraktion nur eine Wahrheit in H//.nden: daft er ein dreidimensionales Kontinuum/st.
1 . 4 D O G M A OF T H E C O N T I N U U M IN PHYSICS
29
by being able to account for the magnetic dipole currents p r o d u c e d by r o t a t i n g m a g n e t i c dipoles. T h e finite differences Ax and At of space and time resolution provide a second m e c h a n i s m to avoid infinite energies. This second m e c h a n i s m is based strictly on q u a n t u m theory. Consider the interaction of an electron with itself. A p h o t o n is e m i t t e d and reabsorbed within a time At. T h e u n c e r t a i n t y relation AF_At > h p e r m i t s a large energy of the p h o t o n if A t is very small. For At --+ dt one obtains a possible infinite energy for the p h o t o n and t h u s an infinite self-energy for the electron. This infinite self-energy is c u r r e n t l y overcome by renormalization, which leads to the s u b t r a c t i o n o o - c ~ . If A t c a n n o t be smaller t h a n a finite observation or resolution time we can never get infinite self-energy and we never have to use renormalization with a s u b t r a c t i o n of infinite from infinite. T h e success of a mechanism t h a t avoids infinite energies and is based on q u a n t u m mechanics suggests applying it to charge r e n o r m a l i z a t i o n too and t h e n to the unsolved problem of energy r e n o r m a l i z a t i o n in a q u a n t u m gravitation theory. We want to show here generally t h a t all this is derived from a basic t h e o r e m of information theory t h a t applies to observed, processed or t r a n s m i t t e d information:
Information is always finite. For e x p l a n a t i o n consider a m e a s u r e m e n t t h a t yields a possible largest result Y and a possible smallest result AY, usually referred to as the resolution. Let Y and A Y be chosen so t h a t the ratio Y / A Y = N is an integer; the choice N = 2 n+l - 1, which yields N = 1 1 1 . . . 11 (n digits 1) in b i n a r y notation, works best. T h e information obtained by one m e a s u r e m e n t is at most 4 equal to the n u m b e r of b i n a r y digits required to write N = 2 n+l - 1 as a binary number. For 2 n <_ N < 2 ~+l we need n binary digits 1 or 0 to write N and t h e y represent the information log2 2 '~ = n [bit] or n / 8 [byte] for N = 2 '~+1 - 1. We note t h a t the units bit or byte look like physical units such as m e t e r or kilometer, but n is a pure n u m b e r without a physical dimension. A n u m b e r 10011 with n = 5 b i n a r y digits may stand for 19 meter, Joule, Volt, . . . . T h e i m p o r t a n t fact is t h a t Y can be arbitrarily large but finite and A Y can be arbitrarily small b u t finite. I n f o r m a t i o n theory excludes the infinite as well as the infinitesimal from any t h e o r y based on observation. W h e n e v e r we come across s o m e t h i n g infinitely large or infinitesimally small, t h a t is not merely based on writing convenience like y --, oo or y --+ dy r a t h e r t h a n y >>> 1 or y <<< 1, we know t h a t the most basic law of information t h e o r y is violated. Infinite i n f o r m a t i o n is no b e t t e r t h a n infinite energy or infinitesimal time intervals in a science based on observation and experimentation. One s o m e t i m e s reads t h a t time in m a t h e m a t i c s can run forward and back4This m a x i m u m of the information is obtained if the result y of a m e a s u r e m e n t has equal p r o b a b i l i t y to be in any one of the intervals 0 _< y < A Y , A y _< y < 2 A Y , . . . , (N-1)AY<_y
30
1 INTRODUCTION
ward equally well while this is not so in physics. Such s t a t e m e n t s ignore t h a t the concept of time does not exist in pure m a t h e m a t i c s since variables in m a t h ematics do not have a physical dimension like second. A time variable indicates t h a t m a t h e m a t i c s is applied to physics. In this case we m u s t satisfy not only m a t h e m a t i c a l axioms and rules but in addition physical laws. T h e causality law is such a physical law. It introduces the universally observed distinguished direction of time since an effect comes after its cause. If we can find an example in physics where a cause comes after its effect we would observe t h a t the time runs opposite to its usual direction. 1.5 CONCEPT OF SPACE BASED ON FINITE DIFFERENCES T h e ancient Greeks did not clearly distinguish between m a t h e m a t i c s as a science of the thinkable and physics as a science of the observable. T h e y t h o u g h t in t e r m s of the unobservable continuum and this had an e n o r m o u s influence on our thinking a b o u t space. T h e continuum was not seriously challenged until the t w e n t i e t h century. Two quotes from Schr6dinger show the change of thinking: In Einstein's theory of gravitation matter and its dynamical interaction are based on the notion of an intrinsic geometric structure of the spemetime continuum. The ideal aspiration, the ultimate aim, of the theory is not more and not less than this: A four-dimensional continuum endowed with a certain intrinsic geometric structure, a structure that is subject to certain inherent purely geometric laws, is to be an adequate model or picture of the 'real world around us in space and time' with all that it contains and including its total behavior, the display of all events going on in it. (SchrSdinger 1950, p. 1) Q u a n t u m theory and the general theory of relativity were the two great physical theories of the twentiethy century. B o t h were developed with the help of differential calculus but q u a n t u m theory was always against the concept of infinitesimally a c c u r a t e observations or measurements. Six years after the q u o t a t i o n cited above SchrSdinger wrote as follows: We must not admit the possibility of continuous observation . . . . The idea of a continuous range, so familiar to mathematicians in our days, is quite exorbitant, an enormous extrapolation of what is really accessible to us. (Schr5dinger 1956) This was the time when information theory was developed. I n f o r m a t i o n could be m e a s u r e d in bit or byte. T h e t h o u g h t of observing, processing, or t r a n s m i t t i n g n o n d e n u m e r a b l y infinite bit of information was preposterous. Besides the concept of the continuum our usual g e o m e t r y assumes t h a t we can took 'from outside' on geometric structures drawn on p a p e r or c o n s t r u c t e d in physical space. This is quite a d e q u a t e except if we s t a r t thinking a b o u t the universe. We c a n n o t look at the universe from the outside since we are 'in it' or p a r t of it. To make this point more u n d e r s t a n d a b l e consider the equilateral
1.5 CONCEPT
O F S P A C E B A S E D ON F I N I T E D I F F E R E N C E S
31
2
a
b
FIC.1.5-1. Two equilateral triangles. triangle of Fig.l.5-1a. It has three vertices 1, 2, 3 and three equally long edges 1. Figure 1.5-1b shows again three vertices and three equally long edges. We have no difficulty seeing a difference between the two triangles. However, if we plot the three edges in Fig.l.5-1b in planes that are perpendicular to the paper plane, the figure looks exactly like Fig.l.5-1a if we look from above at the paper plane and see the curved edges projected onto the paper plane. We can see different things if we can look from different directions. This is geometry viewed from outside. Let the triangles of Fig.l.5-1 be implemented 'in space' by three rods or thick wires and make them very long. If we put an ant schooled in mathematics and physics on these structures it is 'in the structure' or part of the structure. The ant will observe that there are vertices where two rods come together. It can also observe t h a t the rods are equally long by pacing, which means it counts how often it has to put its left front foot forward when walking from one vertex to the next. Finally, the ant will observe t h a t there is a forward and a backward direction; it can crawl in either direction indefinitely. The ant will first assume it lives on an unlimited structure according to Fig.l.5-2. Giving it some more thought, it may make a scent mark at one of the vertices and run either in the forward or the backward direction. It observes that it returns to the starting point after having run along three edges. This indicates t h a t it lives in a limited or closed structure according to Fig.l.5-1, but it cannot tell which of the two. Consider a triangle connecting Earth, Alpha Centauri, and Sirius. We can measure the distances from Earth to Alpha Centauri and Sirius directly, but not the distance from Sirius to Alpha Centauri. There are various methods to measure such distances; they all use electromagnetic waves. These waves do not propagate along a straight line in the sense of Euclid since they get bended by the masses in the universe and we do not know the distribution or 1We use t h e w o r d edges r a t h e r t h a n sides since t h e w o r d sides t a k e s on a different m e a n i n g if we c o m b i n e four t r i a n g l e s into a t e t r a h e d r o n in s p a c e l a t e r on.
32
1 INTRODUCTION -2
-3
0
-I
2
1
3
Frc.l.5-2. Open-ended or unlimited two-dimensional structure t h a t resembles the closed triangles of Fig.l.5-1 if one does not see the triangle from outside but lives on it.
3
2
FIG.1.5-3. Three-dimensional equilateral triangle with four vertices and three edges meeting at every vertex.
the amount of these masses. Since we cannot measure the distance from Sirius to Alpha Centauri we know from direct observation less about the geometry of our universe than the ant that dwells in a universe that we call a triangle. The two triangles of Fig.l.5-1 are usually called two-dimensional objects. If we bend the page according to a cylinder we obtain three-dimensional objects, unless we use cylinder coordinates. The dependence on a coordinate system is avoided if we classify according to the maximum number of edges that meet at a vertex. This becomes particularly simple in Fig.l.5-1 since two edges meet at every vertex. Let us advance from a two-dimensional equilateral triangle to the threedimensional one that is usually referred to as a tetrahedron. Figure 1.5-3 shows a drawing of this object. To enhance the perception that this is an object in space we represent the edges by bars of finite width rather than as lines as in Fig.l.5-1, and the vertices are represented by small spheres. An ant living in a structure according to Fig. 1.5-3 will observe that forward and backward direction have no meaning but it could introduce a red, a green, and a blue direction by painting the rods. A new difficulty occurs because a scent mark at vertex 1 will not prevent the ant from running arbitrarily often around the triangle with the vertices 2, 3, 4. The six rods of Fig.l.5-3 may be
1.5 CONCEPT OF SPACE BASED ON FINITE DIFFERENCES
33
FIG.1.5-4. Four-dimensional equilateral triangle with five vertices and four edges meeting at every vertex. 3
1
2 FIG.1.5-5. Five-dimensional equilateral triangle with six vertices and five edges meeting at every vertex. bent in planes perpendicular to the paper plane. If they are equally long one cannot tell whether Fig.1.5-3 is a generalization of Fig.l.5-1a or 1.5-1b. Let us advance to the four-dimensional equilateral triangle of Fig.1.5-4. There are five vertices and four edges that meet at every vertex. One may call the edges red, yellow, green, and blue to characterize their direction. In Fig.1.5-3 all edges could be straight in the Euclidean sense but this is no longer so in Fig.1.5-4. We advance to the five-dimensional triangle of Fig.1.5-5. We recognize six vertices and five edges t h a t meet at every vertex. We would now need five colors to distinguish the direction of the edges. The navigational problem for an ant starting at vertex 1 and wanting to return to it is quite complex. A different scent mark at every vertex solves the problem. It is evident t h a t we could go on to triangles of ever higher dimension
34
1 INTRODUCTION
V
0
Z WOY
= ZOX
= ~r/2
Z WOX
= XOY
= YOZ
,2. = ZOW
= rr/2
FIG.1.5-6. Four axes x, y, z, w with angles 7r/2 between them. Some of the angles are measured on a conical surface rather than on a Euclidean plane. except for the increased difficulty of drafting. A discrete structure with any number of dimensions can be implemented 'in space' and drafted 'on a surface'. We can choose the surface even though it will usually be a Euclidean paper plane. But a paper plane can be bended into a cylinder or a conical surface. There is no such choice for the space in which the structures of Figs.l.5-3 to 1.5-5 can be implemented. The claim that Figs.l.5-4 and 1.5-5 represent four- and five-dimensional structures t h a t can be implemented in space deviates from the usual thinking. One frequent objection is that we can only have three mutually perpendicular vectors, axes, or directions in space. Consider Fig.l.5-6 to see where this belief comes from. It shows four axes x, y, z, w with angles of 90 ~ between any two of them. This is possible because four of the angles are measured on conical surfaces rather than planar ones. It is evident that the principle of Fig.1.56 can be generalized to more than four dimensions. Since a Euclidean plane can be bent into a conical surface without changing the Gaussian curvature it would be difficult for inhabitants of the surfaces of Fig.l.5-6 to distinguish between the planar and the conical surfaces. An outside observer has no such difficulty. A second problem of Fig.l.5-6 is the existence of surfaces on which one can define angles. Our Figs.l.5-1 to 1.5-5 require the paper surface for drafting but we always pointed out that the structures could be implemented in space with rods and vertices. Since at least some of the rods in Figs.l.5-4 and 1.5-5 must be bent one cannot use the structures to define planar surfaces as one does with the tetrahedron of Fig.l.5-3 in Euclidean geometry. We shall show
1.5 CONCEPT OF SPACE BASED ON FINITE DIFFERENCES
1
35
2
FIC.1.5-7. Making a third dimension invisible by wrapping the plane L with the vertex 4 around the edge between vertices 1 and 3.
later on that the lack of surfaces is an important feature of a geometry based on finite differences. The concept of hiding dimensions has attracted attention lately. To see how it works in a geometry of finite differences consider Fig.l.5-7. We again have the triangle 123 of Fig.l.5-1 that shall be implemented in space with three rods. In addition, we have a triangle 134 that shall be drafted on a sheet of paper L. Since three edges meet at the vertices 1 and 3 we must call this structure a three-dimensional structure. Let the sheet L of paper be wrapped around the edge connecting the vertices 1 and 3. The third dimension disappears from view but it is, of course, still there. For a generalization of this concept refer to Fig.1.5-8. We have the threedimensional triangle of Fig.l.5-3. Three planes L1, L0, L-1 are shown in which the vertices 3, 1, and 4, as well as 2 are located. We compress these three planes into one plane L0. Then we replace the plane L in Fig.1.5-7 by the plane L0. The compressed structure of Fig.l.5-8 may be wrapped around the edge 13 of Fig.l.5-7. Eureka, the three dimensions of Fig.l.5-8 have disappeared from view. W i t h o u t compression and wrapping the combination of the structure of Fig.l.5-8 and the triangle 123 of Fig.1.5-7 would have yielded a five-dimensional structure. The use of ever higher unobservable spatial dimensions reminds one of the use of ever more epicycles for the representation of planetary orbits before Kepler. In a differential theory a particle moves from x to x + Ax in infinitely many steps of length dx. The theory can describe the particle at any one of these steps. The assumption that the particle does not move in one dimension but in many provides one with degrees of freedom that make the theory reconcilable with observation. This is quite different in a geometry based on finite differences Ax. We may observe a particle at the location x t h a t disappears. Shortly later an equal or related particle appears at the location x + Ax. Since ....
36
1 INTRODUCTION
-~- --:--_-':. . . . . .
I ~ i II \\
/~!
II
/// //
\\
//
i
/11]
J/
-[-1l
9
~11:
111
I t---d ....... L1~ ' -
~o~
F I O . 1 . 5 - 8 . Compression of three dimensions into one paper plane L0 that is then wrapped around the edge from vertex 1 to 3 in Fig.1.5-7 instead of the plane L shown there.
3 3
2 oP 0 o0
1
2
x / x ---, a
3
0
1 2 3 x l x ----, b
FIC.1.5-9. A two-dimensional continuous coordinate system (a) and a corresponding discrete coordinate system (b). we can n e i t h e r observe the particle continuously nor h a n g a license plate on it we are never sure t h a t the same particle d i s a p p e a r e d at x and r e a p p e a r e d at x + A x . Hence, a first a s s u m p t i o n is required to decide w h e t h e r the particles are the s a m e or not. Since we c a n n o t observe w h a t h a p p e n e d to the particle d u r i n g t h e t r a n s i t i o n from x to x + A x we can make as m a n y a s s u m p t i o n s as needed to m a t c h t h e t h e o r y to the observation. We do not need u n o b s e r v a b l e spatial d i m e n s i o n s since the t r a n s i t i o n from x to x + A x is unobservable. We t u r n to c o o r d i n a t e systems. T h e continuous c o o r d i n a t e s y s t e m of F i g . l . 5 - 9 a defines a point for any real variable x / X and y / Y , which m e a n s it defines a surface. This is not evident since t h e p a p e r plane used for the drafting and the defined surface are not readily discriminated. We will r e t u r n
1.5 CONCEPT OF SPACE BASED ON FINITE DIFFERENCES
y/ y I ~ 1 9
3
37
0 ~
~
1 z/X "~
r
3
a
b
FIG.1.5-10. A closed two-dimensional discrete coordinate system with 16 points (a) and the surface obtained by replacing the 16 discrete points by nondenumerable many points (b).
to this point presently. In Fig.l.5-9b we have a discrete c o o r d i n a t e system t h a t defines points for integer values x / X and y / Y . T h e r e is no suggestion t h a t a n y t h i n g more t h a n a finite or a d e n u m e r a b l e n u m b e r of points is defined. We use rods and small spheres to show the discrete c o o r d i n a t e system. Such a c o o r d i n a t e s y s t e m will be called n-dimensional if no more t h a n n coordinate 'rods' intersect at any coordinate point. T h e c o o r d i n a t e systems in Fig.l.5-9 have a limited extension. We can either e x t e n d t h e m a r b i t r a r i l y f a r - - b u t not infinitely f a r - - b e y o n d their limits or we can t u r n t h e m into dosed systems. T h e closed discrete c o o r d i n a t e system is shown in F i g . l . 5 - 1 0 a and the closed continuous systems in Fig.l.5-10b. T h e r e is no surprise with Fig.l.5-10a; it clearly shows a m a n - m a d e structure. This is not so for Fig.l.5-10b. We see a surface with c o o r d i n a t e lines painted on. T h e difficulty we had in Fig.l.5-9a with the distinction between the p a p e r plane and a surface defined by the coordinate system has been resolved. T h e c o o r d i n a t e lines in Fig.l.5-10b are clearly m a n - m a d e but the surface appears to be m a t h e m a t i c s - m a d e . We note t h a t most points in F i g . l . 5 - 1 0 b are 'on' the surface, but there are edge points and four corner points. No such edge or corner points exist in Fig.l.5-10a; all 16 points are equal. Let us advance to three dimensions. Figure 1.5-11 shows the extension of F i g . l . 5 - 9 b to the three dimensions x, y, z. We recognize 8 interior points where three c o o r d i n a t e lines or rods intersect, 24 surface points where two rods intersect and one t e r m i n a t e s , 24 edge points where one rod goes t h r o u g h and two t e r m i n a t e , and finally 8 corner points where three rods t e r m i n a t e . T h e extension of Fig.l.5-11 to a continuous c o o r d i n a t e system would produce a cube with c o o r d i n a t e lines painted on the surface and t h r o u g h its interior. We shall use the word space--without the modifier t h r e e - d i m e n s i o n a l - - f o r this extension of the t e r m surface 2. T h e r e is no reason why the physical con2The connection between space and surface is not so close in everyday language. We say, e.g., "I have space" but not "I have surface".
38
1 INTRODUCTION
9
t
. . . . .
X --'--*
FIG.1.5-11. Three-dimensional discrete coordinate system with 64 points of which 8 are interior points, 24 surface points, 24 edge points, and 8 corner points. cept of space should not be distinguished from the mathematical concept. The problem of distinguishing between the physical concept of space and the mathematical concept of a three-dimensional continuum does not exist in a discrete geometry. Neither Fig.l.5-9b nor 1.5-10a suggests the concept of 'surface'. The coordinate system of Fig.1.5-11 can be transformed into a closed one but the drafting problem becomes overwhelming. We show Fig.1.5-11 again in Fig.1.5-12 but there are now three additional lines shown: dotted, dashed, and dashed-dotted. These lines connect the points 0,1,1 and 3,2,2 of the open coordinate system. Furthermore, the points 1,0,1 and 2,3,2 as well as 1,1,0 and 2,2,3 are connected. There are a total of 24 + 24 + 8 = 56 non-interior or outside points in Fig.l.5-12, which would require 28 connections rather than the three shown. If this seems complicated one might consider the equivalent of Fig.1.5-12 for a continuous, three-dimensional coordinate system. The advancement from the three-dimensional coordinate systems of Figs. 1.5-11 and 1.5-12 to four dimensions is a challenge to one's drafting skills. First we reduce the coordinate points or spheres from 43 = 64 in Fig.l.5-11 to the absolute minimum 33 = 27, which becomes 34 = 81 in four dimensions. This is a large number but much better than 44 = 256. Figure 1.5-13 shows such a four-dimensional coordinate system with the variables x, y, z, w. To help one understand it we show in Fig.1.5-14 the corresponding three-dimensional coordinate system with 33 = 27 coordinate points. Figure 1.5-13 consists of three structures according to Fig.1.5-14 shifted in the direction w from w = 0 to w = 1 and w = 2. We can draft such a structure on a surface and we can
1.5 CONCEPT OF SPACE BASED ON FINITE DIFFERENCES
" ......
......
"'"-i-0..1.1
I0 ~
"'"-. "[-..... I ~ l
39
I
=
~.
k..
,aj
----'" 3..-- .....
v/Y'+
FIG.1.5-12. Three-dimensional closed or bounded coordinate system. Three typical connections of surface points that change the open to a closed coordinate system are shown by dotted, dashed, and dashed-dotted lines. c o n s t r u c t it with rods and small spheres in space. To make Fig.1.5-13 more u n d e r s t a n d a b l e consider an inspector for hidden things who is restricted to the three dimensions x, y, z. H e / s h e can visit only the 33 = 27 points with w = 0 b u t not the 2 x 33 = 54 points with w = 1 or w = 2. T h e fourth dimension a p p e a r s to be the ideal hiding place. B u t it is easier to restrain m a t h e m a t i c a l inspectors to the dimensions x, y, z t h a n physical inspectors. F u r t h e r m o r e , even t h o u g h our inspectors c a n n o t walk to the points with c o o r d i n a t e w = 1 or w = 2 they can still see t h e m , since the p r o p a g a t i o n of light is i n d e p e n d e n t of the c o o r d i n a t e system. We have shown t h a t a g e o m e t r y based on finite differences p e r m i t s one to i m p l e m e n t s t r u c t u r e s with more t h a n three dimensions in physical space. An e q u a l l y surprising result is t h a t a discrete c o o r d i n a t e s y s t e m can always be reduced to a one-dimensional c o o r d i n a t e system. F i g u r e 1.5-15a shows the two points A a n d B in the usual two-dimensional c o o r d i n a t e system. T h e c o o r d i n a t e distances x = 5 - 2 = 3 and y = 5 - 1 = 4 yield t h e P y t h a g o r e a n distance 7/32 + 4 2 - 5 . In F i g . l . 5 - 1 5 b the 64 c o o r d i n a t e points of F i g . l . 5 - 1 5 a are shown again b u t t h e y are now n u m b e r e d consecutively from 0 to 63, which eliminates y
40
1 INTRODUCTION 2.2.2.2
0.2,2.2
0.2.2.0
0.2,0,{ 2,0.0.0
"
,
,
,
~
a ....... -'~;
).0.0.0
z , y,.z, w = o, o, o, o
FIO.I.5-13. Four-dimensional, discrete Cartesian coordinate system with n" = 34 = 81 grid points and r n , - 1 = 4 x 33 -- 108 rods. and leaves us with only one dimension x. It becomes difficult to calculate the P y t h a g o r e a n distance, b u t it m a y no longer be of much interest. An ant r u n n i n g along the spatial axis x will s t a t e the distance from A to B as 4 5 - 10 = 35, and this will be the distance of interest. As a subtle aside we note t h a t no square root is needed. F i g u r e 1.5-15c shows a different way to connect the c o o r d i n a t e points of Fig.l.5-15b. T h e distance between A and B along the c o o r d i n a t e axis is now
1.5 CONCEPT OF SPACE BASED ON FINITE DIFFERENCES
41
2.2.2
2.1.2 2.2
2.1 .I
2,0,
2,1.0
2.0 2.0. .1.0
x, y, z = 0, 0, 0 FIc.1.5-14. Three-dimensional version of Fig.1.5-13. Figure 1.5-13 shows three such structures shifted in the direction w from w = 0 to w -- 1 and w = 2. 3 3 - 26 = 7, again w i t h o u t a square root. M e a s u r i n g t h e d i s t a n c e according to Fig.1.5-15a makes good sense to an outsider looking at the 64 c o o r d i n a t e points. An insider having to r u n along the c o o r d i n a t e axis will not find the P y t h a g o r e a n d i s t a n c e very helpful. A possible e x t e n s i o n of Fig.1.5-15c for ' t h r e e d i m e n s i o n s ' is shown in Fig.1.5-16. We need only one dimension x to connect all discrete c o o r d i n a t e points in t h r e e dimensions. F i g u r e s 1.5-15 and 1.5-16 make it clear t h a t t h e concept of ' d i m e n s i o n ' is flexible in a g e o m e t r y of finite differences. B u t t h e illustrations also show the i m p o r t a n c e of t h e concept of topology. If we consider Fig.1.5-15a to represent an area of t h e surface of E a r t h we will consider it plausible t h a t t h e t e m p e r a t u r e
1 INTRODUCTION
42
7 ----
63,,~
6s~---
.
~
~
36 35 21 20 10 g
56
47L--~%,Bo--.--o--o--T4o 31
2 ---
1
1
1
~
1
~
48
3 ~
0-0
~
2
3
4
5
6
7
55
~ 6
-
-
,5
~
o
I'
0
1
a
2
Z4
\
-
.
x" x
_
I A"
3 2
Z3
'~n~---- l, 8 3
4
5
6
0
7
1 5 6 14 15 27 28 x ------+
b
FIG.1.5-15. Two-dimensional, discrete Cartesian coordinate system (a), its replacement by a one-dimensional coordinate system (b), and the generalization of the principle to coordinate systems with denumerable many points.
34r i I
33
',
Ig' % I I
, '
! , I
20
x
'
} 'l
2g
28
iz4
FIG.1.5-16. Replacement of a three-dimensional, discrete Cartesian coordinate system with denumerable many points according to Fig.l.5-14 by a one-dimensional coordinate system.
1.5 CONCEPT OF SPACE BASED ON FINITE DIFFERENCES
43
in point B at the time t + At will depend on the t e m p e r a t u r e at the time t in point B as well as in the neighboring points x, y = 1, 5; 2, 4; 2, 6; and 3, 5. The topology of Fig.1.5-15b does not appear to be useful in this case. However, the situation changes if we consider Fig.1.5-15b to represent a rope or string folded up in the way shown. The topologies of Figs.1.5-15b, c and 1.5-1.6 appear more useful for a folded string than the topology of Fig.1.5-15a. We hope this short discussion of discrete geometries with inside or outside observers demonstrates the difference between mathematical geometries based on the continuum and geometries restrained by the physical requirement of finite resolution of any observation. Physics is not a branch of mathematics, we can never demand that mathematical assumptions and their results must apply to physics. We must always demand that the m a t h e m a t i c a l assumptions match the physical requirements.
ADVANCES IN IMAGING AND ELECTRON PHYSICS, VOL. 137
2 Modified Klein-Gordon Equation
2.1
DIFFERENTIAL
EQUATION
WITH
MAGNETIC
CURRENT
DENSITY
The modified Klein-Gordon differential equation is obtained by adding a magnetic (dipole) current density term to Maxwell's equations. The modified Klein-Gordon difference equation is produced by replacing differential operators with finite difference operators. We start with the Hamilton function of the usual Klein-Gordon equation without magnetic dipole current density 9s
c[(p -- eArn) 2 -k m02c2] 1/2 + eCe
(1)
which may be rewritten as follows:
(p
_eAm)
2 - - ~ -1~ (~}~ __ e C e ) 2
=
_Trt02C2
_ e A m . ) 2 - . . ~1 (9~_er
(px_eAm~)2+(pv_eAmy)2+(p
2 _m2c 2 =
(2)
Using the substitutions hO p~ ~ -
hO
hO
'p" ~ -
i-5~z'
9s ~
hO lot
(3)
and applying the operators to a function 9 yields the conventional KleinGordon equation:
~( J=~
h 0 i ozj
eAmz~ )2
l(h0
+ eCe)21 ~/= -m2c2ff~
- "J X 1 - - X~ X 2 ~
y,
X3 -- Z
(4)
The addition of a magnetic current density term to Maxwell's equations calls for a Hamilton function with three components 9/~, ~u, 9r in Eq.(1). Equations (1.1-42) to (1.1-44) show a first-order approximation in a of these three components. We copy 9/~" 44 ISSN 1076-5670/05 DOI: 10.1016/S1076-5670(05)37002-9
Copyright 2005, Elsevier Inc. All rights reserved.
2.1
DIFFERENTIAL
EQUATION
WITH
9 ~ . ~ = c [ ( p - e A m ) 2+m20c2]
MAGNETIC
CURRENT
(2CAe)
1/2 l + a ~ Q
9
45
DENSITY
+eCe-Lcx
(5)
--Q
(6)
e
(~-~x -- eCe -3t- f-.Jc:r) 2 = C 2 [ ( P -- e A r n ) 2 -k- ?Tt2c 2]
1 +
a
""
e
The terms Q and Lcx are defined in Eqs.(1.1-45) and (1.1-27). They require the potentials A~, Am, r and era, which are defined by Eqs.(1.1-15) to (1.1-22). We may replace [1 + in Eq.(6) by 1 + since Eqs.(5) and (6) hold only in first order of a. Equation (6) can be written as follows in first order of a, using Eq.(1.1-47) for factors that may not commute:
a(2AcAe/e)Q]2
(p
_
eArn) 2
~1
-
2a(2AcA~/e)Q
( ~ _ eG)2 +Oilr/4AcA, e [(p-eArn)2 + +
_ c2
ol
(~
- ~r
m2c2]Q
+
= -m02~ 2
(7)
~.
A solution as an expansion in powers of a is obtained if we replace the function 9 in Eq.(4) by the function ~ " II/x = lI/x0 -}- al,I/xl
(8)
The following equation in first order of a is obtained if we apply Eq.(7) to the function ~ " (p -
eArn) 2
[]4AcAe 1 (9s --- ~ eCe) 2 + o~L e "[(P - cAm)2 + __
-~-
C2
Ol
.... (~-C x -- e C e ) -3t- f'jc2_..__~ x OL Ol
m2~
(~I/x0 ~- O/l~./xl )
= --~0~ ~(~0
Lc~/a
+ ~V~l)
(9)
L2x/a
We assume that and are of order O(I) or less. This assumption will have to be checked later on. Equation (9) may then be separated into one equation of order 0(1) and a second one of order
O(a)"
( ( p - - e A r n ) 2 -~-~1(9s x - eCe) 2 + m2c2) ~xo = .0
( ( U - - ~ h m ) ~ - - ~ ( 9 (1~ - - ~ r 1((9s x -
c2
}
~+",]c ~ ~1
= --
eCe)Lcx+ Lc, ( ~ Ol
(~
(10)
[4,kcAe[(p_ehm)2+,m2c2]Q - ~r
L2z)] ~:~0
+ --
(~
'
(11)
46
2 MODIFIED KLEIN-GORDON EQUATION
We recognize in Eq.(10) the usual Klein-Gordon equation of Eq.(4). Equation (11) is the same equation with an inhomogeneous term added. The factor Q in Eq.(ll) has a term [1 + ( p - eAm)2/m2oc2]-3/2 according to Eq.(1.1-45). We want to replace the moment p by the operators of Eq.(3). This requires an explanation of what to do with Q. A straightforward approach would be to multiply Eq.(ll) with [1 + ( p - eAm)2/m2oc213/2 , shift all terms multiplied with a square root to one side, and square both sides of the equation. The terms Lcx cause a problem. A significant simplification is possible if one restricts the calculation to small values of (p - eAm)2/m2oc2: (p - eAm)2/m2c 2 << 1
(12)
The following series expansion may then be used; the factor k rather than - 3 / 2 is shown since the same expansion will be needed for k = - 1 / 2 later on: [1 + (p - etm)2/m2oC2]k " 1 + k(p - eAm)2/m2oc2
(I3)
This takes care of part of Q in Eq.(1.1-45), but we still have a factor Qo2 = [A~. (p - e A m ) ] 2 A2(p - c A m ) 2
(14)
to deal with. Only the factor ( p - eArn) 2 of Q in Eq.(1.1-45) can be rewritten with the substitutions of Eq.(3) without any explanation. The factor Q02 of Eq.(14) can be eliminated if we replace the vectors Ae and ( p - eArn) by matrices of rank 3 whose components are vectors:
Ae --
p - cAm =--
(
Ae~ex 0 0
(px - eAm~)ex 0 0
0 Aeyey 0
0 / 0 Aeze~
0
(15)
0
0 (py - eAmy)e u 0 (Pz - eAm~)ez
)
(16)
The substitution of Eqs.(15) and (16)into Eq.(14) yields Q0~ = 1
(17)
This is a satisfying simple result but it replaces the usual Klein-Gordon equation by a matrix equation of rank 3. The form of Q in Eq.(1.1-45) is reduced to 1
(p - eArn) 2
Q = m]c2 [1 + ( p - eAm)21m]c2lal~
(18)
2.1 DIFFERENTIAL EQUATION WITH MAGNETIC CURRENT DENSITY
47
The denominator of Q can be eliminated with Eq.(13) and Q is brought into a form that permits the substitution of the operators of Eq.(3)"
m~c2
1
2
m~c2
(19)
We note that there is no difference whether we multiply (p - eArn) 2 on the right with m2oc2 - (3/2)(p - e A m ) 2 as done in Eq.(19) or on the left, since ( p - eArn) 2 has only to commute with a constant or itself to transform a multiplication on the right to one on the left. The term Lcx in Eq.(11) must be written explicitly in a form that permits the substitutions of Eq.(3). This term is excessively long and we must break it into the five components shown by Eqs.(1.1-27) and (1.1-32). We refer to the literature for their derivation1:
lf~cxl --~-
ol
Ze(Aex9Aey,Z)Jc oLc Ze olm2c
2orm2c
[Aez(p - eAm)y - Aey(p - e i m ) z ] ( p - e h m ) x 1 T (p -- earn) 2/m2o c2 [Aez(p - eAm)y - Aey(P - eAm)z](p - eAm)x
• (1 -- (p -- eAm)2
(P -- eAm)2
)
• [Aez(P -- eAm)y - Aey(P - eAm)z](p - eAm)xJ
(20)
1Lcx2 - Ze / ( ~OCm'~- ~-~ 9 dx ef(O~)m -. Zamoc"
O~)m
~
(p - eArn) ~ - ~
(p - e A m ) v
• (lnu (p-eArn)2)
m2c2
9 Ze / [((0r
-- 2otmoc
x (1 \
\ Oy
(p - earn)2
2m2oC2 x
(0r
(p - e h m ) z - ~
-~y(p-
(p - e i m ) y
(P - e i m ) 2 )+(1-2m2c 2 eAm)z - - ~ z ( p
1Harmuth et al. (2001), Eqs.(3.3-53) to (3.3-57).
) -1/2
dx
)
)
- eAm)y
dx (21)
48
2 MODIFIED KLEIN-GORDONEQUATION
1Lc~3 = Z_..~e/ (Aez9 - Aev2,)dx ot o~c Ze Ae~ amoC 0-t [1 + ( p - eA-~27m2c211/2 9
-
-
0 ( ) dx - Aey~ [1 + (p - eAm)2/m]c2] 1/2 ( p - e A m ) z
o oc/ {Aez (9
9 Ze -
-
2m2c 2 + (p
- A.z-~
1-
y _
+ 2,
eAm)y (1
2m2c 2 + (p
1Lcx4--_ Z e c / ( O A e y o~ a Oz
--
--
)(p-
e g m ) v
(p
-
--
eAm)2
2m2o )
( p - egm)~
eAm)z (1
--
(p
--
earn)2
(22)
OAez)d x Oy
(23)
(Aez9- Aey2,)dx
Ze
(p - eAm) 2
4am]c ff { [ ( 1 -
2m]c 2
) (p - eAm)y
+ (p -- eAm)y (1 - (p
- earn)2
0
2m2oC2 ) ( p - eAm)z + (p - egm)z (1 - (p - eAm)2 x
( 1-
- 2 (p 2m]c
0
[Aez(p - eAm)y - Aev(p - eArn)z]
+ [Aez(p - eAm)y - Aey(p - eArn)z] (1 - (p 2m2oc2 - eAm)2 ) ] dx
(24)
Having explained the terms Q and Lcx in Eq.(11) we may turn to writing the simpler Eq.(10) in the matrix form of Eq.(15) as well as substituting the operators of Eq. (3):
2.1 DIFFERENTIAL EQUATION WITH MAGNETIC CURRENT DENSITY
hO ): -i-~x - eAmx
0
0
0
-~ -~y - enmy
0
o
0 hO
7-5
49
(ho "( -~z -
eAmz
2 + er
0
0
~-~-7 + eCe
0
1
C2
o
o
+ m 2c2
1 0
0
~xo.~
0
0
0
1
0
0
~xOy
0
0
0
1
0
0
~xOz
--0
(25)
Since only the terms in the main diagonals are not zero this matrix equation is essentially three times Eq.(4). The summation sign of Eq.(4) has disappeared but the index j retains the values j - 1, 2, 3 that it has in Eq.(4):
h 0
7ox-- -
eAmx~
-
+ eCe
+ m2c2 ~xOxj = 0
(26)
Equation (11) can be written in matrix form like Eq.(25) with ~x0x, ~x0y, ~x0z replaced by qJxlx, ~xly, qJxlz and another matrix instead of 0 on the right side. The shorter notation of Eq.(26) is a better choice. We begin by rewriting the first term on the right side of Eq.(11) with the help of Eq.(18): 4ACAe -------- [(p -- eArn) 2 + m2oc2]Q e - 4AcA._.____.A[1 ~ + ( p - eAm)2/m]c 2] (P - eArn) 2 e [1 + (p - eA2)/m2c2] 3/2
( . 4AcA.....__...~e(p_ eArn) 2 1 e
1 (P-- earn)2) 2 m~c 2
(27)
50
2 MODIFIED KLEIN-GORDON EQUATION
Equation (11) may now be rewritten in analogy to Eq.(26). Note the change of the variable L c x / a to s explained in detail in the following Eq.(29):
2_ 1
i Oxj = -
e
0
eee)
~ Ox--~. - eAm:r.~
+ m2oc2] ~xlx~
1
2m2oc2
i Oxj
_,,~I} Vx0~ (28) + -~cx~j)(~.0~ + er ) + ~~r
+ (~cx~j + ~cx2j + -~cx3j+ ~r
The last term (-o2 J~CX ~ a(L CX /a)2 may be dropped because of the factor a. The operators ~r to s introduced here follow from Eqs.(20) to (24) with the operators of Eq.(3) and the substitution 1 L cxkj~s
k=l,
2 3, 4, 5
(29)
The font Euler Script Medium is used for the symbol L but Euler Fractur Medium is used for the symbol s The matrix s using Euler Fractur Bold, has the terms s215 along its main diagonal and zeroes everywhere else:
~Ocxlj
20lTYt2oc{[Aez(h ~
x
+[1-
1
m
~ZZ
-i-~z-eAm~:
0
i Oxj
j = 1, 2, 3;
1-
m20c2
eAm~j) 21
xl = x, x2 = y,
m
iOxj
_
_
i ~y
X3 ~-
Z
(30)
The mixed notation x, y, z, and xj in Eq.(30) requires clarification. The terms with x, y, z
Ae z
~y - e A m y , Ae y
~ z - e A mz ,
~ -~x - e A m x
(31)
form matrices of rank 3 with equal values for all elements in the main diagonal like the second and third matrix in Eq.(25). On the other hand, the terms
2.1 D I F F E R E N T I A L E Q U A T I O N W I T H M A G N E T I C C U R R E N T D E N S I T Y
51
)'
(hi Ox~0
eAmx~
(32)
are the terms of a matrix with rank 3 like the first matrix in Eq.(25) with the terms along the main diagonal varying according to j -- 1, 2, 3. The remaining expressions for s162 to s162215 assume the following form:
0era ] - __~Z "cx2j--2OL'0CI{[0+m( 0[W ~"~ZZ-'Amz/ (~y_~yO_.Amy) Ze
[
(h 0
1
• 1
)'] + [1
eAm~.~
2~I. = ~ o..~
OCm 0 • __~y(h Y-~z -
1
2~Ic'
h 0 eAmx~)'] ( Yo..-W. -
~ OCm 0 eAmz/ -- -~Z (~'~y
cArnY)]}dx
"cx3j--2"=0"/ Aez:{ 1 2m2a2(~~xj O _eAmxj)] 7 ~O 0
-
0{[
-
Ae=-~
1
(h 0
1 2m2oc2 i Oxj
eAmy)
(h 0 _eAmx~)21 2m~c2 \ Y o..--U j eAmx~)2](hO ) ~~z -eAm=
~= _,,..) [1 2m2ci 2 \ i Oxj
(34)
-
~ c x 4 j --"
Zec/(OAey a Oz
Zc
~cxsJ = 4~mlc l
(
{[1
OAezoy I dx
(35)
1
2~2oC2( -mhi Ox~ 0
eArn=j)2](~~- eAmy)
_ _ ,~.....) [1_ ~ 1 ( hi ~OXj {[ 1 (~0 7N
+(ho
2m2c
2
+ 1 2m2oc2 i Ox~ +(hO
~,z -''m.) [1
(33)
2~I~~ Yo~-W.-
~Am=~
--
)'1}'Oy
- ~Am~)
52
2 MODIFIED KLEIN-GORDON EQUATION
x{
(
1
1 li 0 2m2c 2 i Oxj
eAm.a
)2]
0 X [dex(~--~y-egmy ) -dey(~ -~z0 - e A m . ) 0 "~[Aex(~"~y eAmy)-Aey( ~-0 x [ 1 02too x2c21 - (hO..i - J
(36)
eAm~)2]}dx
-
Equations (30) to (36) make all terms in Eq.(28) defined for known values of the potentials Am, Ae, Ce and the rest mass m0 of a charged particle. The equations are unique provided we accept Eq. (1.1-47) as necessary. Only the success of the derived results can determine that. The complexity of Eqs.(30) and (33) to (36) makes one hesitant to replace Eq.(1.1-47) by a more complex relation. Equation (26) can be solved for certain initial and boundary conditions. The solution of Eq.(28) requires a particular solution of the inhomogeneous equation that contains the solution 9• of Eq.(26). The homogeneous part of Eq.(28) equals Eq.(26). When the notation of Eqs.(25) and (26) is used one must rewrite Eq.(8) with a longer subscript 9•
= 9x0xj + a g x l ~
(37)
as was already mentioned in the text following Eq.(26). These long subscripts will often be shortened as follows: ~xO~j = 90,
9xlx~ = 91
(38)
2.2 Modified Klein-Gordon Difference Equation For the derivation of the Klein-Gordon difference equation including the modified Maxwell equations we start from Eq.(2.1-2):
(p - eArn) 2 (Px -- eAmx) 2 + (Py -
1
= -m0 1 (x-
enmy) 2 + (Pz - eAmz) 2 - -~
c 2
(1)
Instead of the differential operators of Eq.(2.1-3) we use difference operators. Since the transition from first- to second-0rder difference operators is not as simple as in the case of differential operators we must define them both according to Eqs.(1.2-1), (1.2-3), and (1.2-7):
2.2 MODIFIED KLEIN-GORDON DIFFERENCE EQUATION
P'~ ~ --* i /~xj 9{~ ~ p2j
h A~
z =
(2)
2Axj
h ~(t + A t ) - ~ ( t - At)
i z~t z _h: zi~v _ _n: ~(~
53
2At + a~j) - 2v(~j)+
(3) v(~j - axj)
9{2~ ~ - h 2/~2~ _- - h 2 ~ ( t + At) - 2~(t) + ~ ( t Zt~ (At) ~
At)
(4)
(5)
For the normalized variables
0 = t / A t , 4j = x j / c A t , N = T/At O<_t<_T, O<_xj<_cT, O<_O<_N, O<_Q <_N
(6)
we follow the simplified notation of Eqs.(1.2-3) and (1.2-7):
h h - -:[v(r i /~Q - 2i h
Ao
-- - ~ [ ~ ( 0
2~
- n ~A~v- = - n ~ [ v ( r
acy
+ 1) - v ( ; j - 1)1
(7)
+ 1) - ~ ( O - 1)1
(8)
+ 1) - 2~(r
+ v(r
- 1)]
- h 2z~2v -- -n2[V(0 + 1 ) - 2V(O) + V(t9- 1)] Z0 2 -
(9) (10)
These definitions suffice to rewrite all equations in Section 2.1 from Eq.(2.13) to Eq.(2.1-20). From EQ.(2.1-21) on we encounter integrations in addition to differentiations. To remain consistent we have no choice but to substitute summations for integrations as shown in Table 1.2-1: x
X
(11) We may write the matrix equation (2.1-25) by substituting A//~x for O/Ox to ,4/~t for O/Ot, but must observe that the squares A2/A~ 2 to Z2/At ~ ~ replaced according to Eqs.(4) and (5). Equation (2.1-8), ~x -- ~x0 + a ~ x l , applies from here on.
54
2 MODIFIED KLEIN-GORDON EQUATION
Using the notation xj rather than x, y, z we rewrite the differential matrix equation of Eq.(2.1-25) with the help of Eqs.(2) to (5):
hA
7 "~--x - eAm~
)2
0
0
1
i ~-x2
eAm~=
o
0
o
hA
0
o
)2
7~+~r
o
rno2c2
0
(h,~
c2
+
eAmza
)2
7~--~ + eCe 1
i ~x3
o
o
N+~r
1
0
0
~xO~l
0
0
0
1
0
0
~x0~2
0
0
0
1
0
0
~xOx3
= 0 (12)
The three equations represented by Eq.(2.1-26) assume the following form as difference equations:
~ 7li ~-~ -
eAmx~
- V1
~Z + ~r176 + m~
Vxo~, = 0
(la)
We shorten ~xOxj to ~o as well as xj to x and rewrite this equation explicitly rather than in operator form. To simplify the notation further we do not write a variable that is not changed, e.g., we write ~o(x + Ax) - ~ o ( X - Ax) rather than ~o(x + Ax, t ) - ~ o ( X - Ax, t). The first term in Zq.(13) becomes:
~I/0 ---- ~I/xox~ ~
A
X -----Xj
2 O0
= --h 2Z~2~~ + 2iheAmx
A~'2
Z~
+
e Amz + ihe
.
Az
~0
2.2 MODIFIED KLEIN-GORDON DIFFERENCE EQUATION
= _h 2 r
+2ineAm~q!o~,x+Ax,--~o,x--Ax,(
(Ax) 2 (2
2
+ e Amz +ihe
55
~ ( 2Az
Zmx(xq-Ax)-Amx(x-Ax)) 2Ax
~
~o (14)
The second term of Eq.(13) assumes the form: ~I/o- ~I/x0xj
A
"-'-'-7 zSt -
Bt + e r - ihe At" Oo
=_h2 ~ o ( t + A t ) - 2 ~ o ( t ) + ~ o ( t - A t ) (At) 2 (2
2
+ e Ce - i h e
~o(t+At)-~o(t-At)
-2iner
2At
r162
2At
~o
(15)
Substitution of Eqs.(14) and (15) into Eq.(13) yields a difference equation instead of the differential equation (2.1-26). We write here both variables xj and t of ~o, Amxj, Ce: ~I/0 -- ~x0x~
~,o(zj + Axj, t) - 2r t) + ~,0(z~ - Azj, t) (ZXx~)2 1 ~'o(~j, t + At) - 2 ~ , o ( ~ , t) + ~"o(~j, t -
c~
e(
-- 2i~ Amx~(xj, t)
At)
(ht)~
9 o(Z~ + Axe, t ) -
'I'o(X~ -- A z j , t)
2Ax~ + c~:tr
t)~o(zj, t + a t ) - ~o(zj, t - at)'~
) 2At e2 [ 2 (xj t) 1 2 h ( a m ~ (xj + Axj, t ) - Am~j ( x j - Axj, t) - -h 2 ,.Amx~ ' - ~ r (zj, t) + - e \ 2Ax~ l r i c2
+ At) - r 2At
At))
m2c 2 ] +
e2
~o(xj,t) = 0
(16)
Turning to Eq.(2.1-11) we avoid again writing a matrix equation and use the space-saving notation of Eq.(13). Equation (2.1-27) remains unchanged since the operators p and 9f can stand for differential or difference operators, but Eq.(2.1-28) must be rewritten for difference operators:
56
2 MODIFIED KLEIN-GORDONEQUATION
[(
li Z~ i Axj
= --
{
)2 l(h )2 ] ( )E ( )
eAmz~
- ~
7 N + ~r176 + m~
+x~
2
4AcA~
e
+~
h z~
"~ ~---~j eAmz~
1
1
h /~
2m2oc2
~/~x-----]
eAm~
)2]
-
7 X / + ~r176(L~j + L=~j + t_~j + L~x~j+ L~j)
+ (k~x~j + Lcx2j + 1-r
+ Lcx4j + I_~j)
~
+ er
+ aI_~•
~I'xo~j (17)
As in the case of Eq.(2.1-28) we drop the last term aL2x because of the factor a. The operators Lcxt to [r (Euler Roman Medium font) introduced here follow from Eqs. (2.1-20) to (2.1-24) with the operators of Eqs.(2), (3), and (1.2-23) as well as the following substitution based on Eq.(2.1-29): 1 --s
~ Lcxkj,
C~e
k = 1, 2, 3, 4, 5
(18)
The matrix [-cxk, using the Euler Roman Bold font, has the terms kcxkj along the main diagonal and zeros everywhere else:
Lcxkl Lcxk --
0 0
0
0 ) k = 1, 2, 3, 4, 5
Lcxk2 0 , 0 Lcxk3
(19)
In analogy to Eqs. (2.1-30) and (2.1-33) to (2.1-36) we obtain:
Ze c { [ A ~ ( ~ Al--~ ~ - e A m y ) _ A e y ( h ~ ~Z ~ - eAm~ ) Lcxlj -- 2am2 X 1 + 1 m~d
- - eAmz Ax hi , ~z~j
1
m2od
eAmzj
i Azj
Aez
eAmz~
,hy - eAmy
-Aey(~ z~_.-eAmz) -z-Az (~ Z~Ax~ - egmx)} j=l,
2, 3; x l = x ,
x2=y,
xa=z
(20)
2.2 MODIFIED KLEIN-GORDON DIFFERENCE EQUATION
57
As previously in the case of Eqs.(2.1-31) and (2.1-32) the terms with the variables x, y, z
A~z 7h G AA _ eAmy ' A~u
A _ eAmz ' A-~
7 ,~''x - eAm~
(21)
form matrices of rank 3 with equal values for all elements in the main diagonal like the second and third matrix in Eq.(12). But the terms
hA i~xj
2
eAmx~)
(22)
are the terms of a matrix with rank 3 like the first matrix in Eq.(12) with xj = xl, x2, x3 = x, y, z. The four expressions for l-r to l-cxhj are written according to Eqs. (2.1-33) to (2.1-36) with the help of Eq.(1.2-23)" X
Ze [_cx2J _-- 2a m oc ~ { [ filCym ( 7AZ h
1
[
x 1
-eAmz)
/~AZ (/)m (~Z~y /7~ -eAmy)
[
2m2c 2 7 Ax---~-- eamx~
C
(
hA
2m2oc2 7/~x---~- eAmx~
Ze
A~Z {
2o~'rnoc fil
-Aez-~{ z~ +
kcx4j - -
1
ACm h A _ eAmz - Az ,~V 7a-~
x
Lcx3j - -
+ 1
[1
C
&
- eAmy
Ax
(23)
(h
1
( ,h~ x z] j
( h'Tl
I~ A-~'~ eAm~) 1
zec 8( Aey
hA
2m~c2 ~ ,4xj-eAmxj) l ( ~ - ~ y - e A m y ) [1
1
(
eAm~) 21 } 2
1 (h /]
2m~c 2 i ~ x j
/~
7 AZ - ~Am~)
)2]})
eAmx.~
Ax
(24)
(25)
58
2 MODIFIED KLEIN-GORDON EQUATION
Lcxh~ =
cJ
1
eAmxj
~
Ay-
c
+(~.~-~m~) ~~ ~~ ~~)}~y + + ~z
•
1
~xj - edmz
1
eA~
-'=-Zlz- eA~z
2m2oc2 i ~xj
1 _ i z~xj eAm~) 2m0~c~( h ~
[ (~)
eAm~
~z
~]
(_)] + [A~ (~ ~,~y -eAmy ) - A~y (--~.)] h A i Az • A~ ~,~y-eAmu
x [1
-A~y hiAzA-eAmz
1
~o~(~
~]
(26)
Equations (20) to (26) make all terms in Eq.(17) defined for known values of the potentials Am, Ae, Ce and the rest mass m0 of a charged particle. The equations are unique if one accepts Eq.(1.1-47). Equation (16) can be solved for certain initial and boundary conditions. The solution of Eq.(17) calls for a particular solution of the inhomogeneous equation that contains the solution ~x0x~ of Eq.(16). The homogeneous part of Eq.(17)is equal to Eqs.(13)or (16). Our calculation contains so far the simplification of the Hamilton function and the term o~(2)~cAe/e)Q in Eq.(2.1-5) that go back to Eqs.(1.t-42) to (1.1-46) rather than the exact Hamilton function of Eq. (1.1-37) without the restriction v << c. A second simplification was introduced by Eq.(2.1-9) for small values of c~. But our equations are still too complicated and we introduce further simplifications for the potentials Am and r
Am = Am0 + ~Aml(r, t), Amxj = (~e -- (~e0 + C~)el
(Xj, t),
Xj
Amoxj + ~Amtx~(xj, t)
--" X, y, Z;
IX/x -- lI/x0 -{- oLlX/xl
(27)
We first rewrite Eqs. (2.1-10) and (2.1-11) with the approximations for Am and Ce:
2.2 MODIFIED KLEIN-GORDON DIFFERENCE EQUATION
59
( p - eAmo) 2 - 1 ( ~ , _ eCeo)2 + m02c2) ~xO = 0
(28)
( ( p - eAmo) 2 - c~(9{7x- eCeo)2 + m2c2) ~xl
=-[4A:Ae[(p-eAmo)2+m~c2]Q . . . .
C2
OL
+
OL
(9{7x - eCe0) +
+ (p -- eAmo) Aml] + ~'g[r
-- e[Aml" (p - eAmo)
-- eCe0) + (gt:x -- eCe0)r
~xO
(29)
Equation (28) corresponds to Eq.(2.1-2). If we replace Am by Amo we must replace Am~j by Amo~. in Eqs. (2.1-4) and (2.1-28). This implies the same replacement of Am~j in Eqs.(13) to (16). Hence, Eq.(16) is reduced to the following form by the approximation of Eq.(27):
(Azj)~ 1 ~xO~j (x~, t + At) - 2VxO~ (xy, t)
c2
- 2i e.h
Amozj
+ Vxo~j (xj, t -
At)
(~xt)= (zj + Azj, t) - Vxo~ (xj - Azj, t)
r
2Axj
+ ~r
r
(xj, t + At)2At-r
(xj, t - At) /
m~c2)~P• e2
e2( 2 1 2 hg. Amo~ - ~ r
t)=0
(30)
The difference equation representing Eq.(29) must be written in a more compact form by means of the definitions of Eqs.(2) and (3): ~
2
-- ~
i Axj
= {4 CAe( + ~z
7i
~
"~- e.r
2E1 + ~r
Jr" ?gt2C2 ~xlxj (Xj, t)
1
( t ~ . + L~x~;+ Lcx~ + tc~ + r ~ ; )
+ (tcx. + Lcx~j + tcx~j + tcx,j + t c ~ )
~ + ~r
1
+ ~L~x
60
2 MODIFIED KLEIN-GORDON EQUATION
-e[Amlx./(hi~-xjA ~2 r
i
eAm~ + ~r
ff~ eAmoxj~Amlxj]
h
+
7i
+ ~r
r
Cx0~(~j,t)
(31)
As in Eq.(17) we ignore the term ak2x because of the factor a. The terms [-cxlj to kcxsj still have to be rewritten to correspond to the simplifications of Eq.(27). We have to add two more approximations: Ae = Ae0 + aA~l (r, t),
Aex~ -- Aeox~ + aAelzj (xj, t)
Cm -- Cm0 3r" aCml (Xj, t)
(32)
Only the terms Ae0 and era0 are needed in Eq.(31); the terms multiplied by a are neglected. The term kcxlj in Eq.(20) is not changed much optically, but all terms Ae0, Ae0., Am0. are now constants:
Ze
-
]1
- " = - - e A m o v ) - A e o v ( li •
--=-" -- eAmox Ax
+ 1 - . ~1
B ~ - eAmo~ 7,~-~
-
j=l,
1 - -----rn2oc2
- _-z---- eAmoxj i Axj
"Aeoz
li z~z /~ _ eAmoz A~oy "( 2, 3; x l = x ,
~~: y - eAmoy
-= - eAmox Ax (33)
x2=y, x3=z
The term I-r of Eq.(23) vanishes since the differences Z~r etc. of a constant is zero. This implies that our results will not depend on the existence or nonexistence of a magnetic charge according to Eq.(1.1-22): kcx2j - 0
(34) For kcx3j of Eq.(24) apply the same comments as for Eq.(20):
kcx3j--2~rnoc ~c
_[_ lli. . . .A
eOz'~
i ~xj
1-2mo2C2
enmoy) [ 1
1
eAmoxj
i~xxj
eAmoxj)
Ay'="-eAmo }
2.3 SOLUTIONOF THE DIFFERENCEEQUATIONOF tI/x0 /~
[1
1 ~oc~ ~~
7-~z ~mO~j)](~
61
eAmoz)
9
kcx4j -- 0,
(35)
AA.0~/z~x-- 0, AAeoz//~y = 0
Lcx5J 4~-~2c ,(1
_jr.(~Ay
+{[~
[1 2Tn-~C2(hAA-Xj 7
(36)
eAm~ ] (~/~y
1
eAmoy) 1 27.n,02c2()~ Z
1 ~c~(~~~
(
)E
li /~ + 7Ai -eAm~
1
1
2
A
~o~) ~(-~i Alz ~ - ~mO~) 1
(h A
2mo2C2 iAx~. -
h Z
eAmoxj
~zz
)
u] J
• Aeo~ ~y-eAmoy
-Aeoy lii AzZ~"
eAmoz
_ A Jr Imeox(h~-emmoy)-Aeoy(~ z~z-eAm~ i Ay
~[1
1
~
2m~c2 ( hi Zxj
eAmo~j)
~]} A x
(37)
2.3 SOLUTION OF THE DIFFERENCE EQUATION OF t~x0 We start from Eq.(2.2-30). The normalized variables 0 and r are used instead of the nonnormalized variables t and xy. The number of spatial variables is reduced to one"
O=t/At,
r
O
(1)
The function ~x0x~ is shortened to ~o. The potentials Ae0z~ and Amox~ have the components Aeo~, Aeoy, Aeoz and Amoz, Amoy, Amoz which are constants. They are not affected by the normalizations of Eq.(1). In the special case of Eq.(2.2-30) we could write Amor and not decide whether Am0; represents Amox, Amoy, or Am0z since the choice of ~ or xj = x, y, z in Eq.(1) will decide what
62
2 MODIFIED KLEIN-GORDON EQUATION
Am0r stands for. This simplification would not be possible when we advance from Eq.(2.2-30) to Eq.(2.2-31) in Chapter 3. Hence, we will not use it. The following normalized partial difference equation with one spatial variable is obtained from Eq.(2.2-30):
9xOzj = 9x0 = ~ 0 , Xj = x
[90 (~ + 1, 0 ) - 29o (~, 0)+ 9o ( ~ - 1, 0)1- [9o (r 0+ 1 ) - 2 9 o (g, 0)+ 9o (~, 0-1)] -iA1 { [ 9 o ( r 1 6 2 [9o (~, 0 + 1 ) - 9 o (r 0-1)] }-A2 9o(r O)=0
Ce0
ecTAmo~,
A1 =
Nh
,
A3
c 2A2mo~+e 2A2mo~
=
~2 _ ~2 _ (T/Nl~)2(m2c4 _ e2r ~+~=(T/Nh)~(.~ 2 2 A1A3 . A22 .
eTr
cAmoz'
A1A3
-fib
Nh '
T
N = At
(m2c 4 - e 2Ce0+e2c 2 2 Amoa: 2 )
> 0 for mo c2 > eCeo, A2 >> A2
~. +~ ~~~Amos), ~+~-~=(mo~T/Nh) 2 (T/Nli)2(m20c4 2 + e2c2 Amox) . . 2e 2CeO
~
(2)
We look for a solution excited by an exponential ramp function as boundary condition. The exponential ramp function avoids the 'instant jump of a step function and is thus better suited for equations that describe particles with a finite mass m0. The constant ~ cannot be chosen. It will turn out to be imaginary:
r
o) = r
- ~-~o) = o = r
- e -~e)
for0< 0 for 0 > 0
(3)
An initial condition is needed for the time 0 = 0 and all values of r We choose it to be r 1 6 2 o) = o
for o = 0, r > o
(4)
Since Eq.(2) is a second-order difference equation with respect to 0 we need a second initial condition in addition to Eq.(4). We proceed as follows. Equation (2) requires 9o(~, 0), and 90(r 1) to compute 90((~, 0) for 0 > 2. Since 90(r 0) is defined by Eq.(4) we may choose a second condition 90(r 1 ) - d '
(5)
2.3 S O L U T I O N OF T H E D I F F E R E N C E EQUATION OF ~I/xo
63
but the connection with 0 9 o ( ~ , 0 ) / 0 0 for/9 = 0 of the differential theory becomes clear if we use the "right" difference quotient of Eq.(1.3-1) instead of Eq.(5)" 9o(r 1) - 90(~, 0) = d
(6)
We use the following ansatz (Habermann 1987) to find a general solution from a steady-state solution F(r multiplied by ( 1 - e -~~ plus a deviation u ( ~ , O ) from ( 1 - e - " ~ 9
Vo(r o) = Voo[(1 - ~ - ' ~ 1 6 2 Substitution of 900(1 - e - ~ ~ 1 6 2
9 o(ff,0) = ~oo(1 - e - ' ~
(7)
+ ~(r 0)], r 0 >__0
into Eq.(2) produces the following relations
~o({,0 + 1) = ~oo(1 - e - ~ ( ~ 1 6 2
90(~ + 1 , 0 ) = 900(1 - e - ' e ) F ( r
+ 1)
and we obtain:
(1 - e - ~ ~
+ 1) - 2 F ( { )
- iA1 { (1 - e - ~ ~ 1 6 2
e~)F({) A3e-~~ - ~ - e~)F(r
+ F ( ~ - t)] + e - ~ ~ + 1) - F ( r - 1 ) 1 -
-~ - 2 +
- ~(1-
~-~~162
= 0
(8)
We collect all terms multiplied with 1 - e -~~ on the left side and the others on the right side:
(1 - e-~~ {F(r + 1) - 2 F ( ~ ) + F ( r
1) - i ) ~ l [ F ( r = -e-~~
+ 1 ) - F ( r - 1 ) ] - ,k2F(r }
-~ - 2 + e ~ + iA1A3(e -~ - e~)lF(r
This equation must hold for all values of/9, which is possible only if both sides are zero:
F ( r + 1) - 2F(r
+ F(r
1) - i A I [ F ( r + 1) - F ( r
e -~ - 2 + e ~ + i)~1)~3(e -~ - e ~) = 0
For the solution of Eq.(9) we may use the ansatz
1 ) ] - )~2F(r = 0
(9)
(10)
64
2 MODIFIED KLEIN-GORDON EQUATION
F(r = Alv ~, F(~ • 1) = Alv ~+~
(11)
and obtain an equation for v:
(1 -- i/~ 1)v 2 -- (2 + ~2)V + 1 + iA1 = 0
I+iA1
{ 2 + A 2+[4(A22 . A 2 ) + A 4 ] 1/2}
= 2i; -(I+iA1)[I•
2-A2)1/2],
A22, A2 < < 1
(12)
vl " (1 + iA1)[1 - (A 2 - ~2)1/2] - exp[_(A2 - A2)l/2]ei~l v2 " (1 + iA1)[1 + (A2
_
(13)
A2)1/2]
-- exp[(A2 - A~)l/2]e'~l We obtain for F(r
(14)
in Eq.(ll)"
F(~)=Aloexp{[()~2-)~21)1/2+iA1]r
exp{[-(A22-A2)l/2+iA1]r
(15)
According to Eq.(2) the difference A22- A2 will be positive except for extremely large values of Ceo. 2 We restrict the calculation to this case A2 - A2 > 0. In addition, we want F(0) to be 1, which is achieved by the choice All - 1 and Alo - 0 . T h e function F(r becomes:
F(r
= e x p [ - ( A 2 - A2)l/2r162
)%2_A12>0, F ( 0 ) = I ,
Alo=0,
Al1=1
(16)
E q u a t i o n (10) yields a quadratic equation for e ~"
(1 -
iA1A3)e 2~ - 2e ~ + 1 + iA1A3 = 0
e~_ (I+iA1A3)(I+iA1A3) -1+22 ~1,~3
. I+i(A1A3+X1A3)
(17)
A trivial solution e ~ - 1, L - 0 and a nontrivial solution are obtained" e ~ = 1 + 2iA1A3, ~ " 2iA1A3
(18)
2.3 SOLUTION OF THE DIFFERENCE EQUATION OF ~x0 Equation (7) must satisfy the boundary condition of Eq.(3). yields F(0) = 1 we get:
65
Since Eq.(16)
0>0
'I.'oo[1 - e -~~ + u(O, O)] = ~oo(1 - e-~e),
(19)
~(0, e) = 0
Hence, u(~,0) has a homogeneous boundary condition, which is the goal of this method of solution. The initial conditions of Eqs.(4) and (6) yield with
F(0) = l: Vo(r o) = Voo~(r o) = o,
9 oo[(1 - ~ - ~ ) F ( r
r
(20)
> o
+ ~(r 1)] - ~ o o ~ ( ~ , o) = o
~(~, 1) - ~(~, o) = - ( 1 - ~ - ~ ) F ( r
(21)
r > 0
Substitution of u(~,O) into Eq.(2) yields the same equation with ~0 replaced by u:
[~,,(~+ 1, o ) - 2~,(r o)+ ~(r ~, o)]- [~(r o + 1)-2~(r o)+~(~, o - 1)]
-iA~{[u((+l,0)-u(r162162162
(22)
Particular solutions u~(~', O) of this equation can be obtained by the extension of Bernoulli's product method for the separation of variables from partial differential equations to partial difference equations: ~(r
= r162162
(23)
= r162162
We observe that the notation r162 instead of r162162 is a generally made simplification that will be acceptable in Chapter 2. The substitution of u~(r 0) for u(r 0) in Eq.(22) yields:
[r162+ 1 ) r
- 2r162162
+ r162 - 1)r
- [ r 1 6 2 1 6 2 + 1) - 2 r 1 6 2 1 6 2 + r 1 6 2 1 6 2 - 1)]
iA1 {[r162+ 1)r
- r162- 1)r
+ Aa[r
+ 1) - r162162 - 1)1} - ~r162
= 0
(24)
In analogy to the procedure used for partial differential equations we multiply with 1/r162162 and separate the variables:
66
2 MODIFIED KLEIN-GORDON EQUATION
1
r162 <~[,(r + 1) - 2r162 + r162- 1 ) ] - ~ [ , ( r = r
1
<~[r176+ 1) - 2 r
r
+ 1) - r162- 1)]}
+ ~~[r
+ 1) - r
- 1)1} + ~
=-(27rp,c/N) 2 A constant -(2~p,~/N) 2 is written at the can be equal to a function of 6t for any a constant. The divisor N will permit the length N rather than 1 later on. Equation equations for r r
[r162+ 1) - 2r
+ r
- 1)] - iAI[r
end of Eq.(25) since a function of r and /7 only if they are equal to use of an orthogonality interval of (25) yields two ordinary difference
+ 1) - r162- 1)]
+ (27rp~/N)2r162 = 0
[r
+ 1) - 2r
+ r
- 1)] + iA1A3['r
(25)
+ 1) - ~(0 - 1)] + [(2;rp~lN) 2 + A22]r
Equation (26) can be solved like Eq.(9) for F(r alent to Eq. (11)'
(26)
(27)
by using an ansatz equiv-
r162 = A2v ~, r162-t- 1 ) = A2v r
(28)
The following equation is obtained for v"
(1 - iA1)v 2 - [2 - (27rp,~/N)2]v + 1 + iA1 = 0
v~,~ = 2(1 + ~ )
2-
N + 2i 27rp~ [ 1 (27rp~) 2 A21 ] N 1 - -~ N + (2rp-"~/N) 2
1/2}
(29)
We shall need v3,a only for real values of the square root. Furthermore, we need v3,a only up to the first order of At. Let us note that A2 in the square root is of order O(At) 2 according to Eq.(2)"
2.3
67
SOLUTION OF THE DIFFERENCE EQUATION OF tI/xo
4A2 for (2~rp~/N)2 < 4 + (2~rp~/N)2 = 4 + O(At) ~ 1 2 v3--1--~
~
--
A
1-
1 N
4
[ 2~rp~ ( 1- (2~rp"/N)2) ~/2+A~ ( 1- (2~P,,/N)2)] +O(At) 2 +i N 4 4 v4--
1-- ~1
~
_i[2~
l -
+A1 N'"
(l_(2~rp~/N) 2 4
)
1/2
(30)
4 -AI(1-
(2~p~/N)2)] +O(At) 2 (31) 4
We use Eqs.(29)-(31) to write v~ and v4~ as follows, ignoring terms of order O(At) 2 or higher: 4A~ for (27rp~/N)2 < 4 + (27rp~/N)2 = 4 + O(A) 2 { ~,4=(1+i~)
v3r
=
1-~
(1+iA1)r
1 ( N2 ) 2 ~
• ~ 2~p,,[ 1 - 1(2~p~)2] ~/2} N ~ N
_ 1
~(27rp~) 2] 2 + ( ~ - 5 ) : ~ [ 1 N
= (1 + i ~ ) c { 1 }c/2 ~ . . ~
v4r
1
~ 9 (1 + i ~ ) ~ '*'c "
(2"xP'~)2]} r
(32) e~O~r
N
r
= cos(cp~ + Ax)~ + i sin((p~ + A1)~
(33)
(1 + i)~)~e -~~ -~(~o -~)~ = cos(~p,, - A1)( - i sin(~o~ - )kl)~
(34)
~o~ -
arctg
(2wp~/N)[1 -(2~p~/N)2/4] ~/2
1 - (27rp,,/N)2/2
, A1 =
ecnmox At
h
(35)
The function r is written in complex form using Eqs.(33) and (34) with A1 and ~ defined by Eq.(35): r
= A3ovr + A31vr = A30e'(~+x~) r + A31e-'(~~162
(36)
The boundary condition u(0, 0) = 0 in Eq.(19) requires in Eq.(36) the following relation between the constants:
68
2 MODIFIED KLEIN-GORDON EQUATION
A31 - - A 3 o
r
= A30 (e i(r162
= 2iA3oe i~'r sin ~ r
- e-i(v~-~l)r
(37)
E q u a t i o n s (36) and (37) will be used with a Fourier series. T h i s requires t h a t we choose ~ = ~ ( p , ~ ) so t h a t we get an o r t h o g o n a l s y s t e m of sine a n d cosine functions in an interval 0 <__ ~ <__ N = T / A t with a m a x i m u m of N periods. We m u s t choose ~ as follows:
N ~0~ = 0, -t-1 927r, =t=2.27r, . . . , %-~- 92~T
~o,r = 21r~/N,
~ = 0, =t=1, i 2 ,
...,
=t=N/2
0 <__~ <_ g = T / A t
(38)
T h e variable ~o~ can a s s u m e N + 1 values. T h e r e are N o r t h o g o n a l sine functions, N o r t h o g o n a l cosine functions, and a c o n s t a n t defined by ~ = 0. The function r c o n s t a n t is a solution of Eq.(26) for ~o~ = 2~rp,~/g = O. In o r d e r to o b t a i n the eigenvalues (27rp,~/N) 2 associated with the angles ~o~ we m u s t solve Eq.(35) for (27rp,c/N) 2. W i t h the relation 1 1 + tg 2 qo~ = cos2 qo~
(39)
we o b t a i n from Eq.(35) two solutions:
(27rp.) N
2
= 2(1 + cos ~o~) = 4 cos 2
r
= 2 ( 1 - c o s ~ , ~ ) = 4sin 2 ~---2-~ 2
(40)
(41)
In o r d e r to see which solution to use we take the inverse of Eq.(41)" ~o~ = 2 arcsin
27rp~/N 2
(42)
At least for small values of ~ we are back to Eq.(35). Hence, we use Eq.(41) to define p~ with the help of ~ "
N
~
p~ = - - sin ~
7r
-
9
2
N 27r ~ = 1 r
. - - -
N
~
7r
N-'
= - - sin
for~<
= O, +1, +2, . . . , + N / 2 (43)
2 . 3 SOLUTION OF THE D I F F E R E N C E EQUATION OF ~I/x0
69
0.8
LO.6 --~ o. 4 0.2 o.s
~
1.5
2
FIG.2.3-1. Plot of the excitation function ~0(0, 9) according to Eq.(46).
0.8
%-06 T
0 2 0.5
1
1.5
2
(2~1~3/~)e --~ FIG.2.3-2. Plot of the excitation function ~I'0(0, 9) according to Eq.(47). We want to derive a plot of the b o u n d a r y condition or excitation function of Eq.(3). S u b s t i t u t i o n of ~ from Eq.(18) produces a complex function. E i t h e r the real or the i m a g i n a r y c o m p o n e n t represents an acceptable b o u n d a r y condition:
9 0(0,9) = ~ 0 0 S ( 8 ) ( 1 - cos2A1A39)
(44)
v0(o, e) = v00s(e)~i. 2~1~0
(45)
These are (causal) cosinusoidal and sinusoidal functions t h a t are zero for 9 __ 0. One m a y readily derive step functions with a s m o o t h e d step from them"
+ s(e - ~/2~:~)[1
- cos(2:~e
- ~)]}
(46)
9o(0, 8) = ~oo { S(9) sin 2A1Aa9 + S(8 - 7r/4AIA3)[1 - sin(2AiA38 - 7r/2)1 }
(47)
Plots of Eqs.(46) and (47) are shown in Figs.2.3-1 and 2.3-2. T h e tangent at O = 0 is zero in Fig.2.3-1 and larger t h a n zero in Fig.2.3-2. T h e r e is no j u m p or kink at (2A1A3/Tr)9 = 1 in Fig.2.3-1 or at (2A1A3/Tr)O = 0.5 in Fig.2.3-2.
70
2 MODIFIED KLEIN-GORDON EQUATION
Boundary conditions with any time variation other than those of Eqs.(46) and (47) must be represented by a superposition of these inherent boundary conditions. The choice d - 0 in Eq.(6) yields Fig.2.3-1, while d - - ( 2 A l , k 3 / T r ) ~ 0 0 yields Fig.2.3-2. 2.4 T I M E - D E P E N D E N T SOLUTIONS OF tX/x0(~,~ ) Equation (2.3-27) is the time-dependent difference equation of r separation variable p~ is defined in Eq.(2.3-43)'
r
+ 1) - 2r
+ r
- 1) + iAIA3[r
+ 1) - r
The
- 1)]
+[(27rp~/g) 2 + A219(9)= 0
27r,~ cp,~= N , ,~=O, +1, + 2 , . . . , N Substitution of r
=4sin 2
N +2
, p~=--sin 7r
(1)
2
= A2v ~ into Eq.(1) yields an equation for v
(1 + iAIA3)v 2 - [ 2 - (2rp,~/N) 2 - A22]v + 1 - iAIA3 = 0
(2)
with the solutions
Vs,0= 1 - ~ + A~A3
1--
2
)+
1
2
l [(
+A22
N 2
2
)
(3) /
The square root is imaginary for
-4
N
+ A2 -4A~A2< 0
(4)
If we replace the sign < by the equality sign - we obtain a quadratic equation with the solutions 2 31/2 9 (27rp,~/g) 2 = 2 - A2 =t=2(1 + AIA3) 22 - 9 4 - A2 + A1A 3 9
- -~
- ~i
2 2 2 - A2 + (2 + AIA3)
22 for + (2 + A1A3)
for - ( 2 + ~ 12 ~3)
(5)
Using Eq.(2.3-2) we see that the square root in Eq.(3) is imaginary if the following two conditions are satisfied:
2.4 TIME-DEPENDENT SOLUTION OF II/xo
71
4 - A~ + A21A32 > (2zrp,~/N)2 = 4sin2(9~/2)
4 --
(m20c4 --2C 2 Ce0 2 -Jr-e2 C2A2 .[-Im0x) > (27rp,~/N) 2
(6)
We may rewrite these equations as follows:
q0~=
27rN ( ~2 _ /~1~ 322)1/2 N <2arcsin 14
N (A22-A2A2) < t~0 = - - a r c s i n 1 7r 4
1/2
(7)
In addition to Eq.(6) we get from Eq.(5) a second condition:
- A 2 - A21A32< (27rp,~/N) 2 = 4 sin2(~p~/2) -
(-~)2(/Tt02C4nt-e2 c 2 A m20 ~ ) <
(271ptr /N) 2
(8)
The condition of Eq.(8) is always satisfied since the range of interest of
(27rp,~/N) 2 = 4sin2(99~/2)is restricted as follows' - N / 2 <_ K, <__N/2,
- 1 _< V)~/Tr <_ +1,
0 _< 4sin2(99~/2) < 4
(9)
On the other hand the condition of Eq.(7) may not be satisfied. Both the real and the imaginary root of v5,6 must be analyzed. We consider the imaginary root. (2~p,~ -~--~-~ 1 V5'6 ---~ 1 + A1A3 2
N
{ 1[(
I
-
1+
iAIA3
+A
ii
N)
2 2 i/2e+i~,~ 9 (1-iA1A3)e +i~'~ (I+A1A3)
2
i/2
.e-i~e
•
2 t g f l ~ = [(27rP'~/N)2+ A~]1/2 { 1 - [(27rp'~/N)2+ A~]/4 + AI2 Aa/[(27rP'~/N)2+ A~]} 1/2 1-[(27rp,~/N) 2 + A~]/2 (11)
72
2 M O D I F I E D K L E I N - G O R D O N EQUATION T 0.4 "~0
V
2
-0.4
-0.2
0.5 1 ~/Ir ---*
FIG.2.4-1. Plot of fl,~/Tr according to Eq.(14). Equation (11) for ~,~ can be simplified with the help of the following identities 1 + tg 2 ~ = COS2/~,, 1
sin ~ = (1 - cos 2/3~) 1/2
(12)
2np,~/N
from Eq.(1)
/3__5~_ _1 a r c s i n / / 1 - [ 1 - A 2 / 2 - 2 sin2(~/2.)]2'~ 1/2 ~k 2 23 ] ~ 1 + AIA ~=2~/N, ~=0, +I, ..., +N/2
(13)
which are solved for cosfl~ and s i n ~ . brings'
Substitution of
For small values of A12 and A2 one obtains ./3~ . 1 arcsin . . 1 . 1 . 2sin 2 ~ 7r 71"
7r
'
A~A~' A2 << 1
(14)
A plot of this equation is shown in Fig.2.4-1. The plot of Fig.2.4-1 needs some details close to p ~ / ~ = 0, +1 where the simplification A2 << 1 is not justified since s i n 2 ( ~ / 2 ) approaches zero. For ~ / T r equal to zero we get from Eq.(13):
~__2_~= I arcsin (1 _ (1 - ~-2-/2-_)2/ 1/2 ~ 1 +A1A 2 ' --0.0225
f o r A 2 = 0 . 0 0 5 , A1A322=0
~=0 (15)
2.4
TIME-DEPENDENT
SOLUTION
OF
73
~x0
0.04
TO 02 .
-0.04
.
.
.
,
.
.
0.02
-0.02
.
.
J
0.04
~o~/~r - - , /
/
-o.o4
FIG.2.4-2. Plot of fl~/~ according to Eq.(13) in the neighborhood of ~ / ~
~
= 0.00~, ~
= 0 for
= o.
Figure 2 94-2 shows a plot of #~/Tr for A2 --- 0.005 , "~1)~3 2 2 = 0 in the interval - 0 . 0 4 < ~p~/~ < 0.04 according to Eq.(13). The important feature of Fig.2.4-2 is that ~ does not become zero for W~/~ = 0 but assumes a value defined by
Eq.(15):
1 arcsin(i (I- ~~2) I/2_
7r
7[
1+
A_s
(16)
71"
For the neighborhood of ~ / T r = 1 we obtain from Eq.(7)
2 s i n T~
<(4-A
2 <arcsin
2+A2A32) 1 / 2 . 2 [ 1 - ~ (1A 2-A21A2) ] 1
-
2 [ .(P~ < . 7r arcsin . .1
g(A2 1
2 2 - AIA3)
(A~
2 = 0.9775 A21A3) for A2 -- 0.005, A21A23 -- 0
(17)
Plots of/5~/7r close to ~ / ~ r = +1 are shown according to Eq.(13) in Figs.2.4-3a and b. The function v 5,6 ~ according to Eq.(10) can be written in the following form if the conditions of Eqs.(7) and (8) are satisfied"
The function r
v ~ = e -i(~l~3+#")e
(18)
v ~ = e -i(~l~3.#")e
(19)
of Eq.(1) becomes:
74
2 MODIFIED KLEIN-GORDON EQUATION 0.08
-0.02
0.06
-0.04
0.04
-0.06
0.02
-0.08 -i
-0.98
-0.96
-0.94
-0.92
0.92
a
0.94
0.96
0.98
1
b
FIG.2.4-3. Plots of ~ / z r according to Eq.(13) in the neighborhood of ~o~/Tr -- - 1 (a) and qo~/zr = +1 (b)for A2 = 0.005, A12A32= 0.
~b(O) = A4o vg d- A41 vg = A4o e-i(xlAa+/3'~)O d- A41 e-i(AIAa-fl'~)O
(20)
The particular solution u.(r 0) of Eq.(2.3-23) assumes the following form with the help of Eqs.(2.3-37), (18), and (19)'
u.,(~, 0) = {A1 exp[-i(A1A3+l~,~)O]+A2exp[-i(A1A3-~,~)O]}e i)'1~ sin ~,~< (21) The solution u~(~, 0) is usually generalized by making A1 and A2 functions of a real variable n and integrating over all values of n from zero to infinity. This would imply nondenumerably many oscillators or photons. It is usual in q u a n t u m field theory to reduce the nondenumerably many oscillators to denumerably many, using box normalization to accomplish this reduction. We follow the spirit of this approach but (a) the box normalization is avoided since it is not based on a physical principle but strictly on the success of the calculation and (b) the denumerably many oscillators or photons are replaced by a finite number since we have no means to produce or observe more. We introduce arbitrarily large but finite time and space intervals 0 <_ t _< T and 0 <_ xj <_ cT. The normalized variables 0 and ~ cover the intervals
O <_ O = t / A t <_ T / A t , O <_ ~ = y / c A t <_ T / A t ,
T / A t = N >> I
(22)
Equation (21) is not generalized by a Fourier integral but by a Fourier sum (Harmuth et al. 2001). The notation n > - n 0 in the following equation means the smallest integer larger than - n 0 while n < n0 means the largest integer smaller than n0'
2 . 4 T I M E - D E P E N D E N T S O L U T I O N OF ~ x 0
u(~,O) -- E
75
{Al(t~)exp[-i(A~A3 + fl,~)O] + A2(t~)exp[-i(AiA3
-/3,~)0]}
t~:> --t~o
x e~
sin r
(23)
The summation is symmetric over negative and positive values of ~, while the differential theory always yields nonsymmetric sums over positive values of ~. Substitution of Eq.(23) for u(~,0) and u((, 1) in Eqs. (2. 3-20) and (2.3-21) yields: <~0
u(~, 0) =
~
[A~(~) + A:(n)le ~
sin ~,~ = 0
(24)
to> --t~o <~0
u((,1) - u(r 0) = ~
[Al(a)(e-i('x~'x'+~)-l)+A2(a)(e-i('xl"xs-~')-l)]
t~>--~O
• e ix~c sin~p~ = - ( 1 - e-2~
" -2iA1A3F(~)
(25)
To solve these equations for AI(~) and A2(~) we use the Fourier series in the form of the sine series: N/2-1
9s(a) = ~2
fs(~) sin 27r~d{, N
f~(~) =
E gs(~) sin ~=-N/2+1
0
27rN,__~
N
(26)
The factor e ~1r in Eqs.(24) and (25) can be taken in front of the summation sign. With 99~. = 2~r~/N the substitution of Eq. (26) into Eqs.(24) and (25) yields: AI(~) + A2(~) = 0
(27)
Al(t~) (e -i('~as+f~') - 1) + Az(t~)(e -i(a~as-~'~) - 1) N
_ _
4iA1A3/ F(~)e - i ~ s i n 27rt~ N d~ = -2iA1A3IT(~/N) N
(28)
0
N
IT(a/N)=--~
N
/ F(~)e -~
sin
N
0
0
2 (27ra/N){1 - exp[-(A~ - A~)I/2N]} N
27r~4 de N
-
+
(29)
76
2 MODIFIED KLEIN-GORDON EQUATION
0.1 0.05
-0.05
-0.I -0.4
-0.2
0
0.2
0.4
~/N FIG.2.4-4. Plot of IT(x/N) according to Eq.(29) for N = 100, A~ = 0, and A22 = 0.005 in the interval -0.5 < ~ / N < 0.5.
0.i 0.05 -0.05
-0.i -0.04
-0.02
0
0.02
0.04
~/N FIG.2.4-5. Plot of IT(x/N) according to Eq.(29) for N = 100, A2 - 0, and A22 -- 0.005 in the interval -0.05 _< ~/N <_0.05. A plot of IT(R/N) is shown in Fig.2.4-4 in the interval - 0 . 5 < ~/N <_0.5 for N = 100, A~ -- 0 and A22 - 0.005 . The same plot is shown expanded in the interval - 0 . 0 5 < ~/N < 0.05 in Fig.2.4-5. The solution of AI(~) and A2(~) in Eqs.(27) and (28) may be written in the following form with the help of the function IT(x/N)"
A1 (~) = -A2(~) = A1A3IT(~/N) ei~,~,3
(30)
sin ~ T h e substitution of AI(~) and A2(~) into Eq.(23) yields the function u(~, 0)"
u(~,O) = -2iA1A3e~
~176
E
IT(~/N) s i n ~ O s i n 2~-~__._.~ sin ~,~
N
(31)
2.4
TIME-DEPENDENT
SOLUTION
77
O F ~I'xO
I
I I -N/2 -N/2 + 1
-2
!
-1~
. _
0
1
2
i
!
N/2-1 N/2
tr ----o
FIG.2.4-6. Step function in the interval - N / 2 <_,~ < N/2 with steps of width 1. The area under this step function is determined by the values of tr from n = - N / 2 + 1 to = N / 2 - 1. The functional values for tr = :kN/2 are not needed. We note t h a t fin does not become zero for ~Pn = tr = 0 according to Fig.2.4-2 if the exact formula of Eq.(13) for/~n is used, but the a p p r o x i m a t i o n of Eq.(14) yields zero. This is not necessarily a m a t t e r of concern since IT(n/N) of Eq.(29) is zero for n = 0 and the term sin flnO in Eq.(31) becomes zero for fin = 0 too, but c o m p u t e r s tend to get confused. T h e s u b s t i t u t i o n of F(~) of Eq.(2.3-16) and of Eq.(31) into Eq.(2.3-7) produces the solution ~I'0({, 0) for Eq.(6): for 0 < 4 s i n 2 ( ~ n / 2 ) < 4 ~0(~, 0) = ~00 [(1 - e-2ixl)~a~
+ u(~, 0)]
= ~00 I(1 - ~ - = ~ , ~ 0 ) e x p [ - ( a ~
<no
-- 2iAiA3ei~l)k3ei~l(~-~30) E
n> --no
n0=--arcsin 7r
A~ + A2A32
1-
-
a~)~/=r162 I T ( n / N ) sinfln0sin sin ~n
AI A3 4
27rn~ N
(32)
2 3. 2 It seems that anew We turn to the case 4sin2(~n/2) > 4-A22+AIA solution of Eq.(1) is required to close the gaps between -N/2 and -to0 or no and N/2, but this is not so. According to Eq.(2.3-2) the coefficients A2~and A2IA23 are of the order O ( A t ) 2. Hence, In0[ of Eq.(32) is of the order N / 2 - O ( A t ) 2. T h e sum of Eq.(32) represents the area of the step function shown in Fig.2.4-6. If the sum runs from n = - N / 2 to n = +N/2 the first and the last a m p l i t u d e needed are for n = - N / 2 + 1 and n = + N / 2 - 1. Hence, the s u m m a t i o n signs in Eq.(32) m a y be replaced
N/2-1
<no
E tc>-no
-+
E n=-N/2+l
and we do not need the solution for Inl _> N/2.
(33)
78
2 MODIFIED KLEIN-GORDON EQUATION 2.5 HAMILTON FUNCTION AND QUANTIZATION
The Klein-Gordon differential equation defines a wave whose energy density is given by the term Too of the energy-impulse tensor 1" 1 09* 09 m~c 2 Too= c 2 0 t Ot b v g * " v g + h 2 9 " 9
(1)
To produce the dimension J / m 3 for Too the dimension of 9 * 9 must be J / m or V A s / m in electromagnetic units. For a planar wave propagating in the direction x we have V = 0 / 0 x . The Fourier series expansion in Eqs.(2.4-31) and (2.4-32) permits an arbitrarily large but finite time T and a corresponding spatial distance cT in the direction of x, using the intervals 0 < t < T and 0 < x < cT. In the direction y and z we use the intervals - L / 2 < y < L / 2 and - L / 2 <
J i
U =
,o,.o, Ot
e2 0 t
i
o,.o, Ox Ox
..o..,.9
~- ]22
dx
(2)
dy dz
-L/2 -LI2
T h e dimension of U is VAs. Equation (2) is rewritten with the normalized variables 0 = t / A t , ~ = x / c A t of Eq.(2.3-1)'
L/2cAt
i
U = cat
L/2cAt
N
i [i(o~ oo ___~176176176 },. + r~-----f-rg--f +
-L/2cAt -L/2cAt
-
0
xd(
y
9
)
d~
z
N = cat
00 00 ~- 04 0r
~-
h2
9
(3)
d~
0 We obtain from Eq.(2.4-32) for 9 = 90'
~9
0 = 920[(1 -- e2iXlx3~
+ u*(r 0)][(1 - e-2ixlx3~162
aBerestezki et al. 1970, 1982; w10, Eq. 13.
+ u(r 0)]
2.5 HAMILTON FUNCTION AND QUANTIZATION
79
1 = 4~o2o ~(1 - cos2AiA30) exp[-2(A 2 - A2)1/2~]
_ 2/~1/~3 CO8A1/~3 sin A1A30 exp[_(A2 _ ,~2)1/2~] N/2-1
x
E
IT(n/N__......._~) sin fl~O sin sin fin
,~=-N/2+1
22(
N/2--1
+ A1A3
IT(lq,/N )
E
sin flnO sin
sin fin
~=-N/2+l
2rc~r N
2
(4)
The differentiation of ~((, O) = ~0((, O) of Eq.(2.4-32) with respect to 0 or ~ produces the following results with the help of Eqs.(2.3-16) and (2.4-31)'
Oq~o0= 0~o00 = ~00 2iAiAae-2~176 ~ Ou _
o0
exp[-(A22 -/~2)1/2~] _~_~
(5)
--2A1Aae -v'l ha(0- I) eiA1 N / 2 --1
•
E
2 "ffn ~
t~=-N/2+l
0/I/0~---- 0~It00<~-~ /I/00 ((1
IT(n/N)sinfl,~ (AIAa sinfln0
+ ifln cosfl~0) sin ~ N
/~2)1/2 _}_i/~11
_ e-2iA1A30)[_(,~2 _
• exp[--(A 2 -
07 = 2A1A3e-iA1Aa(O-1)eiAIr
A2)1/2(]ei'xlC + -.~
N/2-1
IT(~/N)
n=-N/2+l
sin fin
o~
E x
(
Alsin
27r~ N
27tin 2rr~) N cos N
o~,~ Omo O0 O0
= e~o
- 2~a~a~~~
x
- ~ ' < ~ x p [ - ( a ~ - a ~ ) ~ / ~ ] + oo )
2iA1x 3e - 2 ~ x ~ ~ ~
(7)
sinfln0
The first term in Eq.(3) becomes: 0~* 0 ~ 00 00
(6)
exp[-(,X~ -
a~)~/~~l +
N
(8)
80
2 MODIFIED KLEIN-GORDON EQUATION
= 4A2A2~o20[exp[-2(A 2 - A2)1/2r N/2-1
--2 exp[--(A2- A21)1/2~1 E ,~=-N/2+l
IT(a/N)[A1A3sinA1A3(O+l)sinp'~O
-}- fl~ COS/~1~3({9 + N/2-1
+(
E
27rM 1) cos#~0] sin T
IT(~/N)AIA3
2~-t~r sin fl,,Osin N
sin fl~
,~=-N/2+I
N/2-1
sin fl,~
cos fl,~0sin N
(9)
~=-N/2+l
For the second term in Eq.(3) we get:
0r 0r
a~or O~0or = ~~176 ((1 - e2iXlx3~162
2 - ,~12)1/2 -- i~1]
• ~xp[-(a~ - a~)~/~r + or 2 X
((1 -e-2~x~~162
- a~)1/2+ i,Xl]
,%
• ~ x p [ - ( ~ - ~/~/~r
+ K
After considerable time one obtains the following result: O~or O~Oor= ~2~ 2A~(1 -c~
A2)1/2r
- 8 / ~ l A a s i n A 1 A a 0 e x p [ - ( A ~ -- ~21)1/2r ]
N/2-~ IT(~/N)
E
sinfl. sinfl~0
a---N/2-t- 1
x cosA1A3 A ~ s i n - - 7 - -
N
(
+sinA1A3 Al(A2-A2)l/2sin
22[(
+ 4A1A3
/21
E
~=-N/2+1
sin fl~
cos N
27rM 27r~1 27rM/l N + N cos N sin fl~0 sin N
(10)
2.5
81
H A M I L T O N F U N C T I O N AND Q U A N T I Z A T I O N
+(
N/2-1
27raIw(a/N)Nsin13~sin/3~0c~ 27rn~)2] } N
z
(11)
~=-N/2+1
Equations (9), (11), and (4) have to be substituted into Eq.(3). The integration with respect to r is straightforward but lengthy (Harmuth and Meffert 2003, See. 6.8). The energy U of Eq.(3) is separated into a constant part Uc and a variable part Uv(0) that is composed of sinusoidal functions of 0 and has the time-average zero:
u=g~+gv(O)
(12)
We obtain the following three constant parts from Eqs.(4), (9), and (11)"
N
Ucl = ~
h2
~;~od~
L 2 ,mo2c4(At) 2 '~oN = cAt li2
0
x (2 1 - exp[-2(A~ - A~)I/2N]
_qt_
2 (A~ - A12)1/2 N
A2IA23
NI2-1 E
~=-N/2+1
I~(t~lN) "-' sin -g/3"-s )
2 N/2-1 (IT(n/N)~2 L 2 m~c4(At)2 2 forN>>l, A217~A2 (13) tltooNA1A] E .... cat h2 ~=_N/2+lk, sin fl~ J
N
Uc2 = ~-~
00
00
0
L2
N/2-1
9
2
2
- ~Zxt~o0N~l~]
Z
IT(,c/N) ) 2 sin fl~
(A21A]+ ~2),
N >> 1 (14)
~=--N/2+1
N
0
L2
N/2-1
(IT(a/N))2[A21+(~-~) --" -cAt~o NAIA3 2 2 E ~=--N/2+1 The sum of these three components yields Uc:
, N >> 1
(15)
82
2 MODIFIED KLEIN-GORDON EQUATION
Uc -- Uc2 -qu Uc3-~ Ucl
-- c'At ~2~
E
IT(n/N)
sin/3~
~;=-N/2+l 2 2 + a~ + • ~I + a~a~
A1, A2, A3 Eq.(2.3-2),
+
N/2-1
m2c4(At) 2
u~(~) E ~=-N/2+1
h2
IT(n/N)Eq.(2.4-29),
/3,, Eq.(2.4-13)
(16)
For the derivation of the Hamilton function ~ we need only the constant energy Ur since the average of the variable energy Uv(O) is zero. We normalize Uc in Eq.(16): cAtUr L2~20N
-
cTAtUr L2~20TN
cTUr
9s
L2~020N2
N2
N/2-1
cTUc
(~7)
N/2-1
L~0~0=x= ~ = -~N / 2 + 1 x~= e ~ =~2 - N / 2 + l d~(~) sin~------7- ~ + x~a~ + ~ +
--~
+
h~
(18)
The component 9s of the sum of Eq.(18) may be rewritten as follows:
d(n) (sin 2~'n0 + i cos 27rn0) d(n) (sin 2zrn0 - i cos 27rn0) 2--~ - _27rin27rnd(n) e 9.,~i,~o2~nd(n) - - e - 2 ~ i , o = -27rinp,~(O)q,~(O) v/27vin x/'27rit~ _
p,~(o) =
27rnd(n) 2,~i,~o " e v/27ri~
(20)
Ap,~ _ 2~,~d(~) e2~o ~ ( j ~ A o
z~0~/27ria 2A0 27rnd(n) 2,~i,,oi sin 2~rnA0 x/'ff~n e
(19)
AO
_ ~-~~o) i sin 27rnA0
= p'~
AO
q~(O) = 2 ~ d ( ~ ) ~-~,~o
-- 27rit~p,~
(21)
(22)
v/27ri~
z~q~ _ O~ = A o
--
27rnd(~) -27ri~oisin27r~AO 42~i-----T~
i sin 27rnA0 = -q'~
AO
-- -27cinq,~(O)
Ao
(23)
2 . 5 HAMILTON FUNCTION AND QUANTIZATION
83
The relations for the finite derivatives/~gC~/z~q~ and z~J~/z~p~ have the form known from the differential theory but use finite differences instead of differentials:
~ = -27ri~p,~ Aq~ ',,.
Ap~
=-27riq~
= -27ri~p,~ " -fg,~
2Aq~
" +(l~
(24) (25)
We introduce the definitions
a , ~ = ~d(~) e2,~i,~o,
a * = ~ ed(~) -2,~,~o
(26)
to rewrite the Hamilton function of Eq.(18)"
N
~r
27rer,hb
N
N
~=0
to=0
b~ = (27rht~T)~/2 a,~,
N
(27)
to=0
b* = ( 2 7 r ~ T ) l / 2 a,~, *
(28)
Equations (27) and (28) are written in a form that permits one to follow the standard ways of quantization. It can be used both for the Heisenberg and the SchrSdinger approach. Let us consider the Heisenberg approach first. The terms b~ and b~ are replaced by operators b~- and b+"
1( lO)
1( lO)
(29)
1(lO)
(30)
b,~ --~ b~- = - ~
a ( + --aO-~
Alternately one may interchange b* and b,~"
b~ ~
b+ =
1(lO)
-~ ~r
-
b~ -~ b2, =
-~ ~r +
-~
The two ways of quantization are a well-known ambiguity of the theory (Becker 1963, 1964, vol. 2, w52; Heitler 1954, p. 57). If we use first Eq.(29) we obtain from Eq.(27):
b_~b+ = 9(,~T _ E~T 2~~h - 27r~h
(31)
84
2 MODIFIED KLEIN-GORDON EQUATION
which leads to the eigenvalues F_~=F_~,~=
(1)
T
n+
,
n=0,
1, . . . , N
(32)
as shown in some detail by Becker (1964, vol. II, w15). On the other hand, Eq.(30) leads to rc~T b+b~- = 27r~
(33)
and the eigenvalues F_~=rc~n=
2~( T
1)
n-~
,
n=0,
1, . . . , N
(34)
We prefer here the use of the Schr6dinger approach mainly because it leads to differential equations that we may readily replace by difference equations. We replace the differential operators of Eq.(29) by difference operators:
b: -~ b~+ = ~
~r
-~
,
b,~ ~ b-s = --~
0/~ + --~-~
(35)
The products b+b-s or b-s + produce a second-order difference operator: !
b-2 b + = b + b-2 = -~
(36)
a2 z~r
Substituting into Eq.(27) and applying the operator to a function ~ yields in analogy to Eq. (31):
a2 Ar
~ = 27r~h
= 27r~h
(37)
For the solution of Eq.(37) one makes first the substitution
{=~r
r
--
1 -{
A2
__~ 0/2
A2
(38)
that produces a difference equation with a variable coefficient {: z~2~
A{2
+ ( 2 ~ - {2)~ = 0
The substitution 2 2For details of the calculation see Harmuth and Meffert 2003, Sec. 3.6.
(39)
2 . 5 HAMILTON FUNCTION AND QUANTIZATION
85
= e-~2/2X(()
(40)
leads to the equation
A~2
- 2~ , S. zx~( ~ )
+ (2A,~ - 1 ) x ( ~ )
-- o
(41)
which can be simplified by the further substitution x = ~ / A ( , A~ = 1
(42)
to the equation
x(m + 1) - 2X(z) + X(z - 1 ) - m [X(z + 1) - X(z - 1)] + (2A~ - 1)X(~) = 0 (43) This difference equation with variable coefficient x can be solved by a factorial series (of the second kind)
X(x) = bo + bl
( x - 1)(x- 2) x--1 1! + b2 2~
§ bj
, o .
(x - 1)(x - 2 ) . . . ( x - j)
j~
+ ...
(44)
whose coefficients bj are determined by a recursion formula with three terms: j+l bj+2 = (j + 2) 2 + 1 {(2A,~ - 1 - 2j)bj + [2A,~ - 1 - 3(j + 2)]bj+l}
(45)
A polynomial solution with highest term bn exists if the following two conditions are satisfied:
n
bn+l = (n + 1) 2 + 1 {[2A,~ - 1 - 2 ( n -
1)]bn_ 1
+[2A,~ - 1 - 3(n + 1)]b,} : 0 2n=2A~-1,
n=0,
(46)
(47)
1, 2, . . . , N
Substitution of Eq.(47) into Eq.(37) yields the eigenvalues known from the differential theory or from Eq.(32) of the Heisenberg approach, except for the limit N of n: [~---=E~'~-
T
n§
,
n--0,
1, 2, . . . ,
N
(48)
86
2 MODIFIED KLEIN-GORDON EQUATION
The recursion formula of Eq.(45) is a difference equation of second order. Hence, two of the coefficients bj in Eq.(44) can be chosen. Our first choice is b0, but we leave the value of b0 undefined so that it remains available for normalization. The second choice is not bl but bn+l = 0, which is Eq.(46). The eigenvalues of Eq.(48) are the same as in the differential theory except for the finite value of N. The associated eigenfunctions in the differential theory are the Hermite polynomials that are defined by a recursion formula with two terms. Equation (45) is a recursion formula with three terms. The eigenfunctions are thus different from the Hermite polynomials. 2.6 PLOTS FOR FIRST-ORDER APPROXIMATION In the differential theory we had derived the probability p(s) of a photon with a period number ~ (Harmuth and Meffert 2003, Eq.4.5-14)" p(~) = 1/1.20205~ 3
(1)
This function is plotted for the interval 1 < ~ < 49 in Fig.2.6-1 and for the interval 1 < s _< 500 in Fig.2.6-2 using a logarithmic scale for p(~). The plots hold for integer values of ~. In the difference theory the energy Uc~(s) for the period number ~ is defined by Eq.(2.5-16)'
u~(,~) =
L2m2 nr2~2 [ ~oo . . . . ~~ I ~ ( ~ / g )
~T
(~~_)2
s-]-~Z-- Z . + ~ ] + ~ +
(moc2At]2] + h
fl~=arcsin ( 1 - [ 1 - A 2 / 2 - 1 +2sin2(~s/N)]2) 1/2A 2 1Aa2 ~i~ ~ #~ = 1 - [1 - : ~ / 2 - 2 sin~(~-,~/N)] ~ 2 2 1 + AIA 3
2 (2~/N){1
~ = ~
(
1-
:q - ~
~,~
r
- exp[-(A~ - A~)Z/~N]}
-~0~ ) (~t) ~ ~ ~~ = h~ ( m ~ 4 - ~ , o 0 + ~ c Am0~) m0x m0x A12----e2c2(At) ~.Amox/h 2 = (At)~(mo~ ~ ~Ce0)/h, ~ ~A~
+ ~A~
--
e
= ~ ( A t ) ~ WeO/ ~ ~h ~ , 0 < t ~ < s o = ~ a r N csin -
-A1A 3 -----
~
- 2e Ce0 +
C
1 - :~
-
~
~
4
Amox)/h 2, N = T/At
~/~
(2)
2.6 PLOTS FOR FIRST-ORDER APPROXIMATION
87
10 -1
T 10_2
".
~" 10-a
"..
10-4
"'....
10-5 .
". . . . . . . . . . . 9
0
i0
~0
a0
4'0
/~ ----+
50
Fio.2.6-1. Probability p(e;) of a photon with period number ~ according to Eq.(1) of the differential theory in the interval 1 <__~ _< 49. 1
9
10 -2 l l0 4 10-6 10-s 0
100
200
300
400
500
FIG.2.6-2. Probability p(~) of a photon with period number ~ according to Eq.(1) of the differential theory in the interval 1 < ~ < 500. T h e energy of a p h o t o n with period number ~ and a certain value of n is defined by Eq.(2.5-48)'
E~
T
n+~
,
n
0, 1, . . . , N
(3)
T h e average value of E~,~ for all N + 1 values of n becomes:
E~=
2rrnh 1 T N+I
~(
1) n+
1 = ~(N+I)
2rrnh 1 2rrnh - -2 N ~T T
(4)
t~--0
For a specific value of n the energy Uc~(~) requires the n u m b e r photons:
Uc,~(~)/F-,~nof
88
2 MODIFIED KLEIN-GORDON EQUATION
10-I
I 10_2 <1 10-3
"'''''''''..,,.....o,..o.
10-4
0
1'0
2'o
I,r -----+
~eoeooO
oOOOO. "~ I
30
40
~;0
FIG.2.6-3. Probability p~x(,~) according to Eqs.(9) and (14) for N - 100, A2 - 10 -4, A1A3 - 0 . 1 A 2 in the interval 1 < ,~ < 49.
10-I
10-2
.."..
~- 10 -3
,
I
920
40
'eeee'e~176
10-5
eee~oeooeeoeoeeeeo J
eoeooeoooo~~176176176176
60
80
100
FIG.2.6-4. Probability pA(a) according to Eqs.(9) and (14) for N 5 X 10 - 5 )~1)~3 -----0.1A2 in the interval 1 --< a _< 99 9
Ur
--DrnrA(g)
E,~n
s~.~.~,~ 2~; Z~ + ~ , ~ + .~ + Drn -
. . . . 1"3
ch
200, A2 -
I
n + -~
7-
+
h~. (5)
If p h o t o n s w i t h various values of n are equally frequent we o b t a i n t h e following r e l a t i o n in place of Eq.(5)'
Uc~(~ E~ Dr -
= D~r~(~) 2 2
2 2
2L ~I'I NA 1A3 ch
(6)
2.6 PLOTS FOR F I R S T - O R D E R APPROXIMATION
89
1
10-1 10-2
I 10_3
••10 -4 10-5 10-6
'o
i0
100
1;0
2;0
2'50
Fie.2.6-5. Probability pA(~) according to Eqs.(9) and (14) for N = 500, A2 = 2 x 10 -5, Al,Xa = 0.1A2 in the interval 1 _< ~ _ 249. 1 . I
i0_2
9 .
10 -4
10-6
0
1;0
250
3;0
450
5;0
FIG.2.6-6. Probability pa(~) according to Eqs.(9) and (14) for N = 1000, A2 = 10 -s, A1Aa = 0.1A2 in the interval 1 < ~ < 499. T h e total n u m b e r of photons equals the sum over ~ in Eqs.(5) or (6)'
-
-
N/2
#o: ~=02
N/2--1 1 Vc~(~;) __ D r n Z TA(N;) ~=0
N/2-1
~=o
(7)
N/2-1
(8)
E~
e~--O
T h e probability p a (~) of a photon with period n u m b e r ~ is the same for Eqs.(5) and (6) since the constants Dr,~ and Dr drop out" rzx(a)
p A (~) "-- X..~N/2_ 1
(9)
For the c o m p u t a t i o n of rA (~) and pA(~) one can make several simplifications in Eqs.(5) and (9). First we deal with the special case ~ = 0. We see
90
2 MODIFIED KLEIN-GORDON EQUATION
from Fig.2.4-2 that ~ and thus sin ~ is not zero at ~p~ = 27r~/N = 0 or ~ = 0. The function I~r(~/N ) decreases like ~2, the factor 1 / 2 ~ increases like 1/~, and the terms in brackets in Eq.(5) become constant. Hence, rA (0) and p~ (0) approach zero. For discrete values of ~ we get I T ( ~ / N ) 2 / 2 ~ : 02//0 for ~ = 0, which makes rA (0) and pA (0) undefined. This was an argument in terms of differential mathematics. In terms of mathematics of finite differences one has to state that ~ in Fig.2.4-2 is defined for ~ . . . . , 3, 2, 1 but not for intermediate values and particularly not for --~ 0 or ~ = 0. In terms of physics a photon with period number ~ = 0 or ~ -~ 0 would be a strange thing indeed. It would have trouble producing interference phenomena. We leave out the terms ~ = 0 in Eqs.(7) to (9). The following approximations hold for 1 _< ~ < N/2:
n0 = - - a r c s i n
--A1)~ 3 4
1-
1<_~<_N/2
--
1<~o
~ - A21A32 << 1
(A2 A2)l/2N = NAt(m~c 4 -
( r~(~) =
-
NTr N = -~2 2'
forA~2 - - ~ l A23 2 for At <<
e2r
2~/N
<
}2
(10)
<<1
h/mo c2
:>> 1,
)~2 + ( 2 ~ / N ) 2
Eq.(2.4-7)
1 1 sin 2/3~ 2 ~
(11)
for T >> [
li/rnoc 2
(2N~/2 ] /32 +
(12) (13)
We obtain the following values for A1A3 = 0.1A2" N/2-1
rA(~) : 98.743
for
N = 100, A2 = 10 - 4
t~--1
= 390.914
N = 200, '~2 ---5 X 10 -5
= 2430.32
N=500,
= 9706.03
N = 1000, A2 = 10 -5
A 2 = 2 X 1 0 -5 (14)
Equation (11) states that At can be arbitrarily small but finite while Eq.(12) states that T can be arbitrarily large but finite. This is in line with our assumptions. We have the numerical value h/moc 2 = 2.95241 • 10-23 s for pions r + and ~ - . Figures 2.6-3 to 2.6-6 show p~(~) plotted with a logarithmic scale for the four cases N = 100, 200, 500, 1000 of Eq.(14). The plots drop rapidly with increasing values of ~, similarly to the plots of Figs.2.6-1 and 2.6-2. The conspicuous difference is the increase of p~(~) when ~ approaches N/2. To show the practical identity for ~ < 0.2N of the plots of pA (~) for N = 1000 in Fig.2.6-6 with the plots of p(~) in Figs.2.6-1 and 2.6-2 we plot the difference p~ ( ~ ) - p ( ~ ) in Figs.2.6-7 and 2.6-8. According to Fig.2.6-7 the function PA (~)
2.6 PLOTS FOR F I R S T - O R D E R APPROXIMATION
91
o -0.oo02
I -o.oo04
"-~-o.ooo6 I
,~
-o.o008 -0.001 -0.0012
;
;
8
1'0
FIG.2.6-7. Difference p ~ (~) - p(~) b e t w e e n t h e p l o t s of Fig.2.6-6 for N = 100 a n d Fig.2.6-1 for t h e i n t e r v a l 1 < ~ <_ 10. T h e p o i n t at ~ = 1 is b a r e l y visible since it h a s t h e value - 0 . 0 0 1 2 .
:-.....
4x 10 -7 3x i0
.
'.%
t~ -7 I 2xlO
I x 10 .7
s'0
1oo
,
~5o
i
200
-ix 10 -7
FIG.2.6-8. Difference p A ( ~ ) - p(~) b e t w e e n t h e plots of Fig.2.6-6 for N = 100 a n d Fig.2.6-1 for t h e i n t e r v a l 10 _< ~ <_ 200.
is slightly smaller t h a n p(~) in the interval 1 <_ ~ _< 10. For larger values of we get p(~,) > pA(~) in the interval 11 _< ~ _< 200 according to Fig.2.6-8. B u t the difference is of the order 10 -a or less in Fig.2.6-7 and of the order 10 . 7 in Fig.2.6-8. T h e increase of the plots of Figs.2.6-3 to 2.6-6 for ~ ~ N/2 is a typical result of a finite difference theory. By limiting the interval of interest to a m a x i m u m value of ~ = 49, 99, 249, 499 we introduce a b o u n d a r y condition at the upper limit of ~ t h a t causes pz~ (~) to increase when approaching this upper limit 9 We see in Fig.2.6-3 at the upper limit pzx(49) " 10 . 2 while Fig.2.6-6 shows pA(499) " 10 -a. One will expect t h a t this value decreases to zero for N --, oo in analogy to Figs.2.6-1 and 2.6-2. In a finite difference theory one must specify t h a t one wants the k most significant digits of pA (~). T h e n one must choose N > 2~, so large that a further increase of N does not change these k digits 1. 1In a differential theory we always have ~ << c~ if we make a plot as function of ~. The corresponding condition ~ << N is introduced by Eqs.(2.5-21) and (2.5-22) for the difference theorie. We hope to discuss the matter in more detail in a book on the Dirac difference equations.
ADVANCES IN IMAGING AND ELECTRON PHYSICS, VOL. 137
3 Inhomogeneous Difference Equation
3.1
INHOMOGENEOUS
TERM
OF
EQ.(2.2-31)
We have developed solutions of the homogeneous difference equation (2.230) for Cx0xj (xj, t) in Sections 2.3 to 2.6. The next step is to find a particular solution of the inhomogeneous Eq.(2.2-31). The homogeneous part of that equation equals Eq.(2.2-30) as is readily seen from Eqs.(2.2-28) and (2.2-29). Hence, the homogeneous solution is obtained by replacing ~x0xj = ~0 in Section 2.3 by ~xlxj = ~1. This implies that Eq.(2.4-32) is both the solution of Eqs.(2.2-28), (2.2-30) and of the homogeneous part of Eq.(2.2-29), (2.2-31) if we replace ~xOxj(xj,t) = ~o by ~xlxj(xj,t) = ~I/1. We copy ~o for xj ~ and t ~ 0 from Eq.(2.4-32). The notation ~IH(~,0) means that this is only the homogeneous solution of ~1 (~, 0):
9o(r o) = ~H((, o) = ~oo[(1 - ~ - ~ ' ~ 1 6 2
O)]
+ ~(r
_ ~)~/~r
9oo [(1 - ~-~,~o)e•
N/2-1
-- 2iA1A3ei)~l)~3e~)~l(~-)~3O)
IT(~/N) sin/3~ sin fix0 sin 27r~4 N [7
E
.1
~=-N/2+1
IT(~/N) see Eq.(2.4-29)
(1)
The next task is to write Eq.(2.2-31) with the normalization of Eq.(2.3-1) and reduce the number of spatial variables from three to one:
O= t / A t ,
~= xj/Ax, j=l,
O A t = t , ~Ax = xj,
A ~t
xy = x
1 A A ~ At AO' A x j
1 ,5
Ax/~r
(2)
Equation (2.2-31) assumes the following form:
(hA iAx /~
)~ eAmox
2
1(hA - -~
~r
iAt /~0
] + "~02~~ ~xX~(r 0)
92 ISSN 1076-5670/05 DOI: 10.1016/S1076-5670(05)37003-0
Copyright 2005, Elsevier Inc. All rights reserved.
3.1 INHOMOGENEOUS TERM OF EQ.(2.2-31) {4AcAeo( =
-
~
h iAx
/~
)2[ ~Amo~ 1
&
1 ( h z~ ~..~c: i ~ ~-~
)
+~
iAt SO + echo (Lc• +e kcx21 + Lcx31 + Lr
+ (LCx~ + Lr
+ kr
--e Amlz iAx ~ e c2 r
93
eAmox) 2] + L~xs~)
+ L~x4~+ LCx~) iZX---7Zi--0+ ~r eAmox +
( h A -iAtzSO --+ echo +
iAx ~
iAt
eAmoz Amlx
2 8 ~- eCe0 r
IIIxox(r 0)
(3)
Ae = Ae0 + aA~l (~, 8), Aex = A~o~ + ctAelx(~, 8) Am = Am0 + olAml(~, 8), Amx = Amox + ctAmlx((, 8) Ce -- ~)e0 -~- O~r (~, 8), Cm -- era0 nt- O/r 8) (4) Using Eqs.(2.2-14), (2.2-15), and (2.3-2) we may write the homogeneous equation defined by the first line of Eq.(3) as follows:
eAmox h2 v ~ ( r (A~)~
-- " ~
iAt
28
eq~e~
'[- 77t02C2 lI/xlx(~' 8)
+ zxr o) - 2 v ~ ( r o ) + V x ~ ( r (ZXr
Ar 0)
]~2 ~i/xlx(~ ' 8 q-- A 0 ) - 2~I/xlx(~, 8) + ~I/xlx(~, 0 - A8) (cAt) 2 (AS) 2 2g~ Vx~(r + Ar 0 ) - V ~ ( r Ar 4- --~'x A mox 2A ~ 2~h c2At r
~x~(r o + A0) - ~x~(r o - A0) 258 --
Amox-- C'2r
-~- e2
~I/xlx(~, 8) ----0
(5)
With Ax = cAt and Ar = A8 = 1 we get:
(
[~1 (~+ 1, 0)-2~1 ((, 8)+ ~1 (~- 1, 8)] - [~1 (~, 0+ 1)-2~1 (r 8)+ ~1 (~, 8-1)]
-iAl{ [~1(~ + 1,8)- ~1(~- 1,0)] + Aa[~l(~,8 + 1)- ~1(~,8- 1)] }
~1 = qJxlx, At, A2, A3 see Eq.(2.3-2)
(6)
94
3 INHOMOGENEOUS DIFFERENCE EQUATION
The second line in Eq.(3) is rewritten as follows'
4AcAe0[( h A e iz~z Ar
eAmoz) _ -
-
2[
1 z~ 2m2c2 ( lAx h ~
1
4AcAeoh 2 [
)21
eAmox
ieAXAmoz
z~ 1(
B
x 1+-~ mocAz
ieAXAmo~) 21 ----g--
)2(z~
Ar
(7)
If Ax is significantly larger than the Compton wavelength
),c =
27rh 'rrto c
=
h
(8)
?Tt0C
we may ignore the second term in brackets on the right side of Eq.(7). For pions 7r+ and 7r- we have
Ac =
h
moc
6.626 x 10 -34 = 8.88 x 10 -15 [m] 273.2 x 9.109 x 10 -31 x 2.998 x l0 s
=
(9)
If the condition Ax >> h/moc is satisfied we get a second-order difference equation according to the first term on the right side of Eq.(7), otherwise we get a fourth-order difference equation according to the whole right side of Eq.(7). With the help of Eq.(2.2-14) we may rewrite the first term on the right side of Eq.(7) as follows with ~x0x((, 0 ) - ~0(~, 0):
4ACAeo h2
m~
(-
Ax >> Ac
A
= h/moc,
)2 ieA.________x Amoz ~o(~, 0)
~0 = ~x0x
4ACAeo h2 [~o(~ + A~, 0) - 2~o(~, 0 ) + ~ o ( ~ - A~, 0) ~(/,,~)2 [ (~xr
- ---~
+
A m ox
4AcAe~ ~(A~)2
[
2 /X ~
A ,o) +
~ --~
A m ox
)
%(r
Vo(r + 1,0) - 2Vo(r 0) + %(r - 1, 0)
- ieA----~XAmozfl2o. ~[ (
+ 1 0) - Vo(~"- 1 0)] +
eAx .
\2
m
]
---7-dmo~/ qo(~',0)
(10)
3.1 INHOMOGENEOUS TERM OF Eq.(2.2-31)
95
For the other extreme, Ax << h/moc, we ignore the term 1 on the right side of Eq.(7) and use only the last term, which is multiplied by (h/mocAx) 2"
A x << Ac = h/moc, ~0 = ~x0x -
ieAX Amox I
4AcAe0h2( A
~(A~)~
/~r
2
h
1(
h )2(Z mocAz Ar
• -2
ieAXAmox) 2 ~o(r ---i-4
--
e
~
mocA-------~
~Amo~
/~r
~o(~, {9)
li
(11)
Since we have not yet shown explicitly difference operators/~n//~{n for n larger than 2 we must postpone the further analysis of Eq.(11) to Section 6.1. However, comparison of the first line of Eq.(10) with the last line of Eq.(ll) shows that completely different equations are obtained for Ax >> h/moc and Ax << h/moc. Nothing comparable can exist in a differential theory since dx is never large or small compared with Ac but is always infinitesimal. We return to Eq.(3). The last two lines may be rewritten in the following form:
{[
e Amlx
eAmox
iAx/~
+
e /li 'A + -~ r ~l ~ ~i A ~t /~{9 + eCe0 +
iAx z ~
eAmox Amlx(~,{9)
N
-- e Amlx(r {9) (
h
iAt ~0 + eCe0 r
(r {9)
~0(r {9)
9o(r + Ar O)- ~o(r Ar O) ~) 2A~ ' -- eAmox~o(~, {9)
' k iAx / ~I/0( ~ -Jr- A~', {9)Amlx(~ --[- Ar {9) - ~I/o( r - A~,{9)Amlx( ~ - A~, {9)
iAx
2Ar
-- enmoxnmlx(~, {9)1I/0 (~, 0)] + ~ r162 O) iAt §
h II/0 (~, {9 -~- n{9)r iAt
2ZXO
+ ~r162
(~, {9 -~- n{9) -- II/0(~, {9 - n{9)r 2A0
O)
(~, {9 - n{9)
+ er162162 e)~o(r e)]
3 INHOMOGENEOUS DIFFERENCE EQUATION
96
"- e[Amlx(~ 0)( 2iAx h [~g~ + l ' O) - q2~
I ' O)]
h + 2iAx[~O(( + 1, 0)Aml~(( + 1 , 0 ) - ~ o ( ( -
1, O ) A m l x ( ( - 1,0)] 1
-
-
2eAmoxAmlx(~, 0)~o((, 0)[ J
+ ~-~ r
2iAt [~o(~,0 + 1) - ~o(~,0 - 1)] + eCe01I/0(~, 0 )
' h
+ 2--~-~[1~/0(~, ~9 + 1)r
+ 1) -- II/0(r ~9- 1)r
~9- 1)1 1
+ 2~r162 (r O)~o(r o)[
(12)
.d
Consider next the terms kcxlx to kcx51 in Eq.(3). They are shown for j = 1 by Eqs.(2.2-33) to (2.2-37). We rewrite Eq.(2.2-33) for one space variable xj -- x, j = 1:
Ze _ 2o~o~c{ LCXll[Aeoz(~~ h~ eAmo~)-Aeo~(h ~ x(h~ ~~ ~mO~)[1_1mo~( ~ 9 ' ~x-~~
1
9[1- ~o~ (~
A
- Aoo~ (~ ~z A
2
]
A
- ~mO~)] (~. ~
The differences z~//~y and z~//~z are zero but the constants Amos, Amoy, Amoz are generally not zero:
Ze
(
2o~m2c~e(-AeozAmoy + AeoyAmoz)
kcxll ~
x (~,~lx-eAm~ +
1 - m02c2
z~z -eAmox
2(li Ax
2
e(-AeozAmo~ + AeoyAmoz)
~(-~~~ -~mO~)} (~4~ We note that the terms in lines 3 and 4 can be written in reverse order since this requires only that (-ili/~/z~z-eAmox) commutes with the constant 1 and itself. We obtain for x = Ax ~ and Ax = AxA~:
3.1 INHOMOGENEOUS TERM OF EQ.(2.2-31)
Lcxll
Ze 2
-
97
h
i -amo - mocAx (AeozAmoy- AeoyAmoz)
-
)2]
ieA----XXAmox
(15)
h Substitution into Eq.(2.2-31) yields with 0 =
t/At and thus A0 = At~At = 1:
-~ c iAt ~0 ~ echo Lcx,1 = 7Lcx,, iA--t,~--0+ eCeo Ze2(h) 2 (/;~ieAt) = a mocA--------x (A~ozAmoy- A~oyAmoz) - ~ + h Ce0
X (AX~ ~ieAXAm~
hCAx) 2Z(A~ ~ieAXAm~2] (16)
As in Eq.(7) we obtain again difference operators of higher than second order. As before we may distinguish between Ax >> h/moc and Ax << h/moc:
1( h ,,~
)
Ax >> h/moc
1
( h /~
~- iAt ~ + ~r L~xl,+ ~Lc~l~ iAt _.2z~2( h ) 2 Ol m0cA---------~ (AeozAmoy- AeoyAmoz)
X(_~A -'~'~----'~r ~ ieAt =2Ze2( a
XO +
~r
)]
~o(r O)
~ iceAt~Amox--=-A-Z~O /~ ~ce2(At)2 ~) /I~A h 2 Ce0Amo ~0(~ ,0)
h )2
mocAx
(AeozAmoy-
(1 {[~o(~'+1 0 + 1 ) - ~ o ( r
AeoyAmoz) 0-1)]+[~o(~'-1 0 + 1 ) - ~ o ( ~ - 1
0-1)]}
1 ieAt
+ ffTCeO[~O(~ + 1 , 0 ) - ~ o ( < - 1,0)] 1
2
ce2(At)2
ieAXAmox [~o.~ ( 0+1)-~o(~,0-1)]+ ~r162 h ' h2
)
(17)
98
3 INHOMOGENEOUS D I F F E R E N C E EQUATION
Ax << h/moc
[1(~~ iAt ~0
) + -~L~x~ izxt ~o ~ ~r Vo(r _2Ze2( h )4 a mocA--------~ (AeozAmoy - AeoyAmoz)
~-eCeo
I-cxll
x ( z~
ieAt
~o 9~--r (~,
ieA'-'-"-~XAmox
3
~o(~, 0 ) ( 1 8 )
h
As in Eq.(11) we obtain difference operators of higher than second order and refer to Section 6.1 for further analysis. The terms kc• Eq.(3):
in Eq.(3) are zero according to Eq.(2.2-34) and we get in
-~ iAt ~0 t- eCeo
1 (h~
)]
(19)
kcx21 + ;5Lcx2~ iAt Z0 ~*r176176 ~o(~,0) = 0
The term kcx31 is shown for j = 1 by Eq.(2.2-35). We rewrite Eq.(2.2-35) for one space variable xj = x, j = 1:
~x~ = ~o~oc
e0,
~{
9(~~-~mO~) z~
+
[1
7 z~-~ -
1 (~~
Amos/
2.~
~o~ ~-~m0~) }
1 ~o~ ~-~m~
2,~c~
] ( -i~z~z -- eAmoz/ 7 A~ -~Amo~
ZX~ (2O)
Again, the differences z~/~y and ~/z~z are zero but the constants Amox, Amoy, Amoz are generally not zero. We obtain with x = ~Ax:
3.1 INHOMOGENEOUS TERM OF EQ. (2.2-31)
99
Ze 2 l-~x31~o(r O)=
'
2arnoc
- AeozAmo~ A t 1
2m2c 2 i ~ x
eAmox
a
-- AeozAmoy Z
2mo2c2 ~z~x - eAmox
+ A~o=Amoz~ 1 + Aeo=Amoz~ 1
Ze 2 r 2amoc
1
(
h z~ _ eAmox
1 (h_ A -eAmo= 2m2oc2 \ i Ax
) ~l}Vo(x,O)~x
2Aeoz(Amoz - Amoy) e
• At
2.~o~ ~ i z ~ - ~Amo~
~o(~.0)ZX~
(21)
We could write Eq.(21) in the form
Ze 2 Lcx31 @o(~, 0) = ~ c2amo Vc~x31(~' 0)
(22)
but this way of writing would give Vc~x31(r 0) a physical dimension. We prefer to use a dimension-free function Vcx31(r To achieve this we note that the square of the mechanical momentum moc in the first line of Eq.(3) has the electromagnetic dimension (VAs2/m) 2. Since the factor li/c2At of the terms Lcxil in that equation has the dimension VAs3/m 2 we infer that the terms Lcx~l have the electromagnetic dimension VAs. The factor m0c 2 has the dimension VAs. Hence, we rewrite Eq.(22) as follows
Ze2 Vi'~ (r 0) 0) = re~ 2~ Vcx31(~, 0) -- 2~.~o----~
L~~o(r
(23)
and obtain
Ze 2 gcx31 (~, 0) -- ?Trio3 g:x31 (~, 0) (Ax
Ze2 8
= m2c3
2Aeoz(Amoz - Am0y)
e
x --=At 1 - m2c 2 -i /~x - eAmox
~0 (x, 0 )Ax
(24)
100
3 INHOMOGENEOUS DIFFERENCE EQUATION
Equation (24) written explicitly assumes the form
r Ze2 8 z~[ 1 V~xal (~, {9) = m2c3 2Aeoz(Amoz- Amoy)~--~ 1 -~ 2m2c 2 c
• ( _ h~ ~o(~ + Ax, o) - 2~o(~, o) + ~o(~ -/',~, o) (/,,a:)2
+~,~Amo~'O(~+~'~,O)-~'o(~-~'~ 2Ax
I]
= ' O/ +e2Amoxq2o(x, {9 Ax
(25)
From Eqs.(3) and (23) we obtain the following relation for L~x31~o(r {9) and Vcxal (r {9)
-~ C
iAt
X{9 }- gee0 Lcx31~o(r {9) ==
--i
Lax31 iAt X{9 ~ eOeo ~o(r {9)
?TtOh Ycx31(r {9) ~-~ A -[- -ieAt - ~ r ) ~10(r [9) (26) 20lAt
which yields: N
kcx31 --k ~-~Lcx31 iAt ~
+ er
~o(~', {9)
1[~o((, {9+ 1 ) - ~o(r {9- 1)] + T r
{9
iA---tA~--0+ er
.,~oh(A
~xt
)
mob -- --7"~-~ Vex31(~, {9) 9
•
(27)
According to Eq.(2.2-36) the term Lcx41 is zero and we get in Eq.(3):
[1(~~
i/xt ~-e + ~r
)
1
(h~)]
LCx4~+ ~'~Lcx41 iAt ~0 +
er
~0(~, {9) : 0
(28)
The final term Lcx51 in Eq.(3)is defined for j = 1 by Eq.(2.2-37):
ze x({[ 1 2m2oc2 Ax - eAmo~ 21(h7 ~y - eAmoy/ +(~ [11~~(~.~ ~~ - ~mO~) ~ -~o~) ~]}~
Lcx51 = 4~m2oc ~ c
3.1 INHOMOGENEOUS TERM OF EQ.(2.2-31) +{[1
+
x{[1
(
101
1 (~~ 2 -eAmoz) 2m2oC2 Ax - eAmox) ( ~ Az A
li z~ -i z~Z -- eAmoz
2m2c 2 i ~ z
(
1
1
1% ,'~
2m02c2 i" Z~a~--
e A mox
~zz
eAmox 7Z~y-eAm~
[ (~, z~
+ Aeox ~ y - e A m o y
[
) -Aeoy
7Z~'-'~
) -Aeoy (--'t%i Az=A ---enm0x)] 1
x 1
(29)
i-z~x-
2rn~c2
The differences z~/z~y and z~/~z are again zero and all terms multiplied by them are removed. This means primarily lines 1-4 but as a consequence lines In analogy to Eq.(23) we 5-8 too, which leaves a summation constant C~x51. / get:
moc 2 L~x51 =
Ze
4c~ C~•
- 4c~m~'-"--'~Clcx51
(30)
W e obtain from Eqs.(3) and (30)"
[l(h
z~
)
1
(B
z~
)
-~ iAt AO ~-eCe~ Lc• + ~Lcx51 iAt AO F eCeo 9o(r
.~oh
( A i~zxt )
mob
h r ( ~11%(r ~ 1) - % ( r ~ 1)]+ ~zxt
= -i2c~AtC~x51 ~ = - i 2c~At Cc•
+ ---~r
~o(~,0)
o))
(31)
With the help of Eqs.(6), (10), (17), (28), (31), and (12) we may rewrite Eq.(3). The lines are numbered for easier reference later on. We remove the factor - h 2 / ( A x ) 2 of Eq.(6) by multiplying Eq.(3) with -(Ax)2/B 2'
Ax >> Ac - h / m o c
[v~(r
o)-2v~(r o)+v1(r 1, o)]- [v~(r o+1)-2v~(r o)+v~(r o- 1)] ~
-iAl{[~l(~+1,0)-~l(~-l,0)]+A3[~l(~,O+l)-~l(r162
2
102
3 INHOMOGENEOUS DIFFERENCE EQUATION
_- 4AcAe(~',_eO) [~o(( + 1, O) - 2~o((, 0 ) + ~ o ( ~ - 1, O) eAx .
3
/ eAx .
- i - ~ A m 0 ~ [ ~ 0 ( ~ + 1,0) - ~o(~ - 1, 0)1 + L--~Am~
\2 ~0(~, O)] 4
2Ze 2 + am2oC2 (AeozAmoy -- AeouAmoz) X
~ (1{[~o(~.+1
5
~0 + 1 ) - ~ o ( ~ + 1 , 0 - 1 ) ] + [ ~ o ( ( - 1
_ 1)]}
~0 + 1 ) - ~ o ( ~ - 1 , 0
1 ieAt ~- ~ T C e o [ l I / o ( ~ + 1, O ) - ~ o ( ( -
1, 0)]
7
ce2(At)2 - ieA---~XAmox.~o_(,O[ ( + 1) - ~ o ( ~ , 0 - 1)] + ~ r
2h
. mo(AX)
h2
'
O)
)
8
2
-- z o z h , A ~ Vcx31(~, O) ~1.~o_(,0 + 1) -
6
mo(Ax) 2
(1
i 2o/hA----------~Ccx51~[lI/0(~, 0 -~- 1) -
--e(-'~)2 Amlx(~'O)( h [q2~
-
r
, 0 - 1)] + ----~r
q~o(~, 0 - 1)] ' O)-~~
, {9)
ieAt
+ TCeOlI/o(~,
0)
9
)
10
,
11
h
+ 2 l A x [~0(( + 1, O)Amlx(~ + 1, 0) - ~0(~ - 1, O)nmlx(~ - 1,0)]
12
- 2eAmox.Amix(~, 0)~o(~, 0)] 13 .J
e
C2
(
r
+ '2iAt[qlo(~,O + 1)r
~[~I'0(r
0+1) --~0(~', 0--1)] -1t- eCe01I/0(~,{9)
O + 1) -- ~I/o((, O - 1)r + 2eCe0r
O -- 1)]
(~, 0)~I/0 (~, 0)/
J
)
14 15
16
(32)
3.2 EVALUATION OF EQ.(3.1-32) Equation (3.1-32) needs to be simplified before attempting to find a solution. A first step is to make as many as possible of the constant components of Ae, Am, Ce, and Cm in Eq.(3.1-4) zero. The two constants Amox and Ce0 must be retained to keep Eq.(2.3-2) valid but the following components may be chosen equal to zero: Ae0x = Ae0y = Aeoz = 0,
Amoy - Amoz -
0,
Cm0 = 0
(1)
103
3.2 EVALUATION OF EQ.(3.1-32)
The first two lines of Eq.(3.1-32) remain unchanged. We note that they are equal to the left side of Eq.(2.3-2) except that ~0 is replaced by ~1. Lines 3 and 4 are multiplied by Ae((~,0), which according to Eqs.(3.1-4) becomes ~Ael (~, 0). These two lines are of order O((~), they are left out. Lines 5-8 are zero because of Eq.(t). Line 9 is zero according to nqs.(3.1-25) and (1). Line t0 remains unchanged. We shorten the coefficient Ccx51 to Ccx: Cr = Cr
(2)
The remaining lines 11 to 16 do not permit any significant change. Hence, our first simplification of Eq.(3.1-32) leads to the elimination only of lines 3 to 9:
[~ ((+ 1, 0)- 2~t (~, 0)+ ~ (r 1,0)]- [~ (r 0+ 1 ) - 2 ~ ((, 0)+ ~t ((, 0-1)] - iA~ { [~t(r + 1,0)-- ~ t ( r 1, 0)] + &3[~1 (r 0 + 1) - ~ (~, 0 - 1)] } - &2~1((, 0) = G~(r o) (3)
c~(r
(2 [~o(<, 0+ 1 ) - ~ o ( r 0 - 1 ) ] +- ieAteeo ~o(r 0)) m~ h 2ahAt 9eAx -- z - ~ { [ A m t z ( r + 1,0) + Amlx(r 0)]~0(r + 1,0)
= -i
-- [Amlx(~, 0) + A m l x ( r - 1, 0 ) ] ~ o ( r
- 2 --i-
- 1, 0) }
Amo~Am~(r162
9c A z
- ~-ff2/- { [ r 1 6 2 o + 1) + r 1 6 2 1 6 2
+ 1)
-- [eel (~, 0) -4- eel (~, 0 -- 1)]~0(r 0 - 1) } "4- k ' - ~ ]
ee0r162162
0)
(4)
We need the general solution of the homogeneous Eq.(3) plus a particular solution of the inhomogeneous equation. The boundary condition of Eq.(2.3-3) is used again 9
9 ~(0, 0) = v 0 0 s ( 0 ) ( 1
forO < 0
- ~-~) = 0
= 900(1 - e -~~ The initial conditions of Eqs. (2. 3-4) and (2.3-6)
are
for 0 > 0
also used
for
~1(~,0)"
(5)
104
3
INHOMOGENEOUS DIFFERENCE EQUATION
91(~,0)=0 9 1(~,0 + 1 ) - ~1(~,0) = 0
for 8 = 0 , C > 0 or ~I'I(C, 1) = 0
(6)
for0-0, ~>0
(7)
The ansatz of Eq.(2.3-7) is also used once more but we write 5(~, 0) to emphasize the difference with u(r 8) in Eq.(2.3-7) or (2.4-23)" ~1 (r 8) = ~00[(1 The determination of F(r to (2.3-18):
e-~~162 + fi(r 8)]
(8)
and~ follows the calculation from Eq.(2.3-8)
F ( ( ) = exp[-(A 2 - A~)1/2r
(9) (10)
L = 2iA1A3
The boundary and initial conditions for ~(~,0) are the same as for u(~,0) in Eqs.(2.3-19) to (2.3-21)'
~(0,0) = 0 ~(~, 0) = 0 5(~, 1) - 5(~, 0) = - ( 1 -
e-'~)F(~)
8>_0 ~> 0 ~" > 0
(11) (12) (13)
Substitution of ~2(r of Eq.(8) into Eq.(3) yields the same equation with ~1 (~, 8) replaced by ~(r 0):
[~t(( + 1,8) - 2~,(r 8) + ~(r - 1 , 0 ) ] - [~(~', 0 + 1) - 2~(r 0) + fi(r 0 - 1)] -iA1{[4(r + 1,8)-4(~1,8)] + A3[~(r + 1 ) - ~(C~, 0 - 1)]} - A2fi(r 0) = G1 ((, 8) (14) We write the solution of Eq.(14) in the form ~(r 0) = u(r 0) + ~3(r 0)
(15)
where u(~, 0) is the solution of the homogeneous equation" and 73(~,0) is a particular solution of the inhomogeneous equation. The homogeneous equation equals Eq.(2.3-22). We solved it with Bernoulli's product method:
~.(r o) = r162162 which yielded Eq.(2.3-36) for r
r162
(16)
3.2 EVALUATION OF EQ.(3.1-32)
r
= ei)'~r
105
iv'~; + Aa~e -ivan) = ei)'~;(A32 cos ~0,~ + iAaa sin q),~) ~
= ~,~/u,
and Eq.(2.4-20) for r
,~ = o, + 1 ,
...,
•
(lZ)
r = e-ia~'~~
r
~'~~ + A~oe - ~ ' ~
(18)
Using the approximation/3~ = ~ of Fig.2.4-1 in the interval -7r/2 < ~o~ < 7r/2 we may rewrite Eq.(18) in the form
r
- e-i)'~)~~
i~~ -t--A4oe -i~~
= e-~)'~~
cos (p,~0 + iA43 sin (p,:0)
(19)
We turn to the inhomogeneous solution ~3(~, 0). Let ~(~, 0) be substituted for ~,(~, 0) in Eq.(14). Writing in some detail we obtain in analogy to Eqs.(16)(19)"
9,r
0) = r162
-- e/'X~r
cos qo,~ + iA33 sin qo,~)
x e -i'xl"x3~(A42 cos q)~0 + i-443 sin (p~0) = e-i,x~,xao (A.32 A42 cos qo,~0 + ifi-32-A43 sin (p,,0) e i'xl ~ cos qo,~ e-i)~ ~30 (A33A43 sin ~0~0 - i-A33A42 cos ~o,~0)ei'x~r sin qo~r
_
In the spirit of the method of the variation of the constant we make the constants -A42 and Aa3 functions of ~ and write ~3~(4, 0) in a shorter form:
~(r
0) = s~(o)~ ~ r cos
27r~r 27r~ N + T'c(O)ei)'~r sin
(20)
For a more general solution 1 ~3({, 0) we may sum over all values of ~. To shorten a number of formulas we use the notation r162 and Cs(~) for e ixl; cos(27r~4/N) and e i~1~ s i n ( 2 ~ { / N ) " <~o (
"b(~, O) = ~
2r~r
S,~(O)e i~'~ cos N
+ T'~(O)ei~'~ sin 27rNt~r )
t~>--t~o
N/2-1
:
E
[S~(0)r
T~(0)r
(21)
~----N/2+l
1See Smirnov (1961), vol. II, w169 for the use of this method for partial differential equations.
106
3
INHOMOGENEOUS DIFFERENCE EQUATION
Substitution of ~(r 0) for ~t((, 0) into Eq.(14) yields: N/2--1 ~=-N/2+l
[
S~ (0) [4~ ((+ 1 ) - 2r162 + r162 - 1)] + T~ (0) [r (r + 1) - 2r (r + r ( ( - 1)]
- r162 (r [S~ (0+ 1)-2S,~(0)+S,~(0-1)]-r -- i/~1
(S~r
+ A3{r162
[r
-]- 1) - r
1)-2T,~(O)+T,~(O-1)]
- 1)] + T~(0)[r162 + 1) - r
+ 1) - S,~(O- 1)1 + Cs(r
+ 1) - T,~(O - 1)1}]
)~2[S~({9)r162 -[- T~(0)r162
-
- 1)]
-- Gl(~,{9)
(22)
With 27r~ 27r~r 2r~ r162 • 1) = e+i~e i~1r cos ~ cos g =]=s i n - 7 sin
(23)
27rn 27rn~ 27rn cos ~ sin N + sin--if- cos
(24)
r
1) =
ea:i~te i~r
we get
r162 + 1) - r162 - 1) =
2e i~''r
-
27r~ 2r~r cosA1 sin ~ sin 27r~ 27r~r ) + i sin A1 cos - 7 cos N
Cs(~+ 1 ) - r
1)-- 2e ixlr
27r~ 2r~r cos Al sin ---~--cos g 27r~ + i sin A1 cos - 7 - sin
r162162162162162
(25)
ixlr
cosAlcos-7-1
-
(26)
cos g
27r~ 27r~ ] i sin A1 sin - - ~ sin - N
(27)
3.2 EVALUATION OF EQ.(3.1-32)
Cs(~ + 1) -
2r
+ Cs(~ - 1) = 2e i~r
[(
107
27rs; ) 27rtr cosA1 c o s - - ~ - 1 sin
(2s)
+ i sin A1 sin - - ~ cos Equation (22) assumes the following form' N/2-1
E
~=-N/2+1
{2S,,(O)ei'x~r[( cOSAlCOS--~---1 cos
+2T,,(O)ei'xlr _
cosAlcos--~--1 e i A 1 ~ COS
- - e i)'~r s i n
sin
(
-
N
27r,~r N [T,r + 1) - 2T~(0) + T,~(O- 1)]
+2T,~(0)e i)'~r cos Al sin - ~ ~X~cos
+isinAlsin-~cos
" 2NX~( [S,~(O+ 1 ) - 2S,~(0) + S,~(O- 1)]
-iA1 2S,~(O)ei~r - c o s A l s i n - - ~ s i n
+A3
N -isinAlsin-~sin
cos N
+isinAlcos--~cos
N'
+isinXlcos--~sin
[S,~(O+I)-S,~(O-1)]+e ~r sin
A~ ( S,~(O)e~162cos -27r~ 7 - + T,c(O)e~ sin 27rtc~)}-Gl(~,O) N
(29)
This equation may be reduced to the following form:
N/2-1
E
(
-(I+iA1A3)S,~(O+I)+
2~
)
2 ( c o s ) ~ l + A l s i n A 1 ) c o s - - ~ - - A 2 S~(0)
,r
2~
-- (1 -- i)~IA3)S~(0 - 1) + 2i(sinA1 - A1 cos)h) sin N
]
2~r
~(0) e '~'r cos N
N/2-1[ ( 27r,~ _ A2) T,,(0 ) + E -(I +iAIA3)T,~(O+ I)+ 2(cosAI+ Aasin AI) cos --~ ,~=--N/2+1
- (1 - i ~ 3 ) T ~ ( 0 - 1) - 2i(sin ),~ - ),1 co~ ~ ) sin
27r,~ _ ] --y-S~(O) ~
sin
2~,r162
.!
= G1 (r 0)
(30)
108
3
INHOMOGENEOUS
DIFFERENCE
EQUATION
The left side of Eq.(30) is essentially a Fourier series in terms of ~. The right side may be represented in the same form" N/2-1
c~(r
~)e i>'~r cos 2zrnr N
,~=-N/2+IE(Gs,~(O, ~,)ei>'~; sin 27r~r + Gr
(31)
Multiplication with 2 N - l e -~>'1r sin(2zrv(/N)or 2 N - l e -i>'~; cos(27rv~/N) and integration over the orthogonality interval 0 < ~ < N yields Gs~(0,~) and
a~(o, ,~). N
2 f al(r o)e - ~ a~(o,,~) = -~
cos.21r~r N de for v -- ec
(32)
0 N
G~,~(O, ~) = -~
a l (~, 0)e -~)'~ sin 2zr~;~d~ g
for v = t~
(33)
0
Equations (30) and (31) yield for every component ~ of the two sums the following two equations:
- ( 1 + i/~1/~3)S~(0 -~- 1) +
2(cosA1 + A1 sinA1) cos ~2 ~ _ ~ ) s~(0) 27rt~
- ( 1 - iA~A3)S~(O - 1)-2i(sin A~-A~ cos A1) sin --~-T~(O)=Gr
-
(1 + iA1Aa)T,~(O + 1) +
(34)
2~,~ _ ~)T~(O) 2(cosA1 + A1 sinA1) cos--~-
27rt~ - ( 1 - iAIA3)T,~(O - I)-2i(sin A1-A1 cos A1) sin ---N--S,~(O)=G.~,~(O,~,)
(35)
The following substitutions and simplifications are made:
A~ = ecTAmo~/Nh, cosA~ = 1 + O(At) 2, A~ sinA1 = O(At) 2 2 A 2 = (T/Nh)2(m20 c4 - eCe20q- e 2 c 2 ~Am0x) = O(At) 2
~l)k 322
_..
(eTr
= O(At) 2,
T = NAt
1 + iA1A3 " e iA1A3, 1 - iA1A3 " e -iA1A3,
(36) (37)
(38)
3.2 EVALUATION OF EQ.(3.1-32) 2 (cos A1 + A1 sin ~1
) COS
2r~
109
-- ,~22 " 2 COS
27[t~
2(sin A1 - A1 cos A1) sin - 7
~2r~
(39)
= O(At)
(40)
The small value of Eq.(40) tempts one to leave out the fourth term with T~(0) in Eq.(34) and with S~(0) in EQ.(35). This is not justified mathematically since we do not know how large the sum of the remaining four terms--including the inhomogeneous term--is. From the standpoint of physics one would eliminate the coupling of the two processes represented by Eqs.(34) and (35). We may solve Eq.(34) for T,~(O) and substitute T,~(O), T~(O:hl) into Eq.(35) to obtain an ordinary, inhomogeneous difference equation for S.(8):
e 2v'~ ~'~S,~(8 + 2) - 4 cos
- 4 cos
2~e'~3S~(0+ 1) + 2
27r~e-~'~x~S,~(O-
1) +
e-~~S,~(O-
27rt~
= -e~
4~-~) 2 + cos ~
+ 1,~) + 2cos --~--Gc~(8, ~)
-
S~(8)
2)
e-~
-
1,~)
(41)
Similarly, we may solve Eq.(35) for S,~(0) and substitute S,~(8), S,~(0 4- 1) into Eq.(34) to obtain an ordinary, inhomogeneous difference equation for T~(O):
- ~ T,~(0) e2i~'X3T,~(O + 2) - 4cos 27rtCei'X~'X3T,~(O+ 1) + 2 2 + cos -47rtr
2~-~X~T~(O- 4 cos --if= -e~
1) + e-2~T~(O-
2)
27[g
+ 1,~) + 2cos--~-Gs~(8, ~) -e-~
1,~)
(42)
The homogeneous parts of Eqs.(41) and (42) are equal; their only differences are the subscripts c and s of the inhomogeneous terms. For the solution of the homogeneous Eqs.(41) and (42) we make the usual
ansatz
0
or
T,~(O)=d,~v~
(43)
and obtain an equation for v~:
e2i~1~3v~2 _ 4 cos ~
+ 2 2 + cos -
27[~s iAIA3 -4 cos --N--ev~ 1 +
e-2iAiA3
v~-2 = 0
(44)
110
3 INHOMOGENEOUS DIFFERENCE EQUATION
This is a fourth-order equation. Because of its symmetry it can be reduced to two equations of second order:
27r/~ (ei)~l)~3v~ + e - i x ~ v 2 1 ) 2 - 4cos --ff-(e i x ' ~ v~ + e -i~' ~ v21)
+2(1 +cos~
v~ + e -i~l~av~l) = 2 cos ~ 27rg
= 2cos "N
•
-~) = 0
(45)
4 cos 2 --if- - 2 1 + cos double root
(46)
The second quadratic equation
ei~'IX3v,~ + e-iXl)'3v= 1 -
27rt~
2cos --if- = 0
27r~ ~ e2iX~x3v~ - 2 cos --if- e i x~v~ + 1 = 0
(47)
yields two single roots
2~'~ I cos 2 --if2~'~- 111/2
e ~1~3v~ = cos ---if- • v~l = e -i~lx3
(
v,~2 = e -i9'1"~3(
cos -2r~ 7 + i sin -27rS -~ )
= e_i~,i X3e2,~i,~/N
(48)
2zrtr 27rtC)=e_i,Xl,X3e_2~ri,~/g cos - - ~ - i sin --if-
(49)
and we obtain for v~0 in Eq.(43) the first two solutions:
o V~l
=ei(2,,~/N-~l~3)O=cos
0 V~2
=e-i(2,~,~/N+X~3)O=cos
--~-,kl)~30+isin "-~'+)~1)~30--isin
~-,~l,~a '-ff-+)~1,~3
The double root of Eq.(46) calls for two more solutions:
(50)
0 0
(51)
3.2 EVALUATION OF EQ.(3.1-32)
111
OV01 = Oei(2~r~/N-~l)~3)O
: 0[ o:
1
(52)
Ov02 : Oe-i(2~r~c/N+AiA3)O
--- 0[COS ( -27r~ ~ + A1A3) 0 - i.sin ( 27r~ -if-+ The general solution S,~(O) in
AIA3)0]
(53)
Eq.(43) becomes:
02 + c,~30v,r01 + C~40V02 S,~ (0) = c,r v,r01 + c,~2v,~ = Ctclei(27r~c/N-AIA3)O _}. C,r "~- C~30ei(2r~
For T,r
A3)O Jr" C~40e -i(27r~/N+k1A3 )O
(54)
we obtain from Eq.(43)" 0 + d,c4Ov,r02 T,~(O) - d,c~v,r01 + d,~2v,~02 + d, c3OV,r -- d~l ei(27r~r AxAs)0 + d~2e - i(2~r~/N+Jk11k3)0 (55)
'}- d~3Oei(2~r~'/N-)~lAz)O + d~r
We turn to the method for this task available books 1 but c~ and d~, with i c~i(O) and d,~,(O)'
inhomogeneous solution of Eqs.(41) and (42). A general goes back to Lagrange (1736-1813). It is discussed in the we present it in Section 6.3 in more detail. The constants t, 2, 3, 4, in Eqs.(54) and (55) are replaced by variables
s~(0) = ~(0)~o~ + ~(0)~o~ + ~ ( 0 ) 0 ~ o + ~(0)0v% T,r
0
0
0
0
-- d,r (O)v,r 1 + d~2(0) v~2 "+"d,c3(O)Ov,ct + d,c4(O)Ov,r
(5.6) (57)
Equations (6.3-23) and (6.3-30) define d,~i(O) and c,~(O) in a form suited for computer use. The solution derived here for Eq.(3) is extremely general. The only significant requirement is that the functioa Gt (~, 0) of Eq.(4) permits a cosine and sine transformation in terms of r according to Eqs.(32) and (33). Convergence of the Fourier series of Eq.(31) is not required since ~ does not approach infinity but only i ( N / 2 1). 1N6rlund 1924, p. 396; 1929, p. 22, 125; Milne-Thomson 1951, p. 374.
112
3 INHOMOGENEOUS DIFFERENCE EQUATION
Substitution of Eqs.(56) and (57) into Eq.(21) yields the particular solution of the inhomogeneous equation (14). We must still satisfy the boundary and initial conditions of Eqs.(ll) to (13). Using Eqs.(2.3-19) to (2.3-21) and Eq.(15) we obtain the following boundary and initial conditions for 9(r 0) from Eqs.(ll) to (13):
9(0,0)=0
forO>_O
(58)
~3(~,0)=0
forr
(59)
for(>O
(60)
~(~,1)-~3(~,0)=0
The boundary condition of Eq.(58) is satisfied if we discard the first term in Eq.(21)" N/2- 92
9 O) =
E ,~=-N/2+l
(61)
T,~(8)ei)~; sin 27r~r N
The initial condition of Eq.(59) is satisfied for the third and fourth term of T~ (0) in Eq.(57) due to the factor 0. The first and second term have the coefficients d~i(0) for 0 = 0' N/2-1
~(r 0) =
27r~r
Idol (0) + d~2(0)]e ' ~ s i n N
E
= 0
(62)
,~=-N/2+l
From Eqs.(57) and (61) we get with Eqs.(48), (49), (60), and (59) the relation
N/2-1
'b(#, 1) =
E
{[d~1(1) + d~3(1)lval + [d~2(1) + da4(1)lv~2}
~=--N/2+1
x e ixlr sin 27r~ = 0
N
--e
N/2-t ( ~
[d~ (1) + d,~2(1) + d,~3(1) + d~4(1)] cos
27r~N
~=-N/2+1
+ i[d~i(1) - d~2(1) + d~3(1) - d~4(1)] sin 2N~)eiXl~ sin We multiply Eq.(62) with N-le thogonality interval 0 <_ ~ < N:
-~162sin(27rv~/N) and
d~l (0) + d~2(0) -- 0
for v - a
2~ N
=0
(63)
integrate over the or-
(64)
113
3.2 EVALUATION OF EQ.(3.1-32)
Equation (63) is multiplied with N - l e -~1< sin(27ru~/N) and integrated over the interval 0 < ~ < N. This time we obtain two equations:
d~l (1) + d~2(1) + d~3(1) + d~4(1) = 0
for v = ~
d~l(1) - d~2(1) + d~3(1) - d~4(1) = 0
(65) (66)
We get from Eqs.(64) to (66)"
d~l(0) = -d~2(0)
(67)
d~l(1) = -d~3(1)
(68)
d~2(1) = -d~4(1)
(69)
E q u a t i o n (6.3-24) yields"
d~l (1) = d~l (1) - d~l (0) + d~l,
d~l (0) = d~l
(70)
d~2(1) = d~2(1) - d~2(0) + d~2,
d~2(0) = d~2
(71)
From Eqs.(67), (70), and (71) we get d~2 = - d ~ l
(72)
There is no corresponding link between d~3 and d~4 due to the factor 0 in the third and fourth term of Eq.(57). From Eqs.(6.3-23) and (6.3-22) as well as Eqs. (6. 3-17) to (6.3-21) we obtain the following relations for d~1(1) and d~3(1):
D~I(1) -t- d~l D~0(1) D~3(1) + d~3 d~3(1) = Ad~3(0) + d~3 = D~0(1) d~l (1) = Ad~l (0) + d~l =
(73)
(74)
Substitution into Eq.(68) yields' d~3 - - d ~ l - D ~ I ( 1 ) + D~3(1) D~0(1)
Similarly
we
obtain for d~2(1) and d~4(1) of Eq.(69)'
(75)
114
3 INHOMOGENEOUS DIFFERENCE EQUATION
D~2(1) ~- d~2 D~o(1)
(76)
d~4(1) = Ad~4(0) + d~4 = D~4(1) + d~4 D~0(1)
(77)
d~2(1) = Ad~2(0) + d~2 -
D~2(1) + D~4(1)
d~4 - - - d ~ 2 -
(78)
D~o(1)
Substitution of Eq.(72) brings: d~4 = d~l - D~2(1) + D~4(1)
(79)
D,~0(1) The constants d~4, d~3, and d~2 are now expressed in terms of d~t. The sums d~l + d~3 and d~2 + d~4 are further developed from Eq.(6.3-53) on in Section 6.3, after the evaluation of the determinants D,~ (O)-D,~4(O) and D,~o(O). We may now write the solution for ~(~',0) of Eq.(3). Starting with Eqs.(8), (9), (10), and (15) we obtain the following expressions: II/1(~, O) --- ~oo{(1 - e-2'x~a3~ exp[-(Ag - A2)l/2~]e '~r
+ ~(r o) + ~(r o) }
(80)
IT(n/N) sinfl~0sin 27rn._~
(81)
N/2-1
~(r
= -2i~a~'~(r
+~~
~ ~=-- N / 2 + 1
IT(n/N)
see Eq.(2.4-29);
9 O) =
N/2-1
E
,,=-.,'v/2+ 1
sin fl~
N
A1, A2, A3 see Eqs.(36), (37)
T,~(O)ei~r sin
27r n~
(82)
N
Ta(O) = [dal(0)+ Oda3(O)lv~l+ [d~2(0)+ Od~4(O)]v~ (8a) Vnl , Vn2 see Eqs.(48), (49); din(O) see Eqs.(6.3-23), (6.3-47)-(6.3-50) d~2, d~3, d~4
see Eqs.(72), (75), (79)
Comparison of Eq.(80) with Eq.(2.4-32) shows that ~1(4,0) can also be written in the following form:
N/2-t
'/~(~, 0) -'-
E
([d~l(0) -[- Odtc3(O)]e i(2~rt~/N-A1)~3)O
t~=-- N/2+ 1
+ [d~2(O) +
Od~4(O)]e-i(27r~/N+X~x3)~ ix~Csin 27rn{ (84) /
N
3.3 QUANTIZATION OF THE SOLUTION FOR A x ~ h/rnoc
115
3.3 QUANTIZATION OF THE SOLUTION FOR A x >> h/moc
In Section 2.5 we derived the Hamilton function of ~o - ~x0~ according to Eqs.(2.1-37), (2.1-38) and carried out the quantization with Eq.(2.5-29) or (2.5-30). We follow the process used there, but with ~1 = ~ x l ~ of Eq.(3.2-80) we carry the approximation to the next order in a. Using Eq.(2.1-38) we write
-- ~0 -F a ~ l -- ~xOz~ + a~xlz~.
(1)
Equations (2.5-1) to (2.5-3) remain unchanged but from Eq.(2.5-4) on we must replace ~ = ~o with 9 = ~o + a ~ l '
r
= (~; + ~r162
oo 0o = oo oo 0r 0r 0r
0r
+ ~)
~a\oo (0r
= r162 + ~(r162 + r162 + o(~ 2) (2) oo
~ oo 0 o / + O ( a 2 ) 0r 0 r
0r
0r 0r = 0r 0r ~-a. 0r 0r t 0r 0r _ +~
(3) (4)
The function Vo(r 0) is defined in Eq.(2.4-32) and used in Eq.(2.5-4)" ~o(~, 0) =~oo[(1 - e-2i:~X3OF(<) + u(<, 0)] -- ~oo [(1 -- e -2''xl"x30) exp[-(A 2 - A2)l/2~]e"Xl( N/2-1
"-i;~;~3e~;~lCe-i;~;~3~ -- 2iA1A3~
E
~=--N/2+1
IT(tc/N) sinfl'~Osin 2~]~~] sin fl~
(5)
The function ~1 (r 0) is written in the form of Eq.(3.2-84):
r (r o) = r162 o) + r162
o)
(6)
N/2-1
~(~,0) = ~ ' ~ , ( ~ - ~ )
~
{[a~(o) + oa~(Ol]~ ~'~~
tc---N/2+l
+ [d~2(0)+
Od~4(O)]e-2~''~~ } sin N27r,;~
(7)
With the help of Eq.(6) we may rewrite Eqs.(2)-(4) as follows, using the notation [Re(... ) for the real part of the expression in parentheses:
9 *r = r162 + ~[2r162 + r 1 6 2 = r162 + 2~[r162 + r 1 6 2
+ ~*r (8)
116
3
INHOMOGENEOUS DIFFERENCE EQUATION
0v~ 0~o
0~* O~ O0 O0
00
00
[ 0~ 0~o ~-aL2 00
00 + ~ o o
O0 O0 § O0 O0
o~,~ O~,o + 2~ o ~ O~o oo oo oo oo o ~ O~,o [ o~,~O~o +
0~* O~
or 0r
(9)
0r Os t 0r O( O~ O~0 Fo~ O~o + ~oo9~e ( 0 ~ ; 0~3 or or +2~L0r or k
o, o )1
(~0)
The terms ~ o , (0~/00)(0~o/00), and (O~/O()(Oq2o/Or are shown in Eqs.(2.5-4), (2.5-9), and (2.5-11). We still must evaluate the additional terms from ~;7) to (Oqt~/O~)(OO/Or in Eqs.(8)-(10) with the help of Eqs.(5) and (7). Since we are interested in the effect of the inhomogeneous term G1 (~, 0) in Eq.(3.2-4) we must show more clearly how ~)(r 0) depends on G1 (~, 0). From Eqs.(6.3-1) and (3.2-33)we obtain: N
H,~(O,,~)=-~2
~'~'~ ~' C~ (r 0 + 1) + 2 cos .--~- G~ (r O) - ~-'" ~~" V ~(r e - 1) 0
x e-i)~r
27ra(dr N
0 - 1 ' 2, ... ' N - 2
(11)
The four functions d~l (0) to d~a(0) in Eq.(7) are defined by Eqs.(6.3-22) and (6.6-23). We use them in the form of Eqs.(6.3-59)-(6.3-62)"
0-1
d,~l(O)- ~Hs,~(n, a){nFl (n, x.)+F3(n, a)+i[nFh(n, a)+ FT(n, t~)]-+-d,~l}
(12)
n--0 0--1
d,~2(O)= ~-~Hs,~(n, ~){nF2(n, ~)+F4(n, ~)+i[nF6(n, ~)+Fs(n, ~)]+d,~2}
(13)
n--0 0--1
H,,~(n, a) { FI (n, a) + iFh(n,a) + d~3}
(14)
d,,4(O)-- Z Hs,,(n,a){F2(n,a) + iF6(n,a) + d,,4}
(15)
d~3(0) = - ~ n---0 0--1
n'--O
The functions Fl(n,a) to Fs(n, to) are defined by Eqs.(6.3-51)-(6.3-58). With the help of these equations we may rewrite the terms
[d,.;1(0) + Od,,3(O)]e2~i'`e/g + [d~2(0) + Od,~4(O)]e-2"~''r
(~6)
3.3 QUANTIZATION OF THE SOLUTION FOR Ax >>h/moc
117
of ~(~, 0) in Eq.(7) into the form of Eq.(6.3-75) and obtain the expression
r
O) = e i'x~(r176
N/2-1 E
27r~ [Jr(O,~) + iJi(O, ~)] sin
,~=--N/2+l 2~r~0 Jr(O,n) = [Jl(0, ~) - OJ~(O,~)]cos N -
2~rn0 [Ja(0, n) - OJ4(O,n)]sin----f-
2~r~O 2~rnO Ji(O, n) = [Jb(0, n) - 0J6(0, ~)] sin N + [./7(0, ~) - OJs(O, n)] cos N (~7) The sums over n shown in Eqs.(12)-(15) have been separated in Eq.(6.3-75). We obtain ~;({, 0) from Eq.(5) by changing the sign of i. The product ~o0~;(~, 0) 9 0) becomes:
t9009;(~, 0)~9(~, 0) = 9~0 e-i'xlx3~ ((1 - e2{xlxa~ exp[-(A~ - A2)1/2~1 "t- 2i,~1/~3eiA1A3(O-1)
N/2-1 E
,~=-N/2+l x
IT(K/N) sinfl~ sinfl~0sin
N/2-1 E
2"x~r ) N
[Jr(0 ~) + iJi(O, ~)] sin
~=-N/2+l
'
27r~ N
(18)
We have a product of two sums. The terms sin(27r~/N) are the important parts as will soon become clear. We write u for ~ in the second sum and separate the products sin(27r~(/N)sin(27ru()/N for ~ = u and for ~ 7~ u:
~00~;(~, 0) 9
O) = ~ 0 e-i)'~'xa~ [(1 - e 2i)'l"xa~
N/2-~ 27r~ E [Jr(O, ~) -+-iai(o, ~)1 exp[--(A22 -- ~2)1/2r sin N ,~=-N/2+l N/2-1 27rt~r IT(n/N) + 2iA1A3e~176 E sinfl~ sin(fl,~O)[Jr(O,~)+iJi(O,~)]sin 2 U ~=-N/2+l N/2-1,•v N/2-1 + E E IT(~/N--------~)sinfl'~O tc=--N/2+l u---N/2+I sinfl~ 27rk'~ 27r~;~) ] (~9) x [Jr(O,u) +iJi(O,u)]sin N sin N x
118
3 INHOMOGENEOUS DIFFERENCE EQUATION
The integral over ( of this expression taken from 0 to N will be needed according to Eqs.(2.5-3) and (8). Only the terms N
i exp[-(A - A2)lt2(]sin
/sin22-~'~d(
2"~'r162 and N
N
0
0
contribute to the integral: N
g2ooi t~((, 0)~((, O)d(
0
= 92o
~
(
E ,~=-g/2+l
2 sin(A1130)
(2~/N){1 - exp[-(a~ - a~)~/~g]}
12- 12 + (2r~'/N) 2 x [J~(O, ,<.) - i Jr(O,
~)]
ST(~/N)
+ NA113 sin fl'-------~sin(fl~0){ Jr(O,~,)sin 1113 - Ji(0, ~) cos 1113
+ i[J~(O,~.)cos.X~.xa + Ji(O,~)sin.X~.Xa]}) We turn to Eq.(9). Eq.(2.5-9). For the term
It requires (Ot~;/O0)(O~o/O0), which 0~/00 we obtain from Eq.(17)"
05(r 0)
~/2-~
O0 = e"X';
E
(20)
is found in
[Jp,.(O,~) + iJi(O, ~)1 sin 27r~._.~N
,r
j~(0,~)= (~176 (90
-k 1 1 1 3 J i ( 0 , t~)
) cosA1A30
"t-(OJi(O'l';) O0
Ji(0, ~ ) = -
( OJr(O, n) + 00 +
( OJi(O,l';) O0
-- 1 1 1 3 J r ( 0 , ~)
A1A3Ji(0,~)
)
) sin 11130
sinA1A30
)
- 1113Jr(0, ~) cos 11130
(21)
The factor e i~1r is not written as cosAl( + isinAl( since it will be cancelled in the following equation for (Oq2~/O0)(O~/O0)by e - ~ ' ; . The term 0 ~ / 0 0 is obtained with the help of Eqs.(2.5-5) and (2.5-6):
3.3 QUANTIZATION OF THE SOLUTION FOR A x >> h/moc
119
~oo 0~(~,00 0) 0~(~,000) _. ~20 [ _ 2iAIA3e2i'x~'xa0 x
Nt2--1 ~ [JR(O,~) + iJI(O, t~)]exp[-(A~ - A2)1/2~] sin 27r~_~ N ~r N[2-1 -
2A1Aaei'X~x3(~ (
Z
IT(t~/N)sin fl,~ (A1A3sin fl,,O
to----N/2+ 1
- ifl~ cos fl,~O)[JR(O,~) + iJI(O ~)] sin 2 27rt~r '
N/2-1,~tv
+
N
N/2-1
~
Z
tc----N/2+l v=-N/2+l
Iw(x~/N) [JR(O, v) + iJi(O, v)] sin fl~
2~rt~r 27rvr x (A1A3sinfl~0 - ifl~ cosfl~0) sin 'N sin N JR(O, u) = JR(O, ~), Ji(O, v).= J~(O,~) for t~ ~ v
(22)
As in Eq.(20) we integrate this expression over r
N
/ 0 ~ (~, 0)&)(~, 0) ~oo. O0 ~ d ~ = ~ 2 o
(2
N/2-1
AIA3 Z
0
{[JR(O,~)+iJI(O,~)]sin2A1A30
~:=--N/2+1
+ [Ji(O tc)--iJR(O ~)]cos2A1A30} (27r~/N){1-exp[-(A2-A2)l/2N]} , , A2_ A~ + (2zr~/N) 2 gl2-a - A1AaN Z IT(x/N) {[JR(0, t~)+ iJi(O,t~)] sin fl,, ~r X [A1A3sinfl~OcosA1A3(O - 1) -k-fl~ cosfl~OsinAtA3(O - 1)] -
-
x [AIA3sinfl~0sin A1A3(0- 1 ) - fl~ cosfl~0 cos A1A3(0- 1)]}~ /
(23)
Turning to Eq.(10) we recognize the need for the term 0~/0(. Equation (t7) yields:
120
3 INHOMOGENEOUS DIFFERENCE EQUATION
0r
0)
--ei)~(~-)'3~
0~
N/2-1 E [Jr(0, ~) + iJi(O,m)] ~;=--N/2+l
x
cos N
+iAlsin 27r~N)
(24)
The term (0~/0~)(0~/0~) in Eq.(10) is obtained with the help of Eqs.(2.5-7) and (2.5-8):
~ooOV~(r o~(r o) : or
or
N/2-1
kI/020( _ 2[/~1 _ i(,~2 __ ~2)1/2]
27rt~exp[_(A2_A2)l/2r ] cos 27r~r g [J,.(0, t~) + iJi(O,~)] ---if-
~=-N/2+1
+ iAlexp[-(A 2 - A2)l/2r
+ 2iA1A3e-i)~)~3
sin/3,,
• A2 sin2 N/2--1,ytu
E
sinA1A30
N/2-1 Z IT(t~/N) [Jr(0, ~)+ iJi(O,t~)]sini~O ,~=-N/2+I
+ 2A1A3e-ixla3
g
+
cos 2
N/2-1
IT(~/N) [Jr(O u) + iJi(O, u)] sin/3~0
E
~=--N/2+1 u=--N/2+l
sin/3~
x ( A1 sin 27r~ N + i---r;-~ 2~-~ iv cos 2~-~{) N (~_~ cos 27ru4 N + i A1 sin 2~u() N )
(25)
This expression is integrated over r following Eqs.(20) and (23)' N
~oo
f ov~(r o) ~o~(r d ~ " o) 0r
0r
= ~20 { 2A2 sin A1A30
0
N/2-t
• ~
~-'--U/2+l
(27rt~/N){ 1 -exp[-(A 2 - A2)1/2N] } /~2 __ A2 + (2~-~/N)2
[J,(O,,~)--~Jr(O,,~)]
N/2-1
-NA1A3
E ~=-N/2+l
- i[Jr(O, ~) cos A1A3+ Ji(0, ~) sin A1A3]} Iw(~/N)sin/3~sin/3~0}
(26)
3.3 QUANTIZATION OF THE SOLUTION FOR Ax >> h / m o c
121
Let us turn to the text following Eq.(2.5-11). In order to allow for the generalization of the terms of Eq.(2.5-3) by Eqs.(8)-(10) we define the energy U by the sum of the following three components
0
= ~1 + 02 + 03 = 0c +
Ov(O)
(27)
where 01 to 03 are obtained by the substitution of Eqs.(8)-(10) into Eq.(2.5-3). We use the notation U to Uv(0) to distinguish the terms from the approximations U to Uv(0) in Eq.(2.5-12): N
N
L2 m2c4(At)2 ] qJ*~dr = L2 m2c4(At)2 ((1 + 2 a ) / ~ ; ~ o d { U1 = cat r-i,2 cat li2 0
0 N
+ 2c~oo / ~Re(~?)d~)
(28)
0 N
N
02
L 2 / 0~* 0~ - -o0 - - d eo0 = ~
= ~
(1 + 2c~)
00
0---O
0
0
N
+ 2C~$oo/ J~e( 0 ~ 00 oo
(29)
0
L2 /
N
0a:~-~
09* 09
~~dr
or or
0
L2
N
= ~-~ (1 + 2c~) f 0 9 ; 0~I,_____sd~ or or 0
N + 2olli/00/:~e (01I/~ 0~
(30)
0
As in Eqs.(2.5-13)-(2.5-15) we want the time-invariant part ~rc -" brcl + ~rc2 ~- gc3
(31)
of U1, U2, U3 and we ignore the time-variable part Uv(0). Again we write Uc to U~3 to distinguish the terms from the approximations Ur to U~3 in Eq.(2.516). When deriving Eqs.(2.5-13)-(2.5-15) we simply left out terms containing sin ~ 0 or cos fl~0. The time variations of Eqs.(28), (29), (30) are not so obvious and we must write integrals over 0. Using Eqs.(2.5-13)-(2.5-15) we may write the time invariant part of Eqs.(28)-(30) as follows:
122
3 INHOMOGENEOUS DIFFERENCE EQUATION
0~1 =
UldO = cAt;
h2
(1 + 2ol)~2oNA2A32
0 N
U:2 =
j
sinDI.
N
,,>
-,:-.( Iw(~lN)si):n/7, 0
"{
U2dO= ~
0
Z ~=-Nt2+l
(1 + 2,)tIs020NA~A2
0
(~,,X,'' + n,)'
Z
~----N/2-+.l
N
N
(33) 0
i
"(
0~= 0 03dO=7A7 (I+2~)@~176
0
"~ '-" N
0
tIT(a/N)si7[n/7"~+= ("'71 --N N
0
We recognize on the right side of Eqs.(32)-(34) the energies Ucl, Uc2, Uc3 of Eqs.(2.5-13)-(2.5-15) multiplied by 1 + 2c~ -'- 1.0146. This difference of 1.5% is barely visible in a plot. The interesting terms are the integrals multiplied by 2o~00. We shall analyze them in the following Section 3.4. A comparison of Eqs.(31) and (2.5-16) shows that the only difference is the symbol ^. Hence, we may use the results of Section 2.5 from Eq.(2.5-16) on. The energies E~ of Eqs.(2.5-32), (2.5-34), and (2.5-48) are obtained. One could write F.~ instead of E~ and obtain the result F_~ - E~, but there is little incentive to do so. 3.4 EVALUATION OF THE E N E R G Y Uc
We start from the energies De1, Uc2, Uc3 in Eqs.(3.3-32)-(3.3-34). Their first terms equal the energies Ur Ur Ur in Eqs.(2.5-13)-(2.5-15), except for the factor 1 + 2c~. The results of Chapter 2 apply to these terms. Here we are interested in the second terms that are multiplied by 2a"
L2 O~ = 2a~oo~-~
( m~
2
N
N
(1) 0
0
3.4
0~2 =
123
E V A L U A T I O N OF T H E E N E R G Y 0 c N
N
0 N
0 N
0
0
2~9oo~
O0 O0
9~e
L2
O~ o~)d~]dO
(3)
The sum Uac of these terms is ultimately wanted as function U ~ ( a ) of as shown by Eq.(2.5-16) N/2-1
0~=0~i
+0~2+0~3
=
~
0.~(~)
(4)
,r
N/2--1
ooi: ~
N/2-1
N/2-1
a~(~), oo~: ~
.=- N/2§ ~
o~(~), <~: ~
~=- N/2+ ~
~=- N/2+
o~(~) (5)
and Eqs.(1)-(3) must be written correspondingly. We write 01,~(~;) with the index ~ after the integration over { that indicates that the summation over in Eq.(3.3-20) was not carried out: N
h
01~(~) = 2a~oo~-~
N
ire 0
~}-~1~(/,~)---
q2;(~,O)~((,O)dr dO 0
2aqj2o(L2/cAt)(moc2At/h)2 N
1 /
= ~oo'
N
ire
0
(/
,
)
~o(~,O)~((,O)d( dO
(6)
1%
0
Dividing Eq.(3.3-20) by 92o and leaving out the summation sign yields the following expression for ~1,~(~;)" [KI~(~) =2 (2Try/N){1 - exp[-(A 2 - A2)l/2N]}.
~
- ~ + (2~/N)
2 N
x / iRe[Ji(O,~)- iJr(O,~)]sinA1A3OdO 0 N
+NAIA3IT(~/N)./ iRe{Jr(O~;) sin AIA3 sin ]3~
Ji(O, t~) cos AIA3
0
+/[Jr(0, ~) cos A1A3 + iJi(O, ~) sin A1A31} sin fl~0 dO
(7)
124
3
INHOMOGENEOUS DIFFERENCE EQUATION
TABLE 3.4-1
FUNCTIONS fl~, IT(to~N),... AND THE EQUATIONS EQ.(2.4-13), EQ.(2.4-29), ... DEFINING THEM, WHICH ARE REQUIRED FOR THE CALCULATION OF ~-~ln(t~), ~]~2~(~), AND ~J~3~(~) OF EQS.(2S), (39), AND (45). THE NUMERICAL VALUES OBTAINED FOR EQUATIONS SHOWN WITH BRACKETS, E.G. EQ.[2.4-13], SHOULD BE REMEMBERED BY THE COMPUTER TO REDUCE THE COMPUTING TIME.
fl~ Eq.[2.4-13] Ami~((,0h) Eq.(3.4-14) Amox Eq.(3.4-9) Gi+(r Eq.[3.4-201 G~,,i+(O~,~) Eq.[3.4-23] F2(n, x,) Eq.(6.3-52) F~(n, ~) Eq.(6.3-55) Fs(n, x~) Eq.(6.3-58) J1 (0, ~) Eq.[6.3-75] J4(O, ~) Eq.[6.3-75] JT(O, ~) Eq.[6.3-75] Ji(O, ~) Eq.[3.3-17] Iii(0,~) Eq.(3.4-27) K1 (0, ~) Eq.[6.3-75] K4(O, ~) Eq.[6.3-75] KT(O, ~) Eq.[6.3-75] OJi/O0 Eq. [3.4-37] I2~ (0, ~) Eq.(3.4-32) /24(0, to) Eq. (3.4-35) I3i (0, e~) Eq.(3.4-43)
IT(n,/N)
Eq.[2.4-29] Ce~(~,0~) Eq.(3.4-15) Gi(4,O,~) Eq.[3.4-18] G~,,i(O~,x,)Eq.[3.4-21] H~,~(O,~,x,) Eq.[3.4-24] F3(n, tr Eq.(6.3-53) F6(n, ~) Eq.(6.3-56) d3r(~) Eq.(6.3-73) ,12(0,~,) Eq.[6.3-75] J~(O,~,) Eq.[6.3-75] ds(0, ~) Eq.[6.3-75] AI~(~) Eq. [3.4-25] 9f1~(~) Eq.[3.4-28] g2(o, ~) Eq. [6.3-75] K~(O,~) Eq.[6.3-75] Ks(O, x,) Eq. [6.3-75] JR(O,K,) Eq.[3.3-21] /22(0, x,) Eq.(3.4-33) 9f2~(~) Eq. [3.4-39] I32(0,x,) Eq. (3.4-44)
~0(r 0,~) Ce0
Eq.[3.4-8] Eq.(3.4-10) Gl_(r Eq.[3.4-19] G~,~i_(O,,~)Eq.[3.4-221 Fi(n,~) Eq.(6.3-51) F4(n, ~) Eq. (6.3-54) FT(n, ~) Eq.(6.3-57) d3i(a;) Eq. (6.3-74) d3(0, ~) Eq.[6.3-75] J6(0, ~:) Eq. [6.3-75] Jr(0, ~) Eq.[3.3-17] Bi~(tr Eq. [3.4-261 K3(0, ~) K6(0, ~) Jr(e, ~) I23(0, ~)
Eq.[6.3-75] Eq. [6.3-75] Eq. [3.4-36] Eq. [3.3-21] Eq. (3.4-34)
9Q~(~)
Eq.[3.4-45]
OJr~00
We observe that Jr(O,~) and Ji(O,u,) are not real but complex functions. An attempt to separate real and imaginary components analytically creates unnecessary complications since the computer instruction :Re will do this separation numerically. In order to integrate Eq.(7) over 0 we need to evaluate integrals containing Jr(0, to) and Ji(0, t~). For the determination of Jr(O, to) and Ji(0, ~) we start with Eq.(3.1-1) for ~0(r and choose ~00 = 1. The variable O of Eq.(3.1-1) will become the variable n in H~,,(n,n,) in Eqs.(6.3-47)-(6.3-50). For this reason we rewrite Eq.(3.1-1) with 0 replaced by 0~ and shorten O,~ tb either O or n as required: ~o(r Or~) = (1 - e -2''x~'xae'~)exp[-(A 2 - A2)i/2r162 g/2-i
- 2iAiA3e';~ae'~(r176
E t~=-N/2+l
IT(n/N)sin fin sinfl~0n sin 2 ~ r
(8)
3.4 9 EVALUATION OF THE ENERGY Uc
125
Table 3.4-1 lists the sequence of functions required to obtain 9Q~(a) and the equations that define them. The function ~o(r is needed for G1(r of Eq.(3.2-4). A number of definitions is introduced to make Eq.(3.2-4) suitable for a computer:
eAx ecAt Amox - ~ Amox --- ~ A m o x h h eat eAx
(9)
.
r
= --g- r
= -~
(10)
r ~o
(11)
eAx . ecAt Aml~(r 0n) Amlx(~,On)- --~-Amlx(~,On)h r162
eAx
-~r
e~) =
GI(~, On) - -iAm~
/2
(r
e~) =
eAt
--/-r (r
(12)
e~)
§ 1 ) - ~0(~, 0,~ - 1)]
+ ir162 i
0~))
{[Amlx(~ + 1,0n) -{- Amlx(r 0n)]~0(~ + 1, 0n)
e~)+ Amlx(~
--[Aml~(r
--
i
-- ~{[r
- 1, 0n)]~0(~ =- 1, 0n)} 2Amo~Amlx(~, On)~o(~, On)
0n -~- 1) -~- eel(~, 0n)]lX]0(~, 0n ~- 1)
-[r162
o~) + r162
0~ - 1)]~0(r o~ - 1)}
+ 2r
On)~O(r On) (13)
The space and time variation of the potentials Aml~(r 0,~) and r (r 0,~) must still be defined. A particularly simple choice is to let the potentials be independent of time and vary linearly with the space variable (::
Amlx(~, On) -- Amlx(~) : A l m ~ / N eel (~, On) -- eel (~) -- r
(14) (15)
Here Aim and r are dimension-free constants. The two constants eeo and Am0~ in Eqs.(10) and (9) may be expressed in terms of A1 and A3 according to Eq.(2.3-2):
r
eat = --h--r
= ~3
ecAt Am0x = ----ff-Am0x = A1
(16) (17)
126
3 INHOMOGENEOUS DIFFERENCE EQUATION
The constant Cc~ poses a problem. Its value is arbitrary and we may thus choose it to be zero without causing a mathematical mistake. We do so here but point out that this is not a particularly satisfying solution of the problem. The simplified Eq.(13) becomes: i Gl(~,On) = -~{[Amlx(~" + 1)4- Aml.(r162 --[Amx:~(r -
+ am~((-
1)]~o_(~- t, 0n)} - 2Amo.am~.(r162
iCel (<)[(gO(<,On + 1) -- ffgO(r
-
+ 1,0n)
-- l)] q- 2r162
(<)~0(r 0n)
We shall also need Gl(r 1) and GI(~,O,~ + 1). A not very elegant but lucid method to produce these time shifted functions is to write Eq.(18) for On --* 0, - 1 and On --* 0N + 1" i G~_ (~, On) = -~{[Am~:(r
1) +Am~(r
1 , 0 n - 1)
-[am~(()+Am~((-1)]gto(r162 - ir162
0,~) - ~ 0 ( ~ , 0,~ - 2)] + 2 r 1 6 2 1 6 2
0,~ - 1)
(19)
i GI+((,O~) = -~{[Aml~(~'+l)+ Aml~(()]~o(~'+l, 0.+1) -[Amt~ ( r -
-
(r
1)]~o ( r 1, Or,+ 1)}--2amo~Aml~(~)~o (r 0n + 1)
i~)el (~)[lI/0(~, On ~- 2) -- 1~/0(~, On)] Jr 2r162
(r
0n + 1)
(20)
We can compute Gs~(0,~,a) of Eq.(3.2-33) once GI(~,O,~) of Eq.(18) is obtained. The notation Gs~(0,, a) is changed to Gs~l (0n, ~) to make the use of Eqs.(19) and (20) more lucid. Let us observe that Eq.(3.2-33) treats the function GI((, 0) as a step function with step width t. The integral may readily be replaced by a sum of GI((, 0) taken at { = 0, 1, ... , N - 1: N
2 / i~1r .2zr~r Gs~l(0n, ~) = Gs~(0n, a) = ~ G1(r 0,)esin N de 0 _ __2 ~ 1
-
N ~--'0
(r sin 27r~.__~ Gl"-'On-e-i~"r N
The functions Gs~l-(0,~,t~) and Gs~l+(0,,~) follow from Eqs.(19) and (20)"
127
3 . 4 EVALUATION OF THE ENERGY Vc
2 ~~ Gs~l_(0~,~) = as~(O~ - 1,~) = ~ Gl_(r
2r~r
-i~'r sin N
(22)
r N-1
2 Gs~+(O~,~) = G,,~(On + 1 , ~ ) = ~ ~
2~r~
G~+(r
-i~r sin N
(23)
~=0
With Eqs.(21)-(23) we obtain the function Hs~(0,~, ~) from Eq.(6.3-1)"
27r~_ (On,x) e-i)~lXSGs,~(On 1,~) Hs,~(On,~) = _ei)~'xa as,~(On + l,~) + 2 cos --~Gs,~ -
27r~
= _e~,~a,~l+(O~,~) + 2 c o s - - ~ a , ~ ( o ~ , ~ ) - e - ~ a , ~ - ( o ~ , , ~ )
(24)
Eight products of H.~,~(n, ~) with certain combinations of the functions FI(n, ~) to Fs(n, ~), defined by Eqs.(6.3-51) to (6.3-58), are listed in Eq.(6.3-75) and denoted K1 (n,~) to Ks(n, ~). The functions Ji(O, ~) and Jr(O, ~) required in Eq.(7) are derived from J1 (0,~) to Js(O, ~) in Eq.(6.3-75); they are shown in Eq.(3.3-17). We are now ready to evaluate Eq.(7). The following definitions are made to write it in a more compact form: AI~(~) = 2(2Try/N){1 - exp[-(A 2 - ,X~)*/2N]} ,~2_ ,~2 + (21r~/N) 2
(25)
B~,~(~) = N)u )~3IT( ~/N----'-~)
(26)
sin fl~ Equation (7) without the integration signs may be written as follows: Ii1(0, ~) = A~,~(tc)~e[Ji(O,~) - i Jr(0, ~)] sin,X1)~30
+ B~(~)~in(Z~0) m~{ Jr(0, ~)sin ~ 3 -- J~(0, ~) co~ ~ 3 +~[J~(0,~)cos~ + Yi(0,~)sin~]}
(27)
The equivalence of the integral over a step function with steps of width 1 and a sum permits us to rewrite Eq.(7):
/
N
N-1
0
0=0
(2s)
This ends the calculation of 9s and grl~(~) of Eq.(6). We shall return later to EQ.(28) when we need it for plotting. Here we turn to/)2~(~) of EQ.(5).
128
3 I N H O M O G E N E O U S D I F F E R E N C E EQUATION
Again the index ~ is written after the integration over ~ to indicate that the summation over ~ in Eq.(3.3-23) is not carried out' N
N
O0 O0 0
(29)
dO
0
With the help of Eq.(3.3-23) we get: N
2ae2ooL2/cAt = 9(2~(~)= ~oo
N
Me 0
-~Nd~ 0
dO
(30)
t~
N 1 : ~ e / ( 0 ~ 0~) ) (27r~/N){1-exp[-(A2-A21)1/2N]) ----~ --~--~--~dr ,, = 2AIA3 A22_ A21+ (2;r~/g) 2 0
x { 9~e[JR(0, ~)+iJi(O, t~)] sin 2A1A30+ 9~e[Ji (0, ~)--i JR (0, ~)] cos 2AIA30] }
- NAIA3IT(a/N~){~e[JR(O,a)~ sin/3~
+ iJi(O a)]
• [ ~ 3 ~i~Z~Oco~~3(o- 1)+/~ co~/~o~i~ ~ 3 ( 0 -- J~e[Ji(O, ~) - iJR(O~)]
1)]
x [AiA3sin/3~0sinA~A3(0- 1)-/%cos/3~0cosA1A3(0- 1)]}
(31)
In order to integrate Eq.(31) over 0 we need to evaluate numerically the integrals over the following terms with respect to 0:
I21(0, ~) = A1A3AI~(e;)~e[JR(O,~) + iJI(O,~)]sin2A1A30 /22(0, ~) = A1A3AI~(~)9~e[JI(O, ~) - i JR(O, ~)] cos 2AIA30 I23(0, a) = Bl,~(a)~e[JR(O,t~) + iJi(O,a)]
(32) (33)
• [ : ~ 3 s i . Z . O c o ~ ~ 3 ( o - 1)+ ~co.~p~o~i~~3(o- 1)] (34) I24(0, ~) = Bl,~(~)~e[JI(O,~)- iJR(O,~)] • [)~1)~3sin/3~0sinA1A3(0 - 1)-/3,~ cos /3,~0cos A1A3(0 - 1)]
(35)
The functions JR(0, ~) and Ji(0, ~) a r e defined in Eq.(3.3-21) by the functions Jr(O, ~) and Ji(O, ~). In turn, Jr(O, ~) and Ji(O, ~) are defined in Eq.(3.317) by Jl(0, ~) to Js(O, ~) of Eq.(6.3-75). We need mainly lengthy but straightforward substitutions to express JR(x, 0) and JI(~, 0) by J1 (0, ~ to Js(O, ~). The only problems are the derivatives OJr/O0 and OJi(O, ~)/00 in Eq.(3.3-21). We obtain with the help of Eq.(3.3-17):
3.4 EVALUATION OF THE ENERGY ~re
oj (o, 00
--
( Oal(O, 00
- ,/2 (0, t~) - 0
oa=(o, )
0----0---- cos
129
N
2~r~ [j~ (0, ~) - OJ2(O ~r N
OJ4(O,~)) -
oo
-
J4(O,,~)
-
o
~
00
sin
27r~r sin
2zr~ [,13(0, to) -- OJ4(O ~)] cos N
00
=
00
- J6 (0, t~) - 0
27r~____00 N
2~-tr N
(36)
0"---'~- sin
2zr~0 +--~27rt~[Js(0 ~) - OJ6(O,~)]cos N ,
( OJT(O, t~) +
00
OJs(O, tr ) - Js (0, t~) - 0
00
27r~0 cos
N
27r~
N [JT(O,~) - OJs(O, a)]
sin
27ra0 U
(37)
T h e functions -/1 (0, ~) to Js(O, ~) are defined in Eq.(6.3-75), but we must show w h a t the derivatives OJ1 (0, ~)/00 to OJs(O, ~)/00 mean. We use once more the equivalence of the sum of a step function with the step width 1 and an integral over the step function. The functions g l (n, ~) to Ks(n, t~) in Eq.(6.3-75) are step functions with a step width 1 for n = 0, 1, ... and an unspecified value of ~. Using the definition of Jj(O, ~) in Eq.(6.3-75) we may write:
oJj(o,~) O0
0 0-1
OJ 0
o--g~ Kj(n, ~) = N
n=O
0
gj(On,~)dOn
o
: J OKj(On,oonIg)don--gj(o,g), 0
j--1,
2,...,
8
Using AI~(~) and BI~(~) of Eqs.(25) and (26) we may write 9s in the following form:
(38)
of Eq.(30)
N 9s
= .I[I2~(0, ~) + I22(0, ~) - I23(0, ~) +
I24(0, m)]d0
0 N-1 -- E [121(0'/g) + I22(0'/g) -- I23(0'/'i;) + /24(0' ~)] 0=0
(39)
130
3
INHOMOGENEOUS
DIFFERENCE
EQUATION
We shall return to 9(2~(~) later when we need it for plotting. Here we turn to U3~(n) of EQ.(5). As before the index n is written after the integration over ~ to indicate that the summation over n in Eq.(3.3-26) is not carried out' N
L2
N
(40) 0
0
With the help of Eq.(3.3-26) we get: N
2a~]oL2/cA t = 9Q,~(n) = ~
N
9~e 0
N
I ~ei (0~0v)
~0--~
\ 0~ ~--~d(
--~-~-~d(
dO
(41)
0
(2~'~;IN) {1 - exp[-(A~ - A~)I12N]}
....
A2 _ A2 + (27m/N) 2
0
x 2A2:~e[Ji(0, n) -iJ~(O,n)]sinA~A30 cos
-[Jr(9, n)+iJi(O,~)]sinA1A3} sinfl~9
(42)
In order to integrate Eq.(42) over 0 w e n e e d to evaluate numerically the integrals over two terms: I31(0, n) = Al~(n)9~e{2A2[Ji(9, a) - iJr(O,a)]} sin A1A39
(43)
I32(0, n ) = BI~(~)[A 2 + (2rn/N)2]{9~e[Ji(O,n)- iJr(O,n)]cosA1A3 -
9~e[Jr(O,n) + iJi(O, n)] sin A1A3 } sin fl~O (44)
The functions Jr(O, n) and Ji(9, n) are defined in Eq.(3.3-17). No new problems are encountered as with Eqs.(32)-(35). We may write 9(3~(~) of Eq.(41) in the following form"
PN
N-1
(45) .J
0
8=0
In analogy to Uc in Eqs.(2.5-16) and (2.5-17) we write the energy [T,~c of Eq.(4) in the following normalized form:
3.4 EVALUATION OF THE ENERGY Uc
131
2c~L2~o20N- 2aL2~o20N2 = N2 U~r = 2 a L 2 ~ ~
2c~L2~2~ ' ~
cT =
(46) 9~=~(n)
(47)
cT
2aL2 ~2~ ~K~ ( n)
(48)
m2or At) 2 ~'~c~(t~) : ~-~2~(~)"~- ~-~3~(n) -~-
t~2
~}~I~(K;)
(49)
A comparison of Eqs.(48) and (49) with Eqs.(2.6-2) and (2.5-18) shows a close similarity. In order to produce plots of 9Q~(n), ~2~(n), and ~3~(n) according to Eqs.(28), (39), and (45) we must decide on a useful choice of N , A1, )~2, ~3, r and Aim. The easiest parameter to decide on is N. For N = 100 our computer required about 5 hours to produce a set of plots for ~ l ~ ( n ) , 9~2~(n), and 9~3~ (n). An increase to N = 200 was estimated to require about 40 hours, which was too much. For ~2 we obtain from Eq.(2.3-2) the following approximate relation between ~2 and )h:
moc
At
A2 " AleAmo z = ~/moc2 =>> '~1
(50)
One may use A1 = 0.1 and A2 = 10, 20, 40. For Aa we need large values due to Eq.(2.4-13). We choose A3 = 1000, 2000, 10000. Due to Eqs.(16) and (17) we choose r = +nA1A3 and A i m -" i n A 1 with n in the range from 1 to 100. We can choose larger or smaller values of n but we cannot choose n = 0. With this background we use r = 10A1A3, A i m -- 10A1, A1 -- 0.1, A2 = 10, A3 = 1000 for Figs.3.4-1 to 3.4-3 that show 9Q~(t~), 9/2~(n), and
~3~(~). For the second set of plots in Figs.3.4-4 to 3.4-6 we choose A2 = 20 and A3 = 2000 but leave all other parameters unchanged. The third set of plots in Figs.3.4-7 to 3.4-9 shows A2 = 40 and A3 = 10000. The values r -- 10)~1~3, Aim = 10)~1, and A 1 : 0.1 remain unchanged. At this stage we cannot yet recognize common features of Fig.3.4-1 to Fig.3.4-9. A better resolution than N - 100 might help but it runs into the problem of computer time. We have found here eigenvalues of a difference equation that depend in a wide range only on the structure of the difference equation but not on the numerical values of the parameters. One will suspect that this is a result of
132
3 INHOMOGENEOUS DIFFERENCE EQUATION -40
-20
0
20
40
i0
5 ~d -~-
===, 9149
0
~ 9
o 9149
==========_.* ,
*** 9
o~
==
-5
-i0 -40
-20
0
20
40
FIG.3.4-1. Plot of 9Q~(~) according to Eq.(28) for N -
100, r
-- 10~1)~3, Aim --
IOA1, A1 = 0.I, A2 = i0, Aa = I000. -40 ,
40000
T
-20 ,
20000
0 ,
20 ,
9
::
40 .
o
*'"
,,
~
9 9
.o 9
-20000
9
.0.
9
-40000
-'~o
-~o
o
fo
~'o
F10.3.4-2. Plot of 9~2~(~) according to Eq.(39) for N = I00, r fOAl,
A1
--0.i,
A2 = I0,
A3
= 10AIA3, A~m =
-- i 0 0 0 .
-40
-20
0
20
40
i00
I
50 o
Be
~
lw00~
~
00000
Q0006
-so 9
-i00 -150
!
l
2o
,o
FIG.3.4-3. Plot of 9~3~(~) according to Eq.(45) for N 10AI, A1 : 0.i, )~2 - - 1 0 , ~ 3 --- I 0 0 0 .
100, r
-'~o
-~o
~
~
K; - - - - *
- 10A1A3, Aim -
3 . 4 EVALUATION OF THE ENERGY Uc -40
-20
1
0
,
133
20
,
4O
200
l
lOO
,o
: 9149
,,*
-~
0
9
9
~176149
,.
,
,"
-I00
,0,
9
9
-200
, -20
-40
0i
20
40
FIO.3.4-4. P l o t of 9s according to Eq.(28) for N = 100, Ca~ = 10AIA3, Aar, -1OAt, A1 -0.i, A2 = 20, A3 = 2000. -40 ,
3x I0
-20
0
,
9
.
20
40
,
o
2x 10 ~.~ l x 10
~ 9- - , ~ o
,---, 9
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
,
- I x 10 6 - 2 x 10 6
- 3 x 10 6
-"4 0
- 2'0
0~
20~
4'0
FIG.3.4-5. P l o t of 9s according to Eq.(39) for N - 100, r 10A~, A1 = 0.1, A2 = 20, A3 = 2000. -40 400
I
200
~
o
-20
,
,
0 .
20 ,
,
- 10A~A3, Aim -
40 9
9
-200 -400
.
9
,
,
|
0
20
40
FIc.3.4-6. P l o t of 9s according to Eq.(45) for N 1OAt, AI = 0.I, A2 = 20, A3 = 2000.
100, r
-40
9
-20
.
N ----,
= 10A1A3, A ] m =
134
3 INHOMOGENEOUS DIFFERENCE EQUATION
0.2
-40
-20
0
20
40
.
.
.
.
.
T 0.I ~
o
~-0
_ 9
9 . . . . . . . . . . .
00
9 9149149
9
9
1
.,,
.,,
90 9
_
~
9
.
.
. .
.
9
.
-0.2 -0.3
i
,
-40
9 n,
n
-20
n'
0
9
20
FI0.3.4-7. P l o t of 9 Q ~ ( ~ ) according to Eq.(28) for N IOA1, A1 = 0.1, A2 = 40, A3 = 10000. -40
-20
600000
0
100, r
20
9
'~" 2 0 0 0 0 0
= 10A1A3, Aim --
40 ,
9
I 400000
l
40
,
.9
9
.
,,.
o ,
-200000
9
-400000 -
6
0
0
0
0
0
2o K,
-'--->
FI0.3.4-8. P l o t of 9~2~(~) according to Eq.(39) for N -- 100, r 1OAt, AI = 0.i, A2 = 40, As = i0000. -40
-20
,
I
0
,
20
,
2
40
,
9
4
-- 10A1A3, Aim --
,
9
o9
9
~o
9
9
v
o ........ ..
.- .................... -
.,.,..--
-2 -4
-40
- 0
0
20
40
F I c . 3 . 4 - 9 . P l o t of 9~3~(~) according to Eq.(45) for N = 100, r IOA1, A1 = 0.1, A2 = 40, A3 = 10000.
= 10At),z, Axm =
3.5 PLOTS FOR SECOND-ORDER APPROXIMATION
135
replacing an infinite interval by an arbitrarily large but finite interval. For instance, sinusoidal functions that fit into a finite interval have the form sin 2~nO or cos 2~n0 with n = 1, 2, ... , while there is no restriction on n in an open or half open interval. A restriction of eigenvalues is of great interest in physics, but it is too early to tell whether this particular restriction leads to results of practical interest. We shall explore the matter some more in Section 3.5 with additional plots but we refrain from drawing conclusions at this time. 3.5 PLOTS FOR SECOND-ORDER APPROXIMATION We are presenting 18 more plots to show the effect of variations of r Aim, A1, ,k2, and A3 on 9{1~(~) to 9{3~(~). The parameter N - 100 remains unchanged from Figs.3.4-1 to 3.4-9. Figures 3.4-1 to 3.4-3 are shown with the electric potential r reversed from +10A1A3 to - 1 0 A l ~ 3 in Figs.3.5-1 to 3.5-3. We obtain essentially the amplitude reversed plots, but the vertical scales of Figs.3.4-2 and 3.5-2 are also changed. In Figs.3.5-4 to 3.5-6 we see the plots of Figs.3.4-1 to 3.4-3 with the magnetic potential Aim reversed from +10A1 to -10A1. We obtain essentially the same plots as in Figs.3.4-1 to 3.4-3. Again, the vertical scales of Figs.3.4-2 and 3.5-5 are not the same. A closer inspection reveals other differences between the two plots. In Figs.3.5-7 to 3.5-9 both r and Aim have a reversed amplitude compared with Figs.3.4-1 to 3.4-3. The plots of Figs.3.5-7 to 3.5-9 appear at first glance to be amplitude reversed plots of Figs.3.4-1 to 3.4-3 but a closer inspection of Fig.3.5-8 shows differences for + n = 1 5 , . . . , 20. In Figs.3.5-10 to 3.5-12 the parameters ~1 = 0.05 and ~3 --- 10000 are changed compared with Figs.3.4-1 to 3.4-3. The plots are changed more drastically than by the changes of r and Aim. In Figs.3.5-13 to 3.5-15 the parameter A1 is reduced from 0.05 to 0.02 compared with Figs.3.5-10 to 3.5-12. The plots of Figs.3.5-13 and 3.5-15 look more like those of Figs.3.4-1 and 3.4-3 rather than those of Figs.3.5-10 and 3.5-12. If we look at the products A1A3 = 100 in Figs.3.4-1 to 3.4-3, A1A3 = 500 in Figs.3.5-10 to 3.5-12, and A1A3 = 200 in Figs.3.5-13 to 3.5-15 we recognize that the product )~1A3 is important for the similarity of the illustrations, as suggested by Eq. (2.4-13 ). Finally, we show in Figs.3.5-16 to 3.5-18 the plots of Figs.3.4-1 to 3.4-3 with the changes A1 = 0.01 and )~3 = 10000 instead of A1 = 0.1 and A3 = 1000. The product A1A3 = 100 is the same in both cases and the illustrations look very similar. However, a closer inspection shows differences in Figs.3.4-2 and 3.5-17 around + ~ = 20. We must once more ask the reader for patience until Section 4.4 for an explanation why we show so many plots here without deriving any conclusions from them.
136
3 INHOMOGENEOUS DIFFERENCE EQUATION -40
-20
0
20
40
i0 T
5
"J
0
'.,
I | IRBB~OBLDB
9
9 ....
:__==__
_
.o"
0
_---= ....
-5 -I0
-'4o
" -~o
o
2'o
4'o
FIG.3.5-1. Plot of ~ 1 ~ ( ; ' ; ) according to Eq.(3.4-28) for N ---- 100, r Aim = 10A1, A1 = 0 . 1 , A2 = 10, A3 - 1 0 0 0 . -40 ,
-
-20
0
,
20
,
,
.
--
-10A1A3,
--
-10)~1)~3,
=
-I0~3,
40 ,
60000
40000
T
20000
0
9
9
.
~
...;....
-
-20000 -40000 -60000
~o FIG.3.5-2. Aim
~o
o
;o
~o
Plot of 9f2~(~) according to Eq.(3.4-39) for N -- 100, r
-- lOA1, A 1 = 0.i, A2 = I0, A3 -- I000. -40 150
T
,
-20 ,
0 ,
20
40
,
.
i00 50 .o. 9 ~
000.0
%~
,*~
-50 -i00
F~c.3.5-3.
-~o
-~o
~
I
Plot of ~3~(~) according to Eq.(3.4-45) for
Aim = 10A1, A1 = 0 1 , A2 = 10, A3 = 1000
|
~o
~o N
=
I
100, r
3.5 PLOTS FOR SECOND-ORDER APPROXIMATION -40
-20
,
0
,
20
,
137
40
,
,
io
v
T
5 ;;. o,,,,o,,
9 9
~-5
::::::: .... 9
.o
o
""" 9
9
-i0 I
K;
I
i
I
------~
Fic.3.5-4. Plot of ~ 1 ~ ( ~ ) according to Eq.(3.4-28) for N = 100, r A~m = -10A1, A1 = 0.1, A2 = 10, A3 = 1000. -40
-20
,
60000
0
,
20
,
= 10A1A3,
40
,
.
40000 I
20000
"~
o
99 0000 9 Og 9 ,
,"
-20000
gO 9
9
9
~
ODOOoQIo
,
,
.
9
,
",
-40000 -60000
-ko
o K;
Fio.3.5-5. Alm
!
i
20
---'--~
P l o t of 9~2~(~) according to Eq.(3.4-39) for N = 100, r
= 10A~A3,
= -10At, AI = 0.I, A2 = I0, As = i000.
i00
f
-40
-20
0
20
40
5O o
".~
0 9 ~.~vOg U 9 vIVlOo0 O
e.Oto--
50
-I00 -150
-.. 9 ,,..
o'" 9
9
i
20
o
!
20
4o
K, - - - - ~
FIG.3.5-6. Plot of 9~3~(~) according to Eq.(3.4-45) for N = 100, r Aim = -10A1, A1 = 0.1, A2 = 10, A3 = 1000.
= 10A1A3,
138
3 INHOMOGENEOUS DIFFERENCE EQUATION -40
-20
0
20
4O
i0 5
o
,
o
9..
::-'........
9
.
0 9
.. 9
~-5 -i0 i
-io
' -20
;
9
2'0
i
40
FI0.3.5-7. Plot of 9Q~(~) according to Eq.(3.4-28) for N = 100, r Aim = -10A1, A1 = 0.1, A2 = 10, A3 = 1000. -40
-20
,
0
,
20
,
=
-10/~1,~3,
40
,
,
40000 9
T
20000
.
9
9
9
9
.
. o
"--.. 9 9
....
_::__--:
:::__:::
9
..
.
....
,
. 9
9
. 9
.,,
-2oooo -40000
-;,o
-;o
2'o
o
/,;, ---.--)
4o
FIG.3.5-8. Plot of 9C2~(~) according to Eq.(3.4-39) for N = 100, r Aim = - 1 0 A 1 , A1 = 0.1, A2 = 10, A3 = 1000. -4O
-20
0
20
40
,
,
l
,
= -10A1A3,
150
T ~, N
I00 ".o"
50
~
o
o**
-50 -I00
-;,o
-~o
o
~'o
;o
FIG.3.5-9. Plot of 9C3~(~) according to Eq.(3.4-45) for N = 100, r Aim = -10A1, A1 = 0.1, A2 = 10, A3 - 1000.
= -10A1A3,
3.5
PLOTS
FOR
SECOND-ORDER
-40 .
.
.
.
-20 ..
0
.
139
APPROXIMATION 20 ,
40
.
,
1500 i i000 50O ~
N
9
0
9
......
9
9
9
9149
....
::
,
-500 -I000
-- 4 0
&
-- 2 0
III
210
4I0
K; -----~
FIG.3.5-10. Plot of 9Q~(P+) according to Eq.(3.4-28) for N -- 100, r Aim = 10At, A1 - 0 . 0 5 , A2 = 10, A3 = 10000. -40 .
T
-20
.
0
.
20
.
= 10A1A3,
40
2x 10 8 9
9
9
-2x I08
9
. . . . . . . . . __+
9
,, 9
.
o
9
9
-4x I08 I
t
K; -----*
FIG.3.5-11. Plot of 9~2~(P+) according to Eq.(3.4-39) for N = 100, r Aim
= 10A1A3,
-- 10At, AI = 0.05, A2 -- I0, A3 : i0000. -40 ,
-20 ,
0 ,
20
,--
,
40 ,
500
t
o,
-500 -i000 ,,,
FIG.3.5-12. Plot of Aim :
4i
...-.. 0
~-C3~(g)
10A1, A1 : 0.05, A2 :
,
m
-20
0
|
20
,
according to Eq.(3.4-45) for N : 10, A3 :
10000.
l
40
100,
r
----
10]~1]~3,
140
3 INHOMOGENEOUS -40
-20
..
,
DIFFERENCE
0
,
EQUATION
20
,
40
,
,
2O
T
$
0
v
........
o9 9
9 9
9
20
o **,* 9
,
,
,,o
9
-40
~o
~o
o I
/~
-+
20
40 i
F i o . 3 . 5 - 1 3 . P l o t of ~-~I~(E) according to Eq.(3.4-28) for N A~m - 10A1, A1 = 0.02, A2 -- 10, A3 = 10000.
-20
-40 i.
20
-- 10AIA3,
40
5x 10 6 Ix
I
0
100, r
10'
500000 o
-,,.,,,.
.
-500000 -lx
10 6
-I. 5x
106
. . . . . . . .
.,.,,.,.-
9
9
9
~o
o
9
~o
o
~o
.o
Fz0.3.5-14. P l o t of 9C2~(~) according to Eq.(3.4-39) for N -- 100, r Aim -- 10A1, A1 = 0.02, A2 = 10, A3 = 10000. -40
-20
,
,
0
20
40
,
,
,
-
10A~A3,
100 l
9
9
" 9149149149149
N -100
.
.
" "..
9149 .o*'"
9149
9
-200
-;o
-;_o
!
i
,
o
~o
~o
F i e . 3 . 5 - 1 5 . P l o t of ~3,~(~) according to Eq.(3.4-45) for N = 100, r Aim = 10A1, A1 = 0.02, A2 = 10, A3 = 10000.
= 10A1,%,
3.5 PLOTS FOR SECOND-ORDER APPROXIMATION -40 ,
-20
.
0
,
20
,
,
141
40
.
,
I0 5 o
~
= =j.o 9 o Q ~ 1 4 9 1 4 9 ...
.
.
"%
9
[g
9
5
9
-I0
9
9
I
i
! 40
I 20
&II
2[ 0
4 0
H; - - - - - ~
FIG.3.5-16. Plot of 9 Q , ( , ; ) according to Eq.(3.4-28) for N = 100, r Aim = 10A1, .kl = 0.01, A2 = 10, As = 10000. -40
-20
0
20
= 10A~A3,
40
40000 20000
T
o"
9
.
"0
U,
e 9
-20000
-40000 I
-40
-2o
o
/~ ----+
I
2o
40
FIC.3.5-17. Plot of 9(2,(~) according to Eq.(3.4-39) for N = 100, r Aim -- 10A1, A1 ----0.01, A2 = 10, As = 10000.
i00
T
-40
-20
,
0
,
.
20
= 10A~A3,
40
,
50 ~ Q90O 999
00000 9 ~ o.O"
-50 -I00 9
-150
-~o F~G.3.5-18. Aim
9
-~o
i
o
I
20
I
40
K; "----~
Plot of 9-[a,~(~) according to Eq.(3.4-45) for N = 100, r
= 10A1, A1 --0.01, A2 -- I0, A3 = I0000.
= 10A1Aa,
142
3 INHOMOGENEOUS DIFFERENCE EQUATION 4
3 .-....dj
a
4 "-, ~
~ 2
"" 1
"~" 2 "~
'
'
o
(~
+ 89
"'"" "'~
1[
L
!
I
J
~+1
~+2 ~ g:+l ~+2 ~+3 ~ ----.-4 ~ ------~ FIG.3.5-19. First order (a) and second order (b) polynomial approximation of a physically not defined amplitude 2/0 of 9{(~) = 2 / ( ~ - ~) at ~ = ~ . The plots in Figs.3.4-1 to 3.4-9 and 3.5-1 to 3.5-18 assume values -t-C/O = -1-1/0 for certain values of ~. An infinite amplitude is no more observable than an infinite distance or time. According to Section 1.4 an infinite amplitude implies infinite information, which cannot be observed, processed, or transmitted. A result 1/0 must be treated as physically not defined just as a result 0/0 must be treated as mathematically not defined. Consider the function 9{(~) = 2 / ( ~ - ~) that equals 2/0 for ~ --- ~. We may replace 2/0 by 3 obtained by the first-order polynomial approximation of Fig.3.5-19a, or by 11/3 obtained by the second-order approximation of Fig.3.5-19b. Going to higher order approximations one gets larger and larger results for 9{(~). But for ~ :> 0 we have N / 2 - 1 values ~ = 1, 2, . . . , N / 2 - 1. Let k of them have the value 1/0. A polynomial expansion of order N / 2 - 1 - k will yield k finite values instead of 1/0 from the N / 2 - 1 - k physically defined values. We see here that a finite number N = T / A t of space intervals Ax or time intervals At produces a finite amplitude g{(~,) if the physically not defined value t / 0 is made defined by a polynomial approximation. O b t a i n i n g the equivalent of the constants 3 and 11/3 in Figs.3.5-19a, b for polynomials of high order is not difficult for a computer. Hence, we have found a very general and powerful method to avoid infinite amplitudes in physics if the calculus of finite differences is used. We do not want to discuss it any further until information about the limitations and drawbacks of the method becomes available.
ADVANCES IN IMAGING AND ELECTRON PHYSICS, VOL. 137
4 Klein-Gordon
4.1
Difference
Equation
EVALUATION OF THE DIFFERENCE
for Small
EQUATION
FOR
Distances
Ax ~
h/moc
In Chapter 3 we solved a second-order difference equation based on Eq.(3.110) that held for Ax large compared with the Compton wavelength h/moc. The physically more interesting but mathematically more difficult case is Ax << h/moc. For explanation of this statement consider elementary particles. Within the differential theory they are called point-like. Since mathematics is a science of the thinkable but physics is a science of the observable one must be careful if basic concepts of the one science are defined for the other science. The term "point-like" rather than "point" gets one verbally around this difficulty without overcoming the problem. Let us look at elementary particles from the standpoint of observation. We can distinguish two marks or points x and x + Ax where Ax is the smallest distance resolvable by the available instruments. We call Ax the resolution of our observation. A particle smaller than Ax cannot be resolved, which means we cannot claim about its spatial features anything more than that it is smaller than Ax. This is a long way from claiming it to be like a mathematical point since Ax can be divided into nondenumerably many intervals dx t h a t have the extension of a mathematical point. If we search for mathematical methods that are based on an arbitrarily small but finite distance Ax rather than an infinitesimal distance dx we find the calculus of finite differences. In essence by starting with a finite possible resolution we state the physical situation first and then look for a matching mathematical method. Alternately, if we choose differential calculus first we must then state the physical situation in matching terms. In the one case one uses mathematics as a tool for physics while in the other case one makes physics a branch of mathematics. We have seen in Section 3.1 that the difference equation (3.1-3) becomes completely different for Ax :>'>h/moc and Ax << h/moc as shown by Eqs.(3.110), (3.1-11) and (3.1-17), (3.1-18). For large values of Ax one obtains a difference equation of second order just like the differential theory yields a differential equation of second order. But for small values of Ax one obtains a difference equation of fourth order of the spatial variable. In Section 3.1 we derived Eq.(3.1-32) from Eq.(3.1-3) for Ax > h/moc. Here we shall derive the corresponding equation for Ax << h/moc. 143 ISSN 1076-5670/05 DOI: 10.t016/S1076-5670(05)37004-2
Copyright 2005, Elsevier Inc. All rights reserved.
144
4
K L E I N - G O R D O N D I F F E R E N C E E Q U A T I O N F O R SMALL DISTANCES
For the first line of Eq.(3.1-3), written as a homogeneous equation, we obtain again Eq.(3.1-6):
([#1 (~ + 1, {9)- 2ql (r {9)+ #1 ( ~ - 1, {9)]- [ ~ (~, {9+ 1)-2~1/1 (~, {9)+ ~I/1 ( ~ ,{9-)] 1 -iA1{[~1(~ + 1,{9)- ~ 1 ( ~ - 1,{9)] + A3[~1(r {9+ 1 ) - ~1(r {9- 1)1} h2 :0
~1 = ~xl~,
A1, A2, A3 see Eq.(2.3-2)
(1)
For the second line of Eq.(3.1-3) we obtain with the help of Eqs.(3.1-11) and (6.2-1) for small values of Ax = cAt and Ar = 1:
Ax<
( h
e
A
~ iAx ~
\ eAmos)
• [1 9
h2
2acAe(~,0)(
( /Xx)2
e
h2
h
t, moc/Xz
2~cA~(r
(Ax) 2
2
1
2mo2C2 i~xX~
)(2 /~ Ar
h
eAmox
)4 ieAXAmox
t~o(~,{9)
~o(~,0)
h
[~o(r + 2,{9)- 4#o(r + 1, {9)
\ mocAx
e
~O=~x0~
+ 6Vo(r e) - 4Vo(r - 1, e) + Vo(r - 2, e)]
+2
ieAxAmo~ h
- 6
[~o(r + 2, {9)- 2~o(~ + 1, {9) + 2 ~ o ( r
1, {9)- ~ o ( ~ - 2, {9)]
(ieAxAmox) 2 h
[ ~ o ( ~ + 1, {9) - 2 ~ o ( ~ , {9) + ~ o ( ~ - 1, {9)]
+ 2(ieAxAm~ h
3
[*0(r + 1, {9) - *0(r - 1, {9)]
_ ( ieAxAmo~ 4
(2)
We turn to the terms in lines 3 and 4 of Eq.(3.1-3) multiplied by Lcx11. They are shown for Ax << h/moc in Eq.(3.1-18). This equation is worked out in Section 6.2 and Eq.(6.2-3) is obtained. We choose A{9 = 1, A~ = 1 in Eq. (6.2-3):
4.1 EVALUATION OF THE DIFFERENCE EQUATION FOR Ax <<
Ax<
-~ c
iAt ~0 ~ eCeo
. 2Ze2(
h
- ac
ncx11 + Lcxll
iAt ~0 ~ eCeo
)4
(~
mocA-----~ (A~0xAm0y- A~0~Am0~)
~
~
( Xc z~
+
Vo(r
ieAtCeo) h
ieAxAm~ h ) ~o(r
(deozdmoy-d~o, mmoz)5
OLC mocAx
- ~o(r + 2,0-
1 ) ] - 2[~o(r + 1,o + 1) - Vo(r + 1 , o -
+ 2[~o ( r 1, O+ 1) - ~o ( ( - 1, O-1)]-[~o (r -3
ieAxAmo~ h
O+ 1 ) - ~ o ( r
(ieAxAm~ h
1)l
{[~o(( + 1,0 + 1) - ~o(( + 1,0 -
~
-[~o ((-1, 0+1)-~o ((-1,0-1)]
+
ieAtCeo [1 [~o(~ + h
2
- 3
0+1)-~o(,+1
}-~ieA~Am~ 3[~o (r
'
0-1)]} 0-1)]
0+1)-~o (~, 0-1)]]
2 0 ) - 2~0(~ + 1, 0) + 2 ~ 0 ( ~ - 1 0 ) - ~ o ( ~ - 2 0)] ' ' '
ieAxAmoz
+3
'
1)]
O- 1)]}
-2[~o (r O+ 1)-~o(r O- 1)] + [~o(r 1, O+ 1 ) - ~ o (r +3
145
~0=~x0x
•
--2Z2
h/moc
h
[~I'o(r + 1, O) - 2~I'o(~, O) + ~I'o (r - 1, 0)]
(ieAxAm~ h
2
1,0)-~o(~-1
0)]
_ ( ieAxAmox 3
(3)
The contributions of the terms multiplied with Lcx21, Lcx31, Lcx41, Lcxhl in lines 3 and 4 of Eq.(3.1-3) are worked out in Eqs.(3.1-19), (3.1-27), (3.1-28), and (3.1-31). The same equations hold for large and small values of Ax. The result is shown, multiplied by -(Ax)2/h 2, in lines 9 and 10 of Eq.(3.1-32). The last two lines of Eq.(3.1-3) are worked out in Eq.(3.1-12). Again, the same equation holds for any value of Ax. Lines 11 to 16 in Eq.(3.1-32) show the result multiplied by -(Ax)2/h 2. We now have all the mathematics to permit us to rewrite Eq.(3.1-32) for small values of Ax. First we multiply Eqs.(1), (2), and (3) by -(Ax)2/h 2.
146
4 KLEIN-GORDON DIFFERENCE EQUATION FOR SMALL DISTANCES
This yields lines 1 to 17 of the following equation 1. Then we add lines 9 to 16 of Eq.(3.1-32) as the new lines 18 to 25: Ax << Ac = h/moc [~1((+1,0)-2~1(~,0)+~1((-1,0)]-[~1(r
1
-iAI{[~I((+I,0)-~I(~-I,0)]+Aa[~Pl(~,O+I)-~I((,O-1)]}-A~I(r
2
2/~cAe(~, 0) (
h
e
mocAx
=-
)2[
[r162 + 2, O)- 4r162 + 1, O)
3
+ 6~o(~, O) - 4r162 - 1, O) + r162 - 2, 0)]
+ 2 ieAxAmox h
[~I'o(( + 2, O ) - 2 9 0 ( ( + 1, O)+ 2~I'o((- 1, O) - r
_ 6(ieAxAm~
2
h
+2(ieAxAm~ h
3
[r
+ 1, O)- 2r162 O)+ ~ o ( r
[~o(( + 1, O) - ~I'o(( - 1, 0)] -
(
2Z2o3 mocA-------~ h (AeozAmoy + olm2c - ~o(r + 2 , 0 -
-
AeoyAmoz) ~
2, 0)] 5
1,0)1
6
( ieAxAmox ) 4 ] h
~I'o((, O)
[~o(~ + 2 0 + 1)
1 ) ] - 2[~o(r + 1,0 + 1 ) - ~o(r + 1,0 - 1)]
+ 2 [ ~ o ( ( - 1,0+ 1)-@o(~- 1, 0 - 1 ) ] - [@o(r
0+ 1 ) - ~ o ( ( - 2 , 0-1)]}
- 3 ieAxAmo~ { [~o(( + 1 0 + 1) - @o(r + 1 0 - 1)] h
'
2
7 8 9
lO 11
'
-2[~o((, 0+ 1)-~o((, 0-1)]+ [~o((- 1, 0+ 1 ) - ~ o ( ( - 1,0-1)]}
+ 3(ieAxAm~
4
,0+1)-~o(~+1,0-1)]
12 13
- [ ~ o ( ( - 1 , 0+1)-@o ((-1, 0-1)] }-~ieAx-~Am~ 3[~o(~, 0+1)-@o((, 0-1)]] 14 +
ieAtCeo [ 1 h
~[~o(( + 2, O)- 2~o(( + 1, O)+ 2~I'o((- 1 , 0 ) - ~I'o( ( - 2, 0)] -
+3
(
3
ieAxAmo~ h
)ienxnm~
[~I'o(( + 1, O)
-
2~I'o((, O) + ~I'o((
/ ~21[~o((+1 , 0 ) - ~ o ( ( - 1 , 0)]-
-
1,0)]
~6
(ienxnmox) 3~o(( O)]} ~
15
,
17
lEquation (1) lines 1, 2; Eq.(2) lines 3-7; Eq.(3) lines 8-17; Eq.(3.1-27) line 18; Eq.(3.131) line 19; Eq.(3.1-12) lines 20-25.
4.2 EVALUATION OF EQ.(4.1-4)
.m0(Ax) 2 ah, At
.m0(Ax) 2 2a~At
Vcx31(r 0) (2 [9o(r
- 9o(r
147 +
ieAt ce090(~, O))
Ccx51 1 [9o(~, 0 + 1) - 9o(~, 0 - 1)] + --'-~r
- - e ( - ~ / 2 [Amlx (<' ~9)( h2iAx [9o(<+1 , 0 ) - 9 o ( ~ - 1
0)
19
, O)]-eAmox9o(~,O))
h + 2iz~x [9o(r + 1, O)Amlx(r -{- 1, 0) - 9o(~ - 1, O)Amlx(~ - 1,0)] -
~2 h
+ 2iAt[9o(r
r162
-
18
20 21
2eAmoxAmlx(~, 0)9o(~, O) 22
2 i z x t [ ~ ~ 1 6 2 1 7 6 1 7 6 1 6 2 1 7 6 1 6 2 1 7 6 1 7 6 1 7 6 1236 2 1 7 6
+ 1)r
+ 1)
-
90(r 0 - 1)r162
-[- 2eCe0r
0
-
-
1)]
(~, 0)90(~, 0)1 l
24
25
(4)
A comparison of this equation with Eq.(3.1-32) shows that the transition from Ax >> Ac to Ax << Ac increases the number of lines in the equation from 16 to 25.
4.2 EVALUATION OF EQ.(4.1-4) Before attempting to find solutions of Eq.(4.1-4) we must radically simplify it. A first step is to make as many as possible of the constant components of Ae, Am, Ce, and Cm in Eq.(4.1-4) zero. As in Section 3.2 one must retain the constants Am0x and r to keep Eq.(2.3-2) valid. In addition, the magnitude A~(~,O) of the electric vector potential Ae cannot be ignored now as will be seen presently. Only the following components can be chosen to be zero:
Amoy -- Amoz -- 0,
Cm0 "-- 0
(1)
Lines 1 and 2 of Eq.(4.1-4) remain unchanged. They are equal to the left side of Eq.(2.3-2) except that 9o is replaced by 91. In line 3 of Eq.(4.1-4) we see the factor Ae((, 0) = [Ae(r 0)! that made us discard lines 3 and 4 of Eq. (3.1-32) in the third paragraph" of Section 3.2. This is now different because there is the additional term (h/mocAx) 2 that becomes very large for small values of Ax. Hence, we retain lines 3 to 7 of Eq.(4.1-4). Lines 8-17 of Eq.(4.1-1) are multiplied by the factor AeoxAmoy-AeoyAmo~, which is zero according to Eq.(1). These lines are discarded. Line 18 is equal to line 9 in Eq.(3.1-32) and is discarded. Line 19 equals line 10 in Eq.(3.1-32); it becomes very small for small values of Ax and At = Ax/c.
t48
4 KLEIN-GORDON DIFFERENCE EQUATION FOR SMALL DISTANCES
Lines 20-25 in Eq.(4.1-4) are all multiplied by Ax if we substitute cAt = Ax. These lines become very small for small values of Ax unless at least one of the potentials Aml~(~, 0) or r (~, 0) becomes very large for some value of or 0. We exclude here such large potentials in order to demonstrate the effect of lines 3 to 7 of Eq. (4.1-4). Equation (4.1-4) is reduced more than Eq.(3.1-32) was to yield Eqs.(3.2-3) and (3.2-4). We write ~ to distinguish it from ~ in Eq. (3.2-3): [~1 (~ -Jr 1, 0) -- 2~1 (~, 0) + ~1 (~ -- 1, 0 ) ] - [ ~ (~, 0 + 1) - 2 ~ (~, 0) + ~ (r 0 - 1)]
--i)~1 { [~1 (~ 4-1, 0)- ~1 (~- 1,0)] + A3[~1(~, 0 + 1)- ~1 (~, 0-1)] } -
)k2~l (~,
= a2(r
G2(r
moo/X~
=
0)
(2)
[%(r + 2, o ) - 4r162 + 1, o)
+ 6~0((, 0) - 4~0(~ - 1, 0) + ~0(~ - 2, 0)] +
eAxAmo~ [~Po(( + 2i h
+ 6
2, 0 ) - 2~o(( + 1 , 0 ) + 2~Po((- 1, 0) - ~ o ( ( - 2, 0)]
( eAxAmox ) 2 n
[~o(( + 1, 0) - 2~o(r 0) + ~o(r - 1,0)]
-- 2i/eAX~m0x)3 [~0 (~ ~Let us compare these two equations with the corresponding Eqs.(3.2-3) and (3.2-4) for Ax << Ac = h/moc. The homogeneous Eqs.(3.2-3) and (2) are equal; the only difference is in the inhomogeneous terms G1(~,0) and G2(~,0). A conspicuous difference is the magnitude Ae(~, 0) of the electric vector potential Ae(~,0) that is caused by magnetic (dipole) current densities according to Eq.(1.1-19). It would occur in Eq.(3.2-4) too but there we were able to make it a small contribution that was ignored. The factor (h/m0cAx) 2 makes this impossible now. But the same factor makes it possible to ignore now all the terms of G1 (~, 0) in Eq.(3.2-4) if we exclude extreme values of the magnitudes Amlx, Am0x, r and r of magnetic and electric potentials. This result implies that the electric vector potential Ae introduced by the modified Maxwell equations is of great importance for differences Ax << )~c but of much less importance for Ax >> Ac. Vice versa one might say the usual Maxwell equations may or may not work for Ax >> Ac but they cannot work for Ax << Ac. We substitute Ax = cT/N and h/moc = 2rAc in Eq.(3). Note that Ae(r 0) as magnitude of Ae(r 0) is never negative.
4.2 EVALUATION OF EQ.(4.1-4)
Ac = Ax
h
_-
mocAx
Ac cT/N
149
= Ac N cT
2AcA~(~ ' 0 ) ( h )2 = 1 ACAe(~,0 ) .. e mocAx 27r2
(4)
(Ac) 2N 2
= p~(r e ) N 2 = pN(r e)
eAxAmox h
ecTAmo~ hN
(5)
(6)
1 AcA~(~, 0) ( X c ) 2 ~
(7)
p~ (r o) = 2~2
Equation (3) is rewritten:
a2(r 0) = - p N ( r 0)[(1 + 2 i ~ ) v 0 ( r -
+ 2,0) (4 + 4iA1 6A~ + 2iA~)~0(r + 1,0) + (6 12A21 A41)~0(r 0) - ( 4 - 4iA1 - 6 A 2 - 2iA~)~0(~- 1,0) + (1 - 2iA1)~0(r - 2,0)] -
-
-
(s)
A general solution of the homogeneous Eq.(2) plus a particular solution of the inhomogeneous equation is needed. The boundary condition of Eq.(3.2-5) is used once more:
~1 (0, 0) = ~ooS(0)(1 - e -w) = 0 = '~oo(1 -
for 0 < 0
~-~o)
for 0 >_ 0
(9)
With the initial conditions of Eqs.(3.2-6) and (3.2-7) we again obtain Eq.(3.2-8) but we must write ~(~, 0) rather than fi(4, 0) since these functions will become different. ~1 (~, 0) = ~oo[(1 - e-W)F({) + "g({, 0)]
(10)
The function F(~) and the constant c are the same as in Eqs.(3.2-9) and (3.2-10). The boundary and initial conditions for ~(~,0) follows from Eqs. (3.2-11) to (3.2-13) by the substitution of ~ for ~:
~(0,0) = 0 ~(r 0 ) = 0 ~(~, 1) - ~ ( r 0) = - ( 1 -
e-~)F(r
0>_0 ~> 0
(11) (12)
r > 0
(13)
150
4 KLEIN-GORDON DIFFERENCE EQUATION FOR SMALL DISTANCES
Substitution of ~(C, 0) of Eq.(10) into Eq.(2) yields the same equation but @1 is replaced by ~: [fi(~ + 1,0) - 2~(~, O) + ~2(~- 1, 0 ) ] - [fi(C, 0 + 1) + 2fi(~', O) + 'u(C, 0 - 1)1 -iAl{[~(~
+ 1,0) - ~(~ - 1,0)] + A3[~(r 0 + 1) - f i ( r
1)]} = G2(~,
~)
(14)
The solution of Eq.(14) is written in the following form:
e(r e) = ~(C, e) + v(C, e)
(15) where u(C, 0) stands for the solution of the homogeneous equation and ~(C,0) for a particular solution of the inhomogeneous equation. The homogeneous solution was expressed in terms of Bernoulli's product method in Eqs.(3.2-16) to (3.2-20). We use here the coefficients A~0 to A53 and A60 to A63:
r
= e~'r
~.(r 6) = r162162
(16)
e~v-r + A51 e-~v'~r = e~X'r
cos ~ C + iA53 sin ~o~r • (17)
~p,r = 27ra/N,
r
,r = O, •
= e i:~3~ (ABle ivy~ + A6oe -~~176 = eiXlx3~
(18)
+ iA63sin~,~O)
Consider next the inhomogeneous solution V(~, 0). We go to Eq.(3.2-21) but write ~, S, and T instead of ~, S, and T:
~(r
=
<,~o ( ~5(0)ei~1r cos 27r,r162 E g + T(0)ei'XlCsin 2 N ~ ) N/2-1
=
~
[~(0)r162 +
~(01r162
(19)
,~----N/2+I
Substitution of ~(~,0) into Eq.(2) brings: N/2-1 ~=-N/2+1
[& (0) [r (r
1)- 2r162 (r + r (r 1)] + 7"~(0) [r (r + 1)- 2r (r + r (r
1)]
4.2 EVALUATION OF EQ.(4.1-4)
151
- r (r [S~ (0+ 1) - 2S~ (0) + S~ (0-1)] - r (r [T~ (0 + 1) - 2T~ (0) + 2P~(0-1)] - iA1 ( S ~ ( 0 ) [ r 1 6 2
+ ~a{r162162
+ 1) - r
+ ~) - ~ ( 0 -
- 1)1 + 2 ~ ( 0 ) [ r
~)1 + r
+ 1) - Cs(~ - 1)]
+ ~) - r
A~[S~.(0)r162 (~') +
r162162
- ~)1})
= a~(r o) (20)
The substitutions of Eqs.(3.2-23)-(a.2-2S) apply again. Equation (3.2-29) is obtained with S~, T~, and G~(r replaced by S~, T~, and G2(~,0). We only write the equivalent of the reduced Eq.(3.2-30) derivable from Eq.(20)"
-(1+i,kl)~3)S~(0+1)+ ,r
(
)
2(coss163163163
2 S,~(0)
27r~ ] 27r~r - (1 - iAI,X3)S~(0 - 1) + 2i(sin,Xl - ,~1 cos,X1) sin - ~ ,~(0) e ixxr cos N
+
N/2-1
Z
-(~+~x~)~(0+l)+
(
27r~
9(cosa~+a~si~a~)cos--~-~
)
~(0)
t~=-N/2+l
- (1 - i;~3)T,r-
+ 1) - 2i(sin,X~ - )~l cos)h) sin ~27r~ S ~ -_ (0) e ~'x~r sin N = G2(r 0 ) ( 2 1 )
The left side of Eq.(21) is essentially a Fourier series in terms of ~. The right side can be represented as a Fourier series too:
a2(r
N/2-1(
2~r
(~,~(0, ~)e i~r sin Y
2~r~)
~)e i ~ cos N
+ (~r
(22)
,~=-N/2+I
Multiplication with 2N-le -i~1r sin(27rv~/N) or 2N-le -i~1r cos(27ru{/N) and integration over the orthogonality interval 0 < ~ < N yields Gs,~(O,,~) and
Gc~(O, ~): N
Oc,,(o,,~) = ~
a=(r
o)~-';',~
cos
de
for u = a
(23)
d~
for u = a
(24)
0 N
d~,~(O, ~) = -~
a2(~, 0)e -i'x~r sin 2 0
152
4 KLEIN-GORDON DIFFERENCE EQUATION FOR SMALL DISTANCES
Equations (21) and (22) yield for every component a of the two sums the following two equations:
~~ -~)g~(o)
-(1+iAIA3)g~(0+1)+
2(cosAl+AlsinA1)cos--~
27r~ - ( 1 - iA1A3)g~(0 - 1)-2i(sin AI-A1 cos A1) sin --~T,~(O)=Oc,~(O,~) - ( I + i A 1 A 3 ) T-~ ( 0 + I ) +
(2~)
( 2 ( c o s A l + A l s i n A 1 ) c o s -2- ~~ _ ~ ) r
~~
- ( 1 - iA~A3)T~(0- 1)-2i(sinA1-AlCOSA1)sin N
~(0)=Gs~(0 ~)
(26)
We make the same substitutions and simplifications as made in Section 3.2 by Eqs.(3.2-36) to (3.2-40)' )k I
-
-
ecTAmo~/Nli,
cos ~1 = 1 + O(At) 2, /~1 sin A~ = O(At) 2, T = N A t
A2 = ( T / N l i ) 2 ( m 2 c 4 - eCe20+ e 2C2Amox) 2 - O(At) 2
(27)
A2A23 _ ( e T C e o / N h ) 2 = O(At) 2
(28) (29)
1 + iA1A3 " e ix~x3, 1 - iA1A3 " e-i)'~)~3, ' 27r~ 2~r~ 2(cos A~ + A~ sin A~) cos --if- - A2 " 2 cos --K2(sin/~1
-
-
(30)
27r~ )kl COS ~1) sin ~ -----O ( n t ) 2
(31)
We may solve Eqs.(25) and (26) just as Eqs. (3. 2-34) and (3.2-35) were solved. The equations obtained are like Eqs.(3.2-41) and (3.2-42) except that S, T, and U are replaced by S, T, and (~:
~'~,~#~(o + 2) - 4co~ --~2~~e~a~Xzs~(o+ - 4 cos -27r~e-~X~3S~(0~
1 ) + 2(2 + c o s - ~ )
S~(O)
1) + e - 2 i ~ 3 S ~ ( 0 - 2)
27rtr
= -e'a~a3d~,~(O + 1,~;) + 2cos --~-(~,~(0, ~) - e-'a~a~(~,~(O - 1,~)
d~~(0
(32)
+ 2) - 4co~ --K-2~d~'~(~ + 1)+ 2(2 + cos ~-~/~(0) -
27rt~ 4cos ---~e-':~a~:s
-
1) + e-2'a~:~r
27rt~ = -eiX~a3d~(0 + 1,~) + 2 c o s - ~ ( ~ ( 0 ,
- 2)
~) - e - i ~ a z d ~ ( 0 - 1,~,)
(33)
4.2 EVALUATION OF EQ.(4.1-4)
153
For the solution of the homogeneous Eqs.(32) and (33) we can follow Section 3.2 from Eq.(3.2-43) # ~ ( o ) -- c~v~ - o
or
~(o)
= a,~v~0
(34)
to (3.2-55) if S, T, c, and d are replaced by S, T, 5, and d. For the solution of the inhomogeneous nqs.(32) and (33) we follow nqs.(3.2-56) and (3.2-57)"
~(o) =
a,~(o)v~
+ e~.(O)v~ + e~(o)ov~ + ~,~(o)ov~. -
0
o 2 + ~ 3 ( o ) o v , ~ 0 + a~4(O)Ov02 ~'~(0) -- d~I(O)vO1 + d~2(O)vx.
(35) (36)
Equations (6.4-11) and (6.4-18) define d,~i(O) and 5~i(0). Substitution of Eqs.(35) and (36) into Eq.(19) yields the particular solution of the inhomogeneous equation (10). The boundary and initial conditions of Eqs.(11) to (13) must still be satisfied. We obtain the following boundary and initial conditions for ~(r 0) from Eqs.(ll) to (13) with the help of Eqs.(2.3-19) to (2.3-21):
9(0,0)=0 9(r 9((~, 1) - ~(~, 0) = 0
for0>0
(37)
fore>0 for r > 0
(38) (39)
The boundary condition of Eq.(37) is satisfied if we discard the first term in Eq.(19)"
V(~, O) =
N/2-1 ~
9 27r~ T~(0)e '~1~ sin
(40)
,~=-N/2+I
The initial condition of Eq.(38) is satisfied for the third and fourth term of T~(0) in Eq_(36) due to the factor 0. The first and second terms have the coefficients d,r for 0 = 0:
v(r 0) --
N/2-1
E Idol(0) + d~2(0)]e ix1 r sin tc=--N/2+l
27r~{ = 0 N
(41)
From Eqs.(36) and (40) we get with Eqs.(3.2-48), (3.2-49), (39), and (38) the relation
154
4
KLEIN-GORDON DIFFERENCE EQUATION FOR SMALL DISTANCES
N/2-1
9(~, 1) =
E
{[&1(1) + &3(1)lv~1 + [&2(1)+ &4(1)]v~2}
~=-N/2+1 x e i'xlr sin 2~'n~ = 0 N N/2-1
~ 27rn [d~l(1) + &2(1) + &3(1) + &4(1)] cos N
__ e-iA1A3
~=-N/2+1
+ i[d~1(1) - &2(1) + d~3(1) - d~a(1)] sin ~ - ~ ) e ~'x'r sin 2~n~ N = 0
(42)
Since Eqs.(41) and (42) equal Eqs.(3.2-62) and (3.2-63) if ~) and d~i are replaced by ~ and d~i we may skip from Eq.(3.2-63) to Eq.(3.2-72) and replace Eqs.(3.2-72) to (3.2-79) by the following equations: &2 = - & l
d~1(1)
n0~l(0 ) -[- 0~1
=
=
(43)
b~1(1____.~)+ D~o(1)
d~3(1) = Ad~3(0)+ d~3 -- /)~3(1) ~
-
d~3 = -d~l -
+
o~1
(44)
d~a
(45)
b ~ l ( 1 ) + b~3(1)
D~o(1)
(46)
d~2(1) = Ad~2(0) + d~2 = b~2(1___~)+ s D~o(1)
(47)
d~4(1) = Ad~4(0)+ d~4 = b~4(1) d~4 (1-----~ D~o +
(48)
=
-
-
d~4 = d ~ l -
D~o(1) b~2(1) + / ) ~ 4 ( 1 )
D~o (1)
(49) (50)
We may now write the solution ~1(~,0) of Eq.(2). Starting with Eqs.(10), (3.2-9), (3.2-10), and (15) we obtain the following expressions:
4.3 QUANTIZATION OF THE SOLUTION FOR A x ~ h / m o c
~1 (r O) = ~oo{ (1
-
e -2~176 exp[-(A 2
-
155
A2)l/2~]ei~; + u(4, O) + v(r O) } (51)
N/ 2-1 u(r 0) = -2iAIA3 e'*xl((+,xa0)
'
27~ar I T ( a / N ) sin
E
sin fl~
~=-N/2+1
IT(a/N)
see Eq.(2.4-29);
~(~, 0) =
fl,~Osin
N
(52)
A1, A2, A3 see Eqs.(27),(28)
g/2-1 27r~r E T~(0)e i~1r sin ,~=-N/2+1
(53) ~
(54)
v~l, v~2 see Eqs.(3.2-48), (3.2-49) d~i(0)
see Eqs.(6.4-11), (6.4-28)-(6.4-31)
(~2, d~3, (~4
see Eqs.(43), (46), (50)
Comparison of Eq.(51) with Eq.(2.4-32) shows that ~. 1(~,0) can also be written in the following form'
~,~(r o) = ~o(r o) + ~,oo~(r o) N/2-1
~(r
=
([~1(o) + os
Z
'(~/~-~'~~
~=--N/2+l
+ [d,~2(0) + Od,~4(O)]e-'(2~'~/N+~'~'3)~ e v'~r sin 2 ~ r N
4.3 QUANTIZATION OF THE SOLUTION FOR A X ~
(55)
h/moc
We follow Section 3.3 but are careful to put a tilde - or replace a hat ^ by a bar - where appropriate. Equation (3.3-1) becomes:
Equations (3.3-2) to (3.3-4) assume the following form:
~,*~, = ( ~ + ~ , ; ) ( r
+ ~,1) = r
+ ~(r
+ ~,;r
+ o(~ =) (2)
156
4 KLEIN-GORDON DIFFERENCE EQUATION FOR SMALL DISTANCES
o~,* o~, O0 O0
o~,; OVo)
or30Vo O0 O0 F a
o~,* o~,
o ~ O~o
or or
or or
00
O0
O0 O0
( o~; o~
(3)
+O(~
o~; O~o) or or +~
(4)
The function ~o(~, 0) of Eq.(3.3-5) remains unchanged
~o(~, 0) =~oo[(1 - e-2ix~~
+ u((, 0)]
=~oo ((1 - e -2/x~xa~ exp[-(A~ - )~12)1/2~]e iAz~
N/2-1
IT(x/N)sin/3~0 sin 2 ~Nr )
--2iA1A3eiA1)'3e~)~l(e-i)'l~a~ E
~----N/2+l
sin/3~
(5)
but ~1 (r 0) of Eq.(3.3-6) becomes ~1 (~, 0) according to Eq.(4.2-55)"
~1(~, O) ~- t~O(~, O) "-[-~00~)(~, O) N/2--1
~(~,o) =
~
(6)
([a~(o) + o ~ ( o ) ] a ( ~ . ~ / N - ~ ) o
a---N/2+l + [(~g2(0) -}- Odgn(O)]e-i(2~r~/N+)~13)O)e iAzr sin
N
(7)
We may use Eq.(6) to rewrite Eqs.(2)-(4). The notation :Re(... ) is used for the real part of the expression in parentheses:
~'*~' = ~;~o + ~[2V;r + ~,oo(V;~ + ~*Vo)] = ~;~o + 2 ~ [ ~ o + ~oom~(v;~)]
o~,* o~, F2a
00
00
F~oog~e
00 00
(9)
= 0~ 0~ ~2a
0(
0r
~oo~e
0r 0~
(10)
00 00 = 00
o~* o~ 0r or
(8)
00
We have again the terms ~ o , (Og2~/O0)(O~o/O0),as well as ( 0 ~ / 0 r x (0~o/0r obtained in Eqs.(2.5-4), (2.5-9) and (2.5-11). The terms ~ , (0~;/00)(0~/00), and (0~;/0~)(0~/0r Eqs.(8)-(10) must be calculated with the help of Eqs.(5) and (7). First we show how ~(4,0) depends on G2(r 0) of Eq.(4.2-2). This starts with/~s~(0, ~) of Eq.(6.4-1) and (~s~(0, ~) of Eq.(4.2-24)"
4.3 QUANTIZATION OF THE SOLUTION FOR A x << h/moc
157
N
~(o,~)=~ 2
-~'~'~a2(r162
a2(r
0 X e -i~lCsin 2 ~N~ d ~ ,
0 = 1, 2, "'" , N - 2
(11)
The four functions d~l(0) to d~4(0 ) in Eq.(7) may be written with the help of Eqs.(6.4-11), (6.4-10), and (6.4-12)-(6.4-15)as follows: 0--1 ~1(0)---- E Hs,c(n,~){nFl(n,t~)+F3(n,t~)+i[nFh(n,t~)+FT(n,e;)] + ~ 1 } (12) r~---'0 0--1 0~2(0)= E [-I~,r tc){nF2(n, tc)+F4(n, tc)+i[nF6(n, tc)+Fs(n, ~)] + d~2} (13) n=0 0--1 a~3 (0)= - E / ~ s ~ (n, ~)[F1 (n, ~) + iFh(n, t~) + d~3] (14) n=0 0--1
d~4(0) = - E H~(n, a)[F2(n, a ) + iF6(n,a)+ d~41
(15)
n--0
Equations (6.3-51)-(6.3-58) define the functions Fl(n,a) to Fs(n,~), while Eqs.(4.2-43), (4.2-46), and (4.2-50) define d~2, d~3, and d~4. Using these equations we may rewrite the terms
[d~(o) + od~(o)], ~''~
+ [d~(o)+ oJ~4(o)]~-~'`~
(16)
of ~(~, 0) in Eq.(7) into the form of Eq.(6.4-48) and obtain the expression
'0(~, 0) = e i)k1(~-~30)
N/2-1 E [3r(0' P~) + e~=-N/2+l
ij~(0, ~)1 sin
2"K1r
j~(0, ~)= [j~(0, ~)-oj~(0, ~)] cos 2"n-~0 N -[Ja(0, t~)-0,]4(0, t~)] sin 27r~0
j~(O,~l=[L(o~)-o3~(o,~)ls~2~~ ' U
+ . [L( _ . 0 , ~ - 0)
L(. 0 , ~ )1cos 2~o N
(17)
The sums over n shown in Eqs.(12)-(15) have been separated in Eq.(6.4-48). We obtain ~ ( ~ , 0 ) from Eq_(5) by changing the sign of i. The product {Joo~V of Eq.(8) becomes with ~oo = ~oo"
4 KLEIN-GORDON DIFFERENCE EQUATION FOR SMALL DISTANCES
158
~00~(~, 0)~(~,O) = ~20e-i)'~)'s~((1
- e zi)'~a~ exp[-(A~ - A12)1/2~1
NI~-~
+ 2iAIA3ei~l)'3(~
E
IT(t~/N)sinfl~0sin sin/3~
tc=-N/2+l
2~r~r N
N/2-1
x
E
[Jr(0, ~) +
iJi(O,a;)] sin 27r~ N
~;=-N/2+1
(18)
This is a product of two sums. The terms sin(27rt~/N) are important. We write u for ~ in the second sum and separate the products sin(21r~r162 for~=uand~r
VooV;(r o)~(r o) = Vo2o~- ' ~ ~
[(1 - d ' ~ ~
N/2--1
X
E [fir(0, t~) + iffi(O , t~)] exp[-(A 2 ~=-N/2+l
-{-2iAIA3eiA~As(~ ( N/2-1 E
IT(x/N)
~=-N/2+Isin fl~ N/2-1,#v
-
A~)1/2r sin
2~-,~ N
27rx~r sin(fl~0)[Jr(0, ~)+i3i(0 ~)1 sin 2 N
N/2-1
~;-----N/2+ 1 v---N~2+1
x [Jr(0, u) +
sin fl~ N " sin 27r,~) N ] iJi(O,u)] sin 27ru{
(19)
According to Eqs.(2.5-3) and (8) the integral over ~ from 0 to N of this expression is needed: N
9oo/ 'G(r e)~(r 0)de 0
N/2-1 (
= ~~176 E
e~---N/2-k-1
2 sin(Al,~3 0)
(2Try/N){1 - exp[-(A~ - A~)I/2N]} )~22--
)k2
"+" (271~/N)2
• [j~(o', ~1 - ~j~(o, ~)] + N , ~ I ~ . [T(~,/N) sin~,~,,. sin(fl,~0){3r(0, ~) sin A1~3 -- Ji(0, t~) cos ~1~3 + i[Jr ('~, 0) cos AI Aa + ffi(/~, 0) sin
AI/~3]})(20)
4.3 QUANTIZATIONOF THE SOLUTIONFOR Ax
159
~< h/moc
Let us turn to Eq.(9). The product (Oql;/OO)(Oq2o/O0) of Eq.(2.5-9)is required. Then we need 0~/00, which we get from Eq.(17)"
N/2-1
0~(r O) = e,~,( O0
2 ~r~,____~
-
~
[J~(O ~) + iJI(O, ~)] sin N
~=-N/2+l 00
+ ,~i,~aJi(0,~)
oo
cos~iAa0
- ala~J~(o, ~)
sin
x~a~o
ffi(0, N)---( (0'/~) -[- AIA3Ji(0, N)) sinAiA30 -- 0fir00 _t_( 0Ji(0' N) 00
)
(21)
,XllaJr(0, ~) cos ,Xi~a0
The exponential e i~'~r was not written as cos Air +isinAlr since it wilt be cancelled by an exponential e -~x1r in the following equation for (Oq2~/O0)(O~/O0). The term 0 ~ / 0 0 is obtained with the help of Eqs.(2.5-5) and (2.5-6). We again use ~00 = ~00"
~~176 0 ~ ; 0(~' ) 0 0 0~(~, 0)00 = ~2~ - 2iA1"kae2i)~lx3~ x
N/2-1 E
~----N/2+l
[JR(0, ~) + iJI(O, ~)] exp[-(A22 - A2)1/2~] sin 27r~._~
IV
-- 2AIA3ei)~i~a(~
-
N/2-1,7~v
+ ~ ~=-N/2+1
(
N/2-1 E IT(re~N) ~=-N/2+i sin/3~
(,~1/~3sin/~0
~/~ ~o~ Z~o)[JR(o. ~) + ~j~(o, ~)] ~in 2 2 ~ r N
N/2-1
~ v=-N/2+l
I~(~/y) [j~(o.~) + ~JI(O..)] sin/3~
x ('~1"~3sin/3~0 - i/3~ cos/3~0) sin
27rt~ N
sin
27rt/~)] N
j~(0, ~) = j~(o, ~), Ji(0,.) = Ji(0, ~) for ~ u Following Eq.(20) we integrate this expression over r from 0 to N:
(22)
4 KLEIN-GORDON DIFFERENCE EQUATION FOR SMALL DISTANCES
160
/0
1I/~(~, 0) 0'0(~, 0)
9oo
oo
0
N/2-1
/2
---W-e~=~o
{ [JR (0, ~)+iJI (0, ~)] sin 2A1A30 ~;=-N/2+l
~1~
%-[31(0, ~;)--i JR(O, g)] cos 2,~1)k30} (271"g/N) { 1-exp[-(A22 -A2)1/2N]} A~ - A~ + (2~~:/N) 2 U/~.-I -
N~:~
~ ~=-N/2+l
~(~/N){ [ A ( O , ~ )
+
sin fl~
~(0,~)1
X {[A1)~3sinfl~0cosA1A3(0- 1)+ fl,r cosfl~0 sin A1A3(0- 1)] -
[Yi(o,
,,) -
~Y~(o,
,,)1
x [A1A3sinfl,~0 sin A1A3(0- 1 ) - fl~ cosfl~0cos A1A3(0- 1)]}~ /
(23)
We turn to Eq.(10). The term 0V/0r needs to be derived from Eq.(17)'
0~)(r 0r
N/2-1 .__eiXl(~_AaO) E [fir(0 /~)+ iYi(0,/~)1 ~----N/2+l x
(27rn 21rnr - ~ cos N + iA1 sin 27r~) N
(24)
We may now obtain the term (Oq2;/O()(O~/O()of Eq.(10) with the help of Eqs.(2.5-7) and (2.5-8)'
~ooo~(r o)o~(r o) = or or X
{
~~176 - 2 [ ~ - ~ ( ~
N/2--1
-
~)~/~]
( 27r~
2,'T~,~
E [L(0,~) + iJi(0,~)] \-'~- exp[-(A2-A2)l/2~] cos N t~---N/2+l + iA1 exp[-(A 2 - A2)l/2r
27r~r N sinA1A30
g/2--1 + 2iA1A3e-i~M
E
IT(g/N)[Jr(0 /~) -Jr-iJi(0, g)] sin fl~0
~=-N/2+l
sin fl~
N + (27r~) ~ 2 cos 2 27rt~ N ] • [A2 sin2 27rn~
4.3 QUANTIZATION OF THE SOLUTION FOR A x << h/moc Jr- 2i,,~1/~3e-iAIAa
•
A1 sin
161
N/2-1,#u N/2-1 E E I T ( x / N ) [Jr(0,//)-t-iJi(O,u)]sinl~O sin/3. ~=-N/2+1 u=-N/2+l
+ i---~-cos
cos
+ i A1 sin
This expression is integrated over ( from 0 to N following Eqs.(20) and (23)' N
O~
------~d~
=
~0
2A~sinA1A30
0 N/2-1
X E
[Ji(O'm)-iJr(O'm)] (27rm/N){1-exp[-(A2-A21)l/2N]}
~=-N/2+l
-NA1A3
N/2-1
[
E A12-1to=--N/2-4-1 --
--.~
1
,~22- A~ + (27rm/N) 2
{ Ji(0, m) cos A1A3- Jr(0, m) sin AIA3
i[Jr(0, m) eos~l/~3 -~- Ji(0, m)sinA1A3]}IT(x/N)sin/3~ sinfl~0}
(26)
We turn to the text following Eq.(2.5-11). In order to allow for the generalization of the terms of Eq.(2.5-3) by Eqs.(8)-(10) we define the energy 0 by the sum of the following three components
0 = 01 + 02 + 03 = ~r + Ov(e)
(27)
The terms 01 to 03 are obtained by the substitution of Sqs.(8)-(10) into Eq.(2.5-3). We use the notation Lr to Uv(0) to distinguish the terms from the approximations U to Uv(0) in Eq.(2.5-12) and from U to Uv(O) in Eq.(3.3-27): N
~ r 1 ---
L2 ,m02c4(At) 2 L2 m2~ cAt li2 / ~*~d~= cat li 2
((1+2a)/
N
0
0
*~0d~ 9~
N
+ 2a~oo / ~e(~)V)dr
(28)
0
N
N
L 2 ] o~* o~, - oo - - - doo r u~ = ~ 0
(~+2~)
oo o-7 0 N
+ 2a~oo/ Y,e(O~ 09 0
(29)
4
162
K L E I N - G O R D O N D I F F E R E N C E EQUATION FOR SMALL DISTANCES
N
N
L 2 J 0~* 0 ~
- 0r - - - dor ~
53 = 7~7
= ~-~ (1 + 2cr)
0
0--'~" 0--'~" 0 N
+ 2a~ oof g~e( O,~8 0~ 0
As in Eqs.(2.5-13)-(2.5-15) and (3.3-2s)-(3.3-3o) w~ w~nt the time-invariant part
Ur = Ur + Lr~2+ Crr
(31)
of 01, U2, U3 and we ignore the time-variable part Uv(0). Again we write Ur to Uc3 to distinguish the terms from the approximations Uc to Uc3 in Eq.(2.5-16) and from U~ in Eq.(3.3-31). When we derived Eqs.(2.5-13)-(2.5-15) we simply left out terms containing sin/3~0 or cos/~0. The time variations of Eqs.(28)(30) are not so obvious and we must write integrals over 0. With the help of Eqs.(2.5-13)-(2.5-15) we write the time-invariant part of Eqs.(28)-(30) in the following form in analogy to Eqs.(3.3-32)-(3.3-34)"
- dO = U1
- I = Uc
cAt
Ii 2
E
IT(to~N)
,~=-N/2+1
sin ~,~
(1 + 2 a ) ~ooUA1A3 - 2 2 2
o N
N
(32) 0
N
_ = / ~=ao= ur
N/2-~
~
E
(1 + 2~),i,o~oN~i
~=-N/2+1
0
N
0
( IT(,~/N) ) 2 sin,e,~
(.X~:,~+ ~ )
N
(33) 0
Uc~ =
O~ao= -s 0
( 1 + 2 ~ ) -~ ~~1 7 6 = ~
0
~,1~+
Z
Jc=-N/2+I N
N
(34) 0
0
4 . 4 EVALUATION OF THE ENERGY O~ FOR SMALL DISTANCES
163
We recognize on the right side of Eqs.(32)-(34) the energies Ucl, Ur U~3 of Eqs.(2.5-13)-(2.5-15) multiplied by (1 + 2a){J020/920 " 1.0146~020/q020. The factor 1.0146 implies a difference of 1.5% that is barely visible in a plot. The interesting terms are the integrals multiplied by 2c~q00. We shall analyze them in the following Section 4.4. A comparison of Eqs.(31) and (2.5-16) shows that the only difference is the symbols-. Hence, we may use the results of Section 2.5 from Eq.(2.5-16) on. The energies F_~ of Eqs.(2.5-32), (2.5-34), and (2.5-48) are obtained. One could write [~ instead of F_~ and obtain the result F_~ = E~, but we follow the last paragraph of Section 3.3 and do not do so. 4.4 EVALUATION OF THE E N E R G Y Uc F O R SMALL DISTANCES
Following Section 3.4 we consider only the terms in Eqs.(4.3-32)-(4.3-34) multiplied by 2c~00. We denote them with 0~1 to 0~3'
_
L2
(]a l --- 2OL~I/00~'~
( m ~c 2A t
h
) fN7 ( :Re(q;f~)dr dO 2
0
L2 ]N [ ]"N ~e ( O~ Of~ Oo,2= 2a~oo--~ O0 o0)d~] dO L2
_
0 N
(1)
0
(2)
0 N
(3) 0
0
We want the sum 0~c of these terms as function 0 ~ ( ~ ) according to Eq.(2.5-16)"
N/2--1
(4)
~=--N/2+l N/2-1
0c~l-- E
N/2--1
01~ (t~)' 0(x2-- E
to=-- N / 2 + 1
N/2-1
02~ (t~)' 0(x3-- E
to=- N/2+ 1
[f3~(t~)
~=- N/2+ 1
(5)
Equations (1)-(3) must be rewritten to correspond to this scheme. An index ~ is written at the end of the kernel of the integrals of Eq.(1) to indicate that the summation over ~ in Eq.(4.3-20) is not carried out'
164
4 KLEIN-GORDON DIFFERENCE EQUATION FOR SMALL DISTANCES
N
: Ul~(n) = 2at~oo~-~
h
N
tP;(r162162
:Re o
o N 1 jf~
al~(-)
~1,(~) = 2a~o(L2/cAt)(moc2At/ti) 2- t~00
e o
N _
1
~00
(/N , o
~I
Co({, 0)9(r O)d ,, ,~dO
N
j. :Re (1 t~; ((~,0)9(r O)dr dO, o
dO
(6)
~oo = ~o0
o
Dividing Eq.(4.3-20) by_~o20 and leaving out the summation sign yields the following expression for 9/l~(n)"
~l,~(n) = 2 (2"nn/N){1 -exp[-(A~ - A~)I/2N]} -
+
N
x / :Re[Ji(0, n)
-
iJr(O,~,)lsinA1AaOdO
0 N + N A 1 A 3 IT(m/N-------~) f
sin fl,~
:Re{Jr(0, n) sin A1A3 - .~(0, n) cos A1Aa
o
+ i[Jr (0, n)cos A1A3 + .~(0, n)sin A1Aal } sin fl~OdO (7) We note that Jr(0, ~) and Ji(0, ~) are not real but complex. As in Section 3.4 we avoid an analytical separation into real and imaginary components since the computer instruction :Re will do this separation numerically. For the integration of Eq.(7) over 0 we need to evaluate integrals containing Jr(0, ~) and Ji(0, ~). For the calculation of Jr(0, ~) and Ji(0, ~) we start with Eq.(3.4-8) and observe that the variable Or, will become either 0 or n as required:
t~o(~, On) = (1 -- e -2i'x~'x3~ exp[-(A~ - A~)l/2~]e i'x~( N/2-~ 27r~{ - 2iA1A3ei~l~3ei~l(r176 E IT(co~N)sin fl~0,~ sin (8) ~=-N/2+l
sin fl~
The function ~o of Eq.(8)is needed for G2(~, 0)in Eq.(4.2-8)"
N
4.4 EVALUATION OF THE E N E R G Y [.f~ F O R SMALL DISTANCES
165
G2(r 0) = --pN(~, 0)[(1 + 2iA1)~0(~ + 2,0) -- (4 + 4 i A 1 -- 6A~ + 2iAa)~o(r + 1, 0) + (6 - 12A~ - A4)~o((, 0) - (4 - 4iA1 - 6A~ - 2 i A ~ ) ~ o ( ~ - 1,0) + (1 - 2 i A 1 ) ~ o ( ~ - 2, 0)] /~1 =
ecTAmo~/liN, PN(~, 0) see Eq.(4.2-5)
(9)
We must define the space and time variation of P N ( r On). In analogy to Section 3 . 4 w e a s s u m e that Ae(r 0) and PN(r 0) in Eq.(4.2-5) are independent of time and vary linearly with the space variable ~:
2AcAe(~', 0) = Ae(~', 0 ) = ./~xe(~") = fteo~/N e
1
N2A~(~) = pcN2A~(~)
p~(~, on) = pN(r = 4 7 pc=~
~
=4~2N2
~
'
zXx =A--~=
~T
(10)
Equation (9) becomes:
G2(~,On) = -pcN2Ae(~'){(1 + 2iA1)~o(~ + 2, On) - - [ 4 - 6A21+ 2i(2A1 + A~)]~o(~ + 1,0r~)+ ( 6 - 12A~ -- A4)~0(~, 0n) -[4-
6A~ - 2i(2A1 + Aal)]~o((- 1,0r~)+ (1 - 2 i A 1 ) ~ o ( r
2, 0r~)}
(11)
We shall also need G2({, 0n -- 1) and G2({, On + 1). In analogy to Eqs.(3.4-19) and (3.4-20) we substitute 0 n - 1 or Or, + 1 for 0n in Eq.(11); as before this notation helps with the writing of the computer program"
G2-(~,On) = G2(r
- 1)
-pcN2.Ae(~'){(1 nt- 2/A1)~o(~" -k- 2, On -- 1) - [4 - 6A12+ 2i(2A1 + Aal)]~o({ + 1, 0n - 1) + (6 - 12A~ - A41)~o(~, 0n - 1) -[4-6A2-2i(2Al+A~)]~o(~-1,0n-1)+(1-2iA1)~o((-2,0n-1)} (12) =
a2+(r 0~) = a2(r 0,~ + 1) = -pcN2.~o(r -[4-
+ 2i:~)Vo(r + 2,0~ + 1)
6A~ + 2i(2..~ 1 "-[- /~3)]1I/0( ~" -'['- 1,0r~ + 1 ) + ( 6 -
-[4-6A~-2i(2Al+A~)]~o(r
12A~ - A'~)~o((, 0n + 1)
(13)
We can compute Gs,~(On,tr of Eq.(4.2-24) once G2(r of Eq.(11) is obtained. The notation Gs,,(On,~)is changed to G~2(0n,~) to make the use
166
4
K L E I N - G O R D O N D I F F E R E N C E EQUATION F O R SMALL DISTANCES
of Eqs.(12) and (13) more evident. We observe that Eq.(4.2-24) treats Lhe function G2(r 0) as a step function with step width 1. The integral may readily be replaced by a sum of G2(r 0) taken at r = 0, 1, ... , N - 1"
N
Gs,~9.(O~ ~) = G~,~(O~ t~) = ~ )
G~.(40,~)e -ix~r sin 27r~ d4
~
~
N
0 _
1
2 N
27rI.~__~
G2(~,0,~)e -i'xlr sin
N
(14)
~=0
The functions G~,,2_(O,~, ~) and G~2+(0,~, ~) follow from Eqs.(12) and (13)'
2 ~I 27r~r G.~,~2_(On,~) = d~,~(On- l,~,) = -~ G2-(~,O,~)e-i:~sin----~
(15)
~=0 N-1
2 Gs~2+(0n, ~) = Gs~(0n + 1 , ~ ) = ~ E 52+(~' 0n)e--i'X~ sin 27rt~N (=0
(16)
With Eqs.(14)-(16) we obtain the function/t~(0,, ~) from Eq.(6.4-1)"
.ffIs,,(On,~.) = -e~
271"/,~
+ l;~)+ 2 cos ---ff-d~,,(O,,~) -
e-~AtA3
ds~(0,~- 1,~)
27r~c~ = - e i~l~3'~'t~s~2+(0,~,~) + 2cos ~ s~2(0~,~) - e-i'xl"x3(~.,~2-(8,,~)
(17)
With Hs~(0,~, ~) we can compute the eight functions Jl(0,~) to L(0,~,) of Eq.(6.4-48). These functions in turn yield Jr(0,~) and Ji(0,~) of Eq.(4.317), which yield finally 9Q~(~) of Eq.(7). We use again AI~(~) and BI~(~) of Eqs.(3.4-25) and (3.4-26)
Ate(n) : 2(2Try/N){1 - e x p [ - ( l 2 - A2)l/2N]} 12 _ 12 + (21r~/N) 2
Bx,,(~.) = NA1A3 Iw(x/'N-----~-)~" sin/3~ to write Eq.(7) in the following form"
(18) (19)
4.4 EVALUATION OF THE ENERGY U~ FOR SMALL DISTANCES
N
167
N
0
0
I l l (0, t~) = A l ~ ( ~ ) ~ e [ J i ( O , ~)
-
-
i Jr(0, t~)] sin/~1~30
i12 (0, ~) = BI,~ (m) sin(/3,~O) :Re{ [Jr(0, ~) sin '~1"~3 -- Ji( 0,/g)] cos )~1,'~3 + i[Jr(0, ~) cosA1A3 + Ji(0, t~) sin A1A3}
(20)
The equivalence of the integral over a step function with step width 1 and a sum permits us to write 9q~(~) in the form N-1
N-1
~1~(/~) = E II1(0'tr 0=0
~- E
I12(0,/~)
(21)
0=0
Let us turn to 02~(~) of Eq.(5). The index ~ is written again after the integration over ~ to show that the summation over ~ in Eq.(4.3-23) is not carried out' N
N
~e
U2~(a) = 2a~00~-~
--~-~d~
0
dO
(22)
0
We obtain from Eq.(4.3-23)" N
U2~(~)
_~2~(~)=
2~2oon2/cAt --
~
1 ]" 0
1 / ~:Re ~0
N
0
N
(fOtP~O~) ~e --~-~dr ~dO
(23)
0
(0xI/~ 0~ ) (27rt~/N){1- exp[-(z~22- A2)1/2N]} d~ 2A A3 k, 00 ~ ,~ = 1 A2 _ A2 + (27rt~/N)2
x {9~e[JR(O,a)+iJi(O,t~)]sin2A1A30+~e[Ji(O,a)-iJR(O,t~)]cos2A1A30]} -- NAIA3IT(n/N-------~){~e[JR(O,n) + iJi(O, ~)] sin ~ x [AIA3 sinfl~0 cos A1A3(0- 1 ) + -
-
~e[Ji(0 , ~)
x [A1Aasin/~0sinA1A3(0-
-
fl,~cos~,~OsinA1A3(O- 1)] -
iffR(0~)]
1)-/?~cos~0cosA1A3(0- 1)1 } (24)
In order to integrate Eq.(24) over 0 we need to evaluate numerically the integrals over the following terms with respect to 0:
168
4 KLEIN-GORDON DIFFERENCE EQUATION FOR SMALL DISTANCES
Ai AaAi~(~)J~e[Jrt(O, ~) + iJI(O, m)] sin 2Al ~30 ]22(0, a;) = Ai Aaai~(~)J~e[3I(O, ~) iJrt(O, s;)] cos 2 A i A30 ]23(0, ~) = BI,~(~)J~e[jR(O, to) + ill(O,/'i;)]
(25)
X [A1A3 sinfl~0cosAiA3(0- 1 ) + fl~ cosfl~0 sinA1A3(0- 1)] -[24(0, N) -- Bitc~e[JI(0, N) i,)TR(0, t'i;)] x [.x~.x3 sin Z,~o ~in.X~.X3(0 - i) - ,0,~ cos,0,~0 cos .Xl.X3 (0 - 1)]
(27)
/~21(0, N) =
(26)
-
-
-
(28)
The functions JR(0, s) and Ji(0,~) are defined in Eq.(4.3-21) by the functions Jrl0, ~) and Ji(0, ~). In turn, Jr(0, s) and Z(0, ~) are defined in Eq.(4.317) by Ji (0, ~) to Js(0, ~) of Eq.(6.4-48). We need mainly lengthy but straightforward substitutions to express JR(x, 0) and JI(~, 0) by J1 (0, ~) to Js(0, ~). The only problem is the derivatives OJr/O0 and OJi(O,~)/O0 in Eq.(4.3-21). We obtain with the help of Eq.(4.3-17)'
0dr(0, '~) = (0.]1 (0, ,*) L(o, ,~) o oJ~(O,oo~) ) cos 2~0N 00 \ 00 -
2r~y [Ji (0, ~) - 0J2(0, ~,)] sin
0,.14(0, ~) ~
_ (oL(O,oo ~) - j~(o, ~) - o ~ 00
/ sin
2~~ [ ' ~ ( 0 N ' ~)
oJ~(O, ,~) O0 =
(oJ~(
0,~) _ J . ( 0 , ~ ) - 0 00
~176 00
-
sin
+-N2ra: [J5 (0' a:) -
+ ( os(O,oo _ s (o, - o os(O,oo
2r~0 g
27r~0 0,]4(0, a:)] cos
2"a-s;0 N
(29)
2~0
0'J6 (0' a:)] c~
2;r~0 N
cos
27r~N[A(0, ~) - 007s(0, ~)] sin
2~0 N
(30)
The functions Ji(0,~) to J8(0,~) are defined in Eq.(6.4-48). We must show what the derivatives 0Ji(0, ~)/00 to 0Js(0,~)/00 mean. To this end we use once more the equivalence of the sum of a step function with the step width 1 and an integral over the step function. The functions/TEi (n, ~) to/~s(n, ~) in Eq.(6.4-48) are step functions with a step width 1 for n = 0, 1, ... and an
4.4 E V A L U A T I O N OF T H E E N E R G Y U~ F O R S M A L L D I S T A N C E S
169
unspecified value of a. Using the definition of Jj(0, a) in Eq.(6.4-48) we may write:
oJj(e,.) 08
0
= c90
[gj (n, ~.) = --~
Kj (O,~,a)dOn
n=O
0
0
= f okj(o,~,,~) O0,., dON = kj(O,a),
j = 1, 2,..., S
(31)
0
Using Al~(a) and Bl~(a) of Eqs.(18) and (19) we may write ~2~ of Eq.(23) in the following form' N
0 N-1
= ~ ] {hi(o, ~)+ &~(o, ~ ) - [&~(o, ~1 + &~(o,,~)]}
(32)
8=0
Let us turn to 03~(a) of Eq.(5). Again the index a is written after the integration over r to indicate that there is no summation over a in Eq.(4.3-26)' N
N
L2
0~ 0
(33)
0
We obtain with the help of Eq.(4.3-26)" N
U3~(a)
__~3~(tr
1 0
N
1 ~e/(O~Ov ~o-"-7
)
N
(
/x
(34)
0
(27ra/N){1-exp[-(A~-A~)l/UN]}
\ 0r ~-~d~ ~ =
A2 _ A21+ (2r~/N)2
0
• 2~m~[3i(e, ~) - ~Jr(e, ~)] si. ~ e
-NA1A3IT(~'/N'---~)sin/~[)~2+
\
/J
-[Jr(0,~)+iJi(0,~)]sinA1A3} s i n ~ 0
(35)
170
4 KLEIN-GORDON DIFFERENCE EQUATION FOR SMALL DISTANCES
In order to integrate Eq.(35) over 0 we need to evaluate numerically the integrals over two terms: /:31(0, ~,) = Al,~(~)~e{2A2[Ji(O,~)- i Jr(0, ~)]} sin A1A30 ]32(0, to)= BI,~(a)[A 2 + (2~tc/N)2l{~e[Ji(O,tr -
(36)
i Jr(0, ~)1 cosA~A3
~e[Jr(0, a) + i.]i(0, a)] sin A1A3} sin ~.0
(37)
The functions J~(0, a) and Ji(0, a) are defined in Eq.(4.3-17). We may write 9{3~(a) of Eq.(34) in the following form: P
N
N-1
"-- / [ I 3 1 ( 0 , t~) 0
-
-
I32(O,t~)ldO -- E
.2
[131(0' t~)
-
-
I32(0' t~)]
(38)
8=0
I n analogy to U~ in Eqs.(2.5-16) and (2.5-17) we write the energy U~r of Eq.(4) in the following normalized form"
2aL2~o2o N
=
2aL2~2ooN2
=
N2
0~r = 2aL2~~176 cT -
= 2aL2~2~ ~~" ~ , ~ ( ~ ) cT t~---1 2aj52~~176~a~ (g)
~ m20c4(At) 2 ~ , ~ ( ~ ) = ~K2~(~)+ ~ 3 ~ ( ~ ) + h2 ~1~(~)
(39)
(40) (41) (42)
In order to work out a computer program for 9~1~(~), 9~2~(~), and ~3~(~) we introduce a more compact notation for ~0(~,0n) of Eq.(8) when ~ varies from r + 2 to ( - 2 and 0n varies from 0n + 1 to 0n - 1"
~o,j,m(~,On)=~o(~+j, On+m), j = 2 , 1 , 0 , - 1 , - 2 , m=1,0,-1 ~o,o.o(~, 0) = ~o(~, 0,) (43) Table 4.1-1 uses this notation in lines 2-6. Equations (11), (12), and (13) assume the following form:
171
4.4 EVALUATION OF THE ENERGY U~ FOR SMALL DISTANCES
TABLE 4.4-1 FUNCTIONS ~3~, IT(x/N),... AND THE EQUATIONSEQ.(2.4-14), EQ.(2.4-29), ... DEFINING THEM, THAT ARE REQUIRED FOR THE CALCULATIONOF 9E1,~(~) OF EQS.(6) OR (7) AS WELL AS 97t:2~(~) AND 9{3~(tr OF EQS.(32) AND (38). THE NUMERICAL VALUES OBTAINED FOR EQUATIONS SHOWN WITH BRACKETS, E.G.
EQ.[2.4-14], SHOULD BE REMEMBERED BY THE COMPUTER TO REDUCE THE COMPUTING TIME.
~ Eq.[2.4-14] IT(x/N) 9o,2,1(~,0n) Eq.(4.4-43) ~o,1,1({,0n) 9o,_~,1(~,0,~)Eq.(4.4-43) ~o,-2,~(~,0n) ~o,~,o(~,0,~) Eq.(4.4-43) ~o,o,o(~,0n) ~o,-2,1(4,0n) Eq.(4.4-43) ~o,2,-1(~,0n) ~o,o,-l({,0n) Eq.(4.4-43) ~o,-1,-1(~,0,~) file(~, On) Eq.(4.4-10) G2((, 0n) G2+((, 0n) Eq.[4.4-46] G.~,~2(On,~) Gs~2+(0,~,~) Eq.[4.4-16] /-)s~(0,~,~) F2(n,~) Eq.(6.3-52) Fs(n,~) Fs(n, ~) Eq.(6.3-55) F6(n, ~) Fs(n,~) Eq.(6.3-58) dsr(tr Jl(0, ~) Eq.[6.4-48] J2(0, ~) J4(0,~) Eq.[6.4-48] Js(0, ~) JT(0, ~) Eq.[6.4-48] Js(0, ~) Ji(0,~) Eq.[4.3-17] AI,~(~) /11(19, g) Eq.(4.4-20) i12(0,~) /~1 (0, ~ Eq.[6.4-49] /~2(0, ~) jr(4(0 , tO) Eq.[6.4-49] /(5(0, n) /7/7(0, n) Eq.[6.4-49] /~8(0, n) OJi/O0 Eq.[4.4-30] JR(0,~) /21 (0, ~,) Eq.(4.4-25) [22(0, tq,) ~:24(~9, tO) Eq.(4.4-28) ~2~(~) [32(0,~) Eq.(4.4-37) ~3,~(~)
Eq.[2.4-29] g2o({,On) Eq.[4.4-8] Eq.(4.4-43) ~o,o,1(4,0,~) Eq.(4.4-43) Eq.(4.4-43)~o,2,o(r Eq.(4.4-43) Eq.(4.4-43) ~o,-1,o((,0n) Eq.(4.4-43) Eq.(4.4-43)~o,~,-1(~',0,~)Eq.(4.4-43) Eq.(4.4-43) ~o,-~,-l(~,0n) Eq.(4.4-43) Eq.[4.4-44] G2-(~, 0n) Eq.[4.4-45] Eq.[4.4-14] Gs~2-(0~,a) Eq.[4.4-15] Eq.[4.4-17] Fl(n,~) Eq.(6.3-51) Eq.(6.3-53) F4(n,t~) Eq.(6.3-54) Eq.(6.3-56) FT(n, ~) Eq.(6.3-57) Eq.(6.4-46) dsi(~) Eq.(6.4-47) Eq.[6.4-48] Js(0, ~) Eq.[6.4-48] Eq.[6.4-48] J6(0,~) Eq.[6.4-48] Eq.[6.4-48] Jr(0, ~) Eq.[4.3-17] Eq.[4.4-18]BI~(~) Eq.[4.4-19] Eq.(4.4-20) ~1~(tr Eq.(4.4-21) Eq.[6.4-49] /~3(0, ~) Eq.[6.4-49] Eq.[6.4-49] /~6(0, n) Eq.[6.4-49] Eq.[6.4-49] 0Jr/Oq0 Eq.[4.4-29] Eq.[4.3-21] Ji(0,~) Eq.[4.3-21] Eq.(4.4-26) /~23(0,~) Eq.(4.4-27) Eq.[4.4-32] i31 (0, ~) Eq.(4.4-36) Eq.[4.4-38]pc Eq.(4.4-10)
G2-b(~, ~gn) : -pc.N2./~e(~) {(1 --[- 2iA1)~o,2,1(r
0n)
-- [4 -- 6)~ 2 --[- 2 i ( 2 . ~ 1 "-[-)13)]1I/0,1,1 (r On) -'[- (6 -- 12~21 -- ~4)lI/0,0,1(r
On)
- [ 4 - 6A2 - 2i(2A1 + A~)]~0,-1,1(r (1 - 2iA1)~0,-2,1(r (46) For the representation of the results of this section by plots we start with ~1~(~.) of Eq.(21), ~:2~(a) of Eq.(32), and ~3~(a) of Eq.(38). We choose again N - 100 since the computations according to Tables 3.4-1 and 4.4-1 appear about equally lengthy. Equations (2.3-2) or (3.4-50)
172
4 KLEIN-GORDON DIFFERENCE EQUATION FOR SMALL DISTANCES
moc _ At _ Ax A2 " AleAm0 ~ - li/moc 2 - h / m o c '
h h = Ac, c2 --TC moc mo
(47)
yield Ax
= 21r Ax
~2 = h/moc
)~C"
(48)
The choice moc2/ceAmo~ > 1 leads to the relation
moc A1 (49) e A2 For A3 = Ceo/cAmo~ in Eq.(2.3-2) we continue_ to prefer the simplest value A3 = 1. We choose Ae0 -- 10A1, which makes Ae0 as well as Amos of Eq.(49) vary proportionately to A1. Consider the set of plots in Figs.4.4-1 to 4.4-3. The parameters listed are the same for all three plots. The important parameter is A c / A x -- 104, which yields A2 =- 27r • 10 -4. We see three determined functions, which means there is no sign of randomness, except around a - 4-20 in Fig.4.4-3. In Figs.4.4-4 to 4.4-6 the parameter A c / A x = 5 x 104 produces z~2 - 47r • 10 -5. All three plots show the previous plots somewhat randomized. The plot of Fig.4.4-1 is still discernible in Fig.4.4-4. The randomization is much more evident in Figs.4.4-5 and 4.4-6. We go one step further and increase A c / A x to 105, which decreases A2 to 27r • 10 -5. The resulting plots in Figs.4.4-7 to 4.4-9 are now so randomized t h a t the patterns of Figs.4.4-1 to 4.4-3 are completely unrecognizable. The randomization by a resolution improvement from A x / A c -- 10 -4 to 2 x 10 -5 and further to 10 -5 is what one would expect from the Compton effect. The small values A x / A c = 10 -4 to 10 -5 are explained by Eq.(47) )~1 < ~2,
Amox < moc2/ce,
Am0x -
A2//)~l -- moc2/ceAmox
(50)
which for A1 - 0.1)~2 implies a very large potential Amos. We will expect a reduced binding force for smaller values of Am0x. Unfortunately, the computing time is increased by 50% if one replaces A1 = 0.1A2 with A1 --- 10-5A2 as is done in Figs.4.4-10 to 4.4-18. For this reason we could not investigate still" smaller values of A1/A2. Figures 4.4-10 to 4.4-18 hold for A1 = 10-5A2 . The parameter A c / A x -5 x 10 a was chosen so that one obtains determined func'dons in Figs.4.4-10 to 4.4-12, except for a slight randomization in Fig.4.4-12 around a - +20. The choice A c / A x = 1.25 x 10 a brings some randomization in Figs.4.4-13 to 4.4-15. Full randomization is obtained for A c / A x = 5 x 104 in Figs.4.4-16 to 4.4-18. The plots of Figs.4.4-19 to 4.4-27 have z~3 ~- 1 in Figs.4.4-10 to 4.4-18 replaced by )~3 - - 0 . 1 . W e obtain again the transition from determined to random functions. In addition, the vertical scales are reduced by factors between 10 -2 and 10 -14 .
4.4 EVALUATION OF THE ENERGY U~ FOR SMALL DISTANCES -40
-20
,
2 x
,
0
9
20
,
4O
9
-12
1 0
1
o
9.9
.";
-4x
1 0 -12
-6x
I0 -I
.
.
9
~
i
-20
,i
0
.....
oW
~o
i
-40
.
... 9 .
- 2 x 10
-
.
.
20
i...
40
FIG.4.4-1. Plot of 9T{1~(~) according to Eq.(21) for N - 100, /~e0 10 a, AI = 0.1A2, A2 -'- 2 r A x / A c , A3 = 1. -40 ,
4x 10-141 2x 10 -14
T
-20
o
0
,
20
,
,
o
9
o
" - : _L_,,~
10A1, A c / A x --
40 ,
~'~
99149 o";
0-14
----
9
:::- --~* 9149
-2x 10 -1 -4•
173
,,-,
" 9
.
.N
9
-6x i0 -14 -8x 10 -14 -40
-20
0
20
40
FIG.4.4-2. Plot of ~2~(~) according to Eq.(32) for N = 100, ie0 = 10A1, A c / A x = 10 4, A1 = 0.1A2, A2 = 27rAx/Ac, A3 = 1.
7 . 5 x i0-15~
-40 ' 9
-20 '
0 '
20 '
40 9"
5x 10 -15 2 . 5 x 10 -15 9
o -2.5x
10-15
_...
9
.
"~ ~
- 5 x I0 -Is -7.5x
I0 -1 4"
0"
'
- 2A0
0,
FIG.4.4-3. Plot of C~C3~(~) according to Eq.(38) for N 10 4, AI --O.IA2,
A2 -- 2 ~ A x / A c ,
A3 =
I.
20'
"
"
4'0
100, Ae0 - 10A1, A c / A x -
174
4 KLEIN-GORDON DIFFERENCE EQUATION FOR SMALL DISTANCES -40 Ix I0 T
9
o
20
9
40
-lx
10 -15~
."
"
.'.0"*"
l O -1~
9
9
." 2
.
9
:-o
........
-5x
5x
0
-16
5x I0
-I.
-20
-15
-
9
"
.9
......
9149
". ".
.
9149
-i
I0
-I
-2x I0
-,~o
-;o
~
~'o
o
.._-...+
4'o
FIG.4.4-4. Plot of ~ ( ~ ) according to Eq.(21) for N = 100, i e 0 = 10A1, A c / h x 5 • 104, A1 = 0.1A2, A2 = 2 r A x / A c , A3 = 1.
4x I0 I
x
-40
-20
,
-17
0
,
20
,
=
40
,
,
10 -17 o
9 .
9
o 9 .
-2x 10 -17 - 4 x 10 -17
. 9
.0
9
9
9 9
. 0 9
9 9
-,~o
9
:2o
o
4'o
2o
FIG.4.4-5. Plot of ~62~(~) according to Eq.(32) for N = 100, /l.e0 = 10A1, A c / A x = 5 x 104, AI = 0.1A2, A2 -- 27rAx/Ac, As = i.
-40
2x io
I
-20
0
20
40
ix I0 9
o
9 9" ' ~ ' - -
-Ix I0
0" 9
9
9 9 9o 9
. . 9
9
9 9
9
9e
-17
-~o
9 -"~
9 .
,0
-;o
o
i
20
i
40
FIG.4.4-6. Plot of 9:G,~(~) according to Eq.(38) for N = 100, Ae0 = 10A1, A c / A x = 5 x 10 4 , A1 = 0.1A2, ,~2 = 2 r r A x / A c , ,X3 = 1 9
4.4 EVALUATION OF THE ENERGY 0~ FOR SMALL DISTANCES
6x
-40
i0 -17
-20
,
0
,
20
,
175
40
,
,
4x 10-17[ I 2x i0-17[ "~
t i l l
9
9
9 l9
0
9
9149
~:~-2 x 10-17t 4x 10-17[ 6x
10 -171"
-40
-20
0
20
40
FI0.4.4-7. Plot of 97Q~(~) according to Eq.(21) for N = 100, /~e0 105 A1 = 01A2 ) A2 = 27rAx/Ac, Aa = 1 -40
-20
,
0
,
20
,
10A1, A c / A x =
=
40
9
9
-18
4x i0 S
o 9
2X i0
"~
9
o
o, "0"
0,
9 o, 9149I
o 9 9
-2x I0 -4x I0
9
9
,,
9
9
9
18
-18
o
2'6
40
FIG.4.4-8. Plot of 9762~i8) according to Eq.(32) for N = 100,-4e0 = 10A1, A c / A x = 105, A1 = 0.1A2, A2 = 2 w A x / A t , A3 = 1.
1.5x
-40
i x 10 -18
T ~
-20
,
0
,
,
20
40
10 -18 09
9
9
5x 10 -19 0
99 . 9
~--'...
5x 10-19 lx I 0 -is
-i 5x i0 -18
i
6
FIC.4.4-9. Plot of 9763~(~) according to Eq.(38) for N 10 5, A1 = 0.1A2, A2 = 27tAx/At, A3 - 1.
20
I
100, /]e0 -- 10A1, A c / A x -
176
4
KLEIN-GORDON
DIFFERENCE
-40
EQUATION
-20
FOR SMALL DISTANCES
0
20
40
3 x 10 -24
2X 10 -24 I
ix i0 -24
0 "~
........
..,.." 9
";,._.._:: _:_:_:: o9- ~
i0-24
N -1• -2x
10 -24
-3x
I0 -24
9 9
.,"% 9
9
-40
-20
9
0
2'0
4'0
FIo.4.4-10. Plot of J:Q~(~) according to Eq.(21) for N - - 100, A~0 = 10A1, A c / A x = 5 X 103, A1 = 10-sA2, A2 = 27rAx/Ac, A3 = 1.
2 x 10 -2E 1.5x
10
l x 10 -2~
"~"
5x 10 -27
10
-20
,
9
0
20
,
1
40 9
,
-2E
l
-5x
-40
o -2
,,. 9 "'"~o
%,, 9
,"-"" 9
9149
9
"9 ~ ' ' " ,
9
,,
-Ix 10 .26
-I. 5x i0-26
-40
-20
0
/'C ---'--*
2'0
"4'0
FIG.4.4-11. Plot of ~(:2~(~) according to Eq.(32) for N - - 100, fi.e0 - 10A1, A c / A x 5
x
10 3, AI = I0-5A2, A2 = 2 1 t A x / A t , A3 = i. -40 2x 10 -2
I
,
-20
0
,
20
,
40
Ix 10 -2
~ N
c
Oir.- ...........
OOe 9
-Ix I0
B 9149
-25
-2x 10 -27
-40
-20
0
I
i
20
40
FI0.4.4-12. Plot of ~ 3 , ( ~ ) according to Eq.(38) for N = 100, fi-e0 : 10A1, A c / A x : 5 x 103, A1 = 10-5A2, A2 -- 27tAx/At, A3 ---- 1.
4 . 4 EVALUATION OF THE ENERGY 6[,~ FOR SMALL DISTANCES -40
-20
,
0
,
20
,
177
40
,
,
6 x 10 -25 4 x 10 -25
I
"
'
"
"
" "%,, 9
2 x 10 -25
" "
o
M
~--2X
.............
10 -25
""
4 X 10 -25
"
""" '
6 x 10 -25
" 9
"
9
;
i
2'o
FIG.4.4-13. Plot of ~TQ~(~) according to Eq.(21) for N = 100, Ae0 = 10`kl, 1.25
x
104, `kl = i0-5`k2,
`k2 ~---2~'Ax/`kc,
-40 4 x 10
-27!
`kc/Ax
=
`kc/Ax
-
`kc/Ax
-
A3 = I.
-20
0
20
40
9
3x 10 -2 2 x 10 -2
I x i0
9
-2"~
9
~o, ~
9176 O0011~W~O0 9149
- I x i0
-2• I0
'
-27
-2"~ 9
]
I
o
20
9
I
FIG.4.4-14. Plot of 97~2~(a) according to Eq.(32) for N = 100, Ae0 = 10A1, 1.25 x 104, `kl = 10-5A2, A2 = 2 r A x / ` k c , `k3 = 1.
-40 4 x 10 -28
l
2x 10 -28
.~
o
,
-20
0
.
20
,
9 00
_::::::_..,0 Z ,."~
9
~ -2x 10-28!
40
,
00
,
9
9176 ,,o__ 9 ..... ...... ~
"%
,..-...
9
- 4 x 10 -28
-6x
.
_-28!
iu
-40
-20
0 ~---+
20
40
FIG.4.4-15. Plot of 9~3~(~) according to Eq.(38) for N - 100, Ae0 - 10`kl, 1.25 x 104, `kl = 10-5`k2, `k2 = 27rAx/`kc, ,ks = 1.
178
4 KLEIN-GORDON DIFFERENCE EQUATION FOR SMALL DISTANCES -40 ix I0
l
5x 10
,
-27
-20
0
,
-28
o
.
::::::==_:_,
9
9
9
9
,
40
.
,
9
9
90
9
20
,
.
0-28
-5•
.
0.
.0
,
9 .
. 9
9
.
J
o_ __.:===:= ;"
o
o
-Ix 10 -27 -I. 5x 10 -27
-~o
-;_o
~
~_'o
~'o
FIG.4.4-16. Plot of f Q ~ ( a ) according to Eq.(21) for N - 100, iL0 = 10A~, A c / A x = 5 x 104, Ai = 10-5A2, A2 = 2 r A x / A c , A3 = 1.
6x 10 4x 10 I
-29
-40
r
20
,
~~ *-"
"o-
9o
- 2 x 10 -2
9
o-
-2
- 6 x 10 -2
I
-;,o
-2o
~
T
2o
FIO.4.4-17. Plot of 9:62~(a) according to Eq.(32) for N 5 • 104, A1 = 10-5A2, A2 = 27rAx/Ac, A3 = 1. -40
-20
,
10
l
~.
,
o~
9
0
ix
40
,.
-29
2x 10 -29
-4x I0
0
-20
,
0
,,
|
20 .,
~'o 100, .4e0 = 10A1,
Ac/n~
=
40 ,
-2"
o 0 0
-lx I0 -2~ -2x I0
.... -~o
-;~o
" ~
2'o
4'o
'
F~0.4.4-18. Plot of ~3~(~) ~ o r d i n g to Eq.(38) for N = 100, Silo = 10A~, A c / A ~ = 5 x 10 4, A~ = 1 0 - 5 A 2 , A2 = 2 ~ ' A x / A o ,
A3 -- 1.
4 . 4 EVALUATION OF THE ENERGY U~ FOR SMALL DISTANCES
6x 10-26 'I'
-40
-20
,
4x 10 .26
20
,
40
,
9 9
10 -26
2x
0
,
179
,
o
9
o
o
0
~-2X
o9n ,
I0 -26
- 4 x
10
-26
-6X
10
-26
-~,o
, 9
-~_o
o
o
2'o
4'o
FIG.4.4-19. Plot of 9TQ~(~) according to Eq.(21) for N = 100, ,J-e0 = 10~1, A c / A z = 5 X 103, A1 -- 10-5A2, A2 = 2 = A x / A c , A3 = 0.1.
4x 10-28[
l
-40 ,
-20
0
,,
20
,
40
1o
,
2xl O~O
O90 9
9
.....,----
9149149
0_281.... ~:~-lX2x110_28i 3x 10-28~
-4o
-~o
|
o
9
20
i
40
FI0.4.4-20. Plot of ~}C2~(m) according to Eq.(32) for N = 100, i e 0 = 10A1, A c / A x = 5 x 103, A1 = 10-5A2, A2 - 2 7 r A x / l c , A3 = 0.1.
-40 4x
10
-20
0
20
I 2x 10-29
9
9
9
o..
~
40
-29
9
.oo
o 90,~, o 9
-2x 10 -4x I0
9co 9
9
9
-29
-29 |
-40
i
-20
!
o
2'o
F10.4.4-21. Plot of ~ 3 ~ ( ~ ) according to Eq.(38) for N 5 • 103, A1 = 10-5A2, A2 = 27tAx/At, A3 = 0.1.
40
100, /ie0 -- 10A1, A c / A x -
180
4 KLEIN-GORDON DIFFERENCE EQUATION FOR SMALL DISTANCES
I
3x
10
2x
i0
-40
-2O
r
-2
0
, .
,
20
40
i
40
-2
l x I0 -2
,-4
ow' 9 9 -lx
10
9
-2
- 2 x 10
-2
- 3 x 10
-2
9
i
-;,o
-2o
.
9
;
i
2o
FIG.4.4-22. Plot of A I , ( ~ ) according to Eq.(21) for N = 100, A-,0 = 10)h, A c / A x 1.25 x 104, A1 = 10-sA2, A2 = 2~rAx/Ac, A3 = 0.1.
-4O
T
t
lxl
20
40
'
'19t 3 /
Oljl 9
.~, 5x 10-"" r
9
9
O09O
"%.
~l
-5x 1 -ix
0
-2O
=
i0-29~
,
-40
9
i
i.
-20
94 I
n
0
/g,
20
-.--.~
0
FIC.4.4-23. Plot of ~ 2 , ( ~ ) according to Eq.(32) for N = 100, .2..0 = 10A1, A c / / k x = 1.25 x 104, )h = 10-5A2, A2 = 2 r r A x / A c , A3 = 0.1.
T
-40
-20
i
i
0 -
w
-
20
40
,
,,
l x 10 -3c o
------==-=-0. ,!*
*l -9. o 9~ 9
- l x 10 -3c
0-- -
.co 9176 9
- 2 x 10 - 3 0
-;_o
2'0
4'0
F[c.4.4-24. Plot of ~(:3~(a) according to Eq.(38) for N = 100, ii.,0 = 10A1, A c / A x = 1.25 x 104, A1 = 10-5A2, A2 = 27rAx/Ac, A3 = 0.1.
4 . 4 EVALUATION OF THE ENERGY U~ FOR SMALL DISTANCES -40
6x 10 -30
-20
0
20
181
40
4x 10 -30 T
2x 10 -30
&
9
,
,
9
~
9
9 9
o
,,
"9 m , , % .9
-2x i0 -30
9
9
,, t;.,,
9
9
,
,
9
-_-_-T=::::-
.0
9
- 4 x 10 -30 - 6x
10
-30 i
n
-40
I
-2 0
|
0 O{ ----~
I
20
40
FIG.4.4-25. Plot of 9:6~(~) according to Eq.(21) for N = 100,/]e0 = 10Ax, A c / h z = 5 x 104, A1 = 10-5A2, A2 = 27rAx/Ac, A3 = 0.1. -40
I
2xI0-31~
0
,
,,-"'o~" o"
o
~ _2x10 -31
-20
,
6x i0-311 4x 10 -31
. 9. . . . . ...........
"0
'
20
,
"%
,,
o
o
9
,
40 ,
v
,
" 9
9
7 " *, --" 9o o
" 9
.
4x I0-311 6x I0-311 "
9
-40
i
i
-20
i
0
20
K; - - - - - *
4'0
FIG.4.4-26. Plot of 9:62~(~) according to Eq.(32) for N = 100,/]e0 = 10A1, A c / h x = 5 x 10 4, AI = I0-5A2, A2 = 27rAx/Ac, A3 = 0.i.
Ix i0
-31
-40 ,
-20 ,
0 ,
20 ,
40 ,
5x 10 -32 i
~
o 9
9
9
9
9
o
N -5x i0
-32
-Ix I0
-31 i
-40
o
2'0
4o
FIG.4.4-27. Plot of 9~3~(~) according to Eq.(38) for N - 100, Ae0 - 10A1, A c / A x -5 x 10 a, A1 : 10-5A2, A2 -- 2 ~ A x / A c , A3 = 0.1.
182
4 KLEIN-GORDON DIFFERENCE EQUATION FOR SMALL DISTANCES
FIC.4.4-28. The real part of the function /~s~(tg,~,~) of Eq.(17) for Fig.4.4-1 with ~ c / A x = 104 (a) and for Fig.4.4-7 with .~c/Ax = 105 (b). W h a t causes the transition from a determined to a random function? A comparison of Tables 3.4-1 and 4.4-1 shows that the computations start in both caes with jb~, I T ( n / N ) , and ~0(~,0n). Then they become different until Hs~(0n, ~) in Table 4.4-1 is reached. From there on the computations are again equal or essentially equal. A plot of/~rs~(0n, n) according to Eq.(17) is shown for t~ = 10, 11, 12 and 0n - 0, 1, ... , 100 in Fig.4.4-28a for N = 100, Ae0 = 10A1, . ~ c / A x = 104, A1 = 0.1)~2, )~2 = 27rAx/)~c, )~3 = 1; these are the values of Fig.4.4-1. The plot in Fig.4.4-28b holds for the same values of the parameters, except for a change A c / A x = 105; these are the values of Fig.4.4-7. It is evident that the plot of Fig.4.4-28a changes very smoothly with ~ and 0. The plot in Fig.4.4-28b, on the other hand, changes radically both with ~; and with 0. This makes the sums Jj(0, ~) for 0n = n in Eq.(6.4-48) change like random numbers. We have shown 27 plots in Sections 3.4 and 3.5 but postponed a justification to Section 4.4. These plots did not show any randomization like Figs.4.4-1 to 4.4-27. This is the reason why we produced so many plots there. The long calculations and the many equations in Table 4.4-1 for the computer program make it very hard to produce Figs.4.4-1 to 4.4-27 without error. It will take time and luckless PhD students to verify or correct the plots. This is the bad news. The good news is that the result of randomization requires ~ s ~ ( 0 n , ~ ) , which is the twentysixth equation listed in Table 4.4-1. But one needs 72 equations to compute ~7~l~(g), ~2~(t~), and ff-~3~(/,~). Hence, the result of randomness demonstrated by Fig.4.4-28 is much less error prone than Figs.4.4-1 to 4.4-27.
ADVANCES IN IMAGING AND ELECTRON PHYSICS, VOL. 137
5 Difference
Equations
in Spherical
Coordinates
5.1 C H A R G E D P A R T I C L E IN AN E L E C T R O M A G N E T I C F I E L D
According to Section 1.3 we may write the Lagrange function of an electrically charged particle in the following form using spherical coordinates:
= Lrer + Loeo + L~e~ = ~-~M -~- J~-~c --
(1
-~mv
+ e v • A m - eCe
)(100) 0
1
0
0
0
1
+
0
0)
0
Lcz9
0
0
0
L~
(1)
The functions Lc~, Lc~, Lc~ are shown in Eqs.(1.3-33)-(1.3-35). We shall need relations between the moments p~, pa, p~ and the variables r, ~, ~. The moments are the derivatives of the components L~, Lo, L~. We get from Eqs.(1), (1.3-29), and (1.3-33)-(1.3-35): OL~ Ze Pr = O~ = mi~ + eAm~ + ~(r~A~c Ze i)Lo = m r ~ + eAmo + ~ ( r c Po = O(rO) OL~ P~= 0(rsin~)
- r sin t9 ~bAee)
(2)
sin 0 (pAer -- ?~Ae~)
(3)
Ze = m r s i n O ( D + e A m ~ +~(i~A~c
- r(gAe~)
(4)
The following relations will be used:
r = re,. + r~ea + r sin 0 ~e~ v = i- = §
+ r~e~ + r sin 0 @e~
p = p~e~ + poe~ + p~e~ Am - Am~er + Amoeo + Am~e~ Ae - Ae~e~ + A~oe# + A ~ e ~
(5)
183 ISSN 1076-5670/05 DOI: 10.1016/S1076-5670(05)37005-4
Copyright 2005, Elsevier Inc. All rights reserved.
184
5 DIFFERENCE EQUATION IN SPHERICAL COORDINATES
We solve Eqs.(2)-(4) for ~, r0, r sin ~)~b as functions of p~, p~, p~. shorten the equations we define a common denominator D:
Ze)
D=m
c
i" -- --D
(A2,. + A~ + A2~,) + m21 = m
Zero C
-
eArns) - Ae~(po -
(p -- eAm)r +
eAmo)] + m2(pr -
Zero C
(-~)
1 m2(p -- eAm)~ + D
m
x (p - eAm)]r
--
c
eAm~o)] + m2(p,~ -
]
(7)
Zem
c
] eAm,~)
A e o A e - ( p - eArn) -{-~[Ae c
• (p-
earn)]0
]
(8)
Ae~[Aer(pr -eAmr) + Ae~(p~-eAmd) + Ae~(p~-eAm~)]
Zem + ~ [CA ~ ( p o -- ~
}
2Aeo[Aer(P r - e A m r ) + Ae~(p~ - e A m ~ ) + Aecp(pcp -eAmcp) ]
Zero + ~[Ae~o(pr - eAmr) - Aer(p~o C
rsin0~b = ~
eAmr)
AerAe- (p - eArn) + ~[Ae
1
(6)
A2
Aer[Aer(Pr - eAmr) -[- Aed(P~ - eAmo) 4- Ae~o(P~o - eAm~o)] + ~[Aeo(p~
-- ~
m 2 4-
To
- eAmd) - A~o(p~ -
(p-eAm)~o+
--
c
eAmr)l 4- m2(p~o -
] eAm~o)
Ae~oAe.(p-eAm) + - -Zem - - - [ A ~ x ( p - eAm)l~ 1
(9)
C
Equations (1.3-33)-(1.3-35) contain the second derivatives f, ~, ~ in addition to the first derivatives § 0, ~b. We differentiate Eqs.(7)-(9) with respect
5.1 CHARGED PARTICLE IN AN ELECTROMAGNETIC FIELD
to t, keeping in mind that dependent"
~=~1
m2p~ +
Am, Ae, and
D in Eqs.(5) and (6) are not time
gem (Ae#~9~o - Ae~9o ) C
) 2Aer(Aerpr + Ae~iS~ +
+ Zero
1 m219r + D
185
c
1[
(A~ x p)~ +
Zero
r;~= -~ m2pe +
c (A~r
)
2AerAe
(10)
- Ae~P~o) + (-Z~) 2A~(A~pr + A~p~ + A~P~)]
1 m219~+ D
gem (Ae • C
p) +
)
(11)
2AegAe
r sin 0 ~5 = ~
1[
m2i5~+
Zero
1
m2p~ +
gem (Ae x p)~ + ( - ~ ) 2Ae~Ae " 15] c
D
c (Ae~iS~ - Ae~iS~)
(12)
Since the Lagrange function/5 of Eq.(1) has three components the associated Hamilton function ~ must have three components ~k too: 3 ~:~(pj,~j,t) = ~pj~j
- ~,
k = ~, ~, ~
(13)
j=l
Equations (2)-(4) yield the following result since the terms multiplied by Ze/c cancel: 3
E pjxj -- Pr§ + p#rO + p~r sin ~ j=l
-- m[? 2 + (r0) 2 + (r sin = m i "2 + e A m
9i"
vq ~b)2] +
e(Amri" + AmorO + Am~r sin
v~~b) (14)
186
5 DIFFERENCE EQUATION IN SPHERICAL COORDINATES
The three components ~Kk are obtained from Eqs.(1) and (1.3-33)-(1.3-35); the terms Amr§ + Amor(9 + Am~r sin 0 ~b cancel:
1 .2 + r2~2 + r2 sin 2 ~9~2) + eCe - Lcr = ~1m r .2 + eCe - f~cr ~Kr = -~m(r
(15)
9~0 = ~1m ( ~ 2 + r2(~2 + r2 sin 2 ~9@2) + eCe - f~cO = ~1 mi'2 + eCe - ~cO
(16)
1 2 + r2 ~2 + r 2 sin 2 0 ~2) + eCe -- Lc~o = ~rnr 1 92 + eCe - Lc~ 9s = ~m(~
(17)
The Hamilton function should be written in terms of the momentum p and the potentials r era, Ae, Am. This can be done by the substitution of ?, r0, .rsin~9~b, ~, r~, and r s i n 0 ~ from Eqs. (7)-(12) into Eqs.(15)-(17) as well as into Eqs.(1.3-33)-(1.3-35). Considerable effort is required. Let us first gain some understanding of the Hamilton function. To this end we consider Eqs.(2)(4) and investigate under which conditions the terms multiplied by Z e / c will be small. Equation (2) yields the conditions
(Ze/c)r(gA~ << m§
(Ze/c)rsin 0 (oAeo << m?
(18)
which can be rewritten as follows:
mc 2 >> ZecAe~orO/i"
and
mc 2 >> ZecAe#rsinO~o/i"
(19)
In case § is close to zero while rO and r sin ~ ~b are not we require an alternate condition for Eq.(2)" Amr >> ZAerrO/c
and
Amr >> Z A e o r s i n O ~ / c
(20)
Equations (19) and (20) state in essence that the energy due to the potential Ae should be small compared with mc 2 or the energy due to the potential Am. More detailed statements referring to the terms O/~b, r sinO@/§ rO/c, and rsin0~b/c are not of interest here. Relations equivalent to Eqs.(18)-(20) may be derived from Eqs.(3) and (4) too. Equations (2)-(4) may be simplified in this case:
Pr = m i~ + eAmr, pe = t a r o + eAmo, p~o = m r sin ~9~ + eAm~o D-m 3
(21)
(22)
Equations (7)-(9) become: 1 1 1 -- - - ( p - eArn)r, r~ = - - ( p - eAm)~, r sin ~ ~b = m ( p - eAm)~ m
m
(23)
5.1 CHARGED PARTICLE IN AN ELECTROMAGNETIC FIELD
187
The three components of the Hamilton function are obtained in the form
1
)2
+ eCe -- Lcr
(24)
1 )2 9~0 = ~-~m(p -- eArn + eCe -- Lc0
(25)
1 9(~o = K-~ (P - cAm) 2 + eCe -- Lc~o
(26)
~J-Cr-- ~ m ( p --eAm
~'llt
These are the terms of the conventional Hamilton function of an electrically charged particle in an electromagnetic field plus correcting terms Lc~, LcO,Lc~o. The relativistic variability of the mass m is not taken into account in Eqs.(24)-(26) due to the term mv2/2 in the Lagrange function of EQ.(1). This simplification permits us to obtain the components 9~, 9~o, 9 ~ of the Hamilton function and the correcting terms L ~ , Lc0, Lc~ explicitly. The relativistic variation of the mass m will be taken into account in the following Section 5.2. The Hamilton function can then be represented by means of series expansions only. All the deviations from the conventional values in Eqs.(2)-(4) are due to the potential Ae that is caused by magnetic monopole, dipole, or multipole current densities gm according to Eq.(1.1-19). The correcting terms Lc~, Lr Lc~ of Eqs. (1.3-33)-(1.3-35) contain mainly terms Ae but the hypothetical magnetic charge Pm enters through the terms 0 r 0~bm/0~), 0~bm/OO~ according to Eq. (1.1-22). We turn to theevaluation of Eqs.(15)-(17) without approximation. Using Eqs.(6)-(9) we obtain 9s in vector notation:
1
~1: ~- ~m (p --cAm)2 -k +
~c
~mc
{2[Ae' ( p - eAm)] 2 q-[Ae • ( p - cAm)] 2}
Ae2[Ae" (p - eAm)]2
1+
~c
A
+ eC~e -- ~-~c
(27)
The vector Lc has the three components Lcr, Leo, L ~ of Eq.(1). Each of them consists in turn of five components. For the first component Lcrl we obtain from Eq.(1.3-33):
Ze
Ze (
c
m2c
+ --{Ae~,[Ae • (p - eAm)]O - Aeo[Ae • (p - eAm)]~,} mc
)
188
5 DIFFERENCE EQUATION IN SPHERICAL COORDINATES
[
Ae~Ae" (p -- earn)
X (p -- eAm)r -t- ~cc
IE
e'] '
+--[Ao~ x(p-~Am)]~ 1+ ~
A
(28)
The second component Let2 is defined by Eq.(1.3-33) as follows:
Lr
= -Z-e / ( c Ze /
: m2 c
00Cm)d r
sin 0 ~b0r 00
sin 0 0~
[10r
10Cm(p_eAm)~ - - ~ - ( p -- eAm)~ rsin0 0~
Ze (-- 1 (~r +--mc _-r - ~ - [Ae • ( p - eAm)]~
+ ( ~ c ) 2 (10r" - ~
1
0era [Ae x (p
r sin ~ 0~
-
-
rsinv 1 0 qC~--m 0- A e ~ ) A e . ( p _ e A m )
eAm)]O ]
• [1+ ( - ~ c ) 2 A 2 l - l d r
(29)
For the third component ~cr3 we obtain from Eqs.(1.3-33), (6), (11), and (12):
ze /( rOAe~ -
Lc~3 -- - -
c
-- c"D
r sin ~ (SA~o )dr
m2(Ae x l~)r + ~ [ ( A e C
x p)~ - ( A e x p)~] dr
(30)
The fourth component Let4 is very long. It follows from Eqs.(1.3-33), (6), (8), and (9):
Lcr4 - - -
0
C
Ze
-- m2c +
--
?TtC
+ ~
(rOAe~ - r sin 0 (oAeo )dr
(p - eAm)g +
m
(p - eAm)e[A~ x (p - eAm)]e
{[A~ x (p - eAm)] 2 + 2A~o(p - eAm)oAe. (p - eAm)} + 2 ~cc
A~e[Ae x ( p - eAm)],~A~. (p - earn) +
~cc
A~o[A~" ( p - eAm)]20(vO)
5.2
RELATIVISTIC
189
MASS VARIATION
Ze (p - eAm)o [Ae x (p + (p - eAm)o(p - eAm),r + "--{
eAm)],,o
mc
+ (p -- eAm),,o[Ae x (p - eArn)]o} +
~
{[Ae x ( p - eAm)]o[A~ • ( p - eAm)]~o + A ~ [ ( p - eAm),~ + (p - eAm)~o]Ae- (p - eArn)}
+
~
{Aeo[Ae x ( p - eAm)]~o + A~o[A~ x ( p - eAm)]0)Ae" ( p - eArn) +
~cc
]( OAe~
Ze (p - eAm)~ [Ae • (p (p - eAm) 2~o+ 2 mc
+
+
~c
~cc
-
-
)
eAm)]
{[Ae • (p -- eAm)] 2 + 2Ae~,(p -- eAm)~Ae- (p -- eArn)} + 2 ~c
+
cgAe~
AeoAe~o[Ae"( P - eArn)] 2
Ae~o[A~ x (p - eAm)]~oA~" (p - earn)
Ae2~[Ae'(p-eAm)]2
OAeo O(r sin ~ ~)
1+
~
mc
Ae
dr
(31)
The fifth and last component Lcr5 in Eq.(1.3-33) remains unchanged:
Lcr5=Zec/lO ( sin ( Avqe ~ s i n v0qv)9r0 A e ~ d r O ~
(32)
The components of the correcting terms Leo and Lc~ may be obtained by analogy but we shall not write them here. 5.2 RELATIVISTIC MASS VARIATION Equations (5.1-1) and (5.1-27) show a constant mass m but the Lorentz equation (1.3-1) permits a relativistically variable mass. The introduction of the rest mass m0 yields 1 the equation of motion d'-~(1
v 2 / c 2 ) 1/2 --
-
(
e E + --Vc x H
/
(1)
and the conservation law of energy' d
m0 C2
dt (1 - v2/c2) 1/2 1Harmuth et al. 2001, Sec.3.3.
=
E.v
(2)
190
5 DIFFERENCE EQUATION IN SPHERICAL COORDINATES
A four-vector can be defined with three spatial components p
p=
?-nov
(1 -v2/c2) 1/2
v = ~e~ + r~e~9 + r sin ~9qbe~ p = pre~ + poeo + p~e~
for spherical coordinates
(3)
and the component P4
moc E P4 = i(1 - v2/c2) 1/2 = i-c
(4)
where E denotes an energy rather than the magnitude E of an electric field strength. The connection between energy E and momentum p is provided by the formula
[C2 2 __ --?Tt2C2
p.p
or
[ = (p2c 2 + m2c4)1/2 ~
m ---~
(1
mo
-
v2/c2) 1/2
(5)
The relativistic generalization of the conventional part of the Lagrange function of Eq.(5.1-1) is defined by
f-~M "-" -m0c2(1
-
v3/c2) 112 + e(-r
+ A m " V)
(
-
-
i'2 + r2~2 + r2 sin2 v~(p2 ) 1/2 --too c2 1 -c2 + e(--r
1 v2 + e(-r - - m o c 2 + -~rno
+ Amr/" +
+ Am . v)
Am~r~ + Am~r sin 0 ~)
forv2/c 2<<1
(6)
We adopt this generalization of the part ~--JM of the Lagrange function L of Eq.(5.1-1). The components of the correcting term s are left unchanged from their definitions in Eqs.(1.3-33)-(1.3-35) since the mass m does not occur there and the potentials era, Ae come from a relativistic theory. The relativistic generalization of the Lagrange function of Eq.(5.1-1) is
= - m o c 2 (1 - v 2/c2) 1/2 + e ( - r
+ Am" v) h- ~c
(7)
where in spherical coordinates s has the components Lcr, Lco, Lc~ and s has the components Lr, Lo, L~. The nonrelativistic momentums p~, Po, P~ of Eqs. (5.1-2)-(5.1-4) are generalized to 'relativistic canonical momentums'. They assume the following form:
5.2 RELATIVISTIC
191
MASS V A R I A T I O N
moi" Ze + eAmr + ~(r~)Ae~ - r sin~ ~bAe~) (8) (1 -v2/c2) 1/2 c
OLr Or .=
OL~
mor~
Ze + eAmo + ~c( r sin ~ CAer - §
Ze OL ~ mo r sin ~) ~b + eAm~o + ~ ( i - A e # P~ = 0(rsintg~b) = (1 v2/c2) 1/2 c
- r~Aer)
(9) (10)
-
For the derivation of the Hamilton function from the Lagrange function we have Eqs.(5.1-13) and (5.1-14) but we must use Eqs.(8)-(10) for Pr, p~, p~ 3
p j x j -j=l
m0
(1-v2/c
2)1/2
(§ +r2~2
+
r2
sin2 ~
~2
)
+ e(Amri" + Amor~ + Am~orsin vQ~b) =
m0
i.2 + eArn 9
(11)
(1
and the three components ~ k of the Hamilton function are obtained in analogy to Eqs. (5.1-15)-(5.1-17)' ~]'~r ~
-
TFtOC2
,
[1 -(§
(12)
+ r2~2 + r 2 sin2 vq~b2)/c2J mo c2
~-~z9 ~
r
[1-
-11/2 -~- eCe -- ~'~cz9
mo c2
[1 -
(13)
(§ + r2~2 + r 2 sin2 ~)~b2)/c2J
(~2 + r2~2 + r 2 sin 2 vq~b2)/c2]
1/2
+ eCe
Lc~
(14)
The variables § ~, ~b and their derivatives ~, ~, ~ must be eliminated. The variables § ~, ~b are defined by Eqs.(8)-(10). They differ by the term ( 1 - v 2 / c 2 ) 1/2 from Eqs.(5.1-2)-(5.1-4). As a result we no longer have a system of three linear equations. The replacement of § ~, ~b becomes more difficult than in Section 5.1. As before we strive to get first an understanding by using simplifying assumptions for Eqs.(8)-(10). In Eq.(8) the terms multiplied by Ze/c will be small if the conditions
Ze ~r~)Ae~ <<
too§
and
Ze ~ r sin ~ ~bAe~ <<
too§
(15)
192
5 D I F F E R E N C E EQUATION IN S P H E R I C A L C O O R D I N A T E S
are satisfied. These conditions may be rewritten" moc 2 (1 --
V2/C2) 1/2
m o c2
>> ZecrOAe~/i ~ and
(1 --
V2/C2) 1/2
>> Z ecr sin 0 ~A~o / § (16)
For small values of ~' but not rO and r sin ~ ~b we require alternate conditions for Eq. (8): Amr >> ZrOAev/c
Amr >> ZersinO~Ae#/C
or
(17)
Equation (16) states in essence that the energy due to the potential Ae should be small compared with the energy moc2/(1 - v2/c2) 1/2 whereas Eq.(17) demands that the magnitude of A~ should be small compared with the magnitude of Am. With these simplifying assumptions we obtain from Eqs.(8)-(10): m0§ +eAmr (1 - v2/c2) 1/2
=
Pr
(18)
m0r0
Po = (1--V2/C2) 1/2 + eAmo m0r sin 0 ~b
P~o =
v2/c2) 1/2
(1 -
+ eAm~o
(19) (20)
Solution of these equations for § r0, and r sin 0 ~b yields" + =
(1 -
v2/c2)1/2
(p - eArn),.
mo r~
=
(1 -
v2/c2) ~/2 (p
- eAm)o
(22)
v2/c2) 1/2-'" (p - eAm)~
(23)
m0
r sin 0 ~ = "(1
-
(21)
mo
Squaring and summing § tO, r sin ~ ~ yields:
§ + (r0)2 + (r sin ~ ~b)2 = v 2 = 1 (p - eArn) 2 =
m2ov2 1 - v2/c 2 =m~
-
v2/c 2 m2 (P - earn) 2
( 1 - v 21/ c 2 - 1 )
m 0c2
mo C2
(1 - v 2 / c 2 ) 1/2
1 - (+2 + r2~2 + r 2 sin 2 ~ ~2)/c2 ] 1/2 = c[(p - eArn) 2 + m2c2] 1/2
(24)
5.2 RELATIVISTIC MASS VARIATION
193
The last line of Eq.(24) is substituted into Eqs.(12)-(14):
~I~r -- c[(p -- eArn) 2 -b m2c2] 1/2 "~- eCe -- ~'~cr
(25)
~ 0 -- c [ ( p - eArn) 2 + m2c2]1/2 4c-eCe -- ~cz9
(26)
~
(27)
= ~[(p - ~ A ~ ) ~ + - ~ ] ~ z ~
+ ~r
- ~c~
If we leave out the correcting terms Lcr, Lc#, Lc~ we have the conventional relativistic Hamilton function for an electrically charged particle in an EM field. The assumption we had to make to obtain Eqs.(25)-(27) was that A~ must be sufficiently small. If one wants to leave out the correcting terms Lc~, Lc~, Lc~o one gets more complicated conditions than Eq.(17) since Eqs.(1.3-33)-(1.3-35) contain Cm and its derivatives in addition to A~. Let us turn to the solution of Eqs.(8)-(10) for § r0, rsin0~b without simplifications. The term (1 - v2/c2) 1/2 = [1 - (§ + r2~2 + r2sin 2 ~(02)/c211/2 makes these equations nonlinear while the corresponding Eqs.(5.1-2)-(5.1-4) of the nonrelativistic theory were linear. There is no standard method for the solution of a system of nonlinear equations and we must find a method suitable for the case at hand. As a first step we ignore that v2.is a function of § r0, rsin0~b and treat Eqs.(8)-(10) as a system of linear equations. We note that Eqs. (5.1-2)-(5.1-4) are transformed into Eqs.(8)-(10) by the substitution .~ ~
t o o ~ ( 1 - v 2 / c 2 ) ~/2
and we get our 'first step of solution' of Eqs.(8)-(10) by making the same substitution in Eqs.(5.1-6)-(5.1-9). The common denominator D of EQ.(5.1-6) assumes the following form:
D
he=
__
ZecA~ "~0~ ~ '
I (v2)
m~ 1 + c ~2 1-- ~-~ (1 -v2/c2) 3/2 a~
(V2) 1-
1/2 ~
=
ZecA~ , ~ 0 ~ / ( 1 - v21c2) li2
(28)
(29)
The constant oLe represents the ratio of the energy due to the electric vector potential Ae and the rest energy of the particle. The reference energy should actually be moc2/(1- v2/c2) 1/2 rather than moc 2 but we shall need v as an explicit variable. The term C~e has no physical dimension but it is a variable due to its component Ae. The solution of ~, r~, r sin 0 ~b as functions of Pr, P#, P~o has the following form if the factor ae of Eq.(29) is used:
5 DIFFERENCE EQUATION IN SPHERICAL COORDINATES
194
(1-v2/c2)1/2 mo
1/2 [Ae • (p - eArn)It
[ (v2) (p - eAm)~ 1 + ae 1 - ~
( 7) v2
21--
+~~
rO - (1-v2/c2)1/2 mo
A~(p - eAm)~
~ - p - ~Am)~
][
[
1/2 [Ae • (p - eAm)]9
A~rA~.(p
eArn)
(v2)
(p - eAm)o 1 + O/e 1 - ~ 2 1 _ v2
+OLe
~
mo
[
( )
V2 1/2 A~vA ~ (p
+O/e2 1 - ~
7~
1 + a~2 1 _
Ae2(p- eAm)~
(v2)
(p - eAm)~o 1 + ae
1--
~e
(30)
A~(p - eAm)d
A ~ A ~ . ( p - eArn)
rsin~9~~ (1-v2/c2)1/2
2
1+
1/2[Ae x (p-eAm)]~
1 - ~-
eAm)
Ae(p - eAm)~o 2
1+
Ae2(p - eAm)~
(31)
(
O/e 1 - ~'~
(32)
Squaring and summing Eqs.(30)-(32) yields: §
2sin 2 ~ @ 2 = v 2 _ 1 - v 2 / c 2 -- rn---------~ (p -- eArn) 2 1 ( +2ae
1-~
v 2 ) 1/2 [Ae 9(p - eAm)] 2 Ae2(p_eAm) 2
2 /~ 1 _ V2'~ [ t e x ( p - eArn)]2 + 2 [ t e - ( p +c% ~-~) Ae2(p _ eArn) fi
(
A3(p _ eAm) 2
+ 2a 3 1 - ~-
(
v2)2[Ae'(p-eAm)]2
4 1 -+ c% c'~
eArn)] 2
A~(p
-
[
eArn) 2
(
1+21 ae
-
v2)] -2
(33)
To find an approximate solution of this equation for ae(1 -
v2/c2)1/2 << 1
(34)
we start by using only the first term on the right side of Eq.(33): v2 = 1 - v 2 / c ~
m02
)2
(P -- e t m moc
[m02c2 + (P -- eAm)2] 1/2
(35)
195
5 . 2 R E L A T I V I S T I C MASS VARIATION
This is the same relation as shown in the first line of Eq.(24). We again get the common term of the three components 9it, 9~o, ~ of Eqs.(25)-(27). We call this the zero-order solution in a~ of Eq.(33). To obtain the first-order solution in Ole of Eq.(33) we use the first two terms on the right side of Eq.(33). Since p will eventually be replaced by difference operators we preserve carefully the sequence of the factors: 1 -v2/C2 )2[ ( v 2 ) l / 2 [ A e ' ( p - e A m ) ] 2] m] (p - eArn 1 + 2Ole 1 - ~ X f ( p _--e-~miS-
v2 -
(36)
The term (1-v2/c2) 1/2 multiplied by 2ae is replaced by the zero-order solution represented by Eq.(35). The resulting equation is solved for 1 - v2/c 2 and the following improvement of Eq.(35) in first order of a~ is obtained: 1
v2
c2
=
m02c 2
m~c 2 + ( P - eArn) 2 ( 2~ x 1 [m2c2 + (P -
[Ae 9(P - e A m ) ] 2 ) eAm)2]3/2 "~e2(p'__- e-~m)~ q- O(O~e2)
_
(37)
The first-order approximation in ae of Eq.(24) follows from Eq.(37):
mo c2 • (1+
aem~ [(p
-
[Ae'(p-eAm)]2) eArn) 2 -{- ,m02C2]3/2 t'e2"(p -- ~'~m)2
-
+ O ( a e2)
(38)
The three components of the Hamilton function of Eqs.(12)-(14) become in this approximation: 9t:r -- c[(p -- eArn) 2 +
m20c2]1/2(1 + o~eQ) +
er
-- f~cr
(39)
er
-- f"~cO
(40)
9t~ ---- c[(p -- eArn) 2 + m02c211/2(1 + aeQ) + er
-- f~c~
(41)
9(0 - c[(p
Q __
-
-
e A m ) 2 + m02c2]1/2(1 + aeQ) +
(p - eAm)2[Ae 9(P - e A m ) ] m2oc2 [1 + (p eAm)2/m2c2] 3/2 Ae2(p -
Ol e
"--
ZecAe mo c2
-
Ze 2 h Ae 2h moc e
--" 2 - -
~
2
0
-
-
2 eAm) 2
AcA~ l ~ e
a~ =2.210• 105A~ for electron, a~= 1.204• 102Ae for proton, A~[As/m] a =
Ze 2 h - 7.297 535 • 10 -9 fine structure constant, Ac = 2h moc
(42)
196
5 D I F F E R E N C E E Q U A T I O N IN S P H E R I C A L C O O R D I N A T E S
For the evaluation of the terms Lc~, LcO, Lc~o we need ~, r~), r sin v9qb of Eqs.(30)-(32) with the terms 1 - v2/c 2 eliminated by means of Eq.(37). Only the zero-order approximation in ae is required: ?_
c(p - eArn). +O(ae) [m2c 2 + (P - eArn)2] 1/2
(43)
r(9 =
c(p - eAm)o + O(ae) [m02c2 + ( P - eAm)2] 1/2
(44)
r sin t0 ~b --
c(p - eAm)~o + O(ae) [m02c2 -I- (p - eArn)2] 1/2
(45)
For the first component Lr
of Lcr we obtain from Eq.(1.3-33):
Ze L c ' l = - - i ' ( r O A e ~ - r s i n O (oAeO)= x
ae
Ae~o(p-eAm)o-Aeo(p-eAm)~o [1 9-(p - eAm)2/m2c2] 1/2
(p - eAm)r + O(a2e) (46) [1 + (p - eAm)2/m2c2] 1/2
The unusual way of writing the denominators is due to the replacement of p by operators at a later time and the resulting noncommutability of the factors of a product. This will become important in the following sections. The second component Lcr2 is defined according to Eq.(1.3-33) as follows:
Ze/(
Odflm 00dflm)dr
Let2 = - c
ae J -- A"~
sin vq~b
00
si~ 0 0~o
(0r (p -- eAm)~o O0 r
0era (p -- eArn)o) 0~o r sin 0 .
For the third component Lc~3 of and ~ of Eqs.(44) and (45):
Ze/
Lcr3 - - -
C
dr
m2c2
Eq.(1.3-33) we
require the time derivative of
(rOAe~ - r sin t~ (hAe~ )dr
__/[..~ (oo, (O ' - A~o cot
rn0r
9
(47)
[1 + ( p - eAm)2/rn~)c2] 1/2
]
5.2 RELATIVISTIC MASS VARIATION x
197
(p - eAm)~ [1 + (p - etm)2/m2c2]
dr
(48)
dr
(49)
1/2
The fourth component Lcr4 of Eq.(1.3-33) equals:
Lcr4 = - -
0
Oe/(
-A~rno -
+ ~
(r~)Ae~- r sin ~ ~A~a)dr
(p _ e A m ) a l 0 1 0 r ~ + (p - eAm)~ r sin-------0099
)
x A ~ ( p - eAm)a - n e a ( p - eAm)~
The fifth and last component Let5 of Eq.(1.3-33) remains unchanged since there are no time-derivatives § r~, r sin ~ ~. Note the negative sign of Lcr5:
Lr
- Zec ff
sin l 0
( O(A~0~sin
-- O/eTrt0C2/Ae r sinl ~ ( 0 ( A ~ ~ )sin 00 To obtain the correcting terms of gration differentials
Lea
OAe~) OA~) dr099
(50)
and Lc~ we must change the inte-
dr -, rd~ -~ r sin 0 d99
(51)
Using Eqs.(42)-(45) one may derive Leo1 to Lea5 and Lr to Lc~5 without any new difficulties from nqs.(1.3-34) and (1.3-35). The derivatives ~, 0, and ~5 c o m e from nqs.(43)-(45). The Klein-Gordon and the Dirac equations are derived from the Hamilton functions of Eqs.(25)-(27) without the correcting terms Lcr, Lr Lr The Hamilton functions of Eqs.(39)-(41) with first-order correction in ae will yield first-order corrections to the Klein-Gordon and Dirac equations, while higher order solutions in C~e of Eq.(33) will yield higher order corrections. The term gm in the Maxwell equation (1.1-2) has an effect on the solution even if the transition gm ---* 0 is made at the end of the calculation since this term produces a different differential equation 2 that yields convergent rather than divergent solutions. According to Eq.(1.1-15) the potential Ae represents the magnetic current density gm here. It is prudent to expect that the transition Ae ~ 0 at the end of the calculation may have an effect similar to gm --~ 0. Hence, the terms with a factor (~e in Eqs.(39)-(41) cannot be ignored even if one takes the limit A~ ~ 0 and gm ~ 0 at the end of the calculation. 2See Harmuth et al. 2001, Eqs.(1.3-1) and (1.2-13).
5 DIFFERENCE EQUATION IN SPHERICAL COORDINATES
198
If we want to carry the theory to second order in ae we must use the first three terms on the right side of Eq.(33). For the term a~2(1 - v2/c 2) we must use 1 - v2/c 2 of Eq.(35) while for 2 O l e ( 1 - V2/C2)1/2 w e must use the better approximation 1 - v2/c 2 of Eq.(37). We may proceed in this way to the thirdand fourth-order approximation in ae. But the process does not stop there. For the fourth order approximation we use the value of 1 - v2/c 2 of Eq.(35) for the term O/e4 ( 1 - v ~ /C 2) in Eq.(33), but for the fifth-order approximation we use 1 - v2/c 2 of Eq.(37). There is no end. Every improved approximation yields new terms and perhaps new effects. 5.3 QUANTIZATION WITH DIFFERENTIAL OPERATORS
Following the matrix notation of Eq.(1.3-24) we may write Eqs.(5.2-39)(5.2-41) in the following form:
-- { c [ ( p -
eArn) 2 4-
m2c211/2(1 4- OleQ)4- eCe} x
(lOO) 0 0
1 0
0 1
-
0 0
o)
o
Lc~ 0
0 Lr
(1)
Consider the first line of this equation:
9-Or -- c [ ( p -- e A r n ) 2 4-
m2oc211/2(1 + o~eQ) + er - Lc,.
(~'~r -- gee + f-~cr) 2 -'- C2[(P -- e a r n ) 2 + m2c2](1 + a~Q) 2
(2)
Our calculation holds only in first order of a~. Hence, we replace (1 + a e Q ) 2 by 1 + 2a~Q. The term L2r can be left out since its components are multiplied by a~2 according to Eqs.(5.2-46)-(5.2-50). The rule of Eq.(1.1-47) is used for the product (9(r - eCe)Lcr:
(p
-
-
1 eArn) 2 -- ~-~ (9~r
[ -
-
eCe) 2 4- (Re
2[(p -
eArn
)~ 4- m20c2]Q
c2 (9(, - e e l ) ao + ao ( x , - ~r ae = Z e c A e / m o c 2 << 1
= -~o~ ~
(3)
The operators of Eq. (3) are applied to the function ~I/r ~ XX/r0 4- O/eXX/rl
and one obtains
(4)
5.3 QUANTIZATION WITH DIFFERENTIAL OPERATORS
(p _ eArn)2 -- ~1' ~ ( ~ r -- eCe) 2 4- Ole [ 2[(p - eArn) 2 4I ((~_Cr _ eCe) ~--~cr 4- ~--Jcr (~-~r -- eCe) ) ] } C2 Ole Ole =
199
m2c2]Q
(lI/r0 4- O~etX/rl )
-m2oc2(~,~o+ ~oerl)
(5)
All components of Lr are multiplied by c~e according to Eqs.(5.2-46)-(5.2-50). Hence, we may divide Eq.(5) into one part of order O(1) and a second one of order O(Ote)" 1 (~-~r -- eCe) 2 4- mo2C2) lI/ro = 0
(p - eAm) 2 -
( (p - ~ h m ) 2 -- ~I( ~ { ~ -- ~r )2 + "%~c2) ~
(6)
[
= -- 2[(p -- ~ A m ) ~ + . ~ d ] Q
1 ( ( 9 { r _ eCe) Let 4- Let (~'~r -- eCe) ) l~/rO (7)
C2
Ole
Ole
Equation (6) is essentially the first line of Eq.(2.1-2) while Eq.(7) is an inhomogeneous variant of that equation. For the generalization of Eq.(6) we follow the usual path of using Cartesian coordinates for quantization and then making the transition to spherical coordinates. In analogy to the transition from Eq.(2.1-2) to Eq.(2.1-4) we obtain from Eq.(6):
_ho___ -
j:l
i OXj
- ~
xj=x,
7 5 i + ~r176 v~0 = -m02dv~0
y, z
(8)
With the relations
(h 0 i Oxj (hO
)2 -- -h 2~ 4- 2iehAmx~ ~ 4- ieh OAmxj 4- e2 2 Oxj ~ Oxj Am~j
eAmx3
)2
h2 02 _ 2iehCe O
we may rewrite Eq.(8) into the following form:
~
22
(9)
(I0)
200
5 DIFFERENCEEQUATIONIN SPHERICALCOORDINATES
[ fj=l i (-- ~2 ~C~2 + 2iehAmx~ ~~ + ieh Onmxj Oxj + e2nmxj) 2 c2
0
~-~ - 2 i e h r
- ieh
+ e2r
)]
Vro = -mo2c2V~o
(11)
In the first three terms of the first line we recognize the vector operators V 2, grad, and div in Cartesian coordinates. Hence, we may rewrite Eq.(ll) in vector form:
(h2V 2 - 2iehAm 9g r a d - i e h div Am - e 2 A m_ 2 m2c2)~ro =
1 (h2 02 -'~ -'~ -Jr2iehCe-~0 -~-ieh _ ~
-e
2 2) r ~0
(12)
At this point we change to spherical coordinates by postulating that the vector operators V 2, grad, and div in Eq.(12) can be replaced by the respective operators for component notation in spherical coordinates. The procedure used also works for the homogeneous part of Eq.(7). We only need to replace ~r0 with ~rl. Problems are caused.by the terms Q and Lcr on the right side of Eq.(7). The factor Q in Eq.(7) has a term [1 + (p - eAm)2/m2c2] -3/2 according to Eq.(1.1-45). After quantization the term ( p - eArn) 2 is replaced by Eq.(9) or the left side of Eq.(12) without the term m2c2" (p - eArn) 2 = h2V 2 - 2iehAm- g r a d - i e h d i v Am - e2A 2
(13)
If we restrict the calculation to small values of ( p - eAm)2/m2c 2 as done in Eq. (2.1-12)
(p
-
etm)2/m2c 2 << 1
-
(14)
we may write 3 - ~Am )2 /~n]~ ~ [1 + (p - ~Am)=/-~0~=] -~/~ " ~ - ~(p
(15)
This defines part of the factor Q of Eq.(1.1-45) for spherical coordinates. But we still have a factor [A~. (p - eAm)] 2 =
A
(16)
(p-
to explain. We do this in analogy to Eqs. (2.1-14)-(2.1-19) , replacing the Cartesian coordinates by spherical coordinates:
5.3
201
QUANTIZATION WITH DIFFERENTIAL OPERATORS
1
(p
-- e A r n ) 2
Q = m2oc2 [1 + ( p - eAm)2/lrt2c2]3/2 Q02
(17)
The factor Q02 can be made equal to 1 if we replace the vectors Ae and p - e A r n by matrices of rank 3 whose components are vectors:
(
A~ =
A~e~
0
0
0
A~oee
0
0
0
A~e~
( (pr - eAst)e,. p - earn =
(18)
0
0
(po - eAmo)eo
0 0
)
0
0 (p~ -- eAm~)e~
)
(19)
The substitution of Eqs.(18) and (19) into Eq.(16) yields Q02 = 1
(20)
This is a desirably simple result but it replaces the usual Klein-Gordon equation by a matrix equation of rank 3. The form of Q in Eq.(17) is reduced to
1
Q
( p - eArn) 2
re]c2 [1 + ( p - eAm)2/m]c2t3/2
-
9 (p-eAm)2 ( 3(p-eAm) 2) ~o~ 2 1-~ ~o~ 2
(21)
The term ( p - eArn) 2 is defined by Eq.(13) for spherical coordinates. There is no difference whether we multiply ( p - eAm)2/m]c 2 on the right with 1 3--(p-eAm)2/m]c 2 as done in Eq.(21) or on the left since ( p - eArn) 2 has only 2 to commute with the constant 1 and itself to transform a multiplication on the right to one on the left. We have come so far without specifying p - e A r n and thus the three cornponents of the matrix of Eq.(19). Only the square (p - eArn) 2 had to be defined, which is done by Eq.(13). This is not sufficient for the term Lc~ in Eq.(7). Equations (5.2-46)-(5.2-49) require the components
(p-etm)r
-
pr-eAmr,
(p-eAm)o
--
po-eAmo,
(p-eAm)~o
-
p~o-eAm~o
The terms Amr, Amo, Am~ were defined in Eq.(5.1-5) and the terms Pr, P~, P~ in Eqs.(5.2-8)-(5.2-10). Equation (5.2-3) yields the relations
202
5 D I F F E R E N C E EQUATION IN SPHERICAL COORDINATES
v 2 = § + (r~)2 + (rsin v~~)2 (1
-V2/C2)-1/2
" 1 Jr- "~ 1 [§
2r
+ (r~)~
(22)
(~ si~ ~ ~)~]
Equation (5.2-46) for LcrZ may be written as follows: 2aAc (1 _ (P - eAm)2 ) f~crl " m"~oe m~c2 [Ae~(p - eAm)~ - Ae~(p - eAm)~] • (p
-
eAm)r
-
9 2c~.__Ac(I_(p-eAm)2 ) moe
m02c 2
ze
)
• Ae~ (1 - v2/c2) 1/2 + - C- ( r sin r (flAer - § - Ae~ (1 - v2/c2) 1/2 + x
(i~Ae~- r~Aer)
)]
(1 -v2/c 2)1/2 + --c (r~Ae~~- r sin ~ @A~,~)
(23)
We turn to Lcr2 of Eq.(5.2-47). With the help of Eqs.(13) and (22) L~2 is fully defined:
~-~cr2 -="
-
-
2otAc/(OC/)mp~-eAm~e :i90 r
OCm P~ - eAm~ ) ( 1 ( p
0~ r sin vq 1- ~ m0rsinv~b Ze ) (1 - v2/c2) 1/2 + ~(rAe#c - r~Aer)
. 2aAc/[10r e -~
10Cm ( rsint9 0qo
- eArn) 2) m2c2 dr
mor(~ Ze ) (1_v2/c2)1/2 + ~(rsin~A~r-/'Ae~O)c (
1 ( P - eArn)2)
x 1 - -~
m2c2
dr (24)
Next comes Lcr3 of Eq.(5.2-48): 0
~-~cr3 "2-" 2aAC/{Ae~0-te
(1-~
m2c2 ) ( p o - eAmo)l
1 (p - eArn) 2
0 [(l(p-eAm) 2) - A~o~-~ 1 - ~ m2c2 (pv -eAm~)
dr
5.4 DIFFERENCE OPERATORS _
0
"2"~~176
203
1 (p -- eAm) 2
m0~c: )
( x
(1
o[(
-
morO Ze V21C2)112 + --(rSinc O~A,~ - §
-
1
-A~oN 1-7
)
~o2~~
x ( morsin0~b
Ze
(7 ~ V27-C'~)172-ll--'-~(~'de~
dr
(25)
1
0 )
Equation (5.2-49) defines •cr4'
Lcr4
" 2aAc f (1-(P-eAm)2)((po-eAmo)lOe m2oc2 r
+(p~-eAm~)
r sin--------O0~o • [Ae~(p~ - eAmo) - A,o(p~ - eAm~)]dr
9 2a_ACff(l_ (p-eArn)2) e m2c 2
--
x (1 -
+
v2/c 2)
1/2 + ~ ( r sin ~ @A~,- - § c
( morsinOqo Ze (1 v2/c2) 112 + --(§ c -
-
1 0 r O0
) 01] s i n 0 rOAe~) r 0
x A~o (1 - v2/c2) 1/2 + ~(rsin0qbA~rc - § -Aee
(1 - v2/c2) 1/2 + -(§
- rOAe,.)
dr
(26)
The final term Lcr5 remains essentially unchanged. We only replace ae by the fine structure constant a'
Lcr5=2aAcm~
r sinl~ (0(Ae~~ 7sin )00
OAe~ drOcfl
(27)
5.4 DIFFERENCEOPERATORS We replace the differential operators V 2, grad, and div for spherical coordinates by the respective difference operators ~2, I~rad, and div. The notation z~ for difference operators introduced by Eqs.(1.2-1) and (1.2-7) is used:
204
5
DIFFERENCE EQUATION IN SPHERICAL COORDINATES
z]2~ 2 z~ 1 z~ ~ 2 @ = Ar 2 + _r ~ + - r- ~2 qz~0 2 ~rad~= div A =
- er+ Ar r-~
e~q
cot 0 z ~
rsin
1
z]2~
z~vq + r 2 sin 2 0 z ~ 2
r2
-%o 0 -z~
AA,. 2 + 1_AA,~ cot0 + -A~ + ~Ao Ar r r AO r
(1) (2)
+
1 /~A~ rsin~ z~qo
(3)
In Section 2.2 we replaced the variables xj and t by normalized variables Cj and 0. The variables xj and t were originally defined in the intervals 0 <__ xj <_ cT and 0 _< t __ T. These intervals became 0 _< Q __ N and 0 _< 9 __ N for Q = x j / c A t and 9 = t / A t . Since ~qand 99 are defined in the intervals 0 _< 0 _< 7r and 0 _< ~o _< 27r we replace them by the normalized variables r / = ~/A~ and = ~ / A ~ to obtain the intervals 0 _< 77 __ N and 0 _< ~ _< N for the normalized angles 0 and ~:
0 = t/At
p = rlhr
~ = OlAO
O<_O<_N
O<_p<_N
0<_71<_N
At = TIN
Ar = cT/N
= ~/a~ 0<4_
AO = 7tiN
(4)
Equations (1)-(3) assume the following form"
9~
1 (z~2~ 2/~ 1 A2~ cot ( r / A 0 ) z ~ = (a~)2 \ Ap2 + -p Zp + p2 A,~ '
1
z~2~)
+ p2 sin2(r/A~) A~2
1 (/i9
~grad 9 = ~rr div A =
lz~
1
A~ )
z~--per + --~-~-eo + p sin(r/A~),4~ --e~o p
(5)
(6)
~rrl (z~A, 2 lz~Ao cot(~?A0) 1 AA~o) --~-p + - A , + + A~+ p p z~r/ p psin(r/A~) z~r
(7)
The first- and second-order difference quotient for p follows from Eqs.(1.21), (1.2-3), and (1.2-7)" ~(p + Ap) ~(p 2Ap Z2~ Ap2
Ap)
J
= =[~(p + 1 ) - ~ ( p 2
~ ( . + a . ) - 2~(.)+ ~(p-a.) (ap)2
= r
1)]
2~(p) + ~ ( p - 1 )
(8)
(9)
5 . 4 DIFFERENCE OPERATORS
205
The substitutions p --. r/---, ~ ~ 0 yield the corresponding equations for r/, ~, and t). Substitution of Eqs.(8) and (9) into Eqs.(5)-(7) yields the final form of the difference operators ~z2 ~rad, and div in spherical coordinates. To shorten the expressions we write ~(p + 1) rather than ~(p + 1, r/, ~, 0) when p is varied:
1
(
1
~ 2 ~ = (Ar) 2 ~(p + 1 ) - 2 ~ ( p ) + ~ ( p - 1)+ p[~(p + 1 ) - ~ ( p - 1)] 1 + ~-~[~(77 + 1)
-
2~(r/) + ~(~7
-
cot(r/A0) 1)1 + ~ 2p 2 [~(~ + 1)
-
~ ( r / - 1)]
1 [~(~ + 1) - 2~(~) + ~(~ - 1)]) + p2 sin 2(77A0)
(10)
1 ( 2 [~(P + 1 ) - 9 ( P - 1)let + ~pp 1 [~07 + 1 ) - ~ ( r / - 1)]e~ grad ~ = ~rr
1
+ 2p~i~(,TA~) [~(~ + 1 ) - ~ ( ~ - 1)le~
1(1
divA -- ~rr ~[A~(p+ 1 ) -
A,.(p-
1)] +
)
(11)
)
(12)
A(p)
+ G1 [Ao(r/+ 1) - Ao(r/- 1)] +
cos(r/AO)
p
Ao(r/)
1
+ 2psin(vZx,9)[A~,(~ + 1) - A~o(~- 1)]
We can now write Eq.(5.3-12) as a difference equation. Since only the specialized case Am -- 0, Ce = Ce(r) = Ce(P) will eventually be used we can simplify Eq.(5.3-12) drastically:
c~(hv2 - m~c2)v~~ =
/~2 02 (At)2 0o~ +
At
O0
This equation is rewritten into a symbolic difference equation. We obtain with the help of Eqs.(5)and (8)-(10):
h2 c 2 ( h 2 V 2 - m02c2)~r0 --
A2
( A t ) 2 Z~0 2
2ieCe(p)h A At
A0
e2 2 )
Ce(P) ~r0
(14)
206
5 DIFFERENCE EQUATION IN SPHERICAL COORDINATES
The function ~r0 = ~ 0 ( P , 0) = ~r0 (p, r/, (, 0) depends on the spatial variables p, 77, ( and the time variable 0. The operator on the left side of Eq.(14) does not contain a term dependent on O, but the operator on the right contains 0 and p. For the elimination of 0 on the right side we follow the spirit of the differential theory augmented by some intuition. First we write Or0 as a product of u(p) and v(O) = v ~
%o(P, O) = u(p)v ~
(15)
then we assume the operator on the right side of Eq.(14) can be replaced by - [ E - eCe(P)] 2, where E is a constant:
u(p)
h2
z~2
(At) 2 z~02~
2ier
~
At
z~0
2 2
e Ce(P)
)v 0
-----U(p)
v0
[ E - e C e ( p ) ] 2 (16)
The explicit difference equation of v ~ becomes:
h 2 (vO+~ _ 2v o + v ~ (At)2
)+
i~rAt (~0+~_ vO-~) _ ~=r =
Multiplication with v - ~
v-2+v
-~ +
-[E
-
0 ~ (17)
~r
2 brings
ier
(v--v-l)-- (eCe(p)At)
+ [E - eel(p)] 2 = 0 (18)
and we obtain a quadratic equation for v:
1+
i~r176
[2+ (~r ~ h
-
-
[E - eCe(p)] 2
+ ( 1 - ier
-
-
v =0
(19)
The solutions vl,2 can be greatly simplified if we rewrite the terms er and EAt~h:
eCe(p)At = eCe(p) h
m0 c2
EAt h
At
eCe(p)
Ar
Ii/moc2 = moc2 h/moc
E Ar rood h/moc
(20)
5.4 DIFFERENCE OPERATORS
207
In the differential theory it is usual to assume er 2 << 1 and E/moc 2 << 1. We follow these assumptions but keep the factor Ar/(h/moc)"
er Ar e Ar << 1, << 1 moc 2 h/moc moc2 h/moc
(22)
We do not assume anything about the magnitude of Ar/(h/moc). The calculation will demand certain minimum values for this term. This is in line with the results of Chapters 3 and 4 where A r / ( h / m o c ) is replaced by Ax/(h/moc). The coarse distinction between A x / ( h / m o c ) >> 1 and A x / ( h / m o c ) << 1 will be replaced by a much finer one. A differential theory cannot have a minimum value for dr. The simplifications of Eq.(22) reduce the solution of Eq.(19) to a manageable form:
1 ( 1 - i e C eIi( p ) A t ) ( 2 • vl,2 . -~ iEAt -iEAt/h --e h iEAt 2ieCe(p)At v2 " l + ~ h li V 1
"
1
(23) (24)
The solution v2 contains the variable p and thus contradicts the assumption made for Eq.(15), but vl is a usable solution that yields the result known from the differential theory" v~ = e - ~ e A t e / h
= e -iet/~
(25)
We return to Eq.(14) and rewrite its left side with the help of Eqs.(15) and (16): (26) as an explicit difference equation:
[u(p + 1, ~, ~) - 2u(p, ~7,~) + u(p - 1, ~, ~)] 1 + - ( ~ ( p + 1, ~, ~) - ~(p - 1, ~, ~)] p
1
+ -~ [u(p, 77+ 1, ~) - 2u(p, 7, r + u(p, ~ - 1, ~)] -4-
cot(~A~) [u(p, ~7+ 1, ~) - u(p, r / - 1 ~)] 2p2
208
5 D I F F E R E N C E EQUATION IN SPHERICAL COORDINATES
+
p2 sin2 (~A~)
[u(p, rl, ~ + 1) - 2u(p, rl, ~) + u(p, ~?,~ - 1)] +
- 1
moc2
](
h/moc
u(p, ~?,r = 0
(27)
We separate Eq.(27) into a function u~,(p, rl) and a function ~ ( ~ ) by means of Bernoulli's product method ~(p,,,r
= ~.(p,~)~.(r
(28)
and obtain: 1 [~=(p + l, ~) - 2~=(p,,) + ~=(p - 1, ~)1 + ~ [~=(p + 1 , , ) - ~=(p - 1,,)1 1
cot(~A~)
+ -7[u~(p, ~+ 1 ) - 2u~ (p, v)+u~(p, ~- 1)]+ ~ 2p 2
[u~(p,~+ 1 ) - u~ (p, v - 1)]
+ \
/
J\
1
p2sin2(r/A~) [ ~ ( ~ + 1)
2~(~) + ~(~-
1)]u~ (p, r/)
(29)
We multiply Eq.(29) with p2 sin2(uAtg) ~(p,n)~(~) and obtain on the left a function of the variables p and 77, while on the right is a function of ~. Such an equation requires that both sides are equal to a constant that we denote - ( v A ~ ) 2-- -(2Try/N)2: p2 sin2 (r/Av~) ~.(p,~)
{left side of Eq.(29) without ~ ( ~ ) }
_ - (~-----~[~(~
+ 1) - 2~o~(~) + ~o.(~ - 1)] = -(~,zx~o) 2 = -
(30)
Equation (30) yields a relation for ~ ( ( ) ~ ( r + 1) - [2 + ( . a ~ ) ~ l ~ ( r
with the solution
+ ~(r
- 1) = 0
(31)
5.4 DIFFERENCE OPERATORS
209
~(~) = g V2 v--
[2 + (vA~)2]v~ + 1
--
0
1 v~, = 1 + ~(vA~o) 2 4-ivA~o
1 + ~1 (vA~o) 2
" 1 + ivAqo " e + i ' ~ ~
~0,,(~) = A l e 2'~u'~/N + A2e -2'~i'~/N = A l e i'~~ + ~2~
Equation (30) without the terms ~ form:
,
=
can be brought into the following
1
[u~,(p + 1,77) - 2u,,(p, ~7) + u,,(p - 1, rl) ] + -~[u,,(p + 1, ~7) - u,,(p - 1,7)] 1
+ -~[u,,(p, ~7 + 1) - 2u,,(p, rl) + u~,(p, 7"1- 1)] + +
moc 2
- 1
cot(~Avq) [u,,(p, 7"1+ 1) - u~(p, 77 - 1)1 2p 2 +
li/moc
p2 sin2 (~7.A~
~(p,~) =0
(33)
We use once more Bernoulli's product method to separate the variables p and 77
~ ( p . ,) = ~ . ( p ) ~ . ( , ) and substitute
u~(p)cp.(rl)
(34)
into Eq.(33)'
1 [u~ (p + 1) - 2u~, (p) + u~, (p - 1)] + p [%, (p + 1) - u~, (p - 1)] 2
= -
(1_
~-~[~,(r/+ 1) - 2~o~,(r/) + ~ , ( r / -
+
cot(r/A0) 2p 2 [~,(~ + 1) - ~ , ( r / -
t?.
2
1)1 (vA~) 2 ) 1)] + p2 sin2(r/Avq) %,(r/) u~,(p, O)
(35)
We multiply Eq.(35) by p2/ut,(p)~t,(rl) and obtain on the left a function of p, and on the right a function of r/. Again, both sides of the equation must equal a constant that we denote -A#"
210
5
DIFFERENCEEQUATIONIN SPHERICALCOORDINATES
1 [ut,(p + 1) - 2ut,(p ) + ut,(p - 1)] + p[Ut,(p + 1) - ut,(p - 1)]
-~-[(EeCe(P)) 2-c2 .no (
1]( AT 1.9.
p2
h/~o~ ~"(P)}.~.(p)
,
= - [ ~ o . ( ~ + 1 ) - 2~o.(~)- ~ . ( r l - 1)] + ~ cot(rlA~) [~o.(~+ 1 ) - ~ . ( ~ - 1)]
(uA~)2 + sin2(7?A~)) ~t,(r/)
) 1 Tt,(r/) = -A#
(36)
Equation (36) yields a relation for ~ ( ~ ) : 1 [~o~(r/+ 1) - 2~o~,(r/) + ~o~,(r/- 1)] + ~ cot(r/A~))[~o~,(r/+ 1) - ~ot,(r/ - 1)]
((~A~)~- ~)~.(~) = 0 +
sin2 (rlAvq)
(37)
This is the difference equation of the discrete spherical harmonic functions. Its solution is much more complicated than that of Eq.(31) for ~ ( ~ ) . In Sections 6.6 and 6.7 we derive the eigenvalues Ae=-/(/+l),
/=0,
(3s)
1, 2, ...
which are equal to those obtained in the differential theory. Equation (36) with A~ from Eq.(38) yields a difference equation for u~,(p) that equals the one of the differential theory except for the replacement of differentials by finite differences:
l(l + 1)
1
[%,(p + 1) - 2u,(p) + ut,(p - 1) + -~[ut,(p + 1) - %,(p - 1)]
-
p2 u~.(p)
[E ' e(;/)e(p)] 2 -- m2c 4 )\*-a [ A r ~ 2) ~ t t t ( P\~
h2C2
(39)
A boundary condition for this difference equation follows from Eqs.(15) and (25) for p = 0. There is no need for a boundary condition for ~o~(~) of Eq.(32) or ~t,(~)of Eq.(37): ~r0(P, 0) = ~ooUt,(p)e -~Eate/n For a causal solution that is zero before 0 = 0 we have two possibilities:
(40)
5.5 SPATIAL DIFFERENCE EQUATION
kI/ro(O , 0) -- ~ooS(O) cos
EAtO
211
for 0 < 0
=0 -- 900 cos
EAtO
9 ~o(0, O ) = t 9 o o S ( O ) s i n ~
EAt•
for 0 > 0
h
= 0
(41)
for 0 < 0
= 900 sin
EAtO
for 0 > 0 (42) h T h e b o u n d a r y condition of Eq.(41) starts at 0 = 0 like a step function while Eq.(42) yields a linear increase. More complicated b o u n d a r y conditions can be represented by making E = En a function of n = 0, 1, 2, ... and using a Fourier series expansion. 5.5 SPATIAL DIFFERENCE EQUATION In the differential theory the equation corresponding to Eq.(5.4-39) can be written in the following form (Schiff 1949, p. 309; Messiah 1962, p. 884) if er is replaced by - Z e 2 / p and t~ is written instead of p. We have already defined p = r / A r in Eq.(5.4-4) and A r has no place in a differential theory. Hence, some other normalization factor has to be used and this creates a different normalized variable iS:
d
2 d
A
1
l(1 + 1 ) - ' ~ 2 / /
moc(
Ze 2
=~,~=-~j,~=2- F
u(/~) : 0
E2 )1/2, i = h ~2E~, z = l ,
1 .~4
2,...
(1)
T h e difference equation (5.4-39) can be brought into a similar form. Note that the hat ^ is replaced by a tilde ~. 1
[%,(p + 1) - 2%,(p) + %,(p - 1)] + p[%,(p + 1) - %,(p - 1)] .
.
.
p
.
4 Ar
h/moc ( a=
2 1
2E~'
E2 ) 1/2 m2oc 4
'
l(1 + p2 1)- ~2). u~, (p) ~ = Z Ze2
h/moc' ~_
~= moc25z '
7 = 1, 2, ... ; Z " 376.730 [V/A]; a = ~z d
= 0
- - ~ = 4;rZa 2Ar[~
hc
Ar
' h/moc
__ 1
A~
(~ h/moc
fine structure constant
(2)
212
5 DIFFERENCE EQUATION IN SPHERICAL COORDINATES
The terms 5 - A~l(lilmoc) = (~Arl(hlmoc), which express the length A ~ - 5 A r in relation to the Compton wavelength, are an important feature of Eq.(2). We cannot choose A~ = li/moc since we have seen in Sections 3 and 4 that Ax >> h/moc and, Ax << h/moc lead to completely different results. The remaining difference between the constants in Eqs.(1) and (2) is due to the use of the international system of units (SI) made conspicuous by the wave impedance Z in Eq.(2). Equation (2) is multiplied by p2 and the terms are rearranged to match the requirements of the method of solution used:
p(p+ 1)uv (p+ 1)+ [gp2+ ~Sp-l(1 + 1)+'~2]uv (p)+ p(p- 1)uv (p-
1) = 0
1
To obtain eigenfunctions of Eq.(3) we use a method based on the Laplace transform (N6rlund 1910, 1915, 1924, 1929; Milne-Thomson 1951; Guldberg and Wallenberg 1911). It is not necessary to use this detour via differential calculus. The same solutions can be obtained by purely algebraic means, but the Laplace transform shortens the calculation. We start with the contour integral
1 f sO_1 (s)ds t
(4)
and take it in the complex plane from the point t~l along a contour g to the point g2. Partial integration yields: 1 (~
t21/
=
)
-
(5)
Let the points tl and t~2 be chosen so that the term 12
(6)
ll vanishes. One obtains:
1/ sPw' (s)ds
-PUt' (P) = ~ i
(7)
t
Further partial integration yields
p(p+
s~
1)u.(p)= ~ s
(8)
5.5 SPATIAL DIFFERENCE EQUATION
213
provided the expression
sP+lw'(s)
(9) tx
vanishes. If one makes the corresponding assumptions about the beginning and end of the contour of integration one obtains the following relations for u t, (p + 1) and u ~ ( p - 1):
( - l l k ( p + 11... (p + kl .(p + 1) =
1 f sP+kw(k)(s)d s t
(-1)k(p-
1 ) . . . ( p + k - 2)uu( p - 1) =
1 ff sP+k-2w (k) (s)ds
(10)
l
We rewrite Eq.(3) so that the coefficients become multiples of the terms on the left side of Eqs.(7), (8), and (10):
[(p + 2)(p + 1) - 2(p + 1)]ut~(p + 1)
+ [9P(P + 1) + (-g + ~6) p - l(1 + 1) + ~'2]ut,(p ) + p(p -- 1)ut.,(p - 1) -- 0
(11)
Substitution of the integral transforms for u~(p), u~(p+ 1), and ut,(p- 1) yields the following differential equation"
(s 2 + gs + 1)sw"(s) + (2s + g - ~6)sw'(s) - [l(1 + 1) - "~2]w(s) -- 0
(12)
This is a differential equation with variable coefficients of the Fuchs-type (Fuchs 1866; Frobenius 1873). We follow the standard method for its solution. Equation (12) has a singular point at s = c~. The other singular points are located at the roots of the equation (s 2 + gs + 1)s = 0 We obtain three solutions for s'
(13)
214
5 DIFFERENCE EQUATION IN SPHERICAL COORDINATES +i
+i 0635
~_____, is1
s3v ] ~
=s_22
-i
b
FIG.5.5-1. Location of the singular points s2 = s2(g) and s3 = s3(g) in the complex plane according to Eq.(14) for - 2 __ g > -c~. (a) shows the integration path from 0 around s3. (b) shows a circle of convergence around s3 for g = -2.0635 that is limited by s3 and does not reach 0.
sl=O,
(g2)1/2
s2=-~+
-~--1
s2+s3=-g,
s2-s3=2
s2 -- s3 = h/moc 1 +
~+ 5
, s3=
-T-1
A~
Ar ~ - 1 = ~ + ~ h/mo
-'1 ~2 h/moc ~
=--1 82
I+T~
9 h/moc : -s3 - -- 83, A ~ - - 5 A t
~
[
-~--1
=5
h/~no
Ar
(g2)lj
g2
~2(
~
Ar
(
= 1 - h/mo"--"~ 1
~ - 1 = o
)2]1/2
E2) ~/2 ,
m2c4
s3 > 0
(14)
The parameter g is always real according to Eq.(3). The loci of s2 and s3 for real values of g are shown in Fig.5.5-1. The value of.g is slightly smaller than - 2 for small values of 5 or A~/(h/moc) << 1 but it is significantly smaller for large values of 5 or A~/(h/moc) >> 1. In either case the root 83 s3(g) is located on the real axis between 0 and +1. The locus of s2 readily follows from the relation s2s3 = 1 in Eq.(14). A formal solution of Eq.(12) is obtained by a power series expansion of w(s) in either the point s3 or s2. Both solutions are required for a general "
-
-
5.5 SPATIAL DIFFERENCE EQUATION
215
solution. We discuss the solution in the point s3. The power series of w ( s ) converges for small values of A~ inside a circle that passes through the point s2 as shown by Fig.5.5-1b: N
w ( s ) = w3(s) = E q 3 ( v ) ( s - s3) p~+" v--0
(15)
The subscript 3 of w3(s), q3(v), and P3 indicates that these terms apply to the solution in the point s3. The solution of Eq.(12) by the series of Eq.(15) is derived in Section 6.8. Here we state only some results. There is a regular solution with P3 = 0. A non-regular solution with p3 = p
~6 p3=p=
~
s3 - s2
= -2~Zo~
--
Af
2E~
1
A~
s3 - s2 l~/rnoc = - m o c 2& 6(1 + 62/16)1/2 h / m o c
E
m2c 4
m0
2
Ar
-
h/m0
1+(1
= -2~Zo~
j 0c)
= p0 + O ( A r )
c2'E
p0 = _27rZc~m0 (1
m2c4
(16)
exists in the neighborhood of s3. From the results of Chapters 3 and 4 we know that A r cannot be made arbitrarily small without risk of obtaining a meaningless result. However, the notation O ( A r ) separates only terms multiplied by A r or powers of Ar, it is not assumed that these terms must be small compared with iP0]. A recursion formula with three terms is obtained for the coefficients q3(v) by substituting Eq. (15) into Eq.(12):
~3,,(~,)q3(~, + 1) + ~3,0(~')q3(~,) + ~3,-~(~')q~(~, - 1) = 0 a3,1 (v) = s3(s3 - s2)(p + v + 1)(v + 1)
c~3,0(v) = (2s3 - s2)(p + v)(p + v ~ _ ~ ( , ) = (p + ~)(p + ,
1) + [3s3 - s2 - (s3 - s2)p](p + v) - l ( z + 1) + ~2
- 1) 82 ~ 83
For v = 0 we obtain a formula with two terms for q3 (1):
(17)
216
5
D I F F E R E N C E EQUATION IN S P H E R I C A L C O O R D I N A T E S
~3,~ (0)q3(1) + ~3,0(0)q3(0) = 0 ~3,1(0) = , 3 ( , 3 - ,2)(p+ 1), ~3,0(0)= ,~p(p+ 1)-l(z + 1 ) + ~ 2 (is) One may choose q3(0) but not q3(1). Since a second-order differential equation should have two choosable constants one would need the second solution in s2 to produce a general solution. Let the line of integration of the contour integral of Eq.(4) begin at the origin, run around s3, and return to the origin as shown by the line t~ in Fig.5.5la. The point s2 remains outside the loop. More details about this integration may be found in the works of Guldberg and Wallenberg (1911), N6rlund (1910, 1915, 1924, 1929), and Milne-Thomson (1951, p. 485). One obtains a factorial series:
r(p)
N
~(_l)~q3(v )
~"~ (P) = *~ r ( p + p + 1) ~0-
(p + 1 ) . . . (p + u)
(P + P + 1)... (p + p + ~)
q3(1) = q~(0)
~3,0(0) ~,~(0)
(19)
The fraction (p+ 1 ) . . . ( p + u ) / ( p + p + 1)... ( p + p + u ) s h a l l always be replaced by 1 for u = 0. This convention avoids the need for writing a special term for u - 0. The upper limit N of the sum is usually replaced by c~ since the calculus of finite differences eliminates only the infinitesimal but not the infinite. According to Eq.(5.4-4) we have N intervals A r -- c T / N in the range 0 <_ r <_ cT or Ap = 1 in the range 0 _< p _< N, which implies N + 1 points 0, 1, 2, . . . , N. Hence, the series in Eq.(19) can have only N + 1 orthogonal or linearly independent functions and the upper limit u = N must be used. This looks like the concept of divergence is eliminated, but there is no such luck. We call the sum of Eq.(19) convergent if it approaches arbitrarily closely a limit for N >> 1; it is called divergent if such a limit is not approached. This replaces the usual requirement that a limit is approached for N ~ oo. In the case of Fig.5.5-1b the power series does not converge everywhere along the line g of integration in Fig. 5. 5-1a. As a result the factorial series of Eq.(19) may diverge for all values of p. There are two ways to obtain a convergent series. First, one may decrease the value g = - 2 ( 2 + 52/4) = -2.0635 used in Fig.5.5-1b to move s3 to the left and s2 to the right so that the circle of convergence goes through sl -- 0 rather than through s2 as shown in Fig.5.5-2b. The distinguished value g = - 3 / x / ~ of Fig.5.5-2a is obtained. For ( g=-
~2) 2+-~- =-
[
1( 2+~
h/moc > 2 - ~ - 2
A~ )2] hlmoc < " 0.69662
3
(20)
5 . 5 SPATIAL DIFFERENCE EQUATION
+i
217
+i
'-]
=- 2
-....
-"
--~
a
b
FIG.5.5-2. The convergence circle of Fig.5.5-1b around s3 reaches both 0 and s2 for g = -3/~/'2 (a). For smaller values of g the singular point s2 becomes unimportant since the circle of convergence is limited by sl = 0 (b). we have convergence of the power series along the integration line l of Fig.5.5la. In essence, as long as the spatial resolution A~ is larger t h a n the C o m p t o n wavelength of the particle we have convergence. For smaller values of AP t h a n defined by Eq.(20) we m a y use a second way t h a t requires conformal m a p p i n g to achieve convergence. T h e circle of convergence of the power series of Eq.(15) according to Fig.5.5-1b is m a p p e d onto a loop t h a t runs t h r o u g h the origin. Except for s = s3 and s = 0 there must be no singularity either inside or on the loop. T h e factorial series obtained in either one of these two ways do not necessarily converge in the whole interval 0 _ p < N unless certain conditions are satisfied. For a convergent power series we must have the following relations according to Eq.(20) and Fig. 5. 5-2a:
[2 )112
A~ &At .A moc( h/moc = h/moc = ~ , , r - - ~ 1 - m~c4 hr
hlmoc
>
0.69962
2
= 0.34981
> 0.69662
for [ ~ m0c 2
(21)
We thus get in Eq.(16) for p the values c~ - 7.297 • 10 -3, P_/m0c 2 ~ 1, [m2c41(m~c4 [2)]2 9 1, and [Arl(hlmoc)] 2 > 0.122. Hence, p is a negative n u m b e r with small magnitude, IPl << 1. We r e t u r n to Eq.(19). If p could be a negative integer the factorial series would terminate. According to Eq.(16) p is negative but small c o m p a r e d with 1. A second possibility is t h a t one can choose the coefficients q3(v) so t h a t
218
5 DIFFERENCE EQUATION IN SPHERICAL COORDINATES
Eq.(19) approaches a polynomial asymptotically. To this end we rewrite s2 and s3 with the help of Eqs.(3) and (14)"
s2=1+~-+~
= 1 + O(Ar)
I+T~
~ ~( ~)~/~= 1 + O(Ar)
sa = 1 + - ~ - -
~
(22)
I+T~
With p0 of Eq.(16) we obtain from Eq.(17)' a3,1(v) = -2nAr(po + v + 1)(v + 1) + O(Ar) 2 a3,0(v) = (Po + v)(po + p + 1) - l ( 1 + 1) + ~2 + O(Ar) a3,-1 (v) = (Po + v)(po + v - 1) + O(Ar)
~(~_~:).~:_~_ 1 (g2)~/~. ~=-
2Ar
=
2Ar
2~Ar " c ~ - - ~ z ~ r = 2 - - ~
-A---~ 1
-~---1 m-~c4
~moc --2
h~
Ar > 0.69662
(23)
The notation O ( A r ) or O(Ar) 2 indicates terms multiplied with Ar or (Ar) 2 as well as with higher powers of Ar. These terms do not need to be small compared with the other terms of Eqs.(22) and (23). The recursion formula of Eq.(17) may be rewritten as a system of linear equations. Cramer's rule yields the coefficient q3(# + 1): q3(# + 1) q3(O)
( - 1 ) "+1
H ~,1(. + 1) v-'0
a3,0(O) 013,-1 (0)
C~3,1(1) 013,0(1)
~,~(2)
a3,_l(v - 1)
~,0(~)
(24)
a3,1 (v + 1)
a3,-l(t~ - 2)
~ , 0 ( ~ - 1) ha,-1 (t~ - 1)
-~,,(,) a3,0(#)
All the terms aa,l(v + 1) for v = 0, 1, ... , # contain terms of order O(Ar) or higher. If we ignore these terms we obtain the determinant without terms of order O ( A r ) "
5.5
SPATIAL DIFFERENCE
219
EQUATION
It
q3(# + 1)
(-1)"+1 H [(Po + v)(po + ~' + 1) - l ( 1 + 1) + ,~2] ~=0
q3(0)
H as,l(~ + 1)
tt
(25)
v~0
Let the last factor for u = # in the numerator of Eq.(25) be zero: (P0 + #)(P0 + # +
(26)
1) - l ( 1 + 1) + ~2 = 0
The # + 1 coefficients q3(0), Arq3(1), ... , (Ar)t'q3(#) are then of order O(1), while the coefficients ( A r ) ~ q 3 ( u ) for u > # contain only terms of order O(Ar). In this case the factorial series of Eq.(19) looks like a polynomial if one ignores terms of order O ( A r ) . The power series of Eq.(15) converges for A~ > 0.69962 according to Eq.(20) and the factorial series of Eq.(19) converges too. The finite upper summation limit N in Eq.(19) is of little consequence if N is made large enough. If the condition of Eq.(20) is not satisfied we get a divergent power series in Eq. (15). This does not mean that the factorial series of Eq. (19) diverges too but only that we are bumping against a limitation of the well worked-out part of the calculus of finite differences. It is known that functions defined by certain divergent but summable power series are also defined by convergent factorial series (NSrlund 1924, Ch. 9, w3). With the change of notation # - n -1 - 1
(27)
we obtain from Eq.(26):
p0 = - n + 9
P01
1+ ~ •
~2
- n + 21 +-----~,
Let # run from 0 to n -
l+ .
P02
_ ~2 ~2
-nt~2l+l,
n' = n-
21 - 1
(28)
1. The index l then assumes the values 1-0,
1, . . . , n - 1
We use the wave number ~ introduced in Eq.(23)
(29)
220
5 DIFFERENCEEQUATIONIN SPHERICALCOORDINATES
?Tt0C( _ ~E2/1/2 ._ ~77~0c ~ = ~
1
m~c4
83 --
2
1
[
li '
( t~-~0~a/2]1/2
F_ = moc 2 1 -
~ mOc A r = 1 - aAr, 2 h
83
(30)
" e -teAr
and rewrite Eq.(16):
P = Po + O(Ar),
Po = "Y---~ 1 -
(31)
--moc
Substitution into Eq.(28) yields with the help of Eq.(23):
( ~1 ~2
nh
1+ ~
9 Zym~ n'h
2 21 + 1
1-
+
~ (
E. - moc 2
p2o
2l+1
"2
~/2 ) - 1 / 2 1 - p2 + Zy2
E1 " moc 2 1
2n 2
n4
2l+1
8
[ ,2 ,~4 ( n, 3)] E2 " moc 2 1 - ~ + ~ - 7 ~ 2/+1 + n'= n- 21- 1
(32)
The values shown in Eq.(32) for ~ and E are the same as obtained from the differential Klein-Gordon equation (1). The positive sign of the square root in Eq.(28) and thus the values ~2 and E2 are usually not considered in order to avoid poles in the eigenfunctions of the differential equation. The recursion formula of Eq.(17) yields the following coefficients q3(u) for the terms not containing Ar
q3(1)_ 1 (Po(Po+l)-l(l+l)+~2 q3(0) - n a t 2(po + 1)
) + O(Ar)
5.5 SPATIAL DIFFERENCE EQUATION
221
q3(2) 1 /' (P0 + 1)(po + 2) - l(l + 1) + ~2 q3(0) = (~Ar) 2 ~, 4(p0 + 2) x
Po(Po + 1) - l ( 1 + 1') + #2 2(po + 1)
) + O(Ar),
(33)
etc. From Stirling's formula for the Gamma function one obtains the relation (Milne-Thomson 1951, pp. 254,255)
r(p + ~) r(p + y)
= pz-y,
p >> 1
which yields
lim F ( p - n + / + 1) (no+n-0 p>>l F ( p + p 0 + l ) = pP(p) = ( p - 1 ) . . . ( p - n + l + 1 ) P ( p - n + 1 + 1) We get from Eq.(28): -(P0 + n - l ) =
(34)
1 [( :/, ]1<,
- ~ q=
l+
_ q2
= u
(35)
Equation (34) yields for p >> 1" (p - 1)(p - 2 ) . . . (p - n + l + 1) = pn-l-1 + O(pn-l-2)
(36)
The factorial series of Eq.(19) becomes for p >> 1:
)
_
= ~,(oI~-"~, ~176 = q3~
q(0,1>p,0+l (I - ~o(~o + I12~<,~-z(l+ ~)+ ~' + ... ) 1 - P~176
Ar ( 1 - -~p~)-l/2,
nAr = 27r h/mo c
+ "'" ) p = r / A r >> l
(37)
Equation (at) contains a polynomial with n - 1 terms of order O(1) and a series with arbitrarily many terms of order O(Ar), which have been left out. The polynomial is the same as that derived from the differential Klein-Gordon equation (1). It may be used as an approximation for the factorial series of Eq.(19), provided the condition ~e/(h/mo~) > o.69662 of Eq.(20) is satisfied.
5 DIFFERENCE EQUATION IN SPHERICAL COORDINATES
222
5.6 QUANTIZATION OF THE SOLUTION lI/r0(P, 0 ) We follow Section 2.5 and calculate the energy U of II/r0(P, 0 ) from the energy-impulse tensor of Eq.(2.5-1), keeping in mind that the operator V is required for spherical coordinates:
10lI/r*0 0~I/r0
m20c2
,
(1)
Ot + V ~ r o ' V ~ r o +
h2 ~*o~ro 0~r0 1 0~Pr0 1 0~r0 ~e~ Or er + eo 4 r 0v~ rsinO O~
Too = c 2 0 t
V~o=grad~o=
(2)
We restrict the calculation to the simplest case where ~ro is independent of 0 and ~. The energy in a sphere with radius cT becomes:
1 (~lI/r*o(r, $) 0~I/ro(r,t )
/ { U=
0
0
0
~
Ot
+
Ot
0~I/:o(r,t ) (~lI/ror +
Or
Or
h2 ~r*o(r, t) ~ o (r, t) r2dr sin~d~
dqo (3)
The dimension of U is VAs. Equation (3) is rewritten with the normalized variables 9 - t / A t and p -- r / A r = r / c A t of Eq.(5.4-4)"
N
U -- 27~cAt / ( O~~
(~) OqlI/r0(P'Oq0~) -[- 01I/r*0(P'op 0) OII/r0,Op (p (~)
0 + m2c4(At) 2 ) h2 " ~r*0(P,0)~r0(P, 0) p2dp
(4)
The function ~r0 is represented by Eqs.(5.4-15) and (5.4-25) if p is replaced by p:
II/r0(P , 0) = ~ooUtt(p)e-iEAtS/h
(5)
For u~(p) we found in Eq.(5.5-19) a particular solution u~3(p)" for the point s3 in Fig.5.5-2:
F(p)
g
(p + 1)... (p + v)
u,a(p) = s ~ F ( p + p + l) ~o(-1)~qa(V) ( p + p + l) . . ( p + p +
(6)
223
5.6 QUANTIZATION OF THE SOLUTION ~I/r0(fl, 0 )
Here s3 is defined by Eq.(5.5-22), p = P3 by Eq.(5.5-16), and q3(u) by Eqs.(5.517) and (5.5-23). The fraction (p+ 1)... (p+ u)/(p+p+ 1)... (p+p+ u) shall always be replaced by 1 for u = 0 to prevent having to write a special term. We obtain:
9 0)~r0(P, 0) -- ~020U/~ ~(p) ~r0(fl,
r2(p) ( N (p + 1)... (p + ~) + p + 1) E(-1)~q3(u) (p + ; + 1) (p + p + ~) ) v--0
= v~176
(7)
The differentiation of Eq.(5) with respect to 0 or p produces the following result: O~rO(P,O)
0O
EAt ~atOln = --iffYoo~eN (p+ 1)... (p+ u) r(p) E(_l)~q3(u ) x s~ r(p + p + 1) ,.,=o (p+p+ 1)...(p+p+u)
O~rO(P, O) =
Op
(8)
r
N 0 (s~ r(p) E(_l)~q3(u ) (p+ 1)... (p+ u) ( p + p + 1) (p+p+v) ) xG r ( p + p + 1)v=O
(9)
The first term in Eq.(4) becomes:
r~(P)
(p+ 1)... (p + u)
(
x S2Pr2(p + p + 1) E(-1)"q3(u) (p+p+ 1) p'-0
,~2
(p+p+ u) )
(10)
For the second term in Eq.(4) we get:
OQ~I/~o O~I/ro __ ~i/020
Op Op
x
s'dr(p+p+ 1)
(p+ 1)... (p+ u)
(- 1)'q3 (u) (p+p+ 1)
(p+p+u)
)
(11)
224
5 D I F F E R E N C E E Q U A T I O N IN S P H E R I C A L C O O R D I N A T E S
Equations (10), (11), and (7) must be substituted into Eq.(4). Since has disappeared from Eqs.(7), (10), and (11) and p is being integrated over we get a constant for Eq.(4). This is different from the result of Eq.(2.5-12). The following three components of Eq.(4) are obtained:
.._,...,/ (:oc..,). h ~*o~rop2dp= 27rcAt~20 (moc2At )2 N
0
7
x @r2(p+p+l) (p+ 1) 1)...(p+ u) v) ~=o(_l)~,qa(u) (p+p+ (p+p+ "" 0
p2dp
(12)
(p+p+v) p2dp
(13)
N
0
(p+l)...(p+v)
(-1)~q3(v)(p+p+1)
• @r2(p+p+l )
)2
y--O
0
N
U3= 27rcAt/ O~r*~ OqYrOop p2dp= 2rcAt~2o 0 N
•
/ [ 07p (s ~r(p+p+l) F(p)
~~o(--1)Nq3(u) (p+p+ ( P + I1)) ' " ( P(p+p+u) +V))] 2p2dp (14)
0
The integrals of Eqs.(12) and (13) are equal. They must be evaluated numerically. In Eq.(14) one can eliminate the differentiation O/Opby the symmetric difference quotient ~/~p to facilitate computer evaluation:
.(
F(p)
oU ~ F(p + p +
~N 1) ~( v(-1)~q3 ) =
(p + 1)... (p + v) (p + p + 1) (p + p + v) /
N
s~ ~o(_l)~q3(v)( p + 1)... (p + v) 2Ap = x
saPF(p+Ap) r(p+Ap+p+l)
~ - a . r ( , - ap)
1 (p+ A p + p + 1 ) . . ( p + A p + p + ~ , )
1
F ( p - A p + p + 1) ( p - A p + p + 1 ) . . ( p - A p + p + u) )
(15)
5.6 QUANTIZATIONOF THE SOLUTION ~I/r0(tO,0)
225
If we use the approximations that led to Eq.(5.5-35) we obtain for the energies U1 to U3 the following expressions:
cT
1(1
po(po + 1) - l ( l + 1) + ~2 -f- . . . ) ] 2r2dr
(16)
2nr
o
u~ = 2~o~o cT
x/
(i_ Po(Po + 1) -2nrl ( l + 1) + ~2 +...
)]2
r2dr
(17)
o
U3 = 27r~20
cT x / ' { ~ r0 [e=~rrU+n_z_ 1 ( 1 - P~176
o 1
~=-~:F
[(1)
2
Z+~
]1/2
_~2
(~8)
Equation (4) may be rewritten: U - U(n) - U1 + U2 + U3
(19)
The parameter n occurs explicitly in Eqs.(16)-(18) but it is contained also in p or P0 of Eqs.(12)-(14) or (16)-(18) according to Eqs.(5.5-16) and (5.531). Using U1 to U3 from Eqs.(12)-(14) we may write U(n) in the following normalized form'
u(.)
Nu(~)
27rcAt~2 ~ = 27rT~020 = Ng{~
u(~) 9f~ = d2(n) = 27rT~ ~
(20)
The variable d2(n) is introduced to avoid having to write square roots of ~ . One may rewrite the component 9f~ as follows:
226
5 DIFFERENCE EQUATION IN SPHERICAL COORDINATES
9s
= (2~-t~)2d(t~) 2--~ (sin 2~-~0 - i cos 27rt~O)~d(~) (sin 2~-t~O + i cos 27rt~O) = -27riap,~(O)q,~(O)
(21)
This is the same equation as Eq.(2.5-19). T h e calculation from Eqs.(2.5-20) to (2.5-48) applies again. 5.7 CONVERGENCE FOR SMALL VALUES OF A r
In Section 5.5 we derived Eq.(5.5-20), which shows that the solution derived for u(fi) of Eq.(5.5-1) converges for
A~
5Ar
h/moc
h/moc
9<
[[2 )1/2
= 2
1
m2c4
Ar -------- > 2 h/moc
3 ~-
)1/2 2
" 0.69662
3
v~ 1
(1)
sa <_--~
The power series converges absolutely inside a somewhat smaller circle. This smaller circle may be transformed by conformal mapping into a loop around s3 t h a t runs through s - 0 in Figs.5.5-1 and 5.5-2. The point s2 will be outside this loop. The transformed series of Eq.(5.5-15) converges absolutely and uniformly along a line of integration g that is inside this loop except at the point s - 0 and that runs from s - 0 around s3 and back to s = 0. One may thus integrate term by term and one obtains a factorial series that converges for p > p0, where P0 is the still to be determined abscissa of convergence. T h e transformation
r
s
,
P=real
(2)
maps the point 83 of the s-plane in Figs. 5. 5-1b and 5.7-1a into the point 1,0 of the ~-plane, and the point s2 into the point s 2P due to the relation s3 - 1/s2 of Eq.(5.5-14). This is shown in Fig.5.7-1. The point s22P is on or outside the circle [ r 11 = 1 around the point r = 1,0 if the condition s2 s-"3
. e_~A ~
> - 2,
In 2 In 2 P > - 2 In 82 " 2 n A r
(3)
5.7 CONVERGENCE FOR SMALL VALUES OF A r s-plane
227
(-plane ,
~'0 s l
-
-/s2
~ g = -2.0635 s3 = 0.77777 s2 = 1.28573
-._ ~ -i
-i
a
b
FIG.5.7-1.Transformation of Fig.5.5-1b from the s-plane (a) to the C-plane (b) according to Eq. (2). is satisfied. In Eq.(5.4-4) we introduced the relation p = r / A r to connect the normalized variable p with the nonnormalized distance r. We introduce in analogy a distance rp: p = r__p_p Ar
(4)
One recognizes from Eqs.(3) and (4) that there must be a m i n i m u m distance rp for which one can obtain convergence. In order to d e t e r m i n e this distance we express s2 with the help of Eqs.(5.5-14) and (5.5-30): s2=--
1 83
" l+~Ar
" e ~Ar
(5)
Equations (3), (4), and (5) yield:
rp > -
ln2 ~
-- In 2
h m0c
(
1
E2 ) -1/2 m02c 4
(6)
For small values of F_ we obtain in essence the result t h a t rp must be larger than the C o m p t o n wavelength. But for large values of t! or n we can obtain much larger values of rp a c c o r d i n g t o Eq.(5.5-32). The inverse transformation of Eq.(2) s = ssr lIP
(7)
maps the circle I ~ - 1] = 1 of Fig.5.7-1b into a loop in the s-plane. The loop is shown in Fig.5.7-2.
228
5
DIFFERENCE
E Q U A T I O N IN S P H E R I C A L C O O R D I N A T E S
0
-0 5 FIG.5.7-2. The circle I r 11 -- 1 in the ~-plane of Fig.5.7-1b mapped into a loop in the s-plane by means of nq.(7); s3 -- 0.77777, P = ln2/(21ns2) - 1.3791, s2 - 1/s3. The power series in the point s2 may be mapped in the same manner as the one in point s3. A complex number P must be used. This solution varies for large values of the distance r like e +~r rather than like e -~r as the solution in point s3. One may infer this from p = r / A r in Eq.(5.4-4), s~ in Eq.(5.5-19), and s2 in Eq.(5). We will not discuss the solution in point s2 any further. For a convergent solution according to Fig.5.7-1a we start with the differential equation (5.5-12) and write it for the point s3 by the substitution s = ( s - s 3 ) + s3. The terms ( s - s3) '~ are rewritten as s~(s/s3 - 1) n and s/s3 is replaced according to Eq.(2) by s/s3 = ~l/P. We now have a differential equation in the r -plane that still has to be transformed some more to make it a Fuchs-type equation. Once we have done that we may solve it with a power expansion in r 1 and obtain the solution OO
w(r 1 / P - 1 ) = E
q ( ~ ) ( ~ - 1)P+~
(8)
v'-O
The calculation is shown in Section 6.9 from Eq.(6.9-1) to Eq.(6.9-24). The final step is to transform w(~ 1/P - 1) into the s-plane by means of the substitution - (s/s3) P. This is done in nqs.(6.9-25) to (6.9-50) and the function w p ( s - s 3 ) is obtained:
N
oo -
=
=
-
v=O
"
-
(9)
v=O
We observe that the upper limit co in Eq.(9) is replaced by a finite number N since an arbitrarily large but finite interval can be subdivided only into a finite number of arbitrarily small but finite subintervals. In other words, there is only a finite number N + 1 of linearly independent functions. The subscript P is used for w and q to distinguish them from w3 and q3 in Eq.(5.5-15), which
5.7 CONVERGENCE FOR SMALL VALUES OF Ar
229
holds for P = I only. We note that the exponent i5 in Eqs.(8) and (9) holds both in the ~-plane and the s-plane, but the coefficients ~(v) and qp(v) are different. The series expansions in Section 6.9 are the bane of the Laplace transform method of solution of difference equations. A simpler and faster method is urgently needed. The operational methods of Milne-Thomson (1951, Ch. XIV) may bring this simplification. The series expansions required here are worked out in Section 6.9. Equation (5.5-15) was transformed into a factorial series Utt3(p) in Eq.(5.519). Replacing ug3(p), q3(v), and p there by up(p), qp(v), and i5 yields the factorial series associated with Eq.(9):
r(p) N (~ + 1)... (i5 + v) up(p) = s~r(p +~ + 1) ~(-1)~qP(v)= o (p +15 + 1)... (p + i 5 + v)
= P
83 -- 82
1 + l = P ( p - 1 ) + l,
Eq.(6.9-16)
~p(1) = _ ( ~ [ s 3 P - l ( ~ - l + 2 P ) - ( s 3 - s 2 ) ( ~ - l ) ( P - 1 ) / 2 P ] - P [ l ( l + P-ls3(s3 - s2)(i5 + 1) 0P(O) s3
2!
'
1 ) - ~ 2]
Eq. (6.9-54)
s2, s3: see Eq.(5.5-14); ~, A, 5: see Eq.(5.5-2); p: see Eq.(5.5-16)
(10)
For P = 1 we obtain i5 = p of Eq.(5.5-16) and ~p(1)/(tp(O) = q3(1)/q3(O) of Eqs.(5.5-19) and (5.5-18). We want to work out a few numerical values. From Eqs.(5.5-23) and (5.530) we obtain the relation:
Ar = 83(82 -- s3) 2~
~3(~ - ~3) ~
2
[2 ) - 1 / 2
1
moc
83(82 -- 83)
h
47r
moc
m20c4 1 m2c 4
(11)
For the limit case represented by Fig.5.5-2a with s3 = 1/v/2 and s2 = x/~ we get Ar > 0"03979--~-hm0c(1 From Eq.(5.5-32) we obtain:
~2 I - 1 / 2
re]c4
(12)
230
5 DIFFERENCE EQUATION IN SPHERICAL COORDINATES
E2
1
--1/2
m~c4 )
__ ,~2
= [1--(1
--1/2
~-n-~n2 )]
. x/~n_15.416 Ar > 0.6134-.
forn
1, Z = l
h
(13)
moc
Hence, the smallest value for Ar is about 61% of the Compton wavelength. If we use s2 = 1.2857 and s3 = 0.7777 of Fig.5.7-1a we obtain E2 ) - 1 / 2
Ar >_ 0"03144--hm0c (1
m2c4
h " 0"48467m0 c
We still have to derive the numerical value of P. Equation (3) yields the relation P> -
In 2 2Ins2
(14)
which yields P _ 1.3791 for 8 2 ---~ 1.2857, 8 3 - - - 0 . 7 7 7 7 in Figs.5.7-1 and 5.7-2. For plotting it is often more convenient to start with a numerical value for P. One obtains then from Eqs.(3) and (5.5-23)" 82 >_ e In 2/2P
(15)
83 ~_ e - l n 2 / 2 P
(16)
s3(s2 - s3) = 2~Ar = e -l"2/v - 1
(17)
The sign _< in Eq.(16) can readily be replaced by = if the exact value of s3 requires more decimal digits than used for plotting. 5.8 PLOTS FOR SECTIONS 5.5 AND 5.7
We start with certain constants obtained in Section 5.5 that are generally applicable. From nqs.(5.5-2) and (5.5-32) we obtain -~
Ar
~ = 2~-~
( ~ 2 1 27tAr = 2~ i - ~ h/,~o~'
= 4~r~/ 1 - ~ ~=47r7a,
k,
E1
" F~2
(1)
k= h/moc
a = 7.297 535 x 1 0 -a,
z = 1,
2,...
(2)
5 . 8 PLOTS FOR SECTIONS 5 . 5 AND 5 . 7
231
Equation (5.5-16) yields: It
P = --2?rZ~
(I
\
It2 (I m20c4) [ 1~- \
It2
2
]
2 --1/2
m2c4) ( Ar
(3)
(1 moc2
mo2C4
(5)
We see from Eqs.(5.7-12) and (5.7-13) that the choice k - 1 will yield convergence for any value of P > 1. But for smaller values of k we must be careful that P is chosen large enough for convergence. For tt we obtain from Eq.(5.5-32)
mOc 2
=
m o c2
=
moc 2
= 1
(6)
2n 2
if terms of order ~4 are ignored. At this point we can choose a value P _ 1. Equation (5.7-10) yields:
= P(p-
1)+ 1
(7)
Further, we obtain from Eqs.(5.7-15) and (5.7-16) with >__ and <__ replaced by the equality sign 82 = e In 2/2P ,
s3 =
e - In 2/2P
(8)
We cain now calculate ~p(1) of Eq.(5.7-10). The free constant ~p(0) is chosen equal to one:
~p (1) = -(p[s3P-
1
(~__ 1 + 2P) - (s3 - s2) (i5-1)(P - 1)/2P] p-ls3(s3
- P[l (l + 1) _921
- 8 2 ) ( p + 1)
+~P-1) s3
2
'
l = 0, 1, ...
(9)
The first term Upl(p) of the factorial series of Eq.(5.7-10) becomes
r(p) Upl(p) = s~ F(p + 15+ 1)'
p=l,
2,..., N
(~o)
while the second term up2(p) equals: 15+1 up2(p) = -up1 ( p ) ~ p ( 1 ) ~ p+l~+l'
p=l,
2, . . . , N
(11)
232
5 DIFFERENCE EQUATION IN SPHERICAL COORDINATES i
0.0014
0 0012
"~ ,.~
0 001 O. 0008
I ~-~ O. 0006
o. ooo4 0.0002 !
i
4
i
p---~
i
I
6
I
8
i
I
I0
FIG.5.8-1. Difference U D l ( p ) - upl(p) according to Eqs.(12) and (10) for u = ul, n - 1, l - 0, Z - 1, P - 1, k -- 1 of the differential and the difference result for
p-l,
2 , . . . , 10.
I
/
0.8
'~0
%,
I"
6
"~.
i
""'"~.%,2
!
I
I
~0.4
t
"..,.
#
I--I
9
"--"
=0.2 I
i
p---~
FIG.5.8-2. Difference U D l ( p ) - upl(p) according to Eqs.(12) and (10) for u = ul, n = 2 or n = 3, l = 0, Z = 1, P = 1, k = 1 of the differential and the difference result forp=l, 2 , . . . , 10. For c o m p a r i s o n w i t h the differential t h e o r y we shall plot UD1 (p) of the first t e r m of E q . ( 5 . 5 - 3 7 ) '
UDI(p) =
~t =
s~p~'pn-z-l, ~tl
"--
1
2
q 3 ( 0 ) = 1, l+
l+
_,~2
S3 = 1/V/'2
(12) (13)
(14)
Let us c o m p a r e t h e result Upl(p) of t h e difference t h e o r y for P - 1 w i t h UDI(P ) of t h e differential theory. F i g u r e 5.8-1 s h o w s t h e difference UDl(P) --
5 . 8 P L O T S F O R SECTIONS 5 . 5 AND 5 . 7
I "-~
233
0.035 0.03 O. 025 0.02
~m 0 9015 I
0.01
O. 005 I
20
!
40
I
p
60
!
80
i
100
FIG.5.8-3. Relative difference [UD1(p)--up1 (p)]/UD1(p) according to Eqs.(12) and (10) for u -- ul, n -- 1, l -- 0, Z - 1, P - 1, k - 1 of the differential and the difference result for p - - 1, 2 , . . . , 100. Upl(p) in t h e r a n g e p = 1, 2, . . . , 10. We n o t e t h a t o n l y t h e integer values p = 1, 2, . . . apply. A p e a k value of a b o u t 0.0014 is o b t a i n e d for p - 2, which is a b o u t 0.5% of UDl(P). T h i s small value increases d r a s t i c a l l y if we replace n = 1 by n - 2 or n - 3 as s h o w n in Fig.5.8-2. T h e difference at p - 2 is a b o u t 45% of UD1 (P) for n = 2 a n d a b o u t 80% of UDl(P) for n -- 3. To see w h e t h e r t h e relative difference w i t h respect to UD1 (P) is m o r e useful t h a n the a b s o l u t e difference of Figs.5.8-1 a n d 5.8-2 we plot t h e relative difference
for n = 1 in Fig.5.8-3 a n d for n - 2, n -- 3 in Fig.5.8-4. In Fig.5.8-3 we get q u i t e r e a s o n a b l e values of up to 3.5%, b u t Fig.5.8-4 shows t h a t this r e p r e s e n t a t i o n has a d r a w b a c k b e c a u s e t h e relative difference is close to 1 or 100% of UD1 (P) for m o s t values of p. T h e reason is t h a t up1 (p) drops faster t h a n UD1 (p) w i t h increasing values of p. T h e set of t h r e e i l l u s t r a t i o n s from Figs.5.8-5 to 5.8-7 shows Upl(p) of Eq.(10) for P - 1, 1.3791, 2, 3, 4. We see t h a t t h e plots for P = 1 a n d P = 1.3791 are essentially equal at p - - 1 b u t s e p a r a t e in Figs.5.8-6 a n d 5.8-7. T h e plot for P = 2 is q u i t e d i s t i n g u i s h a b l e from t h o s e for P = I a n d P = 1.3791 at p -- 1 in Fig.5.8-5 a n d r e m a i n s so in Fig.5.8-6. T h e s u r p r i s i n g p l o t s are for P - 3 a n d P = 4. In Figs.5.8-6 and 5.8-7 t h e y look as one w o u l d e x p e c t b u t in Fig.5.8-5 t h e y are q u i t e different for p - 1, 2, 3. P l o t s for larger values of P yield a p p a r e n t l y r a n d o m values close to p = 0. T h i s is s h o w n in Fig.5.8-8 for P - 10, n - 1, 1 - 0, Z = 1, k = 1. T h e o c c u r r e n c e of r a n d o m n u m b e r s in a result t h a t is s u p p o s e d to be c o n v e r g e n t is baffling w h e n first e n c o u n t e r e d b u t it is p e r f e c t l y correct. W e shall give an e x p l a n a t i o n later in this section. Let us see how s m a l l A r can be chosen before t h e r e s u l t s b e c o m e useless due to r a n d o m n e s s . We have used so far k = Ar/(h/moc) = 1. For k = 0.01
234
5 DIFFERENCE EQUATION IN SPHERICAL COORDINATES i
... . . . . . n. . = . . .3. . . . . . . . . . :::::::::::::::::::::::::::
1
~. 0.8
I'
s
; ,'
"~ 0.6
;/ i:
0.4 i
~. 0.2 5'
' I0
p---,
i'5
2'0
'25
FIG.5.8-4. Relative difference [UD1(p) -- up1 (p)]/UD1 (p) according to Eqs.(12) and (10) for u = ul, n = 2 or n = 3, l = 0, Z = 1, P = 1, k = 1 of the differential and the difference result for p - 1, 2, ... , 100.
we obtain Fig.5.8-9 instead of Fig.5.8-5. The plots appear to be identical but if a larger scale is used, I _< p _< 3 rather than 0 _< p _< I0, one can see a deviation at p - 2 for P - 4. Next we try k - 0.001 and obtain Fig.5.8-10. The deviation of the plot for P - 4 at p - 2 or 3 is now clearly visible. But we also note that the vertical scale has changed, which is due to the negative value of the plot for P - 3 at p- I. Figures 5.8-11 to 5.8-13 show plots like Figs.5.8-5 to 5.8-7 but for k - 10 -4 rather than for k - I. We note drastic differences between Figs.5.8-5 and 5.8II that become less drastic for Figs.5.8-6 and 5.8-12 and still less drastic for Figs.5.8-7 and 5.8-13. We see that the sequence of the plots for P - I, 1.3791, 2, 3, 4 from top to bottom is the same in Figs.5.8-7 and 5.8-13 for p >_ 12. While the reduction of k from 1 to 0.01 shows no noticeable difference between Figs.5.8-5 and 5.8-9 a reduction of k - 10 -4 by one-half to k 5 • 10 -5 in Figs.5.8-14 to 5.8-16 brings drastic changes compared with the plots of Figs.5.8-11 to 5.8-13. One such change is that the sequence of the plots for P = I, 1.3791, 2, 3, 4 from top to bottom in Fig.5.8-16 is no longer restored for larger values of p as it was in Fig.5.8-13. We conclude from this that too small a resolution A r affects Upl(p) p r i m a r i l y close to t h e origin p = 0. B u t the d e v i a t i o n of t h e t h e o r y e x t e n d s to larger values of p for smaller values o f . A r . In order to c h a r a c t e r i z e the t r a n s i t i o n from the s t a b l e plots of Figs.5.8-5 to 5.8-7 to the r a n d o m i z e d plots of Figs.5.8-14 to 5.8-16 we show Table 5.8-1. T h e function Up1 (p) for p - 1 and either P = 1 or P = 2 ret ai ns its first five digits, 0.70713 or 0.49996, in the interval 1 > k >_ 0.7. T h e first four digits, 0.7072 or 0.4999, are r e t a i n e d in t h e second interval shown for k, etc. T h e n u m b e r k - 0.6134 is given in Eq.(5.7-13) for P - 1 as t h e t r a n s i t i o n from s t a b l e results to r a n d o m i z e d ones. We see t h a t this is in line with Table 5.8-1 if we do not look for a s u d d e n change at k - 0.6134 but m e r e l y for a fast change. Let us t u r n to the t e r m v = 1 of up(p) in Eq.(5.7-10). It is r e p r e s e n t e d by
5 . 8 PLOTS FOR SECTIONS 5.5 AND 5 . 7 0.7
235
P=I
0.6 T 0.5
"~o.4
2
m o 3 .
0.2
3
~/1.3791
o.I ..... 2
4
6 p -----~
8
i0
FIG.5.8-5. Plots of up~ (p) according to Eq.(10) for n = 1, l = 0, Z = 1, k = 1, and P = 1, 1.3791, 2, 3, 4 for p = 1, 2, . . . , 10.
O. 02 ,k./.. P = l l o oz5 ~
.3791
9
"~ E
'". \ o . o i "k " '"," " ~ ~ \ \9
3~.
,,
2 ~,~
"~'.
7
~
8
9
p---.
I0
ii
12
FIG.5.8-6. Plots of upl (p) according to Eq.(10) for n = i, I -- 0, Z -- I, k -- I, and P -- I, 1.3791, 2, 3, 4 for p = 6, 7,... , 12.
0.003
i 0.0025
~%--"P--1 \ \
0.002
0.0015 0.001
..? \ , .
0 . 0 0 0 5 4"',.
~ 12
\x 14
16
18
p---~
20
22
24
FIG.5.8-7. Plots of up1 (p) according to Eq.(10) for n = 1, l = 0, Z = 1, k = 1, and P - 1, 1.3791, 2, 3, 4 for p - 10, 1 1 , . . . , 25.
236
5 D I F F E R E N C E EQUATION IN S P H E R I C A L C O O R D I N A T E S
2.1759~ I
0.05 I
1
-0.05
2
I
.,
3
4
I
~
9
p ----,
5
6 i
9
I .
I
7
8
-0.1
-0.15 -0.2
FIG.5.8-8. P l o t of up1 (p) a c c o r d i n g t o E q . ( 1 0 ) for n = 1, 1 = 0, Z = 1, k = 1, a n d P = 10; p = 1, 2 . . . . , 8.
0.7
P=I
0.6 I 0.5
II
,~o.4
~o.3 0.2 0.I
1.3791 4-
.N.._. .......
2
4
p---,
6
8
I0
FIG.5.8-9. P l o t s of up1 (p) a c c o r d i n g to E q . ( 1 0 ) for n = 1, l -- 0, Z = 1, a n d k = 0.01 for v a r i o u s v a l u e s of P .
T
0.6
~0.4 0.2
2-,1,, 1.3791 4
3. . . . . . ..; ..... "2"
4
p---,
6
8
F I G . 5 . 8 - 1 0 . P l o t s of U p l ( p ) a c c o r d i n g t o E q . ( 1 0 ) for n = k = 0.001 for v a r i o u s v a l u e s of P .
10 1, 1 =
0, Z =
1, a n d
5.8 PLOTS FOR SECTIONS 5.5 AND 5.7
0.8
I o-6
.i k . P = 1
::\ 4--'~
-~o4 ~
237
1.3791 2
0.2
i ,,""
~
i
-~,
~
" " .~ ,,.
3 x
...
-. ~ ~ _ ~ _ _
-,,,,.,.-
/ "...;z:" . . . . .
4
p..--+
6
_
8
lo
F I e . 5 . 8 - 1 1 . P l o t s of u p l ( p ) a c c o r d i n g to Eq.(10) for n = 1, l = 0, Z = 1, a n d k = 10 - 4 for v a r i o u s values of P ; p - 1, 2, . . . , 10.
0.I
i
0.08
3
"~.--1.3791
O.O6 O. 04
0.02
P=I '~,~
.......... .
~
"~
.......
4 7
8
~,".,.
~'-.
p----,
9
I0
Ii
12
FIG.5.8-12. P l o t s of up1 (p) a c c o r d i n g to Eq.(10) for n = 1, 1 = 0, Z = 1, a n d k = 10 - 4 for various values of P ; p -- 6, 7, . . . , 12.
0.014 0.012 I a.
0.01 O. 008 0.006 0.004 0.002
"" "' "" 12
"
1.3791
14
16 18 p---,
20
22
24
FIG.5.8-13. P l o t s of U p l ( p ) a c c o r d i n g to Eq.(10) for n -- 1, 1 - 0, Z - 1, a n d k - 10 - 4 for v a r i o u s values of P ; p - 11, 12, . . . , 25.
238
5 D I F F E R E N C E EQUATION IN SPHERICAL C O O R D I N A T E S 2.5
~4 I
1.5
.
p
:
~. i / =
1 1.3791
2
0.5 "~';'-:" . .
-0.5
9 9 ".;
-1
4
9/
p .---~
. .
6
8
i0
FIO.5.8-14. P l o t s of u p l ( p ) a c c o r d i n g to E q . ( 1 0 ) for n = k = 5 x 10 - 5 for v a r i o u s v a l u e s of P ; p - 1, 2, . . . , 10.
0"5I'"'".
I 0, l"
•
.;'-:/
1, l = 0, Z =
1, a n d
3 ..... "-. 9 .
~,,
~0.3 0.2 0.I .
,I '''~
!
7
8
I
!
9 I0 p----~
I
!
II
12
F i o . 5 . 8 - 1 5 . P l o t s of u r n ( p ) a c c o r d i n g to E q . ( 1 0 ) for n = k - 5 x 10 - 5 for v a r i o u s values of P ; p -- 6, 7, . . . , 12.
1, I = 0, Z =
1, a n d
~ 25 1..~--3 0.2 o.15
o
12
14
16
D
18 20 p --.-.,
22
24
FIG.5.8-16. P l o t s of U p l ( p ) a c c o r d i n g to E q . ( 1 0 ) for n -- 1, l k - 5 x 10 - 5 for v a r i o u s v a l u e s of P ; p - 11, 12, . . . , 25.
0, 7 -- 1, a n d
5 . 8 PLOTS FOR SECTIONS 5 . 5 AND 5 . 7
239
T A B L E 5.8-1 IINTERVALS OF k -~ A r / ( h / m o c ) FOR WHICH u r n ( l ) OF EQ.(10) HAS THE SAME FIVE, FOUR, THREE, OR TWO INITIAL DIGITS FOR P -- 1 AND P -- 2.
P-1 Upl(1) P--2 up1(1)
0.70713 1 _> k >_ 0.7
0.7072 0.6 > k _> 0.2
0.707 0.1 _> k _> 0.04
0.71 0.03 _> k >_ 0.002
0.49996 1 >_ k _> 0.7
0.4999 0.6 >_ k >_ 0.3
0.499 0.2 >_ k >_ 0.03
0.49 0.02 >_ k >_ 0.003
up2(p) of E q . ( l l ) . P l o t s for P = I , 1.3791, 2 are shown in Fig.5.8-17 for p = 1, 2 , . . . , t0. T h e plots for P - 1 and 1.3791 are multiplied by 100 to be able to present t h e m t o g e t h e r with the plot for P - 2. Figures 5.8-18 and 5.8-19 show the same plots for p - 6, 7, . . . , 12 and p - 10, 11, . . . , 25. T h e two plots for P - 3 and P - 4 not shown in Fig.5.8-17 are shown in Fig.5.8-20 for p - 1, 2, . . . , 10. T h e y are as odd as the plots for P - 3 and P - 4 in Fig.5.8-5. We do not show t h e m for p - 6, 7, . . . , 12 or p - 10, 11, ...,25. F i g u r e 5.8-21 shows again the plots of Up2(p) for P - 1 , 1.3791, 2 for p = 1, 2, . . . , 10, the ones for P - 1 and P - 1.3791 multiplied by 100, but k - 1 is replaced by k - 0.01. T h e r e are two noticeable small differences c o m p a r e d with Fig.5.8-17. T h e plots for P - 1.3791 and P - 2 are closer t o g e t h e r and the plot for P - 1 has the value - 0 . 3 at p - 1 r a t h e r t h a n - 0 . 6 . We note t h a t for up1 (p) in Figs.5.8-5, 5.8-9, and 5.8-10 we had to go to k - 0.001 before a difference b e c a m e noticeable. Figures 5.8-22 to 5.8-24 show plots like Figs.5.8-17 to 5.8-19 b u t for k = 0.001 r a t h e r t h a n k -- 1. T h e r e are significant differences between Figs.5.8-17 and 5.8-22 t h a t remain significant for Figs.5.8-18 and 5.8-23 or Figs.5.8-19 and 5.8-24. W h a t we have seen in this section is primarily the impossibility of reducing the spatial resolution to less t h a n a b o u t 10% of the C o m p t o n wavelength even for b o u n d particles. This is in line with the results of C h a p t e r s 3 and 4, p a r t i c u l a r l y Eqs.(3.1-10) and (4.1-2). Instead of the coarse distinction between A x >> Ac and A x << Ar we have now the much finer distinction t h a t the resolution can be a b o u t 10% of Ac. T h e conformal m a p p i n g introduced in Section 5.7 does not improve the resolution, it only p e r m i t s us to s t u d y the transition from a result with nonr a n d o m n u m b e r s to one with ( p s e u d o ) r a n d o m n u m b e r s using a convergent series expansion. Since the a p p a r e n t l y r a n d o m n u m b e r s in Fig.5.8-8 were obtained by calculation t h e y are not really r a n d o m n u m b e r s a n d we m u s t use the correct t e r m p s e u d o r a n d o m numbers. T h e situation is the same as in the calculation of a n u m b e r like 7r. Since the n u m b e r is o b t a i n e d by calculation
240
5 DIFFERENCE EQUATION IN SPHERICAL COORDINATES f 100up2(p), P "~Up2(p), P
=
1.3791 2
I
/
/-2--
-
4
p___~
6
,,,
I
i
8
I
10
FIG.5.8-17. Plots of up2(p) according to E q . ( l l ) for n = 1, l = 0, Z = 1, k = 1, and P = 1, 1.3791, 2 for p = 1, 2, . . . , 10; the plots for P = 1 and P = 1.3791 are multiplied by 100.
O. 025
,,, ,~~(p), P=
1.379~
0.02 0.015
0.01
"".. ..~~P2(P), P = 2
0.005
-0.005
FIG.5.8-18. Plots of up2(p) according to E q . ( l l ) for n = 1, l = 0, Z = 1, k = 1, and P -- 1, 1.3791, 2 for p = 6, 7, ... , 12; the plots for P = 1 and P = 1.3791 are multiplied by 100.
0.0015
00up2(p), P = 1.3791
0.001 0.0005
.., ~
up2(p), P-- 2
' 2'2 ' 2'4 -0.0005
FIG.5.8-19. Plots of up2(p) according to E q . ( l l ) for n = 1, l = 0, 7 = 1, k = 11 and P = 1, 1.3791, 2 for p = 10, 11, . . . , 25; the plots for P = 1 and P = 1.3791 are multiplied by 100.
5 . 8 PLOTS FOR SECTIONS 5 . 5 AND 5 . 7 p
241
-.----) l
-I0
I
t. #
/ ~,.
-20
.'
..
" "9 ,~,
'iI"
o~ 0., -40
,
..
/ r / g
-30
i
..."
2."//
ii
-50
"
-60
/
:
.: '. ." :...'
-70
FIG.5.8-20. Plots of up2(p) according to E q . ( l l ) for n = 1, 1 = 0, Z = 1, k = 1, and P = 3, 4 for p = 1, 2, ... , 10. The magnitude of these plots is about 20 times that of the plot for P = 2 in Fig.5.8-17 and 2000 times that of the plots for P = 1, 1.3791.
~- 100up2 (p), P = 1.3791
!
2(p),P=2
100Up2(/9), P = 1
I
6
i
I
8
i
I
I0
0 -----~ FIC.5.8-21. Plots of up2(p) according to E q . ( l l ) for n - 1, l - 0, Z - 1, k - 0.01, and P - 1, 1.3791, 2 for p - 1, 2 , . . . , 10; the plots for P - 1 and P - 1.3791 are multiplied by 100. its digits are not a sequence of r a n d o m digits, but we expect t h a t t h e y a p p e a r r a n d o m if we a p p l y any conceivable test of randomness. Convergence does not exclude p s e u d o r a n d o m n u m b e r s from our results. F r o m t h e s t a n d p o i n t of physics it is quite u n d e r s t a n d a b l e t h a t the Compton effect limits t h e spatial resolution even for a b o u n d particle. This limitation of the resolution is a specific feature of a difference t h e o r y t h a t assumes a finite resolution A x or A r but does not specify the m a g n i t u d e of A x or Ar. Instead, it obtains this m a g n i t u d e as a result of the calculation. We shall discuss this point again in Section 5.9 from a different angle. F i g u r e s 5.8-3 and 5.8-4 d e m o n s t r a t e d t h a t the resu4ts of t h e difference and the differential t h e o r y did not become equal for large distances p or r - p A r . We want to show this analytically with the help of Eqs.(10) and (12). T h e following a p p r o x i m a t i o n of the G a m m a function for large a r g u m e n t s is used ( A b r a m o v i t z and Stegun 1964, p. 257):
242
5 DIFFERENCE EQUATION IN SPHERICAL COORDINATES
i i
[-- Up2(p), P = 2 i i
lOOup2(p),P = 1.379 \~'lOOup2 ,
"
\
(p), P = 1
~ " - .
_. . . .
2
4
6 O --.--~
J
8'
'
i'0
FIG.5.8-22. Plots of up2(p) according to E q . ( l l ) for n = 1, l = 0, Z = 1, and k = 0.001 for various values of P ; p - 1, 2, . . . , 10. 0.03 0. 025 0.02 0.015
' ~ u e 2 (p), P = 2 I
~ , ,100Up2 (p), P = 1.3791
\~,,IOOup2(P), P=I
0.01 0.005 7
8
9 p ----~
i0
ii
12
FIG.5.8-23. Plots of up2(p) according to E q . ( l l ) for n = 1, l = 0, 7 = 1, and k = 0.001 for various values of P ; p = 6, 7, . . . , 12.
o.
ioo.
lOOup2(p),p=1.3791
0. 0015 ~ , 0.001 0. 0005
\~up2(p),p= ,,
\
12 FIG.5.8-24.
(,), p = I
14
16 18 p---,
20
22
Plots of up2(p) according to Eq.(ll) for n =
k = 0.001 for various values of P; p = I0, Ii, ... , 25.
24 i, l =
0, Z
=
i, a n d
5.9 ORIGIN OF THE COULOMB FIELD
r(p) ~ v~e-"e The ratio
UpI(fl)/UDI(P) becomes =
UDI(P)
("-1/2) lnp,
p >> 1
243
(15)
for large values of p:
s~pUpn-l-1
+
1) ~ e~+lp-(~+u+~-0
(16)
5.9 ORIGIN OF THE COULOMB FIELD In Section 5.8 all plots held for p - r/Ar -- 1, 2, . . . . Since the circle of convergence in Fig.5.7-1b goes through the origin, the convergence of the power series in the point s3 is assured in a somewhat smaller circle, which implies that the factorial series of Eq.(5.7-10) converges for p > 0. The convergence on the abscissa of convergence p = 0 is a problem comparable to t h a t of the convergence of a power series on the circle of convergence. It turns out that we do not have to address this problem. For the explanation of such a fortuitous result let us state first t h a t a result up(p) of Eq.(5.7-10) holds not only in the point p but in the interval p - 1/2 < r/Ar < p + 1/2 since we cannot resolve an interval smaller than A t . Hence, we have solutions for the i n t e r v a l s . . . , 5/2 < r/Ar < 3/2, 3/2 < r/Ar < 1/2, 1/2 < r/Ar < - 1 / 2 . In all these intervals except the last one we have a charged boson in a Coulomb field. This situation changes if the boson reaches the interval 1/2 < r/Ar < - 1 / 2 since this is where the charge is located t h a t produces the Coulomb field. The boson cannot be distinguished from whatever carries this charge. Instead of a boson in a Coulomb field we have only a Coulomb field with a changed charge number Z. If the charge + e created the Coulomb field and the boson has the charge - e we get a neutral particle and no Coulomb field. Hence, there is no interest in Up(p) of Eq.(5.710) for p - 0. This is a solution of the problem of convergence for p = 0 that reminds one of the egg of Columbus. The differential theory typically has problems with convergence at r -- 0. We distinguish between weak poles for r -- 0 that are acceptable and the strong poles t h a t are not. To recognize the reason for the different behavior of a differential and a difference theory consider the function 1/r as it approaches r -- 0. In the difference theory we have for 1/r the succession of values . . . , 1 / 3 A r , 1 / 2 A r , 1/Ar, (1/0), but the last value 1/0 was just ruled out as being of no interest. A finite value A r > 0 yields a finite value 1~At. In a differential theory we would have the sequence . . . , 1~3dr, 1~2dr, 1~dr, (1/0). Again, we can ignore the last value 1/0 since the boson would be indistinguishable from the central charge of the Coulomb field at r = 0. But 1/(ndr) exceeds any finite bound for any finite or denumerable number n = 1, 2, . . . . Hence, the problem of divergence arises not only for r = 0 but for any distance n dr from r = 0. We have found a mechanism that avoids or sidesteps convergence problems that works only in a difference theory and not in a differential theory.
244
5 DIFFERENCE EQUATION IN SPHERICAL COORDINATES
The problem with convergence for r ~ 0 is evidently caused by the assumption of infinitesimal distances dr. Since we cannot observe infinitesimal distances, any problem caused by their assumption is an important argument in favor of the calculus of finite differences. 5 . 1 0 UNBOUNDED BOSONS IN A COULOMB F I E L D
We turn to the solution of the differential equation (5.5-12) for which s2 and s3 are located on the circle Is[ = 1 as shown in Fig.5.10-1a rather than on the real axis as shown in Fig.5.5-1a. One obtains for small values of Ar the following relations instead of Eqs.(5.5-14), (5.5-16), (5.5-23), (5.5-30), and
(5.7-5):
s2 - 1 + ia IAr, E. = m o c2
[
1+
83 - 1 - ia IAr,
,
\ moc/
p = ip ~ = i 2 ~ A r ,
s2 - s3 - 2ia ~Ar
~' = --in "
m~c4
Po = iPto = -27riZa moc2 [
1
1
)
1/2
m2c4
(1)
No solution in the form of asymptotic polynomials-meaning the terms of order O(1) are polynomials--can exist since p is imaginary and Eq.(5.5-26) cannot hold; its real and imaginary part do not vanish simultaneously. With the help of Stirling's relation we obtain instead of Eq.(5.5-34) lim F(p) p--*r162 F(p + ip' + 1)
=
1 pl+ip'
=
-K - e - ~ p ' l~p
p
(2)
Equation (5.5-19) yields the following relation in place of Eq.(5.5-37) for the point s3 = e - i x ' A t :
~.3(p) = q'3(o)~-~'~
'~'~" 1 x ~
( 1+ i
iP~o(1 + iP~o) - l(1 + 1) + ~2 +...) 2~ ~Arp
9 (3)
The term exp(-ip~ ln p) is a phase factor that depends essentially on the value of Ar. It disappears if the product u~3(p)u*~3(p ) is used. The function Ct,3(p, 0) : ut,3(P) e-~EAtO/h according to Eqs.(5.4-15) and (5.4-25) is at large distances p an expanding spherical wave:
,,,~ip'olnp
Ct~3(P, 0) ~ u0 ~
1 ei(~,Arp_~AtO/h )
Arp
The series expansion in the point s2 yields a contracting spherical wave.
(4)
5.10 UNBOUNDED BOSONS IN A COULOMB FIELD
Ii
s2(g)lI'i / a
/
T
/
g=0
~/
1/9=2 - 1..-"__ q--+ oo
1
245
"-\\k .
\ s2
/
si 9--+ oo 0
li s2(.q) i t i
.
.
g=O "~.,\\
~
.
.
.
.
~ ~ - _ ~.~.%o~ - ~ ~ s2 = 0.86603 + 0.5i -
\ ~-o/
/
~1 ~
/ y
FIG.5.10-1. Location of the singular points s2 (dashed line) and s3 (solid line) in the complex s-plane as function of g for the Klein-Gordon equation for general values of g and the line ~ of integration (a), and for g = - 2 c o s r r / 6 = -1.73205 (b). These illustrations also hold for the nonrelativistic SchrSdinger equation and the iterated Dirac equations. We recognize the relations s2 = c o s t / 6 + i s i n z r / 6 and s3 = c o s z r / 6 / s i n z r / 6 from Fig.5.10-1b. Equation (5.5-14) yields g = - ( s 2 + s3) >_ - 2 c o s z r / 6
(5)
as a condition for the circle of convergence around either s2 or s3 to reach s = Sl = 0. For A r we obtain from Eq.(1)"
246
5
D I F F E R E N C E E Q U A T I O N IN S P H E R I C A L C O O R D I N A T E S
,2i~i 9 ,, - ~, sin 6 Ar
l/
h/-~0~ = ~
E2
moc
rn~c 4
1
sin r6
/ -1/2
m0~~
1
(61
Hence, A r / ( h / m o c ) approaches infinity for an energy [ ~ m o c 2 of a free boson but drops to smaller values for larger energies; Ar can never be zero. As in Section 5.7 the transformation r
s
,
P=reat
(7)
maps the point s3 of the s-plane in Fig. 5.10-2a into the point 1,0 of the Cplane, and the point s2 into the point s 2P due to the relation s2s~ = s2s3 = 1 of Eq.(5.5-t4). The point s 2P is on or outside the circle [ r 1] = 1 around the point r - 1,0 if the condition
= r 1 P > 6 ~'Ar
> e~/3 r 2i 6 s~ - s3
7r 1 6 ]Jm(s3)]
m
(s)
is satisfied [~]m(s3) = imaginary part of s3]. In Fig.5.10-2 we have ~'Ar =7r/12,
P = 2
(9)
The inverse transformation of Eq.(7) P -- 2
8 - - S3~ 1 / P --" s
(10)
maps the circle I r 1] - 1 of Fig.5.10-2b into a loop in the s-plane that is shown in Fig.5.10-3. For a convergent solution according to Fig.5.10-2a we start with the differential equation (5.5-12). It is written for the point s3 by the substitution 8 = (883) + 83 aS in Section 5.7. The terms ( s - s3) '~ are rewritten as s ' ~ ( s / s 3 - 1) '~ and s / s 3 is replaced according to Eq.(7) by s / s 3 - r We now have a differential equation in the C-plane. It requires some additional transformations that bring it into a form that permits a solution by a power expansion in r 1: CO
~(r
_ 1)= ~ v-----0
~(.)(r
1)~+~
(11)
5 . 1 0 UNBOUNDED BOSONS IN A COULOMB FIELD s-plane
+i
+i
P C-plane
.... m___+ __r
,_+2
82
'
~
)
247
2
g = - 2 cos ~r/12 83 ~- e - i ~ t / 1 2 82 -~- e i~r/ 1 2 a
FIG.5.10-2. Transformation from the s-plane to the (:-plane according to Eq.(7) in analogy to Fig.5.7-1; P = 2, ~'/Xr = n//12.
0.5i
o.~
~
i
---4
~'.~
-0.5i
FI0.5.10-3. The circle I ~ - 11 = 1 in the ~-plane in Fig.5.10-2b is m a p p e d into a loop in the s-plane by means of Eq.(10); s3 - e -in/12 -- 1//82, P - 2, g - -2cosTr/12. T h e c a l c u l a t i o n is carried o u t in Section 6.10. T h e final s t e p is to t r a n s f o r m w(r 1/P - 1) into t h e s - p l a n e by m e a n s of t h e s u b s t i t u t i o n ~ = (s/s3) P. T h i s is d o n e in Section 6.10 t o o a n d t h e f u n c t i o n w p ( s - s3) is obtained" c~
N
u----O
u=O
~ ( ~ - ~) = ~ ( ~ ) = ~ ~ ( ~ ) ( ~ - ~)~+~ 9~ ~ ( . ) ( ~ - ~)~+~
(12)
All this is as in Section 5.7 except t h a t s3 is no longer a real n u m b e r b u t is now a c o m p l e x n u m b e r .
248
5 DIFFERENCE EQUATION IN SPHERICAL COORDINATES
Equation (5.5-15) was transformed into a factorial series u~3(p) in Eq.(5.519), and the same transformation produced the factorial series of Eq.(5.7-10) from the power series of Eq.(5.7-9). If we change ~p(v), i6 to ~p(v), i5 in Eq.(5.79) we obtain Eq.(12), and the same change transforms the factorial series of Eq.(5.7-10) into the one associated with Eq.(12):
r(p) N (i5 + 1 ) . . . (i5 + v) up(p) = s~ r(p + p + 1) ~=o(-1)'~P(~) (p +/~ + 1) ... ( p + / ~ + u) =P
(-- ) A5
83
--
1 + 1 = P ( i p ' - 1) + 1
Eq.(6.10-1)
8~
~p(1) (~[s3P-l(~-l+2P)-(s3-s~)(p-1)(P-1)/2P]-P[l(l+l)-72] ~p(0) = P-ls3(s3 s~)(~ + 1) -
+ i5 P - 1 ) s3 2! ' s3, p'
see Eq.(1),
~, A, 5
Eq.(6.10-14)
see Eq.(5.5-2)
(13)
As in Section 5.7 we work out a few numerical values. From Eq.(1) we obtain for the spatial resolution Ar"
Ar
s~-s3 = - 2i~' - =
s~-s3 h (E 2 i ~47r moc m~c4
1
)-1/2
(14)
From the limit case for convergence shown for P = 1 by Fig.5.10-1b we see the condition s 3 - s3 = s2 - s3 _ i
(15)
and obtain
1 h ( Ar > 47r moc
E2
rn2c4
1
= 0.079577 h
?7%0 C
?7%0 2 C4
1
(16)
which is in line with Eq.(5.7-12). The lowest possible energy E for an unbounded boson is m0 c2, which yields Ar ~ oo. A localization of such a boson is completely impossible. For larger values of E the irresolvable spatial interval decreases to an arbitrarily small but finite one, which implies that a boson with very high energy is not much disturbed by the Compton effect. The equivalent formula for Eq.(5.7-14) is expressed by Eq.(8). If we choose a value of P we obtain an equation for the magnitude of the imaginary part of 83
5.10
r
UNBOUNDED
T
B O S O N S IN A C O U L O M B
35
/
1.6
"/ :'/
.,.
15
"-~ I0
/
,,'Y
5 2
-
i
25
4
249
/
I
P=l.8
r 20 "-2 ~a,
FI(3.5.10-4. P l o t s o f
// //
FIELD
1.4
1.2 6
p----+
[um(plu~l(p)] 1/2 a c c o r d i n g
8
i0
to E q . ( 2 a ) for Z = 1,
A~l(hlmo~ )
k = 1, a n d P = 1, 1.2, 1.4, 1.6, 1.8 for p = 1, 2, . . . , 10.
14
~ 0."
12
P=1.8
.." ." .' ..
10 ~I~,
.'
~, ,-4
.." .'"
.a. 4
...-
2
j:" 4
P l o t s of [upl(p)u~,l(p)l
= k = 0.5, a n d P - -
1.2
.............
2
FIG.5.10-5.
.,,.~"
6
p--+
1/2 a c c o r d i n g
1
7
8
i0
to Eq.(23) for Z = 1,
Ar/(h/moc)
1, 1.2, 1.4, 1.6, 1.8 for p = 1, 2, . . . , 10.
....'"
I 2.5
....."'" 9
2
7 ~. ... .~- ' " ..............
..
. . .
.....
,m 1.5
9. ' " ' "
1.6 . . . . . . . . . .
-
. . . . . .
1.4 - ........
"--'0.5
1 i
,
I
2
,
1.2
"-" " " . . . . |
i
4
,
i
p---,
"2-.5-2K-2-.5"2-. ,
!
6
,
!
,
i
8
,
!
,
I
I0
FIG.5.10-6. P l o t s of [upl(p)u~,x(p)]1/2 a c c o r d i n g to Eq.(23) for Z = 1, -- k -- 0.2, a n d P - - 1, 1.2, 1.4, 1.6, 1.8 for p - 1, 2, . . . , 10.
Ar/(h/moc)
250
5 DIFFERENCE EQUATION IN SPHERICAL COORDINATES
:oo :~;~1/7,6i 8o
~_
:a,
i
/
,
/,,' 1.6
.
/:.4
S ~~ / 2o
~
1.2
........................................... I0
15
....
p---, 20
25
FIG.5.10-7. Plots of [up: (p)u~: (p)] :/2 according to Eq.(23) for Z = 1, = k = 1, and P - 1, 1.2, 1.4, 1.6, 1.8 for p - 5, 6, ... , 25. 25
:' f
2O
."
Ar/(h/moc)
./" P=l.8
s.s"
,.-'"
~- 15
t~
~
5
i.
1.4
------
- ............................
I0
15
1.2
1
:,~.-.,._ ,
p---, 2O
25
FIG.5.10-8. Plots of [up: (p)u~,l (p)] :/2 according to Eq.(23) for Z = 1, = k = 0.5, and P = 1, 1.2, 1.4, 1.6, 1.8 for p = 5, 6, . . . , 25.
5
......
:4
.....
9. . . . .
Ar/(h/moc)
.-'"
F ....... -'""
......'"
~3
....'"
~2
1.6 1.4 .....
,-..___;
.
i0
. . . . --:.--- ,
. . . . ~ "7-~-~
15
O'--*
1.2 -,- ~
t
20
.-,- , -
1
J
,--, 25
FIe.5.10-9. Plots of [up: (p)u~: (p)] :/2 according to Eq.(23) for Z = 1, = k = 0.2, and P = 1, 1.2, 1.4, 1.6, 1.8 for p - 5, 6, . . . , 25.
Ar/(h/moc)
5.10 UNBOUNDED BOSONS IN A COULOMB FIELD
251
I x 1024
T
5x1023
9
,-4
"~
_
I'2
5x 102a
14
'
.
,
!
16
,
|
9
L
18
,
"
20 '
'
9
22 '
'
p---+
1024 _
.
,
9
1 . 5x 1024 - 2 x 1024
FIG.5.10-10. P l o t s of [upl(p)u~,l(p)] 1/2 according to Eq.(23) for Z = 1, =k=l,P=5, andp=ll, 12,...,22.
T
IxlO 9 j
6
,
.
j
|
8
,
i
9
,
~
,
i0 !
'
.
,
i
,
"
i
,
I
Ar/(h/moc)
J
14
12.
~~., _ l x l 09 p---, .1~.,
-2xl
0
9
-3xlO 9
-4x 109 -5xlO
9
FIG.5.10-11. P l o t s of [up1 (p)u~,l (p)]1/2 according to Eq.(23) for =k-0.5, P-5, andp=4,5,..., 14.
Z
= 1,
Ar/(h/moc)
250
r
4
T
-250
~.
-500
,-4
-I000
!
3
.
.
.
.
6 a
,
,
,
9
!
5
.
.
.
i
.
.
.
.
|
.
.
.
.
7
,
8
p
-750 n~
-1250
FIG.5.10-12. P l o t s of [upl(p)u~,l(p)] 1/2 according to Eq.(23) for Z = k = 0.2, P - 5, a n d p - 2, 3, . . . , 8.
1,
Arl(himoc)
252
5 D I F F E R E N C E E Q U A T I O N IN S P H E R I C A L C O O R D I N A T E S
IJ,~(~)l = ~_1
(17)
6P
which is the equivalent of Eq.(5.7-17). The real part of s3 is determined by the imaginary part since s3 is always on the unit circle according to Fig.5.10-1b. We restrict at this time the location of s3 to the interval - 2 < g < 0 but we shall investigate larger values of g in the following section. In analogy to Eq.(5.8-10) we want to plot the first term upl(p) of up(p) in Eq.(13):
r(p) upl(~)=SgF(p+iS+l),
P=I' 2,..., N
(18)
First we determine i5 with the help of Eq.(5.8-1)"
-Ar E Ar A5 = 2 E 7 ~ - = 4n~moc2 h / m o c = 4 r S e ~ k = 47rZa, e, = E / m o c 2, k = A r / ( h / m o c ) a = 7.297 535
x
10-3, Z
--
I, 2,...
(19)
From Eqs.(13) and (17) we obtain:
i5 = P
X5 s3 - s~
= -3p2As-
1 P+
+ 1= P
A5 -2JJm(sa)l
1)+1 (20)
1
71"
Since i5 is real the only complex term in Eq.(18) is s3. W i t h the relations
s3 = x -
iy,
71"
y = I:Jm(s3)l = ~-fi,
X2
+
y2
= 1
(21)
we obtain: s3 = [1 - (~/6p)2] 1/2 - i n / 6 P
(22)
In order to eliminate the complex term s~ in Eq.(18) we may plot up1 (p)u~, 1 (p). The square root of this expression yields a function that is easier to compare with Eq.(5.8-10):
[~p~ (p)~;~(p)]~/2 =
r(p)
r ( p + t + i)
(23)
5.10
UNBOUNDED
2
BOSONS
IN A C O U L O M B
253
:' ",
.'
,
--
","
,
1.2
. ".
,/
.
.
p__+
.
.
.
.
.
.
~ , - 2 2 - 2 ~ - = ~ _. _ _ ( _ . . :
.
-4
1.6
~,
'--:
100~,p2 (p)
- 2
a.,
FIELD
'.
"~ P
1.8
-6 %:
-8 ;i ::
-10
F i c . 5 . 1 0 - 1 3 . . P l o t s of ~P2(p) = [ue2(p)@2(p)] 1/2 according to Eq.(25) for I = O, Z = 1, k = A r / ( h / m o c ) = 1, and P = 1, 1.2, 1.4, 1.6, 1.8; p = 1, 2 , . . . , 10. T h e plot for P = 1 is multiplied by 100.
'. . P = 1, 1 0 ~ P 2 ( P ) x'~- ........ i
T
-i
r
-2
I
,
1.4
]""
;"
P = 1.2, lO0~p2(p) I"--"
- - I - -
4
6
--I--,
,
,,
p ____~ 8
i0
1.6
., ,~
-3 -4
"......................"'... P -
1.8
"'...
-5
P l o t s of ~Pu(P) = [up2(p)u;2(P)] 1/2 according to Eq.(25) for l = 0, 7_ - 1, k -- A r / ( h / m o c ) - - 0 . 5 , and P -- 1, 1.2, 1.4, 1.6, 1.8; p - 1, 2, . . . , 10. T h e plot for P -- 1 is multiplied by 10, t h e plot for P -- 1.2 by 100. FIG.5.10-14.
1
.
4
~
.
.
.
.
.
_~
_ _~ i
T
2 .........
...,..- 9""
r .........
~ .........
p -----~
~.........
i
i0
- o . 2
-0.4
~ ........
-0.6 -0.8
/
P l o t s of ~P2(p) = [up2(p)u~2(p)] 1/2 according to Eq.(25) for 1 = 0, Z = 1, k = A r / ( h / m o c ) = 0.2, and P = 1, 1.2, 1.4, 1.6, 1.8; p - 1, 2 , . . . , 10.
FIC.5.10-15.
254
5 DIFFERENCE EQUATION IN SPHERICAL COORDINATES p ---4, *
,.
~
,
,
,
,
,
J
10
"::: ..........
,
|
,
|
|
15
-o.s ~ - - ' ~
|
|
i
J
i,
25
L:i.2 : 2-~;:2(;,5
'-.~
"~=,,~uu~tp)~":"- ~ s
I
,~
|
20
'"
% "'..... --.:. ~ 0.~ "'.-...
-1.5
-2
~
"..
~b ...:~%
.....
Fie.5.10-16.
Plots of ~P~-(O) = [,~P~(p),4~(0)l 1/2 according to Eq.(25) for l = 0, = 1, and P = 1, 1.2, 1.4, 1.6, 1.8; p = s, 6 , . . . , 25. T h e plots for P = 1 to P = 1.8 are multiplied by 200 to 0.0002. 7 =
1, k = A r / ( h / m o c )
1.5
\ x
T 0.5
""" " ".......
P=
I0 -0.5
.
15
20
.
.
"
-1
500~p2(p)
1.2,
p\
25
:
P=l.6
.-,.
.--.
FIC.5.10-17.
Plots of ~P2(P) - [ u p 2 ( p ) u ~ , 2 ( p ) ] 1/2 according to Eq.(25) for l -- 0, - 0.5, and P - 1, 1.2, 1.4, 1.6, 1.8; p - 5, 6, . . . , 25. T h e plots for P - 1, 1.2, 1.8 are multiplied by 100, 500, 0.1. Z - - 1, k -
Ar/(h/moc)
0.02 0.01
l
.
.
.
.
.
.
10
c~ - 0 . 0 1
,--,
--
--,
. ..~
p---+ 15
--.
-..-
-_,:-_7.--_,:=:_-_v..-=c_-~ 20 25
-0.02 -0.03
-0.04
FIG.5.10-18.
o.O-~,. ,.'" i" ],]
Plots of ~P2(p) = [ u P 2 ( p ) u ~ 2 ( p ) ] 1/2 according to Eq.(25) for l = 0, = 0.2, and P = 1, 1.2, 1.4, 1.6, 1.8; p = 5, 6 , . . . , 25. T h e plot for P = 1 is multiplied by 0.1. Z_ =
1, k = A r / ( h / m o c )
5.10 U N B O U N D E D BOSONS IN A C O U L O M B F I E L D
255
i
ix I026i
l
,
t
.
i-
.
|
|
12
,
i
,
|
14
,
,
|
16
,
!
,
|
18
20
p---,
~GS -Ix 1 026 -2x 1026
F I o . 5 . 1 0 - 1 9 . P l o t o f ~ P 2 ( p ) = [up2(p)u~,2(p)] 1/2 a c c o r d i n g t o E q . ( 2 5 ) for l = 0, Z = 1, k = A r / ( h / m o c ) = 1, P = 5, a n d p = 10, 11, . . . , 20.
2 . 5 x 101~
l
I
-~~. - 2 . 5 x l
,
!
9
8 !
6
01r
.
..
"
IO
i
I
,
l
,
I
9
p .---,
12
O..,
~
-5x 101~ - 7 . 5 x 101~ - I x l O zz
F I G . 5 . 1 0 - 2 0 . P l o t o f ~P2(P) = [up2(p)u~2(P)] 1/2 a c c o r d i n g t o E q . ( 2 5 ) for I = 0, Z = 1, k = A r / ( h / m o c ) = 0.5, P = 5, a n d p = 4, 5, . . . , 12.
30000
T
20000
~-" e,l
I0000
~L
4
. . . .
J
3 -I0000
9
.
.
|
,
9
.
,
!
5
p---+
.
,
,
i
6
.
.
,
t,
7
F I G . 5 . 1 0 - 2 1 . P l o t o f ~P2(P) = [up2(p)u~2(P)] 1/2 a c c o r d i n g t o E q . ( 2 5 ) for I - 0, Z k - A r / ( h / m o c ) - - 0.2, P = 5, a n d p - - 2, 3 , . . . , 7.
1,
256
5 D I F F E R E N C E EQUATION IN SPHERICAL COORDINATES
In Eq.(5.8-6) we used E / m o c 2 = 1 - ~ 2 / 2 n 2 with n = 1. To obtain c o m p a r a b l e values we choose E / m o c 2 -- 1 + ~2/2 in Eq.(19) since E must now be larger t h a n moc 2. As previously, we choose initially k = A r / ( h / m o c ) -- 1 in Eq.(19), but reduce it to k - 0.5 and k - 0.2 rather t h a n to 10 -2, 10 -3, and 10 -4 as in Figs.5.8-9 to 5.8-11. This implies t h a t the transition from nonr a n d o m to ( p s e u d o ) r a n d o m numbers will be much faster for u n b o u n d e d bosons t h a n for b o u n d e d ones. In line with this result we shall also use P = 1, 1.2, 1.4, 1.6, 1.8 instead of the larger values P = 1, . . . , 5 in Figs.5.8-5 to 5.8-11. C o m p a r i n g the plots of Eqs.(23) in Figs.5.10-4 to 5.10-9 with those in Figs.5.8-5 to 5.8-11 shows t h a t large values of the plots no longer occur close to p -- 0. This is to be expected since the eigenfunctions t h a t we derived are used for probability density functions for the location of a boson and a b o u n d boson is more likely to be close to the center of the Coulomb field t h a n an u n b o u n d boson. T h e rapid change of the plots when k is reduced from I to smaller values is conspicuous. One would expect t h a t the location of an u n b o u n d boson is changed more by the C o m p t o n effect than that of a bound boson. In line with this observation are the plots of Figs.5.10-10 to 5.10-12 t h a t show the r a n d o m values obtained from Eq.(23) for P = 5 instead of for P = 10 as in Fig.5.8-8. The narrow and varying ranges of p in Figs.5.10-10 to 5.10-12 are due to the enormous variation of [up1 (p)U~l (p)]l/2 for P = 5. We t u r n to the second term up2 of the factorial series of Eq.(13) and write it in the following form:
r(p) Op(1) ~ + 1 ~'~(P) = - ~ r(p + p + 1) p + p +------f Op(1) see Eq.(13) for Op(0) = 1; s3 see Eq.(22); i5 see Eq.(20)
(24)
In order to obtain for Up2(p ) a formula of the form of Eq.(23) we observe t h a t ~p(1) is a complex number. Hence, we write
[up2(p)u~,2(P)] 1/2 -- UP2(P) = -[qp(1)q~,(1)] 1/2
r(p)
~+ 1
r(p +~ + 1)p+~+ 1
(25)
Plots of Eq.(25) are shown for the range p - 1, 2, . . . , 10 in Figs.5.10-13 to 5.10-15. T h e values of P run from 1 to 1.8 as in Figs.5.10-4 to 5.10-6. The value of k = A r / ( h / m o c ) varies from 1 in Fig.5.10-13 to 0.5 in Fig.5.10-14 and 0.2 in Fig.5.10-15. It is evident t h a t the plots for P = 1.6 and 1.8 are quite different from those for P -- 1, 1.2, 1.4 and erratic. ;i'his behavior is also recognizable in Figs.5.10-16 to 5.10-18, which hold for the range p -- 5, 6, ... , 25. For P = 5 and k = 1, 0.5, 0.2 we obtain the r a n d o m plots of points of Figs.5.10-19 to 5.10-21. They are very similar to the plots of Figs.5.10-10 to 5.10-12.
5.11 ANTIPARTICLES
257
~ =-a//2~
~=-~
a
~'~
b
FIG.5.11-1. Plot of the left half-plane of Fig.5.10-1 in analogy to Fig.5.5-2. The values g -- - 3 / v / 2 and g - - ( 2 + 1/6) are replaced by g -- 3/x/~ and g -- 2 + 1/6. The points s3 and s2 of Fig.5.5-2 become s2 = - s 3 and s3 = - s 2 in this figure. (a) holds for the limit when the circle of convergence around s2 goes through s3 and s = 0, while (b) holds for smaller absolute values of s2. 5.11 ANTIPARTICLES In Fig.5.10-1 we have shown the loci of s2(g) and s3(9) b o t h for positive and negative real values, while only the half for positive real values was shown in Figs.5.5-1, 5.5-2, and 5.7-1. We extend here our investigation to negative real values of the s-plane. Figure 5.11-1 is the equivalent of Fig.5.5-2 for negative real values. We see t h a t g is replaced by - 9 , s3 by - s 2 , and s2 by - s 3 . We s u b s t i t u t e s -- ( s - s2) + s2 into Eq.(5.5-12) in order to solve the differential equation with a power series in the point s2. T h e relation s~ + gs2 -k- 1 = 0
according to Eq.(5.5-13) is needed to obtain:
[(~- ~2) 2 + (2~ - ~ ) ( ~ - ~2) 2 + ~2(~2 - ~s)(~- ~2)]~"(~- ~2)
-[z(z + 1)- ~]~(~- ~) = 0 (1) For the solution of this differential equation we need a power series according to Eq.(5.5-15): N
~(~-
~)= ~ q~(~)(~- ~)~+~
(2)
v--0
W h e n we compare Eqs.(1) and (2) with Eqs.(6.8-1) and (6.8-2), which hold for Fig.5.5-2, we recognize that the substitutions
258
5 DIFFERENCE
EQUATION
IN S P H E R I C A L
COORDINATES
~ -~ ~ , ~ -~ ~ , p~ -~ p~, q~(.) -~ q~(~)
(3)
transform Eqs.(6.8-1) and (6.8-2)into Eqs.(1) and (2), which hold for Fig.5.111. Hence, we can take the results of Section 6.8 and make the substitutions of Eq.(3) to obtain the corresponding results for Eqs.(1) and (2). Let us start with Eq.(6.8-3):
P3 ----P-
A5
83--82 < 0 according to Fig.5.5-2
8 3 -- 8 2
A5 8 2 -- 8 3
A5 . . . .
8 3 -- S 2
P3,
s 3 - s 2 < 0 according to Fig.5.11-1
(4)
For the further evaluation of P3 and p~ we turn to Eq.(5.5-16)'
[ (1
p ' = +27rZa
1+
= -2~'Za
hlmoc) h/moc)
= p~ + o ( ~ ) p~ = - 2 r Z a
E (1 -too 02
E2
mo2~4 )
(5)
The change of sign for p~ - pt in Eq.(4) permits us to write - m 0 instead of m0 in Eq.(5). We postpone a discussion and turn to the rewriting of Eqs.(6.8-5) and (6.8-8). Since we wrote p3 - p and p~ = p' in Eq.(4) we must be careful to replace P3 in Eqs.(6.8-5) and (6.8-8) by p' rather than p~' (3~I 3,1 (0)q~(1)
I + 33, 0 (0)q~ (0) = 0
OL3 , 1 ( 0 ) - - - - 8 2 ( 8 2 -- 8 3 ) ( p t -b 1)
a~,0(0) = s2p'(p' + 1 ) - l(l + 1 ) + ~2
(6)
Finally, we rewrite Eq.(6.8-8)"
' (v)q'3(v + 1) + a3,0(v)q'3(v) ' 33,1 + a'3,-1 (v)q'3(v- 1) = 0 !
33,1 (v) = s2(s2 - s3)(p' + v + 1)(v + 1)
~ , o ( ~ ) = ( 2 ~ - ~ ) ( p ' + ~)2 + [~ _ p , ( ~ _ ~)](p, + ~) = (p' + ~)(p' + ~ - 1)
O/t~,_~(~)
(~)
5.11 ANTIPARTICLES
259
From Eq.(5.5-23) we obtain with the substitutions of Eq.(3) for ~':
s3 (s3 - s2) _" moc ~1 2Ar h \ t~t
s2(s3 - s2) . -moc (
82(82--83).
=
2Ar
E2 / 1/2 m 20c4
-- -
2At
--
h
E2 /
1/2
m2c4
1
(8)
With ~' we obtain from Eq.(5.5-32):
, mock[ 2(n2l + 1 21)] nh 1+ ~
~1
,.-moc~/[ ~2 - n, h
, 9 F_I
~/2 ( n ' 1) 1 - - ~ 2l+1 + "2
[ -moc 2 1
"~2
"~4( n
2n 2
n4
2l + 1 3
E " -moc 2 1-
2 7 + ~
21+1
(9)
From Eq.(2) we get in analogy to the transition from Eq.(5.5-15) to (5.519) a factorial series:
r(p) N ( p ' + 1)... (p' + v) ~(P) = ~ r ( p + p,+ 1 ) ~ (-1)~q~(v)= 0 ( p + p ' + 1)... (p + p ' + v)
q~(1) _ _a~,o(0) q~(0) - ~,,(0)
(10)
The first term of Eq.(10) cannot be plotted like Eq.(5.8-10) since s2 is negative. However, we can plot according to Eq.(5.10-23):
[( = =
r(p) S~F(p §
I~t"
1)
r(p) 9 r(p+p'+l)'
)(~
r(p) )*]~/~ F(p+ p' ~- 1)
p=l, 2,..., N
(11)
We essentially obtain Eq.(5.8-10) except that i5 is replaced by p~. This difference disappears if we substitute E~/(-moc 2) " E'/(-moc 2) or E~2/(-moc 2) " E'/(-moc 2) from Eq.(9):
260
5
DIFFERENCE
E
_
_mOC
2
--
E Q U A T I O N IN S P H E R I C A L C O O R D I N A T E S
E~
_
_moc
2
E~
-- 1 -
,~2
+ O(~ 4)
(12)
_moc 2 --
--
This substitution produces for p' of Eq.(5) the same result as the substitution of Eq.(5.8-6) produces for p in Eq.(5.5-16):
p ' = p = - 2 7rZ a
1-
-~n 2
1-
x
1-
1+
"~ = 47rZa, Z = 1, 2 . . . , n = 1, 2 , . . . ,
2}
-~-~n2
1-
-~/2
(2~k) 2
1-~n-~n2
a = Z e 2 / 2 h = 7.297 535
x 10 - 3
k = Ar/(h/moC)
(13)
Equations (5.8-10) for a particle and (11) for its antiparticle are equal. Consider the second term of the factorial series of Eq.(10). W i t h the help of Eq.(6) we obtain:
q~(1)_ q~(O) -
~'3,0 (0) = p, (p' + 1) a' 3,1(0) (s3-
[l(l + 1) s2)(p'+
- ~2]/s2 1)
(14)
From Eq.(6.8-5) we obtain the result: q3(1) q3(0) =
a3,0(0) p ( p + 1) - If(1 + 1) - ~2]/s3 a3,1 (0) = (s2 - s3)(p + 1)
(15)
Equations (14) and (15) are different. Even though we again have p' = p we recognize in Fig.5.5-2a the values s3 = 1 / v ~ ,
s2 - s3 = x / 2 / 2 , g < - 2
(16)
which must be used with Eq.(15). But Fig.5.11-1a shows the different values s2 = - 1 / v ~ ,
sa-
s2 = - v / 2 / 2 ,
g > +2
(17)
Equations (17) and (16) bring Eqs.(14) and (15) into the following form: q3(1) _ q~(1) q3(0) = q~(0) = x / 2 p - 2 [ l ( / + 1 ) - ,~2]
(18)
The points g = 0 in Figs.5.10-1 and 5.11-1 are of interest since a particle changes to an antiparticle or viceversa at these points. We obtain for g = 0 from Eq. (5.5-3):
5.11 ANTIPARTICLES
g= -
Arc) 2
h/m0
2 + -4
h/moc [2
261
=0 Ar
2
m2oc4)(h/moc) = - 8
= 4 (1
moc 2
h/moc
The energy E at g = 0 becomes strictly an effect of the resolution A r of the observation and says nothing about the observed particle. If one approaches the points g = 0 in Fig.5.10-1 from the particle side g < 0 one obtains
[
moc2 =+ 1+2
h/moc
but if one approaches g - 0 from the antiparticle side g > 0 one obtains
-moc ~
=+
[
1+2
h/moc
(21)
Hence, the finite but otherwise undetermined resolution A r permits a transition between particle and antiparticle without a q u a n t u m jump. This is a result characteristic for a difference theory using A r rather than dr. We note that [Ar/(h/(moc)] 2 in Eq.(21) is not changed if we replace it with { A r / [ h / ( - - m o c ) ] } 2.
ADVANCES IN IMAGING AND ELECTRON PHYSICS, VOL. 137
6 Appendix
6.1 DIFFERENCE OPERATORS OF HIGHER ORDER
The first- and second-order difference operators have been introduced by Eqs.(1.2-1), (1.2-4), (1.2-5), and (1.2-7). For difference operators of higher order we must revisit the definitions. Difficulties are created by the use of symmetric difference operators in Eqs. (1.2-1) and (1.2-7). The nonsymmetric difference operators of Eqs.(1.2-4) and (1.2-5) are easier to extend; this extension is used by mathematicians (Nhrlund 1924; Milne-Thomson 1951; Spiegel 1994). There are a number of reasons to prefer the symmetric difference operators to the nonsymmetric ones in physics (Harmuth 1989, Sec.8.1 and 8.2). Starting with Eq.(1.2-1) we redefine the symmetric difference operator of first order: ,~A(O) A(O + A0/2) - A(O - AO/2) ,~0 = A0
(1)
The second-, third- and fourth-order difference operators become: /~2A(0) A02
{',~A(O)'~
= ,50 \
Ao
A A(O + A0/2) - A(O - A0/2)
) = ,~--~
,,,o
A(O + Ae) - 2A(0) + A(O - A0)
(2)
(/xo)~ AaA(0) Ao~
A (/~2A(0))
= Ao /~o ~A(O + 3 A 0 / 2 ) - 3A(O + A0/2) + 3A(O - A 0 / 2 1 - A(O - 3A0/2)
(/xo)~
(3)
A'A(O) _ A ( A3A(O)) A(O + 2A0)
-
4A(O + A0) + 6A(0) (Ao)~
-
4A(O
-
A0) + A(O
-
2A0)
(4)
262 ISSN 1076-5670/05 DOI: 10.1016/S 1076-5670(05)37006-6
Copyright 2005, Elsevier Inc. All rights reserved.
6.1 DIFFERENCE OPERATORS OF HIGHER ORDER
1
1
0
1
2
1
O
O
1
3
1
4
15
0
- 3 A 0 -2A0
0
-A0
0
0
5
0
20
0
1
0
10
0
6
0
4
0
10
0
1
0
6
0
5
0
3
0
0
1
A/Ao
0
O 0
263
1
0
15 0
A0
0
6
1 z~6//~06
0
2A0
O
3A0
0---, FIG.6.1-1. Arguments and factors of symmetric difference operators from order 1 to 6 using odd and even multiples of A0/2 according to Eqs.(1)-(4).
TABLE 6.1-1 DIFFERENCE OPERATORS FROM FIRST- TO SIXTH-ORDER ACCORDING TO EQS.(1) TO (4).
1 [A(0+ 2--~ )
1 1 - A ( 0 - 2--~)1 ~-~
1 [A(O+AO)- 2A(O)+A(O-AO)]~ (AO)~ 3AO AO _~_ 3AO [A(O+-~)-3A(O+--~--) +3A(0- - - ) - A ( 0 - - - ~ ) 1
1
(A0)--------g
1 [A(O+2AO)-4A(O+AO)+ 6A(O)-4A(O-AO)+A(O-2AO)]~ (2zxo)4
[A(O+5-'~)-5A(O+3-'~')+IOA(O+-~ --)
-10A(0-
Z~-~O)+5A(O-3-~)-A(O-5--~-)](A10)5
[A(O+azaO)-6A(O+2ZXO)+ISA(O-AO)-2OA(O)+ISA(O-AO)-6A(O-2AO)+A(O-az~O)]
(zx0)6
The operators of odd order, z~/z~0 and z~a/z~0 3, contain odd multiples of the difference A0/2 while the operators of even order, z~2/z~0 2 and z~4/z~0 4, contain even multiples of the difference A0/2. Figure 6.1-1 shows what causes this complication. In Eq.(1.2-1) we had sidestepped it by using the arguments 0 4- A0 and the denominator 2A0. The first six difference operators listed in Table 6.1-1 make the connection clear. An alternative way to generalize the symmetric difference operators of Eqs.(1.2-1) and (1.2-7) is as follows: AA(O) Ao
A(O + A0) - A(O - A0) =
2do
(5)
264
6 APPENDIX
o
o
1 o 1
2o
1
40
o
0 o
1
4o
o
1 0
60
150
-3A0 -2A0
2o
A2/A02
2o
o
60
40
0
5o
4o
o
1
A~iAo5
200
150
60
1 0
/~6/z~06
0
A0
2A0
3A0
5
-A0
1
o
1
/~3/z]0a
1
A4/A04
0--~ FIG.6.1-2. Arguments and factors of symmetric difference operators from order 1 to 6 using integer multiples of A0 according to Eqs.(5)-(8).
TABLE 6.1-2 DIFFERENCE OPERATORS FROM FIRST- TO SIXTH-ORDER ACCORDING TO EQS.(5)
To (8).
[A(O+AO)
-A(O-AO)]
1
2A0 1
[A(O+AO)- 2 A ( O ) + A ( O - A O ) ] ~ (AO)2
[A(O+2AO)-2A(O+AO)
1
+2A(O-AO)-A(O-2AO)]2(AO)3
[A(O+2AO)--4A(O+AO) + 6A(O)-4A(O-AO)+A(O-2AO)]
[A(O+3AO)-4A(O+2AO)+5A(O+AO)
1 (A0)4 1
-5A(O-AO)+4A(O-2AO)-A(O-3AO)]--------2(/,,0)5
[A(O+3AO)--6A(O+2AO)+15A(O-AO)-20A(O)+15A(O--AO)-6A(O-2AO)+A(O-3AO)]
A2A(0) AO2
AaA(O) AO3
=
AO A(A2A(O) 2
1 (A016
A(O + A0) - 2A(0) + A(O + A0)
= 2(A0) 1 3 [f~[A(O +
(6)
2A0) - 2A(O + A0) + A(0)]
-.[A(0) - 2A(O - A0) + A ( O - 2A0)] }
= A(O + 2A0) - 2A(O + A0) + 2A(O- A 0 ) - a ( o - 2A0) 2(ZX0)3
(7)
6.1 DIFFERENCE OPERATORS OF HIGHER ORDER
4
z~4A(0) _.z~0 -
z~0 2A2 (A2A(O))z~O 2 =
2[a(0 + A0) =
-
(A0) 41
{[A(O+2AO)-2A(O+AO)+A(O)]
2A(0) + A(O- A0)] + [a(0)
A(O + 2A0)
4A(O + A0)
-
265
+ 6A(0)
-
2A(O- A0)+ a(o- 2A0)] } 4A(O A0) + A(O 2A0) (8) -
-
-
(A0) 4
There are no odd and even multiples of A0/2 as in Eqs.(1)-(4) but only integer multiples of A0. Figure 6.1-2 shows the scheme of amplitudes and arguments. For difference operators of even order, ,'~2n/~O2n, we obtain the same binomial numbers as in Fig.6.1-1, but for difference operators of odd order, z~2'~+l/z~0 2~+1, we obtain different numbers. In analogy to Table 6.1-1 we list the difference operators up to order 6 in Table 6.1-2. In order to decide whether to use the operators of Table 6.1-1 or of Table 6.1-2 consider a finite set of numbers f(0), f(O + A0), . . . , f(O + mAO). These numbers may have been produced by calculation to be checked by measurements or they may have been obtained by measurements to check a theory that should yield the same numbers by calculation. The operators of Table 6.1-2 correspond to this condition better than the operators of Table 6.1-1. One could interpolate the set of numbers by writing
f[O + (2n
+ 1)A0/2] =
1
-~{f(O + nAO) + f[O + (n + 1)A0]}
(9)
to obtain values for integer multiples of A0/2. This interpolation would not add any information if the numbers f(O), f(O + A0), ... f(O + mAO) represent physical measurements, it would only create the illusion of an improved resolution. As an example let us write Eq.(3) with the terms A(O + 3A0/2) to A(O3A0/2) interpolated from A(O + 2A0), A(O + A0), ... , A(O- 2A0):
A(O + 3A0/2) A(O + A0/2)
1 [A(O+ 2A0)
+
A(O + A0)]
= I[A(0 + A0) + A(0)]
A(O-
A0)]
3A0/2) = I [ A ( 0 - A0) +
A(O-
A(OA(O-
=
A 0 / 2 ) =
1 [A(0)+
2A0)]
(10)
If we substitute Eq.(10) into Eq.(3) we obtain Eq.(7). This becomes quite different if f(O),..., f(O + rnAO) represent the values of a defined mathematical function, say the Gamma function F(O). The value of
266
6 APPENDIX
/"(2.5) is not the average of/"(2) and F(3). Hence, the information contained in the set F(0), F(O + AO), ..., F(O + mAO) is increased by specifying the additional values F(O+O.bAO),..., F[O+(m-1/2)A0]. Unlimited information can be obtained from many defined mathematical functions by sufficiently fine interpolation. This is completely different from the finite set of numbers that can be produced by or checked by physical measurements. No mathematical manipulations can increase the information contained in them. The transition A0 --, dO yields the same result for the difference operators of Tables 6.1-1 and 6.1-2, but the inverse transition dO ~ A0 is not unique and we must decide which set of difference operators to use. For a finite set of numbers f(O), f(O + A0), . . . , f(O + mAO) that can be obtained by or for observation in physics the operators according to Fig.6.1-2 or Table 6.1-2 are simpler than the ones of Fig.6.1-1 or Table 6.1-1 plus substitutions according to Eq.(10); we shall use them. This is a good example of how concepts of pure mathematics must be carefully analyzed before using them in physics. 6.2 EXTENSION OF SECTION 3.1 FOR Ax << Ac With the help of Table 6.1-2 we may rewrite Eq.(3.1-11), which holds for small values of Ax, in explicit form:
2AcA~(~,O)h2 (
h
) 2 ( z~
ieAxAmox ~ 4 h
/ ~o(r 0)
2AcAe(~,0)h 2 ( h ) 2 [ Z~4 ieAxAmox Z~3 (/ eA x~Am0x) 2/~2 e(Ax) 2 mocAx A( 4 -- 4 h z~ff------ff -F 6 A( 2 - 4
2),cAe(ff,0)h2 ( -
h
-
( ie/kxAmox ) 3 z~ h
)2
-~ +
( ie/kxAmox ) 41 h
~o(ff, O)
(/xr1 [~o(ff + 2A~ 0 ) - 4~o(ff + Aft, 0)
+ 6~o(~, 0) - 4~o(ff - Aft, 0) + ~o(~ - 2Aft, 0)] +4
ieAxAmo~ h
-6_
1
2(Ar
[V0(r162162162
(r162
(r162
(ieAxAmoz)2h
+4
1 [~0(~ + A~, 0 ) - 2~0(r 0 ) + ~ 0 ( ~ - A~, 0)] _ (A~) 2
( ieAxAmoz ) 3 1 [ffjo(r+ A~, O) _ q~o(( _ "A(, O)] h
)
f ieAxAmox 4
!
~
(1)
6.2 EXTENSION OF SECTION 3.1 FOR Ax << ~c
267
Equation (3.1-18) is another equation that holds for small values of Ax and requires Table 6.1-2 for writing it in explicit form. First we write the term with A/Ar
[ ff53
( /~ ie/kxAm~ ) 3
-
ieAxAm~ -3
~
( ieAxAm~ ) 2
/~C2 ~-3
_ (ieAxAmox
ZC
ff~o(~,O)
1
2(/k(] 3 [~o(~ + 2A~, 0) - 2~o(~ +/k~, 0) + 2@o(~- A~, 0) - ~o(~ - 2A~, 0)1 - 3
ieAxAmo~ h
1
(/,,()2 [~o(r +/',~, 0 ) - 2~o(~, 0) + ~ o ( r
-
h
0)]
r162
(2)
The whole Eq.(3.1-18) becomes:
C~C mocAx
(AeozAmoy- AeoyAmoz) -~ + •
2Ze2( h )4 oLc ?TtoCnX (ne~176176176
(A
ieAzAm~
Xr
h
{ 1 [
~
ieAtCeoh ) ~
~o(~,o)
1 {[~o(~+2A~,0+A0) 2 ( ~ ) '3
- r162 + 2zxr o - A o ) ] - 2[r162 +/',r o + Ao) - r162 + ~xr o - Ao)] + 2[~o(r - zxr o + Ao) - ~o(r - ~r o - Ao)] - [ ~ o ( r
- 2,~r o + Ao)
9 o ( ~ - 2/xr o - / x o ) ] }
ieAxAmoz -3
h
1
(A~) 2 { ['I'o(r + ~Xr 0 + ,~0) - ~'o(r + ZX~,0 - ~X0)]
- 2[~,o(~, o + z x o ) - ~o(~, o~- ~xo)] + [~,o(r
~x~, o + ~xo)
- ~,o(r ~x~,o-,~o)]} +3
(ieAxAm~ h
2 1 { [qgo(~+ A~,O + AO) _ q2o(~+ A~,O _ AO)] 2-~
268
6 APPENDIX - [Vo(r - zxr o + zxo) - % ( r
(ieAxZm~ -
+
ieAtr
" h '
[
- 3
n
- Ar 0 - A0)]
3
J [~o(r o + A0) - ~o(r
0-
ZX0)]
1
2"'iAr 3 [~~176176176
ieAxAmox h
+3
1
(A~)2 [~o(~ + Ar 0) - 2~o(~, 0) + ~ o ( ~ - A~, 0)]
( ieAxZm~ ) 2 1 [q~o(~+ A~ O) - q~o(~ - A~ 0)] h
2-~
'
'
_ ( ieAxAmox
a
6 . 3 SOLUTION OF INHOMOGENEOUS DIFFERENCE EQUATIONS
We have to solve the inhomogeneous difference equations (3.2-41) and (3.2-42). The equations are equal except for the inhomogeneous term. We solve Eq.(3.2-42) for T,~(O) and indicate only at the end the changes required for Eq.(3.2-41) and S~(0). This is done because S~(0) willbe eliminated by a boundary condition while the intermediate steps of the solution of T~ (0) will be frequently referred to. A shorter notation is used. Equation (3.2-42) is brought into the following form1:
p2s(O + 2 ) + p l s ( O + 1)+pos(O) nt-p_18(O - 1) +p_2s(O- 2) = H~,~(O, ~) , Pl=P-l=-4cos~
, po=2
2+cos
27r~
H~,~(O, ~ ) = -ei)'~)'3G~,~(O+ 1, n) +2 cos --~Gs,~ (0, K:)- e-i~'t ~'3G~,~(0 - 1, K:) (1) The general solution of the homogeneous equation
p2s(O + 2) + pls(O + 1) + pos(O) + p-l(O - 1) + p_2s(O - 2) = 0
(2)
is given by the four functions in Eq.(3.2-54):
81(0)
0 = ei(2~ra/N-~l~3)O
- " re; 1
82(0 ) --- V~2 0 __ e - - i ( 2 ~ / g + ) ~ l ) ~ 3 ) O
83(0) -- OvO~l-- Oei(27r~/N-Al~3)O 84(0) -= OvO~2= Oe-i(2~r~/N+Al~3)O
1See N6rlund 1924, p. 396; 1929, p. 22, 125; Milne-Thomson 1951, p. 374.
(3)
6.3 SOLUTION OF INHOMOGENEOUS DIFFERENCE EQUATIONS
269
We use the method of variation of the constant to find a particular solution s(O) of Eq.(1) with the help of these four solutions of the homogeneous Eq.(2)" v(O-2) = d~l(O)sl(O-2)+d~2(O)s2(O-2)+d,,3(O)s3(O-2)+d~4(O)s4(O-2)
(4)
Since we have four arbitrary functions d,~(O) to d~4(0) we can choose three more conditions in addition to Eq.(4). Using intuition we make the following choice that will be justified by its success:
v(O - 1) = d~l (0)sl (8 - 1) + d~2(O)s2(O - 1) + d,~3(O)s3(O- 1 ) + d ~ 4 ( O ) s 4 ( O - 1) v(O) = d,~l(O)s1(O) + d~2(0)s2(0) + d,~3(0)s3(0) + d~4(0)s4(0) v(O + 1)
=
d,~l
(5) (6)
(8)81 (8 nt- 1) -Jr-d~2(0)82(0 nt- 1) + d,~3(O)s3(O + 1) + d~4(O)s4(O -t- 1)
(7)
If we increase 0 of d~l(0), d,~2(0), d~3(0), and d~ 4(8) by 1 in Eqs.(4), (5), and (6) we obtain
v(0 - 1) = d,~1(0 + 1 ) s 1 ( 0 - 1) + d~2(0 + 1)s2(0- 1) + d,~3(O + 1)s3(0- 1) + d~4(0 -+- 1 ) s 4 ( 0 - 1)
v(O) = d,~1(0 + 1)sl(0) + d,~2(O + 1)s2(0) + d,~3(0 + 1)s3(0) + d~4(0 + 1)s4(0) v(O + 1) = d,~l(O + 1)s1(0 + 1) + d~2(O + 1)s2(0 + 1) + d,~3(O + 1)s3(0 + 1) + d,~4+l(O)s4(O + 1)
(8) (9) (10)
Subtraction of Eqs.(5), (6), and (7) from Eqs.(8), (9), and (10) yields with the notation Ad~i(0)=d~(0+l)-dm(0),
i=1,
2, 3, 4
(11)
the result sl ( 8 - 1)Ad~l(0)+ s 2 ( 0 - 1)Ad~2(0) + s 3 ( 0 - 1)Ad~3(0) + s4(0 - 1)Ad~4(0) = 0
(12)
sl (0)Ad~l (8) + s2(0)Ad~2(0) + s3(0)Ad~3(0) + s4(0)Ad,~4(0) = 0
(13)
s1(8 + 1)Ad~1(8) + s2(0 + 1)Ad~2(0) + s3(0 + 1)Ad~3(0) + s4(0 + 1)Ad~4(0) = 0
(14)
270
6 APPENDIX
For v(O +
2) we write with the help of Eqs.(7) and (11):
v(O + 2) = d~l (0 + 1)Sl(0 + 2) + d,,2(O + 1)s2(0 + 2) + d~3(O + 1)s3(0 + 2) + d~4(O + 1)s4(0 + 2)
-- d~l (O)sl (0 +
2) + d~2(O)s2(O + 2) + d~3(O)s3(O + 2) +
d~4(O)s4(O +
+ sl (0 + 2)Ad~l (0) + s2(O + 2)Ad~2(0) + s3(O + 2)Ad~3(0) + 84(0 nL 2)Ad~4(0)
2)
(15)
We substitute the complete solution v(O - 2) to v(O + 2) of Eqs.(4), (5), (6), (7), and (15) for s(O - 2) to s(O + 2) into Eq.(1). Since sl(O), s2(0), s3(0), and s4(0) are solutions of the homogeneous Eq.(2) we get:
sl (0 + 2)Ad~l (0) + s2(O + 2)Ad~2(0) + s3(O + 2)Ad~3(0) + s4(O + 2)Ad~4(0) = Hs~(O, x~)
(16)
P2 Equations (12), (13), (14), and (16) contain the four unknown functions Ad~l (0) to Ad~4(0) and the known functions sl, s2, s3, s4, Hs~(O), P2. Hence, we can obtain Ad~l (0) to Ad~4(0) from these four equations and then the coefficients d~l (0) to d~4(0) by means of a summation according to Eq.(ll) or (1.2-23). The solution of Eqs.(12), (13), (14), and (16)by Cramer's rule for Ad~l(0), Ad~2(0), Ad~3(0), Ad~4(0) calls for five determinants:
D,~o(O) =
81(0- 1) 82(0- 1) 83(0- 1) 84(0- 1) ~1(0) ~2(0) ~(0) ~4(0) s1(0+1) s2(0+1) s3(0+1) s4(0+1) 81(0 -['-2) 82(0 "Jr-2) 83(0 Jr-2) 84(0 -Jr-2) 0=1,2, ...,N-2
0 s2(0-1) 0 ~2(0) D~I(0) = 0 s2(0+l) g~(0, ~)/;2 ~(0 + 2)
s3(0-1) ~3(0) s3(0+1) ~3(0 + 2)
s4(0-1) ~4(0) s4(0+1) ~;(0 + 2)
(17)
6.3 SOLUTION OF INHOMOGENEOUS DIFFERENCE EQUATIONS
81(0- 1) '32(0- 1)
0
"34(0- 1)
'31(0) 81(0 -F 1) 81(0 -%-2)
'32(0) s2(O + 1) "32(0--1-2)
0 0 H~,~(O,~)/P2
84(0 ) "34(0 + 1) s4(O + 2)
81(0- 1) ~1(0) D~4 (0) = s1(0+1) sl(0 + 2)
s2(O- 1) ~(0) s2(0+1) s2(O+ 2)
sa(O- 1) 0 ~(0) 0 sa(O+ l) 0 s3(O+ 2) H.~,~(O,~)/p2
D,~3 (0)
:
271
(20)
(21)
We note that the variable 0 has the values 0 = 0, 1, . . . , N according to Eq.(2.2-6). Hence, the values of si, i = 1, 2, 3, 4, are defined for this range in Eq.(3), while s i ( - 1 ) , si(N + 1) and s~(N + 2) are not defined. As a result the determinants D~i(O) are defined only for 0 = 1, 2, . . . , N - 2. The functions Sl(0) to s4(0) are defined in Eq.(3) while Hs,~(O) and P2 are defined in Eq.(1). We obtain for Ad~l (0) to Ad~4(0):
Dal(0) D~2(0) Ad~3(0)= D~3(0) Add4(0)= D~4(0) Ad.1 (0)= D~o(O) ' A d . 2 ( 0 ) = D~o(0)' D~o(O) ' D~o(0)
0 = 1, 2, . . . , N - 2
(22)
The functions din(O) required in Eqs.(4)-(7) follow from Eq.(22) by summation. A summation constant d~i is required that corresponds to the integration constant of differential calculus:
0--1
dai(O)--E Adai(n)Jr-dai, i---1,
2, 3, 4; 0 = 1, 2 , . . . , N -
2
(23)
n--0 The correctness of this equation becomes evident if we substitute Eq.(11)
d,~i(O) = [d~i(0 + 1) - d~i(0)] + [d~i(2) - d~(1)] + . . . + {d~i[(O - 1) + 1] + d~i(O - 1)} + d~i (24) =d,~i(O)-d,~i(O)+d,~i, i = 1 , 2 , 3 , 4 ; 0=1,2,...,N-2 and choose d~i = din(0). The constant d~(0) is as arbitrary as din. A constant only adds a solution of the homogeneous Eq.(2) to v ( O - 2) in Eq.(4) or to v(O) in Eq.(6). We note that the simplicity of the proof of Eq.(23) by Eq.(24) compared with the proof of Eq.(1.2-23) is due to the elimination of the infinitely large as well as the infinitesimally small.
272
6 APPENDIX
The determinant D'`o(O) in Eq.(17) contains only the solutions of the homogeneous equation according to Eq.(3) while D'`I(0) to D~4(0) contain the inhomogeneous term, which means the determinants have to be recalculated every time the inhomogeneous term is changed. This situation is remedied by an expansion of n,,l(O) to D~a(0):
D'`I (0) -- --Hs'` (0, t~__________~(0) )/:),1 P2
s2(0-1) /:)'`1(0) =
s3(0-1)
s4(0-1)
~4(o) s2(0+1) s3(0+1) s4(O + 1) s2(0)
s3(0)
H~(O, ,~) b~(o) P2 s1(0-1) s8(0-1) s4(O- 1) b~(o) = ~(o) ~(o) ~(o) s1(0+1) s3(0+1) s4(O + 1)
(25)
D'`2 (0) =
(26)
D'`3(0) = -Hs,~(O, ~)/),,3(0) P2
81(0- 1) s2(O- 1) s 4 ( 0 - 1 ) ~(o) 81(0 + 1) s2(O + 1) s4(O + 1) ) D,~(O) = H..~(~ p2 s1(0-1) s2(O-1) ~(o 1) b~(o) = ~1(o) ~(o) ~(o) 81(0 -F 1) s2(O+ 1) s3(o + 1)
(27)
-
(28)
Substitution of the functions d'`~(O)Of Eq.(23) into Eq.(6) yields a solution of the inhomogeneous Eq.(3.2-3) according to Eq.(3.2-43). In order to obtain a solution of the inhomogeneous Eq.(3.2-41) we must replace according to Eq.(3.2-43) the functions d'`i(O) in Eq.(4) by c'`i(O), which is a strictly notational change. In addition we must replace Hs'`(O,~) in Eq.(1) by Hr ~) according to Eq.(3.2-41): 271"/~
Hc'`(0,~)=-ei~l~3Gc'`(0+l,~)+2cos--~-Gc'`(0,~)-e-~)'3Gr
(29)
6.3 SOLUTION OF INHOMOGENEOUS DIFFERENCE EQUATIONS
273
where Gc~(0,~) is defined by Eq.(3.2-32). This change affects Eqs.(25)-(28). Equation (6) becomes:
~(0) = ~.~ (0)~ (0) + ~.~(~)~=(~) + ~.~(0).~(~) + ~.4(~)~4(~) d.~(0) -~ ~.~(~), H..(0) -~ He.(0)
(30)
For the evaluation of the determinants/9~1(0) we substitute Eq.(3) into Eq.(25):
D~(e)
=
e-i(2~/N+)`~ ),3)(e-1) (O_ l)ei(2~r~,/N-),i X3)(t~-l) (O_ l)e-i(2~r~/N+X~ x3)(e-1) e - i ( 2 ~ / N + ), ~),3 )O oei( 2~,c/ Y - ), l ),3 )~ o e - i ( 2~,c/N + ), ~),3 )O e -i(2~'~/N+)`~)`~)(e+~) ( 0 + 1 ) e i(2~'r ( O + l ) e -i(2~'~/N+)`~)`~)(e+~)
(3~) The three factors e -i(2~r~/N+)`~)`3)O, e i(2~ra/N-)`~)`3)O, and e -i(27r~/N+)`~xa)O can be pulled in front of the determinant. Furthermore, we multiply the first column with 0 and subtract the product from the third column:
D,~1(8) :
e -i(27r~/N+3)`l)`3)O e i ( 2 ~ / N + )`l X3 ) •
(0-
1 e-i(27rK,/N+Ai A3 )
1)e -i(27r~/N-)`~)`3)
--e i(2~r'~/N+)`~)`3)
0 (0 + 1)e i(2~r~/N-)`l)`3)
0 e -i(27r~/N+)`~)`3)
(32)
Development of the third column yields: [9~1 (8) --e -i(2~r~/N+3)`l ),3)O • {--ei(2~/N+),l),3)[(O + 1)ei(2~/N-)`1)`3) _ Oe-i(27r~/N+)`l),3)] + e-i(27r~/N+),~),3)[Oe i(2~/N+)`~)`3) -- ( 0 =2e-2iTr~e/Ne-3i),~),ze
20 sin 2 --~
1)e-i(2~/N-),~),3)]}
i sin - - ~
(33)
Following Eqs.(31)-(33) we obtain b~2(0), b~3(O), and D~4(0) from Sqs.(26) to (28):
(
/~2(0) -- --2e2i~r~8/Ne -3i)`1)'30 20sin 2 - ' 7 + / s i n
(34)
274
6 APPENDIX
/)~3(0) = -4e -2i~~
e -3i'xl"x3~sin 2 2 ~rt~ N D~4 (0) = +4e 2ir~O/N e -3iA1"%0sin 2 27rt~ N
(35) (36)
We obtain further D,~I(O) to D~4(0) from Eqs.(25) to (28) with P2 taken from Eq.(1): 2~ 47r~) 20sin2 --~-- -- i s i n - - 7
D~I(0) = -2H~(O,t~)e-2i'~'~~176 D~2(0) = --2H~,,(O,~)e2i'~"~
-i)'~)'3(3~
D,~3(0) = 4Hs~(O, ~)e -2i'~'~~
-i)'~)'a(a~
20sin2--~ +/sin sin 2 2~r~
(37) (38) (39)
N
D~4(0) = 4H~(0, n)e 2irr~O/Ne-iAIAa(30+2) sin 2 27r~
(40)
N
Our next task is the evaluation of D,~o(O) of Eq.(17). A comparison with Eqs.(25)-(28) suggests developing the determinant by its fourth row:
O~o(O) = -81 (0-[-2)./~)~1(O)-'k82(Ok-2)JE)~2(O)-83(O+2)JE)~3(O)-+-84(O--[--2)jE)x~4 (O) (41) Substitution of sl(O + 2) to s4(O + 2) from Eq.(3) Eqs.(33) to (36)yields: D~0(0) = -4e -2/~1~3(2~
and
b~l(0) to /)~4(0) from
sin 2 ~47ra( 1 + 4 c o s ~4N~ )
(42)
The functions Ad~l(0) to Ad~4(0) of Eq.(22) follow from Eqs.(37) to (40) plus Eq.(42):
1
2i,~,,O/Nei~l),aO 20sin2(27rt~/N)
A d ~ l ( 0 ) - -~Hs,~(O,~)e1 Ad~2(0) = ~Hs~(O, ~)e +2i'~'~
-
isin(47rt~/N)
sin2(nTr~/N)[1 + 4cos(47r~/N)]
i~1~3~ 20 sin2(27r~/N) + isin(47rx~/N) sin2(nTr~/N)[1 + 4 cos(47r~/N)] sin2 (27cry/N) Ad~3(0) = -Hs~ (0, ~)e -2i'~'~ e~)'~)~3~ sin2(47r~/N)[1 + 4 cos(47r~/N)] sin2 (2Try/N) Ad~4(0) = -Hsa(O, g)e +2ilraO/g e iA1A30 sin2(nTra/N)[1 + 4 cos(nTra/N)]
"(43) (44) (45) (46)
6.3
SOLUTION OF INHOMOGENEOUS DIFFERENCE EQUATIONS
275
We replace 0 by n in Hs,~(O,to) of Eq.(1) and Gs,~(O,~) of Eq.(3.2-33). The functions d,r to d~4(0) of Eq.(23) may then be written explicitly: 0--1
2n sin 2(2~r~ / N ) - i sin (4~r~/N) +d~ sin2 (4~r~/N) [1 +4 cos(4~r~/N)]
d~(0)= ~ n=0
(~7)
0--1
d~2(0) = ~1 EHs,~(n,tc)e+2i,~,~n/N ei~lA3n rt--0 0--1
~3(0) = - ~
H.~(n,~)~-~'~/~ ` ~ ~
n--0 0--1
~ 4 (0) = - ~
H,~ (~,.) ~+:{'~/~ ~'~ ~
n--O
2n sin 2(2~r~/N) + i sin(4~r ~/N) +d~2 sin 2 (4~r~/N)[1 + 4 cos(47r ~/N)] (48) sin2 (27rt~/N) +d~3 sin 2 (4~r~/N)[I + 4 cos(4~r ~/N)] (49) sin2 (2Try/N) +d~4 sin e (4~r~/N)[1 + 4 cos (4~r~/N)]
(~0)
In order to separate Eqs.(47)-(50) into real and imaginary parts we define eight functions F1 (n, n) to Fs(n, n):
F1 (n, ~) =
sin2(27rn/N) cosn(A1A3 - 27rn/N) sin2(47ra/N)[i + 4 cos(47r~/N)]
(51)
si.~(2../N)co~n(~ + 2~,~/N) F2(n, n) =
sin2(nTra/N)[1 + 4 cos(nTra/N)]
F3(n,a) =
sin(47ra/N) sin n(A1A3 - 21rn/N)
(52)
2 sin2(41r~/N)[1 + 4 cos(47r~/N)]
(53)
F4(n, a) = - sin(47ra/N)sin n(A1A3 + 27ra/N)
(54)
F~(n, ~) = F6(n, ~) =
2 sin2(47r~/N)[1 + 4 cos(47ra/g)] sin2(27r~,/N) sin n(A1A3 - 27ra/N) sin2(4~ra/N)[1 + 4 cos(47rn/N)]
(55)
sin2(27ra/N)sinn(A1A3 + 27ra/N) sin2(nTra/N)[1 + 4 cos(47rn/N)]
(56)
FT(n, a) = - sin(47ra/N)cosn(A1 A3 - 27ra/N)
(57)
2 sin2(47r~/g) [1 + 4 cos(47r~/g)] Fs(n,~) =
sin(47r~/N)cosn(A1A3 + 2Try~N) 2 sin2(47r~/N) [1 + 4 cos(4zr~/N)]
We may rewrite Eqs.(47)-(50) into the following form:
(58)
276
6 APPENDIX
8--1
d,~l(O)-- EH~,c(n, a){nFl (n, a)+F3(n, a)+i[nF5(n, a)+FT(n, a)]+d,r } (59) n--0 8--1
d,c2(O)= EHs,c(n, ~){nF2(n, ~)+F4(n, ~)+i[nF6(n, ~)+Fs(n, ~)]+d,~2} (60) n--0 8--1
d~3 (8) = - E H ~ (n, a){ F1 (n, ~) + iF5 (n, ~) + d~a }
(61)
n--0 8--1
d~4(0) = -
EHs,~(n,a){F2(n,a) + iF6(n,a) + d~4}
(62)
n--0
The terms d,~l to d~4 are summation constants equivalent to integration constants of the differential theory. According to Eqs.(3.2-72), (3.2-75), and (3.2-79) only d,~l is choosable. We choose d,~l equal to zero and obtain the relations:
d~l = 0, d~2 = 0,
da3 -- -[D~I(1) + d~r -- -[D~2(1) +
D~3(1)I/D~o(1) - d3r(a) + id3i(a) D,ca(1)]/D,r = d4r(~;) + id4i(~;)
(63)
Certain terms required in Eq.(3.3-7) may now be written in the form 8--1
d~ (o) + o,z,~(o) = ~ H~(n, ~)[YlO(n, ~) + ~Y~l(n,,~)] n--O 8--1
- OE H.,c(n, a)[F12(n, ~) + iF~3(n, ~)1 n--0
Flo(n, ~) = nFI (n, ~) + F3(n, ~) FI~(n, ~) = nF5(n,~) + FT(n, ~) F12(n, ~) = F1 (n, ~) + d3r(a) F13(n, I,~) -- Fs(n, g) + d3i(tr
0-1
d~2(0) + 8d~4(0) = E
Hs,~(n,a)[F20(n,a) + iF21(n,a)]
n--0 8--1 -
0 E H.~,~(n,~)[F22(n, ~) + iF23(n, ~)1 n--0
(~4)
6.3 SOLUTION OF INHOMOGENEOUS DIFFERENCE EQUATIONS
277
F2o(n, ~) = nF2(n, ~r + F4(n, ~r
F2~ (n, ~) = nF6(n, ~r + Fs(n, ~) F22(n, t~) = F2(n, n) + d4r(~) F23(n, ~) = F6(n, ~) + d4i(~r
(65)
We still have to derive d3r(~), d3i(tr obtain from Eqs.(37)-(40) and (42):
d4r(tr
and d4i(~) explicitly. We
2;ra 47ra) D~1(1) = --2H~,c(1, a)e-2i'~'c/Ne -2i)'~)'a 2sin 2 - 7 + / s i n - - ~
(
D~2(1) = -2H~,c(1, e;)e+2i"~/Ne -2i'x~'xa 2sin 2 - - ~ + / s i n - - ~
N
(67)
-2i)~t)~a sin 2 27ra
(68)
~)e+2i~r~/Ne-2i~1~3 sin 2 27rt~ N
(69)
D,~3(1) = 4H~,~(1, a)e -:i'~'r D~4(1) = 4H~(1,
(66)
D~0(1) = - a s -2i~'1)'3 sin 2 -27~ (
1 + 4cos 4a-~:)
(70)
Substitution into Eq.(63) brings _D~1(1) + D~3(1) = _H~,~(1, tc)[sin(2~TJc/N) + icos(27ra/N)] D.0(1) 2 sin(47ra/N)[1 + 4 cos(4.Tra/N)] _D~2(1) + D,~4(1) = -~ H~(1, a)[sin(27ra/g) - i cos(27ra/N)]
D~0(1)
(71) (72)
2 sin(47r~/N)[1 + 4 cos(47r~/N)]
and we get finally d3r(~), d3i(~), d4r(~), and d4i(~): Hs~(1, ~) sin(27r~/N) d3r(t~) = -d4r(~) -- - 2 sin(47r~/N)[1 + 4cos(4~r~/N)]
(73)
Hs,~(1, ~)cos(27r~/N) d3i(~) = +dni(~) = - 2 sin(47r~/N)[1 + 4cos(47r~/N)]
(74)
Our next task is to separate the kernel of the sum of Eq.(3.3-7) into a real and an imaginary part. This is very tedious but not mathematically difficult:
[d,~l(0) + Od~3(O)]e2~ri'~~ + [d~2(0) + Od,~4(O)]e-2~i'r176 27r~0 27r~0 =[gl (0, ~) - 0J2(0, ~)] cos g - [J3(0, ~) - 0,14(0, ~)] sin - - ~ ( 27r~;0 2;r~0) + i [Jh(0, a) - 0J6(0, a)] sin N + [JT(0, ~) - OJs(O, ~;)1cos, N
278
6 APPENDIX 0-1
Jj(0,~) = E
Kj(n,n),
j -- 1, 2, . . . , 8
n--0
K1 (n, K2(n, g3(n, Ka(n,
~) ~) ~) ~)
= = = =
Kh(n, a) =
K6(n, ~) = gr(n,a) = Ks(n, to) =
Hs,~(n, a){n[F1 (n, a) + F2(n, a)] + F3(n, ~) + Hs~n, g)[F 1(n, ~) + F2(n, a)] g~,~(n, a){n[Fh(n, ~) - F6(n, a)] + FT(n, a) Hs,~(n, a)[Fh(n, a) - U6(n, a)l g~,~(n, a){n[F~ (n, ~) - F2(n, a)] + F3(n, ~) Hs,~(n, ~)[Fh(n, a) - F6(n, t~) + 2d3r(a)] Hs,c(n,g){n[Fh(n,a)+ F6(n, a)] + Fr(n,a)+ Hs~(n, ~)[Fh(n, a) + F6(n, t~) + 2dai(g)]
F4(n, t~) } Fs(n, ~)} F4(n, g)} Fs(n, a)}
(75)
The variable n of Kj (n, ~) sometimes needs to be replaced by the variable 0. The following transformation will do that:
0-1 OJj(O t~) Kj(O,~)= ~=0-1 E Kj(n,t~)= 00 ' ,
j=l,
2,..., 8
(76)
6.4 CALCULATIONS FOR SECTION 4.2
The inhomogeneous difference equations (4.2-32) arid (4.2-33) have to be solved. Except for the inhomogeneous terms the equations are equal. We solve Eq.(4.2-33) for T~(0) and indicate only at the end the changes required for Eq.(4.2-32) and S~(0). Following Eq.(6.3-1) we introduce a shorter notation:
p28(0 + 2) + p18(0 + 1) + pos(O) + p--18(0 , Pl=P-l=-4cos~
-- 1) +
p_2s(O - 2) = H.,~(0, a) , p0=2
2+cos
&~(0, ~ ) = - ~ ' ~ 5 ~ ( 0 + 1, ~)+2 co~ -27r~ 7 0 ~(O, ~ ) - ~ - ' ~ 5 , ~ ( 0 - 1 , ~) (1) Except for the change G ~ (~, H ~ H this is the same equation as Eq.(6.3-1). Hence, Eqs.(6.3-2) and (6.3-3) apply unchanged, but Eq.(6.3-4) must be rewritten for d ---+d and v ~ ~:
'V(0--2) --- d~r (0)81 (O-2)-+-d~2(O)s2(O-2)-k-d-~3(O)s3(O-2)-k-d~4(O)s4(O-2)
(2)
Equations (6.3-5) to (6.3-10) apply again with a tilde over v and d. Equation (6.3-11) must be rewritten to define Adm(0):
6.4 CALCULATIONS FOR SECTION 4.2
AG~(0)=d-~,(0+l)-&~(0),
i=1,
279
2, 3, 4
(3)
Equations (6.3-12) to (6.3-I5) remain applicable if a tilde is written over every v and d. Equation (6.3-16) is rewritten because of the changed inhomogeneous term:
81 (0 q- 2)a&l (0) Jr- 82(0 -Jr-2)A~2(0 ) q- 83(0 -Jr-2)A&3(0 ) q- 84(0 -Jr-2)A~4(0 ) =
K~(o,~) P2
(4)
Following the text after Eq.(6.3-16) we need five determinants for Ad~z(0), Ad~2(0), Ad~3(0), Ad~4(0). We denote them b~i(0) with i = 0, 1, 2, 3, 4. They follow from Eqs. (6 .3-17) to (6.3-21) for H ~ H:
D~o(O) = D~o(O), 0 = 1, 2 , . . . , N - 2 ;
o o o
b~l(O) =
fi~(o, ~)/w
b~2(o) =
see Eq.(6.3-17)
~ ( o - 1) ~ ( o - 1) ~ ( o - 1) ~(o) ~(o) ~(o) ~(o + 1) ~(o + 1) ~4(o + 1) ~(o + 2) ~(o + 2) ~(o + 2)
s1(0-1)
0
s3(0-1)
~(o)
o
~(o)
~(o)
s1(0+1)
0
s3(0+1)
s4(0+1)
83( 0 nL 2)
84(0 q-
81(0 -}- 2) /~s,~(0,~)/P2
s1(O-1) s2(O-1) o ~(o) ~(o) o b~(0) = ~(o + 1) ~2(o+ 1) o 81 (0 q'- 2) 82(0 -Jr-2) /-ts,~(0, ~)/P2
~ ( e - 1) ~ ( e - 1) ~ ( e - 1) ~(e) ~(e) G~(e) = ~(e) s1(0+1)
s2(0+1)
s3(0+1)
(5)
(6)
s4(0-1)
2)
s4(O-1) ~4(o) ~4(o + 1) 84(0 q'- 2)
0 0 0
(7)
(8)
(9)
The functions Sl(0) to s4(0) are defined in Eq.(6.3-3) while Hs~(0) and p2 are defined in Eq.(1). We obtain for Ad~z(0) to Ad~4(0)"
280
6 APPENDIX
Ad~a (0) = b~3(0) Ad,~4 (0) = b~4(0) (0) Aa~2(0)= D,~2(0) D,r A~,~ (O) = L3,~1 D,~o(O)' D,~o(0 )' D~o(O) -
0 = 1, 2, ..., N - 2
(10)
The functions d~i(0) required in Eq.(2) follow from Eq.(10) by summation. A summation constant d~i is required that corresponds to the integration constant of differential calculus:
0--1 d~i(0) = E
Ad~i(n) + d~i,
i = 1, 2, 3, 4; 0 = 1, 2, . . . , N - 2
(11)
n--0 With the help of the determinants b~1(0) to bn4(0 ) of Eqs.(6.3-25) to (6.3-28) we may write the determinants b~1(0) to b~4(0) of Eqs.(6) to (9) in the following form:
b~l (0) = -/~'~ (0, ~._....b~l .~) (0) p2
&~(o ~)
b,~2(0)-- + ~ ' P2 b~a (0) = - / ~ b,~4(0) = +
b,~2(0)
(13)
(0, ~______b~a _____~) (0) p2
(14)
H~ (O, ,~) .
(12)
p2
-
b,~4(0)
(15)
b~l(0), D~2(0), D~3, .V~4(0) see Eqs.(6.3-25)-(6.3-28) Substitution of the functions d~i(0) of Eq.(ll) into the following equation derived for v --+ ~, d~i ~ d~i from Eq.(6.3-6)
~(0) = d~l (0)sl (0) + d~2(0)s2(0) + d~3(O)s3(O) + d,~4(0)84(0)
(16)
yields a solution of the inhomogeneous Eq.(4.2-2) according to Eq.(4.2-34). For the solution of the inhomogeneous Eq.(4.2-32) we must replace according to Eq.(4.2-34) the functions d~i(0) by 5~i(0). Furthermore, we must replace / ~ ( 0 , n) in Eq.(1) by/~c~(0, n) according to Eq.(4.2-32) g-~ (e, ~,) = - ~ ' ~ ~ 0 ~ (e + 1,~) + 2 cos -2-~~ 0 ~ (e,~) - ~ - ' ~ O ~ ( e
1,~)(iv)
6.4 CALCULATIONSFOR SECTION 4.2
281
where t~c,~(0, n) is defined by Eq.(4.2-23). Equation (16) becomes:
~(o) = ~ ( o ) ~
(o) + a~(o)~.(o) + e~(o)~(o) +
~,(o) ~
e~(o)~(o)
a~(o), ~s~(O)~ ~c~(O)
(18)
Equations (6.3-31) to (6.3-36) apply again and b,~(0) to b~4(0) may be written according to Eqs.(6.3-37) to (6.3-40) as follows:
(
]~)t~l(0) :
--2~Is,~(O,n)e-2iTr'~~
-iA~3(30+2) 20sin 2 - 7
]~)~2(0) =
--2IhIs,~(O,n)e2i~r'~~
]~)a3(0) =
+4Hs~(0, I'r -2ircaO/Ne -iAIA3(30+2) s i n 2 2;ra g
--/sin--
-iA1)~3(30+2)( 20sin 2 72;r~ + / s i n 47r~)
27rt~ ]~;4(0) = +4Hs~(0, g)e 2i~raO/ue-iAtA3(30+2) sin 2 N
(19) (20) (21)
(22)
For/),~0(0) = D,~o(O) we retain Eq.(6.3-42):
D,~o(O) = -4e -2i)~lAa(20+l) sin 2 " 7
1 + 4cos--
(23)
The functions AO~l(0) to Ad~4(0) of Eq.(10) follow from Eqs.(19) to (23):
1
~
(O,i,;)e_2ilrnO/NeiA1A3 0 2 0 s i n 2 ( 2 7 r ~ / N ) - isin(47rx~/N)
Ad~l(0) = + ~ H ~ 1 -
nd~2(0 ) --
+~H~,~
(24)
sin2(47r~/N)[1 + 4cos(4~-~/N)] (O,~)e+2i~,~O/Nei~,l~,30 20sin2(27r~/N)+ isin(nTr~/N)
Att~3(0) = --Hs,~(O,l~)e-2i~r'~~
sin2(47r~/N)[1 + 4 cos(n~'~/N)]
sin2(27rt~/N) sin2(47rt~/U)[1 + 4 cos(47rt~/N)]
(26)
si~(2~/N) sin2(4~-~/N) [1 + 4 cos(4~'~/N)]
(27)
i)~lA30
/X~(O) = -fi~(O, ~)~+~,~,~O/U~ ~ 0
(25)
We replace 0 by n in H.~(_0, ~) of Eqz(1) and 0s~(0, ~) of Eq.(4.2-24). One may then write the functions d~l(0) to d,~4(0) of Eq.(11) explicitly:
+~ ~1(o)=~1 ~iYi~,~(n,~)e_2i~,~n/Nei),l~zn sin2 (4;r~/N) [1 +4 cos(47r~/N)] 2n sin 2(2~r~/N) - i sin (4Try/N)
n--0
(28)
282
6 APPENDIX 0-1
2n sin 2(2~-~/N) +i sin(4~-t~/N) sin ~(4~r~/N)[1+ 4 cos(4~r~/N)]
n--0 0-1 E fftsg(n'~')e-2i~r~n/NeiA1A3n
d~3(0)---
sin2 (2711'~/N)
sin2(47r~/N) [1 +4 cos(4~-~/N)] +d~3 (30)
n=o 0--1
d~4 (0)
=
(29)
-- E Hsx~(n,N)e +2irr~n/N eiXl )~an r~--'O
sin2 (2Try/N) +4 cos(4
/N)] (31)
Equations (28)-(31) differ from Eqs.(6.3-47)-(6.3-50) only by the substitutions H and d for H and d. Hence, we can use Eqs.(6.3-51)-(6.3-58) to rewrite Eqs.(28)-(31) into the form of Eqs.(6.3-59)-(6.3-62): 0--1
d~l(0)= Z
[-I~,~(n,tc){nFl(n, tc)+F3(n, ~)+i[nF5(n, e;)+FT(n, ~)]+d~l } (32)
n---O 0--1
d ~ 2 ( 0 ) = ~ [Is~(n,~){nF2(n,t~)+F4(n,~)+i[nF6(n,~)+Fs(n,~)]+d,~2}
(33)
n---0 0--1
d~3(0) = - E fi.~,~(n,lc){Fl(n,tr + iF5(n,~) + d,~3}
(34)
n--0 0--1
d~4(0) = - ~ ' / t s ~ ( n , t~){F2(n, ~)+ iF5(n, ~) + d,~4}
(35)
n--0
The terms d,~l to a~,~4are summation constants equivalent to integration constants of the differential theory. According to Eqs.(4.2-43), (4.2-46), and (4.2-50) only a~,~l is choosable. We choose d~l equal to zero and obtain the relations
b~a(1)]/D~o(1) =
d3r(K;) Jr- id3i(~;)
-[b~2(1) + D~4(1)]/D~0(1) =
J4r(/~) q-- iJ4i(~)
~ 1 = 0,
d~3 ~--- - [ b ~ 1 ( 1 )
J~2 --- 0,
J~4 -'-
nt-
(36)
Certain terms required in Eq.(4.3-7) may be written in the following form: 0--1
d~l(0) + 0d,~3(0) = E [-Is,~(n,~)[Flo(n,~) + iFll(n,~)] n--'O 0--1
--0 Z ~Is~(n,~)[F12(n, ~) + iF13(n, ~;)] n=O
Flo(n, ~), Fll (n, ~), F12(n, ~), F13(n, ~) see Eq.(6.3-64)
(37)
6.4
CALCULATIONS
FOR SECTION 4.2
283
0--1 d-~2(0) + 0d~,4(0) = ~ / - I ~
(n, t~)[F20(n, ~) + iF21(7%,t~)]
n--0 0--1
-0 ~ ~(~, ~)[F~(~,~)+ ~r~(n, ~)l n--0 F2o(n, ~), F21(n, ~), F22(n, ~), F23(n, ~) see Eq.(6.3-65)
(38)
We still have to derive d3r(/~), d3i(~), d4r(/'~), and 6~4i(/~) explicitly. We obtain from Eqs.(19)-(23)'
(
b~l(1) = --2[-I~,~(1, m)e-2i~'~/ge -2~'1~3 2sin 2 - - ~ + / s i n - -
(
/),~2(1) = --2~I.~,~(1,~)e+2i'~'~/Ne -2i~'~'x3 2s in2
--~ 2~ +
isin
-4~) ~
27r~: /),~.3(1) = 4/-t~,~(1, ~,)e - 2 i ~ / N e -2i'x~'x3 sin 2 N 27rn
(39) (40) (41)
D,~4 (1) = 4fI~,~ (1, ~)e +2i'~'~/g e -2i'xl ha sin 2 N
(42)
D,~o(1) = - 4 e -2i)~)~a sin 2 - - ~
(43)
1 + 4cos
Substitution into Eqs.(36) brings
D~0(1)
/~%~(1, ~)[sin(2~-~/N) + icos(27r~/N)] 2 sin(47r~/N)[1 + 4 cos(47r~/N)]
(44)
D~o(1)
H~(1, ~) [sin(27r~/N) - i cos(27r~/N)] 2 sin(47r~/N)[1 + 4 cos(47r~/N)]
(45)
b~1(1) + b~3(1)
b~2(1)+b~4(1) = + and
we
obtain d3r(~), ~i(t~), d4r(t~), ~i(/~) separated: /ts~ (1, ~) sin(27r~/N)
-
d3r(/~) -- -d4r(~) = - 2 sin(47r~/N)[1 + 4cos(47r~/N)]
-
Hs,~(1,~)cos(2~r~/N)
dai(n) = + d 4 i ( ~ ) = - 2 sin(47rt~/g)[1 + 4cos(47rt~/N)]
(46) (47)
We still want to separate the kernel of Eq.(4.3-7) into a real and an imag_inary part. To this end we copy Eq.(6.3-75) but replace d, J, K with d, J, K:
284
6 APPENDIX
[&l(O) + O&3(O)]e 2rri~O/N + [d~;2(O) + O&4(O)]e -2rri~O/N
27r~;0 = [J~ (o, ~) - oj~(o, ~)] cos 27rNO_N- [J3(0, N) - OJ4(0, N)] sin N + i [Js(0, ~) - 0Js(0, g)] sin
2rr~0 2rr~0) + [&(0, s) - 0&(0, ~)] cos N N
0-1
j--l,
2,..., 8
n'-0
/7/1(n, ~) =
[--Is~(n,tc){n[Fl(n,~)+
F2(n, n)] + F3(n, t~)+ F4(n, ~)}
/7/2(n, ~,) = &~(n,~)[Fl(n,~)+ F2(n, ~)] /-7/3(n, ~) =
_fiI.~(n,~){n[F5(n,~)- F6(n, ~)1 +
FT(n, ~ ) - Fs(n, ~)}
k~(n, ~) = &~(n, ~)[F~(n, ~ ) - F6(n, ~)] /~5(n, a) = Hs~(n,a){n[Fl(n,a)- F2(n, t~)] + F3(n, to)- F4(n, ~)} R6(n, ~) = Hs~ (n, ~)[Fs(n, a) - F6(n,~) + 2d3r(t~)]
/7/7(n, a) = [-Is~(n,a){n[F5(n,a)+F6(n, a)] + FT(n, a) + Fs(n, a)} (48) Ks(n, a) =//s. (n, ~)[Fs(n, a) + F6(n, a) + 9~d3i(g)] The variable n of Kj(n, ~) sometimes needs to be replaced by the variable 0. The following transformation will do that in a computer program" /~j(0, t~)=
0--1 Offj(O, l~) E Kj(n,~)= 00 ' n--O--1
j=l
'
2,
""'
8
(49)
6.5 FORMULAS FOR SPHERICAL COORDINATES
We want to rewrite Eq.(1.3-13) in component form like Eq.(1.3-20) for spherical coordinates" Ze d (v• c dt
Ze -- ( -v• c
r
o~ 0v •
[V(v•
C2 curl A e) (1)
We start with the evaluation of the first, second, and fourth term on the right side of Eq: (1)" gradCm-~
0era e,-
1 0r
+r--~e~-+
1 0r --% r sin ~ c3~o
(2)
6.5 FORMULAS FOR SPHERICAL COORDINATES
285
The velocity v of the point re~ + type# + r sin 0 ~pe~ is needed for the three directions er, eo, e~"
v = Or ote~-~ = §
O~ ot eo +
O~p
--~e~
+ r0e~ + r sin 0 ~be~
vxgradCm=
0 0r sinO Oqo
+
rsinOqb Or
(3)
90 r
~ p - - ~ ) e~
sin
OCm r sin 0 0~
0(/)m ~ r0--~r ]e~
r 00
(4)
The second term on the right side of Eq.(1) is evaluated:
Ov
o§ o§
O(r sin ~9~b) 0~b
o(~) oO
0-'-~ = 0-'--r0--'ter -[
00
0~b
0t eo +
-0-Te~
(5) (6)
- ~e~ + r 0 e o + r sin 0 ~5e~
Ae = A~rer + A ~ e o + A~r162
0v
x Ae = (rOA~v - r sin~) ~Aeo)er + (r sin~ ~Aer -/:Ae~)eo
0"--~-
-k (rAe~ - r0Aer)e~
(7)
OAer ) e~ 0~
(8)
The fourth term in Eq. (1) becomes:
c 2 curl Ae --
OA~o)
c 2 (0(Ae~ sin ~) r sin 0 00
+ --c2 r
1 OA~
099
O(rA~) Or
sin 0 0~
er
e~ + r
Or
We turn to the dyadic V(v x Ae) in Eq.(1). Some textbooks show the dyadic V F in component form but it is too complicated for spherical coordinates. However, we find the equation dA
0A
dt = 0-7 + v . (VA)
(9)
286
6 APPENDIX
in the literature I and we may thus write: V. [V(V X Ae)] = d(v • Ae) _ 0(v • Ae) dt
Ot
__0(vxA~)§ Or
0(v x A~) 0 + 0(v x Ar ~b O0 0~
(10)
We obtain with the help of Eqs. (3) and (6):
v x A e = (r(gAe~o - r sin v~@A~o)er
+ (r sin 0 ~ A e r - i'Ae~)eo + (i'Aeo - r0Aer)e~
(11)
0(v orXA~) ~ = § ~rr0[(rOAe~, - r sin 0 qbA~o)e~ - m)Ae,r)e~]
(12)
r~Aer)e~]
(13)
+ (r sin0 CAer - rAe~o)eo -+-(rAeo - r0Aer)e~o]
(14)
+ (r sin 0 qbA~r - §
+ (§
0(v00x Ae)0 = O~---~[(rOA~v - r sin 0 ~bA~o)e~ + (rsin O@Aer - i'Ae~o)eo + (§ O(v x Ae) 0~
-
. 0
= q a ~ [(r0Ae~ - r sin0 ~bAeo)er #
Equation (10) becomes:
v . [V(v x Ae)] =
(o
r~r + 0
+ qo
[(r~)Ae~o- rsinvq~bAeo)er
+ (r sin0 9bAer- rAe~o)eo + (§
- r~)Aer)e~o]
(15)
Equation (1) is brought into the following component form with the help of Eqs.(4), (7), (15), and (8):
d (v x A e ) - d [(r0Ae~o- r sin0qbAeo)er + (r sinvq ~ A e r - rAe~o)eo d-~ d-~ + (/'Aeo - r0Aer)e~o] 1Morse and Feshbach 1953, Part I, p. 295.
6.5 FORMULASFORSPHERICALCOORDINATES =
Q~00
+ -~r+~
[ - ( 0sinto 0q~m sinto99--~ .0r 0qp +~
+
(§ +
-
+r0Ae~-rsintoq3Aeo
e2 (0(Aeqosin to) OAeo~]
~__~)
(rOA~~
Oto
-~ -]j e~
( rsinto~b OCm i" Odpm)+rsinto@Aer_i~Ae~ ~ Or rsinto 0q~
. ff.ff.~ . 0 )(rsintO(oA~ _§ ~rr + to
287
( 10Aer
+ V'~
7
+ . . r. .O0
rO
(0 ~ .0) + i'-~r + (9 + (fl~ (i"Aeo-
O(rA~,))] Or e~
sin tO Oqo
+ fA,o - rOAe~ c2 ( O(rA~o) r Or
rOAer) -- - -
OAer) Oto
e~o (16)
The Lagrange functions Lcr, LcO, Lc~ for Eq.(1.3-13) may be inferred from the factor of e~ using the similarity with Eq.(1.3-20) and the way Eq.(1.3-21) was inferred there:
[ sin 0 qbOfgm ~
L cr = Z___~e
C
( 0
+
~
sin0 tOOCm 0~ ~_rOAe~ -
r sin tO~5Aeo
. O)(rOAe~o_rsinto.fpAeo) r sin to
0to
(17)
0qo
The term multiplied with i@/Or may be pulled in front of the integral and differentiated with respect to § The first term on the right side of Eq.(16) is obtained:
0 [§
- rsinO(oA~o)] = rOA~o - rsinO(oA~o
The functions LcO and Lr in analogy to Eq. (17):
L~
= -Z- e f [
c
§ aCm rsinO O~ +
(o §
+ 0
(18)
follow from the factors of eo and %, in Eq.(16)
r sin 0 .aCm + r sin 0 ~Ae~ - i~Ae~,
~ --~-r
.o)
+ ~
(rsin 0 ~bAe~ - §
c2( 10A,, r sint9 0~o
O(rAr Or
rdO
(19)
288
s
6 APPENDIX
---
Ze f [rOOCm ~" OCm Or r O0 + ~A~o -
--~
+
(o
rOA~
+io c2 ( O(rAeo) r Or
0Aer)] Or9 r sin t9 dqp
(20)
In Eq.(17) one can pull the factor i"(rOAe~ - rsinO(oAeo) in front of the integral, in Eq.(19) the factor 0(r sin # (oAer- § and in Eq.(20) the factor (o(§ - r(?Aer). The terms Lc~, Leo, and Lc~, are shown in this form in
Eqs. (1.3-33)-(1.3-35). 6.6 DIFFERENCE EQUATIONS SOLVED BY POLYNOMIALS We investigate when a second order difference equation with variable coefficients is satisfied by polynomials. Consider the equation P I ( X ) v ( X + 1) + P o ( X ) v ( X ) + P _ I ( X ) v ( X - 1) = 0
(1)
The factors P1 (X) and P-1 (X) shall be polynomials of the same degree, while Po(X) is a polynomial of the same or lower degree"
P I ( X ) = dloX j + d l l X j-1 + . . . + d U Po(X) = dooX j + d o l X j-1 + " " - t - doj P-1 (X) = d _ l o X j + d_ll X j - 1 -t- ' ' ' + d - l j
(2)
dl0, d-10 7~ 0
In differential calculus we like to write polynomials in terms of powers of X since these powers are easy to differentiate. In the calculus of finite differences one simplifies calculations by replacing powers X i by products of the form X ( X + 1)(X + 2 ) . . . (X + j - 1). Hence, we write P I ( X ) , P0(X), and P - I ( X ) in the following form:
PI(X) = Clo(X + 1)(X + 2 ) . . . ( X + j ) + Clz(X + 1)..' (X + j - 1) + ... + Clj Po(X) = c o o X ( X + 1 ) . . . (X + j -
1) + ColX.. . ( X + J - 2) + ... + coj
P - I ( X ) = c-10(X - 1 ) X . . . (X + j - 2) + C - l l ( X - 1 ) . . . (X + j - 3) +...+c-lj
(3)
6.6
DIFFERENCE
E Q U A T I O N S SOLVED BY P O L Y N O M I A L S
289
The coefficients c.. of Eq.(3) can be expressed in terms of the coefficients d.. of Eq.(2)'
C10 = d l o ,
doo, c-lo = d-lo J 1
coo =
C l l "- d l l - c l o ~ / 2 u--1
"- d l l -
~j(j +
1)dlo
1
C01 - -
C--11 =
doo - ~(j - 1)jdoo d-ll -
1 "~j(j -
3)d-lo
j-1
j-1
j
C12 -- d12 -- C 1 1 E
b' -- C10 E E l)lt u=l v=l i t = v + 1
1 = < 2 - -~(j - 1 ) j a ~
-
~,~-
1. 2
~a 0 2 - 1)
elo
~=v+l
1
co2=do2-5(j-2)(j-1)doo-
j-1 ( j-2 E E v#-lj(j-1)2(j-2 ,)doo u=l t t = u + l
I C--12 - - d - 1 2 -
-
~(j-
E E
1)(j - 4 ) d - l l
v#--~(j-1)(j-2)--~j(j-1)(j-3)(j-4
d-lo
(4)
u--1 t t - - u + l
Coefficients of higher order than c12, c02, C - 1 2 will not be needed. The double sums in Eq.(4) may be simplified: d-3 d-2
1 j-3
u---1 t t = v + l j--2
j--1
E
E
v=l
1 v#=K+~(j-1
)2
1 v#=K+~(j-1,
~2 1 (j-2)+~3"2(j-1)
(j-2)
u = l p,=u+l
j-1 j E E ~=1 II=u+l
The integral transformation ~(x)
=
1 / sX_ly(s)ds g
(5)
290
6 APPENDIX
transforms the difference equation (1) into a differential equation, if the line of integration t~ is chosen so that the terms
sXy [~2 ~1' sX+ly, lt2 ~1 = sX+ly(1)lt2~'
...
vanish:
C_lO)SJy (j) (ell s 2 + cols + C_ll)SJ-ly (j-l) -t-(c1282-+Co2S"kc-12)sJ-2y (j-2) . . . . +(--1)J(CljS 2 + CojS+C-lj)Y -- 0 (6)
(el0 s2 -t- c00s +
_
The characteristic equation
(clos 2 + coos + c-lo)s j = 0
(7)
has the simple roots:
1 (-doo + (d2o- 4d-lodlo) 1/2) s2 = 21 dlo 1 1 (-doo -(d20 - 4d-10dlo) 1/2)
s3 = 2 dl0
d-lo 8283
--
(8)
dlo
In addition, there is the j-fold root sl ----0. We make the substitution z = s - s2 in the differential equation (6) and obtain:
cloy(j )
--
ell (z + 82) 2 + c01 (z -t- 82) + C-ll ( j - l ) y
z(~ + ~ ) ( z + ~ - ~ ) Jr- C12(Z "}- 82)2 -~- C02(Z Jr- 82) -]-C--12 ,(j--2) y ..... z ~ ( z + ~2)(z + ~ -
~)
0
(9)
Equation (9) is solved by means of a power series: O0 y =
(10)
b,--0 The following determining equation for the initial power p is obtained if s2 is not equal to sl or s3:
p ( p - 1 ) . . . ( p - j + 2) ( ( p - j + 1)CLO -
CllS~ + COlS2 + c - l l / s 2(s 2 --sT)
= 0
(11)
6.6
DIFFERENCE
E Q U A T I O N S SOLVED B Y P O L Y N O M I A L S
291
Equation (11) has the trivial roots p = 0, 1 , . . . , j - 2 and the one nontrivial root' 1
p =
dloS2(,s2
(12)
(dlls~ + dols2 + d-11) - 1 83)
We interrupt this course of calculations and consider a series expansion in negative powers of X:
P ~ ( X ) = dlo + d l l X -1 + " " + d l j X - j P ~ ( X ) = doo + d o l X -1 + " " + d o j X - j P '--1 ( X )
d-1 0 + d -
--
11 X
-1
nt- " " " -6 d - l j
X -j
(13)
Multiplication by X j yields the polynomials of Eq.(2). The initial power p is independent of the degree of approximation of the series expansion of Eq.(13), since the terms containing j do not show up in Eq.(12). An interchange of s2 and s3 leaves the form of Eq.(12) unchanged. Hence, the same comment also applies to the solution in the point s8, if s3 is not equal to Sl or s2. Let us return to the characteristic equation Eq.(7). In the case of a double root, d00 82 --- 83 - -
2d10
one may solve the difference equation (1) only, if s2 is also a root of the equation cll
82
(14)
+cols+c-11=0
The roots of Eq.(14) shall be denoted s4 and s5, and it shall hold s2 - s3 = s4. One obtains" doo --
s4--
2d10
2dlo ,
s5=
doo
- 3)d-10 2dll - j ( j + 1)d10
2d_11 -j(j
(15)
The determining equation for the initial power p now has the following form:
[(p - j + 2)(p - j + 1)cl0 - ( p - j + 1)Cll (1 - s2S--5)5 +
+
+
The nontrivial values of p are the roots of the equation"
(p-
/ - a) = 0
6 APPENDIX
292
dll 4dlod-11 / 3 _ ~ 1 o .~_ d02o P
p2+
+ 2
do1 1 q d00
dll dlo
do2 doo
+ d12 + 4dlod-12 = 0 ~1o do2o
(17)
Again, p is independent of j. The recursion formula for the coefficients q~ of the power series of Eq.(10) is obtained by the substitution of Eq.(10) into Eq.(6): ao,~,qo + ' "
+ av,~,q~, + a ~ , + l , v q v + l
= 0
(18)
One obtains after lengthy calculations with the help of Eqs.(4), (5), (8), and (12):
a~,+l,~ = d l o S 2 ( S 2 -
j--1
s3)(~' + 1)(p + r, + 1)s 2
(p + ~,)... (p + ~ - j + 3)
t-
~,~ =/d~:~
+ d o : ~ + d _ ~ - (; + . + 1)(d~l~ - d _ ~ )
+ (p + r, + 1)(p + ~ + 2)d10s22 - (p + 1)(dlos22 - d-10) +j(dloS
22 -
(
d-lo)~' v + p
-
2j+~
1)]
s2 (p+~).-.(p+r,
j+3)
(19)
The recursion formula of Eq.(18) may be written as a system of linear equations. The coefficient a~+l,~ vanishes for p = - ( v + 1) and a system of p equations with p - 1 variables is obtained. The necessary and sufficient condition for the existence of such a system is that the determinant of the coefficients vanishes. This condition is simplified if s2 or s2 - s3 is small; one obtains in first approximation: - - p- - 1
I-[ ~ , ~ = 0
(20)
v---0
The condition s2
-
83
~ 1 yields from this product:
2
d12s2 + do2s2 + d-12 - (p + u + 1)(dlls 2 - d - l l ) v=0,
1,...,
+ (v + ~ + 1)(; + . + 2)d~o~ = 0 -(p+l)
(21)
6.7 DISCRETE SPHERICAL HARMONICS
293
6.7 DISCRETE SPHERICAL HARMONICS We start with the difference equation (5.4-37) but rewrite it with the help of Eqs.(5.4-8) and (5.4-9) in operator form in order to connect with previously published more detailed results 1.
( (~A~)~ /~?-]2
/~?.]
)
sin2 (fiAt))
The nonnormalized, standard form of this equation is obtained for r] = ~/Av~
o~ E%(~.4-4). /~2~p~,(~)/~2+ cot t9 z~q~u (O)z~O + ((vA~~ ~ - Ao) ~o~,(~9) = 0
(2)
We substitute c o s O = x,
~o,(~) = ~,(x)
(3)
and transform the difference q u o t i e n t s / ~ / / ~ a n d / ~ 2 / / ~ 2 into difference quotients A/A~ and A~/A~~. A~o,(~9)
1
/~l
z~o,(x) Ax
/~r ~o/z(~))
.at_
sin A~9 z~0,(x)
~
A02
= ~ ~
zx~
2
+ Z-J ........
~x~
(zxe)2
-
- 2 cos ~ 1 - cos ~ A ~ , (x)
--------~ (~x~)
Z~
]
(4)
1Harmuth 1989, Eq.(9.1-6) and Section 12.7. A missing sign + in Eq.(9.1-6) of the English edition is corrected here.
294
6 APPENDIX
Equation (2) assumes the following form:
{ [
(sinA'0/2
(1 - x 2)
A0
x2/1-c~
+
z~2
Avq
-x[21-c~
/~x 2
sinA~
A
ao
~-
(uA~)2 } ~pt,(x)=0
(5)
_-- _1[cos(0 + A0) - cos(O- A0)] = - sin 0 sin A0
(6)
(a~)~
+
~ + l_x2
From the relations 1
2
one obtains Av~ expressed by Ax"
Avq = - sin- 1 "-
-
-
(1
-
Ax
(1 - x2) 1/2 Ax 1
-
X2) 1/2 nt- 2
(Ax) 3 3(1 - x2) 3/2
1.3 (Ax) 5 ) 2-4 5(1 - x2)5/2 + "'"
1 (Ax) 2 sin A0 =1 23(1-z2) AO ) 1 - cos A0 1 Ax ( + 1 (Ax) 2 1 2 (1 x2)l/2 24 1 - x 2 + " " A~ ,
o
,
(Ax)2
<1 1 - x2 -
(7)
Only even powers of (1 - x2) 1/2 appear in the series expansions of the terms (~i~ A ~ ) / A 0 , (~in a 0 ) ~ / ( A 0 ) ~, (1 - co~ A 0 ) / a ~ , ~ d (1 - co~ A~)~/(A0)2. Equation (5) assumes the following form if the explicit difference quotients are substituted for the symbolic ones:
/
x (sinA0
+~
2ao +
1-cosA~)
1
(A0)~
(A~)~
(sinA0)
A0
2 } ~.(z + Az)
6 . 7 DISCRETE SPHERICAL HARMONICS
+ ~(~)
1-cos~
sin~
A,O
-
zx~
+ ( a ~ ) ~.
+
(~)
sin~ A0
_
295
(uA~)2 } 1 - z 2 ~ (x)
-Ao+
ao
~-cos~ A0
x (sinA0 Az
2A~
1-cossA~ ) +
(A~)-
1
(zxx) ~.
A~
~.(~_/x~)
= 0
(8)
This difference equation has transcendental coefficients. They can be replaced by polynomials of arbitrarily high degree by means of the series expansions of Eq.(7) multiplied by ( 1 - x2) k.
12
X3 + xJ--1 ~
+
-2xJ +
[ + xJ-xJ-I+
2(s
X j-2 + . . .
qo,(X + 1)
(A~)2 +'x~ - ~
(7 12
+...
~,,,(X)
j )xJ_2+...]qo,(X_l)=O 2(Ax) 2
(9)
X = x / A X , ~p,(X) = qo,(x), j = 2k + 2
Equations (9) and (6.6-1) become equal if the following values are substituted for the coefficients dij in Eq.(6.6-2): 7
j
dlo = 1
dll = 1
d12 = 12
2(Ax) 2
doo = - 2
do1 = 0
7 j do2 = - ~ + Az + 2(Ax)2 7
d-lo = 1
d-ll = - 1
d-12 = 12
j
2(Ax) 2
(lO)
One obtains from Eqs.(6.6-8), (6.6-15), and (6.6-17):
82 -- 83 --- 84 ~ 1,
2+j(j-3) ss=j(j+l)-2'
lim s 5 = 1 j~ '
p(p + l) = -A~
(11)
296
6 APPENDIX
The solution of Eq.(9) in the point series: F(X)
oo
82
--
83
),
~'(X)=F(X+p+I)~(-1
1 may be written as factorial
:
(p + 1)... (p + #)
C~'(X+p+I)
~---=0
(X+p+#)
(12)
The differential equation (6.6-6) belonging to the difference equation (9) has the singular points s = 1, s = 0, and s = c~. A power series in the point s = 1 converges inside a circle that passes through s = 0. The factorial series of Eq.(12) converges in the half plane X > X0, where X0 remains to be determined. Let ~ run from 0 to ~. The variable x runs then from 1 to - 1 , and X runs from 1/Ax to - 1/Ax. The difference Ax is arbitrarily small. Hence, only factorial series with an abscissa of convergence X0 < -1/Ax are of interest as solutions. Let us assume Eq.(12) converges for X > X0 < -1/Ax. The gamma function F(X) has poles for negative integer values of X, which must be compensated either by zeros in the factorial series or by poles of F(X + p + 1). The gamma functions F(X + p + 1) and F(X) have common poles if p is an integer: p=-(l+l),
p=+l,
l=0,
1, 2 , . . .
(13)
Only nonpositive integer values of p avoid poles for X = 0, - 1 / A x , . . . . The series of Eq.(12) becomes a polynomial if p is a negative integer, and the abscissa of convergence X0 becomes -c~. ,
=
Equations
=
(13) and (11) yield
it--0
c.
l-
'
X =
cot sinA~
(14)
the eigenvalues:
)~ = -l(1 + 1)
(15)
This result is surprising, since it does not depend on A0 or A~. It is impossible for AO and A~ to appear, since the only connection between the equations for ~ ( ~ ) and ~ ( ~ ) with that for %,(p) in Eqs.(5.4-31), (5.4-37); (5.4-39) is provided by the parameter A#. The condition of Eq.(15) is necessary for the existence of polynomials ~ , ( X ) . An investigation of the sufficient condition has so far not been made (Harmuth 1989, Sec.12.7). Let us assume that Eq.(12) does not terminate. The factorial series must then have zeros in the points X = 0, - A x , . . . and the coefficients C~ must be functions of the arbitrary differences Ax. Hence, the polynomial solutions are the only ones that have no poles and that become independent of Ax for small values of Ax.
6.7
DISCRETESPHERICALHARMONICS
297
The difference equation (2) is satisfied in first approximation by spherical harmonics. Let us s u b s t i t u t e (tJA~o) 2 = - m 2 = 0 and replace the difference quotients by means of a Taylor expansion by differential quotients: /~
oo (A0)2~, d2~+1
L]0-- = ( 2 u + l ) ! d @ '+1 /~2
co (A0)2~, d2~,+2
A~ = 2 ~
t]=0
(16)
(2~ + 1)! ~ + ~
One obtains: d2
oo (A0)2v
~-~ + 2 ~
d2V+2
(2~ + 2), d~+~
v--0
COS0(~ + sin 0
~ +
(A0)2u d2u+l ) (2u + 1)'
v"-0
d0 2u+1
1 -
Ao
~~
-- 0
(17)
"
The terms not containing A0 are satisfied by spherical harmonics Pl(cos 0) for A,~ = - l ( l + 1): /3(cos ~) = 2 1 . 3 . 5 - . I l!( 2 / - 1)
1 l c o s ( / - 2)0 cosl0q 1 2 l + 1
1.3 l ( l - 1) ) + 1-2 ( 2 / - 1 ) ( 2 / - 3) c o s ( / - 4 ) 0 + . . .
(18)
Differentiation yields:
d2v+2 __i2v(iAO)22 1 . 3 . 5 . . . ( 2 l - 1) [ (/xO)2dO~+~ez(cosO) 2l-~ - li cos l0 t 1 2l- 1
1
c o s ( / - 2)0 + - - -
(19)
The terms in Eq.(17) that contain A0 may be made arbitrarily small by choosing A0 sufficiently small. The Compton wavelength limits only the spatial resolution, not an angular resolution. Hence, ~0,(0) has the form/}(cos0)+O(A~). Taking the derivative of order m of Eq.(17) with respect to cosO causes the terms not containing A0 to be satisfied by the associated spherical harmonics P~(cos0). The finite number of differentiations does not affect the principle of the vanishing of the terms multiplied by A0. Hence, the difference equation (2) has eigenfunctions of the form P~(cos 0) + O(A0) + O(A~o).
298
6 APPENDIX 6 . 8 S O L U T I O N OF T H E D I F F E R E N T I A L E Q U A T I O N (5.5-12)
T h e differential equation (5.5-12) is to be solved with a power series centered at the point ss. We substitute s - ( s - s3) + s3 and use the relation according to Eq.(5.5-13) s 2 + 983 + 1 = 0
to obtain:
[(~- ~)~
+ (2~ - ~)(~-
~)~ + ~(s~ - ~ ) ( ~ -
~)1~"(~-
~)
+ [ 2 ( s - ss) 2 + (4sa + g - ~ 6 ) ( s - ss) + (2sa + g - ~ 5 ) s s l w ' ( s - ss)
+ 1) - ~ 2 ] w ( s - sa) = 0
-[l(1
(1)
From Eq.(5.5-15) we get: N
N
v=0
v=0
~(~-~)=~ q~(~)(~-~)"~+~,~'(~-~) = ~
q~(~)(;~+~)(~-~).~+~-~
N
w"(s-s3)= E
q a ( u ) ( p a + u ) ( p a + u - 1 ) ( s - s 3 ) pS+u-2
(2)
v--O
Substitution into Eq.(1) yields for u = 0 and the lowest power ( s - s3) p3-1 an equation that determines the initial power P3" [(s3 - s2)(P3 - 1) + 2s3 + g - ~6]psssqs(O)(s - ss) p3-1 = 0 Using the relation s2 + s3 = - g of Eq.(5.5-14) we obtain: P3 -
A5
(3)
83 -- 82
E q u a t i o n (1) is rewritten as follows with s2 + s3 = - g , Eq.(2), and Eq.(3)'
[(~- ~3) ~ +
(2~ - ~ ) ( ~ -
~)2 + ~(~3 - ~2)(~-
~)1
N
• ~
qs(u)(p3 + u)(p3 + u - 1)(s - ss) p3+~-2
v--0
+ { 2 ( s - ss) 2 + [2s3 + (1 - p s ) ( s 3 - s 2 l ] ( s - s s ) + (1 - p s ) ( s 3 - s2lss} N • E q3(/I)(P3 ~- /2)(8 -- 83) pS+u-1 v--0 N -
[l(1 + 1) - ~2] ~ t]--0
qs(u)(s - ss) pS+~ = 0
(4)
6 . 8 SOLUTION OF THE DIFFERENTIAL EQUATION (5.5-12)
299
For u = 0 we obtain from the terms multiplied by (s - s3) p3 the following relation between q3(0) and q3(1)"
aa,1 (O)qa(1) + a3,o(O)q3(O) = 0 0/3,1(0) -- 83(83 -- 82)(P3 -[- 1)
(5)
c~3,0(0) = saP3(P3 + 1 ) - l(1 + 1 ) + ~2
Generally we write Eq.(4) for q 3 ( u - 1), q3(u), and q3(u + 1). The terms in boldface will be explained presently:
9.. + [ ( ~ - ~3) 3 + ( 2 ~ - ~ ) ( ~ - ~3) 2 + ~ ( ~ 3 - ~ ) ( ~ -
x q 3 ( u - 1)(p3 + u -
+ {2(s-
~)1
1)(p3 + u - 2)(s - s3) p3+~-3
sa) 2 + [2s3 - (P3 - 1)(s3 - s2)](s - s3) - (P3 - 1)(s3 - s2)s3 } x q ~ ( ~ - 1)(p~ + ~ -
1)(~- ~)~+"-~
-[l(1 + 1) - ~ 2 ] q 3 ( u - 1 ) ( s - sa) p3+~'-1 + [(~ - ~ ) ~ + ( 2 ~ 3 - ~ ) ( ~ -
ss) ~ + ~ ( ~ 3 - ~ ) ( ~ - ~)1 x q~(.)(p~ + ,)(p~ + . -
1)(~-
~)~+"-~
+ {2(~ - ~ ) ~ + [ 2 ~ - ( w - 1)(~s - ~)1(~ - ~ ) - ( p ~ - 1 ) ( ~ - ~ ) ~ }
x q~(~)(p~ + ~,)(~- ~3)~+~-1 -[l(l + 1) - ~ 2 ] q3(u)(s - s3) p~+~ + [(~ - ~ ) ~ + ( 2 ~ - ~:)(~ - ~ ) ~ + ~ 3 ( ~ 3 - ~ ) ( ~ ~)1 • q3(~ + 1)(p3 + ~ + 1)(p3 + ~)(~ - ~3) "~+"-~
+ {2(~ - ~ ) ~ + [2~ - (p~ - 1 ) ( ~ - 8 ~ ) ] ( ~ - ~ ) - 0 ' ~ - 1 ) ( ~ - ~ ) ~ x q3(~ + 1)(p3 + ~ + 1)(~ - ~ ) ~ + "
-[l(1 + 1) -,~2]q3(u + 1 ) ( s - s3) p~+~+~ + . . . The terms emphasized by boldface are all multiplied by ( s sum must be zero:
} (6)
s3) p3+~. Their
[(pa+u-1)(pa+u-2)+2(p3+u-1)]qa(u-1) +{ (2s3-s2)(p3+u)(p3+u-1)+[2s3-(p3-1)(sa-s2)](pa+u)-l[(l+l)-Z)2]} qa (u) +[s3(s3-s2)(p3+u+1)(p3+u)-(p3-1)(s3-s2)s3(p3+u+1)]qa(u+ 1) = 0 (7) This equation may be simplified:
300
6 APPENDIX
a 3 , l ( v ) q 3 ( v + 1) + a 3 , 0 ( v ) q 3 ( v ) + a 3 , - l ( v ) q 3 ( v -
1) = 0
c~3,1(v) = s3(s3 - s2)(p3 + v + 1)(v + 1) ~ , 0 ( ~ ) = ( 2 ~ - ~.)(p~ + ~)~ + [ ~ - p ~ ( ~ a 3 , - l ( v ) = (P3 + v)(p3 + v -
- ~)](p~ + . )
(8)
1)
6.9 CONFORMAL MAPPING FOR SECTION 5.7
According to Figs.5.7-1 and 5.5-2 we have to resort to conformal mapping if a power series in the point s3 is to converge as far as s - 0 for s3 > 1/v/2. We start from Eq.(6.8-1) and replace the terms ( s - s 3 ) '~ by s ' ~ ( s / s 3 - 1 ) n. This calls for a change of w ' ( s - s3) and w " ( s - s3) in Eq.(6.8-1):
w'(s-
s3) = d w ( s - s3) dw(s/s3 - 1)d(s/s3 - 1) _ 1 w ' ( s / s 3 - 1) ds ~ d(s/ss-1) ds -s~
w"(s-
s3) ~ - ~ 3 w " ( s / s 3 - 1)
(1)
Equation (6.8-1) becomes:
83
+
S_l
+(2s3-s2)
83
2s3
s -1
s _1
+(s3-s2)
+(4s3+g-Ah)
s - 1
s _
w"(s/s3-1)
+ 2s3 + g - A5 w ' ( s / s 3 - 1)
- [ l ( l + 1) -zy2]w(s/sa - 1) = 0
(2)
The substitution s/s3 = ~ l / P transforms the differential equation from the s-plane to the C-plane:
[~3(r ~/P - 1) 3 + (2~3 - ~2)(r ~ / P Jr- [283(r lI P
-
-
1) 2 + (4s3 + g
-
1) ~ + (~3 - ~2)(r ~/P - 1)]~"(~ 1/P - 1) ,~)(r
_ 1) -~- 283 -~. g -- ~ ] W ' ( r lIP -- 1) - - [ l ( t + 1) -z)'2]w(r
- 1) = 0
(3)
In order to obtain a differential equation of the Fuchs-type we must transform the terms ( r 1)n into power series of ~ - 1 or 1 - r This is possible by expanding r _ 1 = - ( 1 - ~l/P) in a binomial series:
6.9 CONFORMAL MAPPING FOR SECTION 5.7
[ 1 1--~ 1/P-- 1--[1-(1--~)]I/P =- 1-- 1 - ~ ( 1 - ~ ) +
ll(p ~
-1
301
)
(1-r
2 --...
_ 1-r X
P P - 1 ( 1 - ~ ) + ( P - 1 ) ( 2 P - 1) ( c~ ( P - l ) . . . ( 3 P - 1) ( l _ r X = I + 2!P ~. P"2 "1-"~2+ 4!P 3 = 1+ b~(1 -r
+ b~2(1 _ r
blj
+ ... + b~j(1 - r
+
.
.
+ 9
9
9
.
(P-1)...(jP-1)
(4)
(j+I)!PJ
--
The series is absolutely convergent in the interval - 1 < 1 - ~ < +1. We can derive the following powers of (1 - { 1 / P ) n from EQ.(4)"
X '~,
(1-(1/P)n=
~l/P-1
=
X",
n = 1, 2 , . . .
(5)
We obtain for X 2, X 3 , . . . , X 6" X 2 =1 + b21(1 - ~) + b22(1 - ~)2 + b23(1 - ~)3 + b24(1 - ()4 + . . . =1 + 2b11(1 - ~) + (2b12 + b121)(1 - ~)2 + 2(b'13 + b11b12)(1 - ~)3 + (2514 + 2511513 + 522)( 1 -- ~)4 + . . .
b21 = 2b11, b22 = 2b12 + b21, b23 = 2(b13 + b11b12),
(6)
b24 = 2b14 -t- 2bllb13 + b22
X 3 = 1 + b31(1 - () + b32(1 - r
+ b33(1 - ~)3 + . . .
531 -- 3511, 532 = 3(5121 + 512), 533 -- 6511512 + 3513 + 531
(7)
X 4 = 1 + b41(1 - ~) + b42(1 - ()2 + . . .
(8)
b41 - 4b11, b42 = 6b21 + 4b12 X 5 =1 + b51(1 - r
+...,
b51 = 5bll
(9)
x ~ =1 + . . .
If we substitute 1 - (~ = - ( ( e.g.,
x = 1 -b,~(r
(10)
1) into Eq.(4) for X we obtain alternating series,
1 ) + 512(r
1) 2 - b , 3 ( r
1) 3 + . . .
(11)
302
6 APPENDIX
Hence, the substitution of (r
_ 1)~ of Eq.(5) into Eq.(3) brings'
{s3p-3(r
1)311 - b31(r 1) + b32(r 2) 2 - . . . ] + (2s3 - s 2 ) p - 2 ( r 1)211 - b 2 1 ( r 1)+ b22(~- 1) 2 - . . . ] -~-(83 -- 8 2 ) p - 1 ( ~ 1)[1 - b l l ( ~ 1)+ b12(~- 1) 2 - . . . ]}'w"(~ 1/P - 1) + { 2 s 3 p - 2 ( ~ - 1)211- b21(~- 1)+ b22(r 1) 2 - ...] + (4s3 + g -
~)p-l(~_
1)[1 - b 1 1 ( r
1) + b12(r
1) 2 - . . . ]
+ 2~ + ~ - ~ } ~ ' ( r
- 1)
- [ l ( l + 1 ) - z~2]w(~l/P - 1) = 0
(12)
This is the differential equation (2) in the ~-plane. Since it is of the Fuchs-type we write its solution w(~ 1 / P - 1) as a power series in r 1. The notation ~(v) and t5 is used in the C-plane instead of the notation q(v) and p in the s-plane: oo
W(~l / P - 1 ) = E ~(v)(~- 1)#+" v-'0 oo
W'(~lIP
-
-
1) = ~
0(v)(i5 + v)(r
-
1)~+'-1
v--0 oo
Wt'(~l / P - 1 ) = E 0(v)(~5 + v)(f + v - 1)(~- 1)p+~-2
(13)
~--0
Substitution into Eq.(12) yields" { s 3 p - 3 ( ~ - 1)311 - b31(r - 1) + b32(~- 1) 2 - . . . ] + (2s3 - s2)p-2(r - 1)211 - b21(r 1) + b22(r 1) 2 - . . . ] + ( s 3 - s 2 ) p - l ( ~ - 1)[1-b11(r 1)+ b12(r 1 ) 2 - . . . ] } oo
x E q(v)(15 + v)(15 +
v
-
1)(r
-
1) p+~-2
~--0
+ { 2 s a p - 2 ( ~ - 1)211- b21(~- 1)+ b22(~- 1) 2 - ...] + (4s3 + g - ~ ) p - 1 ( ( _ 1)[1 - b 1 1 ( ~ - 1) + b12(~ -+ 2~ + g - ~ } ~
1) 2 - - . . .
]
q(~)(~ + ~)(r - 1) ~+~-~
~'--0 c~
-
[l(1 +
1) - ~2] E v--0
~(u)(~ - 1) p+" = 0
(14)
6.9 CONFORMAL MAPPING FOR SECTION 5 . 7
303
The sum of all coefficients of a certain power ( ~ - 1)f+,~ must be zero. The lowest power in Eq.(14) is ( ( - 1)f-1 for v = 0. There is only the coefficient ~(0) for this power: [(83 -- 82)p-1/5(15- 1) + (283 + 9 - X(~)iS]~(0) = 0
(15)
Using the relation - g = s2 + s3 of Eq.(5.5-14) we obtain the following value for the initial power 15: /~=P
83 -- 82
-1
+1
(16)
For P = 1 we get the same value for i5 as for p in Eq.(5.5-16). We derive from Eq.(16) an expression for A5 that will simplify some of the following equations: ~(~ = (83 -- s 2 ) [ p - I ( p -
1 ) + 1]
(17)
The second lowest power in Eq.(14) is ( ~ - 1) f. It contains the coefficients ~(1) and ~(0). It yields ~(1) if we take ~(0) as the choosable constant:
(283 -- s2)p-2q(0)i5 (/5- 1)+ (83 - s2)P-1 [~(1)(/5 + 1)i5- b11~(0)/5(/5- 1)] + (483 + g - i~)p-lq(O) + (283 + g -- X5)~(1)(/5 +
1)
-[l(l+l) ' ~ 2 ] ~ ( 0 ) = 0
(18)
We may rewrite this equation with the help of Eq.(17) and eventually obtain the following simpler form'
~p,~(0)~(1) + ~p,0(0)4(0) = 0
(19)
C~p,l(0) = P - 1 ( s 3 - s2)(p + 1) ctp,0(0) = P - l i 5
(83 P-1 (/5\
1 + 2P)
P-1
) -l(l +
1)+ ~,2
(20)
(21)
For P = 1 and thus 15 = p we obtain:
(22)
ap, l(0) = (83 - s2)(p + 1) ap,o(0) =
s3p(p + 1) - l ( 1 + 1) +
~2
(23)
Equation (23) equals aa,0(0) in Eq.(5.5-18) while Eq.(22) differs by a factor sa from c~3,1(0). We expect that the inverse transformation from the (-plane to the s-plane will provide the missing factor s3.
304
6 APPENDIX
The next step would be to determine the coefficient of ( r t h a t yields in analogy to Eq.(19):
5+1 in Eq.(14)
ap,2(11#(2) + ap, l(11#(1) + ap,o(1)#(O) = 0
(24)
The elaboration of ap,2(1), ap, l(1), ap,0(1) is a challenge but it is not needed here. As ( r 115+1 increases to ( r 115+2, ( r 11P+3, ... the recursion formulas according to Eqs.(191 and (24) increase to 4, 5 , . . . terms. Let us turn to the inverse transformation ~ = (s/s3) P from the C-plane to the s-plane and see what becomes of t5 and ~(~) of Eqs.(16) and (191 for ~ = 1. We write:
w(# I / P - 1)= E O(")(#- 1)f+" = E q(") ,,,=0 ____
s
-
1
u=O
[
(.)(:3)
(8/83)P-1 ~ S __1 Eq(u)(--1)" 1S/S3- 1
(25)
The reason for this notation is that we can change the signs of the first factor:
(~3>~ 11~ (1 1 r~/~3 s/-~3- 1
--
(26/
-
There is no problem with a sign change for integer powers ~ = 1, 2, . . . :
[(~/~3) P - 1]" = (-11"
[1 -(,/~3)P]
(27)
~
We use the binomial series to expand 1 - (s/s3)P:
1-
~ 83
[ (
=1-
1-
+
1-~
P ( P - 1) 2!
83
)] ~
[
=1-
(~)
1-P
1-
s
s
1-
-
3
3!
_~(1_ _ ~) [1 ~_1 ,, (1_;) +
(P-
1)(P3!
21
1-
s
The new variable Y is defined as follows:
-...
=P
1-
s
Y
...t (281
6.9
Y=I
P-l(s-~) 2! 1-
+
(P-1)(P-2)(~3) 3! 1-
. . . + (_I)j (P -1)
2
c1=
( ) (
Cl .
.1 .
s
83
c2 1 . . s . . . . .
, . . . , cj
=
s) j
1 - --s3
( ) 1-B s
cj
83
P-1 (P-1)(P-2) 2--7' c 2 = 3!
+ ...
" " ( P - J) (
(j~: 1)! -. - 1 .
305
CONFORMAL MAPPING FOR SECTION 5 . 7
+""
J _...
83
-(-1
)j ( P - l ) . . . ( P - j ) (j+l)!
(29)
The choice of the negative sign for the coefficients cj will become evident presently. Equation (26) may be rewritten:
( (s/s3)P--1) ~ (1-- (s/s3)P)~ s/s3 - 1
=
1 - s/s3
(30)
= P~Y~
The series of Eq.(29) for Y terminates for positive integer values of P. Hence, the question of convergence is avoided for P = 1, 2, . . . . The expression Y~ in Eq.(30) can again be expanded into a binomial series P ~ Y ~ = P~[1 - ( 1 - Y)]~
(31)
and Eq. (30) becomes:
1 - s/s3
= P ~ Y ~ = P ~ [1-/5(1 - Y ) +
~(p
1)
2[
(1 -
y)2
_
1 [ "'" ]
(32)
The positive integer powers of I - Y will be needed. From Eq.(29) we get:
1- Y =
1- s
c~+c2
1-u
s
-[-c3
1---
83
s
+c4
1-
s
+
83
...
(33)
The reason for the choice of the signs of the coefficients cj in Eq.(29) becomes clear. For higher powers of 1 - Y we obtain:
(1-Y)2
=
( 1/ - 2 [ s - -
c 2+2clc2 ( 1I- s
83
+ (2cl c4 + 2c2c3) 1 -
- -
r92~_
+ ~ - c ~ c 3 + c . ~
( / 2 s1 -
83
+(2clc5+2c3c4+c
-
-
83
2) 1 - u
+... 83
(34)
306
6 APPENDIX
(1 _ y)3 -8 8 ( ~ 3 3 ) 3 1 1 - c3+3c2c2 ( 713- /
+(3c2c3+3ClC22) ( ~13- 3 )8 2
-~- (3Cl2C4 + 6Clc2C3 -}- C3) 1
(1-y)4=
(35)
+ ooo
~
83
( 1 - ~s3 ) 4 [ c4 + 4 c 3c2 ( ~13-3 ) s 22 8 + (4c31c3 -~- 6c lc 2) 1 -
(:3)
(1 - y ) 5
---
(1
-
s s ~33}5[ C l5 .Jr.. ,.~4 . , C l C 2 ( ] 1 _) _-3t- .
+...
]
(36)
(37)
. .
83
( )0[ ( 1 - V)a =
1 - ~ 33
]
(38)
c~+...
Substitution of Eqs.(33) to (38) into Eq.(32) produces a very long formula that we bring into a printable form by defining new coefficients dj:
(1-(s/s3)P) / = [ ( - -~~3) ( s) ( -~3) +"" ] s's3 Pf 1 + dl 1 + d2 1 -- + d3 1 s --
8 3
=P~ 1 -
dl
__s-1
s3
+d2
-- -1
s3
-d3
-1
+
...
dl --/5Cl 1 d2 -/5c2 + ~./~(/5- 1)c~ 1
d3 "- pc3 -}- ~ . i p ( p -
d4 --
1
pc4 -~- ~ . f i ( p -
1 , 1)2clc2 + ,~.i5(/5- 1)(i5- 2)c 3 1 1 1)(2CLC3 + c2) -}- ,~/~(/51)(/~- 2)3c2c2 .+ ~.p...- (~-3)c~ 9
1 1 d5 - ~c5 + ~fi(15- 1)(2clc4 + 2c2c3) + _.,~.'iS(fi-1)(15- 2)(3c2c3 -
3c~c2)
1_ + ~1_ ; . . . (;~- 3)4~3~ + ~.p... (;~- 4)~
6.9
CONFORMAL
MAPPING FOR SECTION 5.7
1
307
1
6CLC2C3
d6 --j6c6 + ~.15(/5- 1)(2CLC5 + 2c2c4 + c 2) + _~.15(/5- 1 ) ( i 5 - 2)(3c2c4 + 1~ + c 3) + ~ p . . .
1_ (15 - 3)(4c3c2 + 6c2c~ + ~ P . . .
+ T h e t e r m s [(s/s3) g Eq.(28) we obtain:
-
8
-1
=(-1)"
1-
For v -
-
1---
s) y,=1+e,,l(l_ s
(~)
Y=
1-
=
1--Cl
e~2
(~) 1---
1 - --
e21 - - - - 2 C l ,
1 -- 2Cl
--C2
( ~) 1 - --
~4(:
1_
(41)
)~ ( s )4 ]
+ ( - 2 c 2 + cl) 2
(
1 - --
83
1 -- - -
83
+...
2CLC3+ c~
-
s
-
1-3cl
1-
s
83
e31 - - - - 3 C l ,
1-
-F ( - - 2 C 4 -~- 2CLC3 -{-C~)
~ : ( )~[ (~) 1-
(
83
+ (-~
\say
(40)
+...
e22 - - - - 2 c 2 + c21, e23 - - - - 2 c 3 + 2CLC2,
e24 = - 2 c 4 +
s__~
yv
1---
--~ - - c 3 , . . .
+(_2c3 +2ClC2)(1_ _.~)3
(1
s
1-
83
83
S3/
~
6 we o b t a i n with the help of Eq.(29):
( ~)~ (~) ~[
\
+
83
e l l - - --C1~ e12 ----- --c2~ r
(1-s~
8
=(-1)'P
--
83
1-
(39)
83
1, 2 , . . . ,
(~)
P
--
83
(1
1 ~.,/~... (15- 5)c61
1] ~ in Sq.(25) also require a series expansion. Using
P
s
(15 - 4)5c~c2
+(3Cl2-3c2)
(42)
( ~)~ 1-
s
+ 6~1~ - 3~) (1 - ~
(43)
e32 - - 3C 2 -- 3C2, e33 - - --C 3 + 6CLC2 -- 3C3
- -s
)4[ (~) 1--4gl 1
83
e41 -- --4Cl, e42 -- 6Cl2 -- 4c2
s
+(6c~-4c2)
+...
( )~ "*'] 1-- s
83
(44)
308
6 APPENDIX
( :,)~ ()~[ 1
w
s
y5
1
- -
s
.
.
.
1
.
551 .
83
()] 1
s
+
.
.
.
83
(45)
e51 - - - - 5 C I
( )~ ( )~ ] 1-
s
y6
1 _s
-~-
1+
~
83
(46)
. . .
83
We m a y rewrite the last sum in Eq.(25) as follows:
[ ()~
E~(v)(-1)"
s
1 -
u=O
= E~(u)(-1)'P"
83
=~(0)P ~
q(1)P
-
1
+~(2)P 2 ~(3)P 3 -
-
1-
1
-
--
1
-
+ ~(4)P 4
s
1
s
1 A- e l l
s
1+e21
s
1+e31
s
-~(5)P 5(1-~3) +~(6)P 6
(;) (:,) (~) (~) 1
1
1 + e41
5 [l+e~l s
--
-
1
y.
--
1+...
( :,)~ ] ( ~)' ] ( ~)~ ] ( )~ ]
+ " ' + e l ~
1 -
s
+ " ' " + e24
1-
s
- t - " ' " - ~ - e33
1 -
s
+ e42
-
(1-
( ~)~ ] 1-
s
-
1
-
--
s 83
yO
( ~)~[ ( :,)~[ ( ~)~[ (~)'[ 1
1-
u--0
1 - -s-
s
+...
s
+
...
+
...
s
+ ...
83
~3) +...]
+0(1-s/s3)
(47)
7
We collect t e r m s m u l t i p l i e d by the same power (1 - s / s 3 ) t' with # - 0, 1, 2, . . . , 6. T h e coefficients e,~ used in Eqs.(40) to (45) m u s t be d i s t i n g u i s h e d from t h e new coefficients e~:
E
q(v)(-1)~'
[ (~)~] ~ [()~ ] (~) ()~ ( )~ () (~)~ ( )~ 1-
s
--
~(v)
s
u=O
u----O
=
eo + e l
1-
s
+e2
-~- 1
83
1-n s
+e3
1-n s
83
= e0 - el
8
-- - 1 83
+ e2
8
--
1
+...
83
-
e3
8 -83
- 1
+ ...
6.9 CONFORMAL MAPPING FOR SECTION 5.7
309
~o = 4(0)
gl ------ ~ ( 1 ) P e2 -- - ~ ( 1 ) P e l l + q ( 2 ) P 2 e3 = - ~ ( 1 ) P e 1 2 + q(2)P2e21 - q(3)P 3 e4 = - ~ ( 1 ) P e 1 3 + ~(2)p2e22 - q(3)p3e31 + q(4)P 4 e5 = - ~ ( 1 ) P e l a + ~(2)p2e23 - 4(3)P3e32 + q(4)p4e41 - q(5) P5 e6 = -c](1)Pel~ + c](2)p2e24 - 4(3)P3e33 + ~(4)P 4e42 - ~(5)Pbe51 + q(6)P 6 (48) E q u a t i o n (25) may be written as follows with the help of Eqs.(39) and (48):
Wp
-
-
-
-
1) -
-
---1
1-d1
[
x eo-el
- (el + e o d l )
---1
+d2---1
-d3---1
83
~-~3-I +e2
~3-1
+
83
.
.
.
-e3 ~-~3-I + . . .
s3
+ (e2 + eldl + eod2) (~3 - 1 ) 2 - (e3 + e2dl + eld2 + eod3) (~3 - l ) 3 + (e4 + e3dl -F e2d2 + eld3 "+"eod4)
(8--1)4 s3
- (e5 + e4dl + ead2 + e2d3 + eld4 + eod5)
s _ 1
+ (e6 + esdl + e4d2 + ead3 + e2d4 + eld5 + eod6)
(;)~ s _ 1
+ o(~/sa
- 1) 7
(49)
W i t h the final transformation s/s3 - 1 = s 3 1 ( s - s3) and w(s/s3 - 1) = w ( s - s3) = w(s) we bring Eq.(49) into the form of Eq.(5.5-15):
310
6
APPENDIX
c~
N
~ ( ~ - ~ ) = ~ ( ~ ) = ~ ~ ( . ) ( ~ - ~)~+~ " Y ] 0~(~)(~ - ~)~+~ v=O
v'-O
~p (0) = + P P s 3 f eo ~p(1) = -qp(0)(el + eodl)/s3 qp(2) = +qp(0)(e2 + eldl + eodl)/s 2
9
(50)
The upper limit of the sum was reduced from oe to N since an arbitrarily large but finite interval can have only a finite number of arbitrarily small but finite subintervals. Let us check our result. The constant qp(0) is of little interest since it can be chosen. For qp(1) we get from Eqs.(29), (39), and (48) the components el
--
-q(1)P,
eo = ~(0), dl =/)c1, cl =
P-1 2!
(51)
Equations (19)-(21)yield:
~(1) ~(0)
~p,o(0) ap, l(0)
p-lp(s3p-I(p-I+2P)-(s3-s2)(p-1)P'/)2'
- / ( / + 1) + ~2
(52)
P - l ( s 3 - s2)(16 + 1) From Eqs.(50) and (51) we get:
4~(~) = -4~(0)~(0)
(~(1) P ~-(65~
~
/hP-1) 2~
(53)
~
We reduce the choosable constant qp(0)4(0) to qp(0) by choosing ~(0) = 1: ~p(1) = _ (/5 [s3 P - 1(/5 - 1 + 2P) - (s3 - s 2 ) ( / 5 - 1 ) ( P - 1 ) / 2 P ] - P[l(l + 1) _,~21 (~3 - s2)(~5 )(f + 1)) ~p(0) \ P-ls3(s3 /~ P - 1 ) s3 2!
(54)
For P = 1, 15 = p we again obtain Eq.(5.5-18): q-p(1) ~p(0)
s3p(p + 1) - l(l + 1) + ~2 sa(sa - s2)(p + 1)
q3(1) q~(0)
(55)
6.10 CONFORMAL MAPPING FOR SECTION 5.10
311
6.10 CONFORMAL MAPPING FOR SECTION 5.10
According to Fig. 5.10-1b we must use conformal mapping if a power series in the point s3 is to converge as far as s = 0 for g > - 2 c o s T r / 6 = -1.73205, We start from Eq.(6.8-1) and use the substitutions of Eq.(6.9-1). In principle we can use all the equations of Section 6.9 that do not specify that s3 is a real number, but we must write i5 and 0 rather than i5 and ~. Furthermore we recognize the relation s2 = s~ from Fig.5.10-1a and Eq.(5.10-1). Hence, in Section 6.9 we go directly to Eq.(6.9-16)
~= p
= P
A5 s3 - s~ - I
+1,
A5 i2~,A r
+ 1
1
9 82 -- 83,
83 -- e--'
itc'Ar
(1)
and Eq.(6.9-17)"
(2)
A5 = - 2 i t c ' A r [ P - l ( ~ - 1 ) + 1] Equation (6.9-18) assumes the form"
(2s3 - s~)p-2~(0)iS(i 5 - 1 ) + (s3 - s~)P-l[0(1)(i 5 + 1)i5- b11~(0)i5(i5- 1)] + (4s3 + g - AS)P-10(0) + (2s3 + g - X5)~(1)(i5 + 1) -[l(1
+
1) -~2]0(0 ) = 0
(3)
bll - - ( P - 1)/2!P Using Eq.(2) we may bring this equation into the following form:
OIP,1(0)0(1) + ap,0(0)0(0) = 0
(4)
ap, l(0) = P - l ( s 3 - s~)($ + 1) ap,0(0) = P - I ~ ( s 3 p - I ( ~ -
(5)
I + 2P) - P2!P - 1 (s3 - s~)(i5- 1) } -
l(l
+
1) + ~2
(6)
As in Section 6.9 we do not a t t e m p t to work out the formulas for the higher coefficients 0(2), c~(3), . . . . The inverse transformation from the ~-plane to the s-plane follows Eq.(6.925). We have only to replace i5, ~ by i5, 0:
312
6 APPENDIX
w(#1/g -
((~/~) s/s3-
~ - 1)
1)=
1
P\s3__I/(S~
(~(v)(__l)~
1__
v--0
(7) 83
For the evaluation of this equation we go from Eq.(6.9-25) to (6.9-39)"
(1-(s/s3)P) ~ [ ( s ) ( s ) 2 ( s ) 1-s/s3 = P P 1 - d l ~--~3-1 +d2 ~33-1 - d 3 dl =/3cl d2
3 73-1
]
+...
see Eq.(6.9-29) for cl, c 2 , . . .
=/~c2 + ~/~(i~- 1)c21 1
(8) The sum of Eq.(7) is worked out in Eq.(6.9-48)'
4(v) ( - 1)" u--0
[
1-
s
=
83
= eo -- el
(~(v)
s s3
-1
u--0
-- 1
+ e2
- - -- 1
-- e3
-- t
+...
83
~o = ~ ( o ) el -
-O(1)P
(9) For the whole Eq.(7) we obtain from Eq.(6.9-50)'
c~
N
~(~ - ~3) = ~(~) = ~
~(.)(~
- ~3) ~§
u=O
~p(0) =
+PPs3Peo
Op(1) -- -qp(O)(el +
"~
O~(.)(s - ~3) ~+~
u=O
eodl)/s3 (lO)
The upper limit of the sum was reduced from oo to N since an arbitrarily large but finite interval can have only a finite number of arbitrarily small but finite subintervals.
6.10
CONFORMAL MAPPING FOR SECTION 5.10
313
The constant qp (0) is of little interest since it can be chosen. For gp(1) we get from Eqs.(9), (8), and (6.9-29)" el = - ~ ( 1 ) P ,
eo = ~(0), dl =15cl, Cl =
P-1 2!
(11)
Equations (4)-(6) yield:
~(1) ~(0)
,~,o(O) c~p,1(o)
g-l~ (s3P-I(P-I+2P)-(s3-s;)(P-1)P/)2'
-/(/+1)+72 (12)
P-l(s3 - s~)(~ + 1) From Eqs.(10) and (11) we get: 0(1) P
/~ P~ - l ) s3 2!
+
(13)
~
We choose ~(0) = 1 to reduce qp(0)~(0) to qp(0)' ~p(1)
~p(0)
//iS[s3P- 1(i5- 1 + 2P) - (s3 - s~) ( i 5 - 1 ) ( P - 1 ) / 2 P ] - P[l (l + 1) -~2] P-ls3(s3 - s;)(~ + 1) i5 P - l ) 83
(14)
2!
For P = 1 we obtain the simpler equation:
s3p'(p' + 1 ) - l(l + 1 ) + ~2 s3(ss- s~)(p'+ 1)
~p(1) Op(0) A5 2~'Ar' s3
*=-2i~
- 83
1At, s3
" 1-i~'Ar,
P=I
(15)
References and Bibliography Abramowitz, M. and Stegun, I.A., eds. (1964). Handbook of Mathematical Functions, National Bureau of Standards, Applied Mathematics Series 55. US Government Printing Office, Washington, DC. Anastasovski, P.K., Bearden, T.E., Ciubotariu, C., Coffey, W.T., Crowell, L.B., Evans, G.J., Evans, M.W., Flower, R., Jeffers, S., Labounsky, A., Lehnert, B., M~sza~os, M., Molngr, P.R., Vigier, J.-P., and Roy, S. (2001). Empirical evidence for non-Abelian electrodynamics and theoretical development. Annales Fondation Louis de Broglie, vol. 26, no. 4, 653-672. Apostle, H.G. (1969). Aristotle's Physics, Translated with Commentaries and Glossary. Indiana University Press, Bloomington. Aristotle (1930). The Works of Aristotle, vol. II, Physics. R.R. Hardie and R.K. Gaye transl., Clarendon Press, Oxford. Barrett, T.W. (1993). Electromagnetic phenomena not explained by Maxwell's equations. Essays on the Formal Aspects of Electromagnetic Theory, A. Lakhtakia, ed., pp. 6-86. World Scientific Publishing Co., Singapore. Barrett, T.W. and Grimes, D.M. (Editors). (1996). Advanced Electromagnetism: Foundations, Theory, and Applications. World Scientific Publishing Co., Singapore. Becker, R. (1963). Theorie der Elektrizitiit, vol. 2 (revised by G. Leibfried and W. Breuig), 9th ed. Teubner, Stuttgart. Becker, R. (1964). Electromagnetic Fields and Interactions (transl. by A.W. Knudsen of vol. 1, 16th ed. and by I. de Teissier of vol. 2, 8th ed. of Theorie der Elektrizits Blaisdell, New York. Reprinted 1982 by Dover, New York. Belfrage, C. (1954). Seeds of Destruction. The Truth About the US-Occupation of Germany. Cameron and Kahn, New York. Copyrighted by the Library of Congress but not listed in the online catalog www.loc.gov either under the name Belfrage, Cedric or the book title. Available at Deutsche Staatsbibliothek, Berlin, Signatur 8-29 MA 265 or at Bayrische Staatsbibliothek, Miinchen, Germany, Signatur BAY'. L 1089. Berestezki, W.B., Lifschitz, E.M., and Pitajewski, L.P. (1970). Relativistische Quantentheor/e, vol. IVa of Lehrbuch der Theoretischen Physik, L.D. Landau and E.M. Lifschitz eds.; transl, from Russian. Akademie Verlag, Berlin. Berestetskii, V.B., Lifshitz, E.M., and Pitajevskii, L.P. (1982). Quantum Electrodynamics, transl, from Russian. Pergamon Press, New York. Blum, J.M. (1965). From the Morgenthau Diaries, vol. 2: Years of Urgency, 1938-1941. Houghton Mifflin Co., Boston. Blum, J.M. (1967). From the Morgenthau Diaries, vol. 3: Years of War'1941-45. Houghton Mifflin Co., Boston. German translation by U. Heinemann and I. Goldschmidt: Deutschland ein Ackerland? (The Morgenthau Diaries). Droste Verlag und Druckerei, Diisseldorf 1968. Bub, J. (1999). Interpreting the Quantum World. Cambridge University Press, Cambridge. Callender, C. and Hugget, N. (2000). Physics Meets Philosophy at the Planck Scale. Cambridge University Press, Cambridge. Cao, T.Y. (1999). Conceptual Foundations of Quantum Field Theory. Cambridge University Press, Cambridge. 314
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Chuang, I.L. and Nielsen, M.A. (2000). Quantum Computation and Information. Cambridge University Press, Cambridge. Cole, E.A.B. (1970). Transition from continuous to a discrete space-time scheme. Nuovo Cimento, vol. 66A, 645-655. Cole, E.A.B. (1973a). Perception and operation in the definition of observables. Int. J. Theor. Phys., 8, 155-170. Das, A. (1966). The quantized complex space-time and quantum theory of free fields. J. Math. Phys., 7,52-60. Dickson, W.M. (1998). Quantum Chance and Nonlocality. Cambridge University Press, Cambridge. Einstein, A. and Infeld, L. (1938). The Evolution of Physics. Simon and Schuster, New York. Elaydi, S.N. (1999). An Introduction to Difference Equations, 2nd ed. Springer-Verlag, New York. Flint, H.T. (1948). The quantization of space and time. Phys. Rev., vol. 74, 209-210. Frobenius, G. (1873). Uber die Integration der linearen Differentialgleichungen durch Reihen. Journal ftir die reine und angewandte Mathematik, vol. LXXVI, 214-235. Fuchs, L. (1866). Zur Theorie der linearen Differentialgleichungen mit ver/inderlichen Coefficienten. Journal ftir die reine und angewandte Mathematik, vol. LXVI, 121-160; (1868) vol. LXVIII, 354-385. Gelfond, A.O. (1958). Differenzenrechnung (transl. of Russian original, Moscow 1952). Deutscher Verlag der Wissenschaften, Berlin. French edition 1963. Ghose, P. (1999). Testing Quantum Mechanics on New Ground. Cambridge University Press, Cambridge. Gradshteyn, I.S. and Ryzhik, I.M. (1980). Tables of Integrals, Series, and Products. Academic Press, New York. Greiner, B. (1995). Die Morgenthau-Legende. Hamburger Edition HIS Verlagsgesellschaft, Hamburg. References 133 books and 39 journal articles. Griffiths, D.J. (1989). Introduction to Electrodynamics. Prentice-Hall, Englewood Cliffs, NJ. Guldberg, A. and Wallenberg, G. (1911). Theorie der Linearen Differenzengleichungen. B.G. Teubner, Leipzig Habermann, R. (1987). Elementary Applied Partial Differential Equations with Fourier Series and Boundary Value Problems, 2nd ed., Prentice Hall, Englewood Cliffs, NJ. Harmuth, H.F. (1986a). Propagation of Nonsinusoidal Electromagnetic Waves. Academic Press, New York. Harmuth, H.F. (1986b,c). Correction of Maxwell's equations for signals I, II. IEEE Trans. Electromagn. Compat., vol. EMC-28, 250-258, 259-266. Harmuth, H.F., (1986d). Propagation velocity of electromagnetic waves. IEEE Trans. Electromagn. Compat., vol. EMC-28, 267-271. Harmuth, H.F. (1989). Information Theory Applied to Space-Time Physics (in Russian). MIR, Moscow. English edition by World Scientific Publishing Co., Singapore 1992. Harmuth, H.F., Barrett, T.W., and Meffert, B. (2001). Modified Maxwell Equations in Quantum E1ectrodynamics. World Scientific Publishing Co., Singapore. Copies distributed to university libraries in Eastern Europe, Asia, North Africa, and Latin Amrica contain one extra page. Harmuth, H.F. and Hussain, M.G.M. (1994). Propagation of Electromagnetic Signals. World Scientific Publishing Co., Singapore. Harmuth, H.F. and Meffert, B. (2003). Calculus of Finite Differences in Quantum E1ectrodynamics. Elsevier/Academic Press (Advances in Imaging and Electron Physics, vol. 129), London. Hasebe, K. (1972). Quantum space-time. Prog. Theor. Phys., vol. 48, 1742-1750. Hellund, E.J. and Tanaka, K. (1954). Quantized space-time. Phys. Rev., vol. 94, 192-195. Hill, E.L. (1955). Relativistic theory of discrete momentum space and discrete space-time. Phys. Rev., vol. 100, 1780-1783. Hillion, P. (1991). Remarks on Harmuth's 'Correction of Maxwell's equations for signals I'. IEEE Trans. E1ectromagn. Compat., vol. EMC-33, 144. Hillion, P. (1992a). Response to 'The magnetic conductivity and wave propagation'. IEEE Trans. Electromagn. Compat., vol. EMC-34, 376-377.
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Hillion, P. (1992b). A further remark on Harmuth's problem. IEEE Trans. Electromagn. Compat., vol. EMC-34, 377-378. Hillion, P. (1993). Some comments on electromagnetic signals; in Essays on the Formal Aspects of Electromagnetic Theory. A. Lakhtakia ed., 127-137. World Scientific Publishing Co., Singapore. HSlder, O. (1887). Uber die Eigenschaft der Gammafunktion keiner algebraischen Differentialgleichung zu genfigen. Math. AnnaJen, vol. 28, 1-13. Hussain, M.G.M. (2005). Mathematical model for the electromagnetic conductivity of lossy materials. J. of Electromagn. Waves and Appl., vol. 19, no. 2, 271-279; VSP Publishers, Zeist, Netherlands. Icke, V. (1995). The force of Symmetry. Cambrdidge University Press, Cambridge. Jackson, J.D. (1999). Classical E1ectrodynamics, 3rd ed.. J.Wiley, New York. King, R.W. (1993). The propagation of a Gaussian pulse in seawater and its application to remote sensing. IEEE Trans. Geoscience, Remote Sensing, vol. 31,595-605. King, R.W. and Harrison, C.W. (1968). The transmission of electromagnetic waves and pulses into the Earth. J. Appl. Physics, vol. 39, 4444-4452. Landau, L.D. and Lifschitz, E.M. (1966). Lehrbuch der Theoretischen Physik (German edition of Russian original by G. Heber). Akademie-Verlag, Berlin. Leader, E. and Predazzi, E. (1996). An Introduction to Gauge Theories and Modern Particle Physics. Cambridge University Press, Cambridge. Lehner, G. (1990). Elektromagnetische Feldtheorie. Springer Verlag, Berlin. Lehnert, B. (1995). Total reflection process including waves of an extended electromagnetic theory. Optik, vol. 99, No. 1, 113-119. Lehnert, B. (1996). Minimum electric charge of an extended electromagnetic field theory. Physica Scripta, vol. 53, 204-211. Lehnert, B. and Roy, S. (1998). Extended Electromagnetic Theory. World Scientific Publishing Co., Singapore. Levy, H. and Lessman, F. (1961). Finite Difference Equations. Macmillan, New York; reprinted Dover, New York. Marsal, F. (1989). Finite Differenzen und Elemente: numerische Lbsung von Variationsproblemen und Partiellen Differentialgleichungen. Springer-Verlag, Berlin. Messiah, A. (1959). Mdcanique Quntique. Dunod, Paris. English transl by G.M. Temmer, Quantum Mechanics. North Holland Publishing Co., Amsterdam 1962. Reprinted Dover, New York. Milne-Thomson, L.M. (1951). The Calculus of Finite Differences. MacMillan, London. Moon, P. and Spencer, D.E. (1988). Field Theory Handbook, 2nd ed., Springer-Verlag, New York. Morgenthau, H. (1945). Germany Is Our Problem. Harper & Brothers, New York and London. See also Belfrage, C., Blum, J.M., Greiner, B. Free download from www.ety.com/berlin/morgenthau.htm . Morse, P.M and Feshbach, H. (1953). Methods of Theoretical Physics. McGraw-Hill, New York. Newton, I. (1971). Mathematical Principles. A. Motte transl., F. Cajori ed. University of California Press, Berkeley. New York Times (1945a). Morgenthau Gives Views on Germany. Oct. 5, p. 7. New York Times (1945b). Morgenthau Volume Sent to U.S. Staffs. Dec. 30, p. 10. New York Times (1945c). Nazi Reform Slow, Morgenthau Says, Nov. 12, p. 3. New York Times (1945d). Engineers Offer Plan for Germany. Sep. 27, p. 14. Nielsen, N. (1906). Handbuch der Theorie der Gammafunktion. Teubner, Leipzig. NSrlund, N.E. (1910). Fractions continues et differences r~ciproque~. Acta Math., 34, 1-108. NSrlund, N.E. (1914). Sur les s~ries de facultSs. Acta Math., 37, 327-387. N6rlund, N.E. (1915). Sur les ~quations lin~aires aux differences finis ~ coefficients rationels. Acta Math., 40, 191-249. N6rlund, N.E. (1924). Vorlesungen fiber Differenzenrechnung. Springer-Verlag, Berlin; reprinted Chelsea Publishing Co., New York 1954. NSrlund, N.E. (1929). Leqons sur les Equations Lin~aires aux Differences Finies, ed. R. Lagrange. Gauthier-Villars, Paris.
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Index
differenciation 18 Dirac equations 197 direction of time 29, 30 discrete coordinate systems 36-43 discrete spherical harmonics 210 dogma of the circle 27 dyadic 21
A Abramovitz 214 Almagest 27 Anastasovski 1 antiparticles 257 Apostle 26 Aristotle 26, 28 Astronomia Nova 27 Ae "--)0, g m --* 0 197
E
egg of Columbus 243 Einstein 28, 30 Eleatic school 26 electric dipole current 2 electric monopole current 2 electromagnetic pulse 2 elementary particle 143 energy-impulse tensor 78, 222 equilateral triangle 31 Euler Script, Euler Fractur 50 Euclid 31 excitation functions 69 exponential ramp function 62
B
Barrett 1, 3, 4, 5, 7, 18, 28, 47, 74, 189, 197 Becker 83 Berestezki 78 Bernoulli product method 104, 209 binary digits 29 bit 29 boundary condition 65 bounded coordinate system 38 box normalization 74 branch of mathematics 143 byte 29
F factorial series 216 Feshbach 19, 286 finite information 29 four-vector 190 Frobenius 213 Fuchs 213, 228
C causal function 1 causal solution 210 causality law 2, 3, 28, 30 character group 27 charge renormalization 29 clocks 28 closed coordinate system 38 Columbus, egg of 243 commuting 8 Compton effect 172 -, random numbers 182 Compton wavelength 239, 297 computer evaluation 124, 171 conformal mapping 226 conservation law of energy 1, 3 contour integral 212 convergence at r = 0 243 Copernicus 27 current densities 4
G Gaussian curvature 34 Gaussian pulse 1 Guldberg 212, 216 gm ---* 0, Ae --~ 0 197 H
Habermann 63 Hamilton 19 Hamilton function 4, 6, 44, 58, 78, 82, 186, 191, 193, 195 Harmuth 1, 3, 4, 5, 7, 10, 18, 28, 47, 81, 84, 86, 189, 197, 262 Harrison 1 HauptlSsung 13 Heisenberg approach 83 Heitler 83 Hermite polynomials 86
D difference equation, one dimension 62 difference operators p, ~-C, ~ , grad, div 53, 204 318
INDEX hiding dimensions 35, 36 Hillion 1 Hussain 3 I
Infeld 28 infinite energies 29 infinite information 29 reformation 29, 265 information theory 29, 30 information transmission 2 reformation, unlimited 266 initial condition 62 interference phenomena 96 invisible dimension 35, 36 iterated Dirac equations 245 K Kepler 27, 35 King 1 Klein-Gordon matrix 49, 54 L lack of development 11 Lagrange 111 Lagrange function 4, 5, 19, 22, 23, 24, 25, 183, 187, 190 Laplace transform 212 left difference quotient 9 Lehnert 1 Leibniz 26 Lifschitz 78 Lorentz convention 3 Lorentz equation 4, 189 M magnetic charge 2, 19 magnetic dipole current 2 mathematical axiom 30 Maupertuis 19 Meffert 3, 4, 5, 7, 18, 28, 47, 74, 81, 84, 86, 189, 197 Messiah 211 Milne-Thomson 12, 13, 16, 18, 111,212, 216, 229, 262, 268 Morse 19, 286 multipole current 2 mutually perpendicular vectors 34
319 Pitajeski 78 Plato 27 polynomial, asymptotic 218 principal solution 13 Ptolemy 27 pure mathematics 26, 266 pure number 29
Q quantum gravitation 29 R
random numbers, Compton effect 182 relativistic canonical momentums 190 relativistically variable mass 189 renormalization 28 right difference quotient 8 Roy 1 S scalar potentials 3 Schiff 211 SchrSdinger 30 SchrSdinger approach 83 self energy 29 separation of variables 65 signal solutions 1 singular points 214,217, 227, 245, 247, 257 Smirnov 165 Spiegel 262 spinning bullet 25 standard change of time 28 steady state theory 28 Stegun 241 Stratton 1 summable power series 219 summation sign 13 symmetric difference quotient 8, 10 T table of sums 17 three components Hamilton function 44 topologic group 27 transient theory 28 transition to random numbers 239 U unlimited information 266 unobservable dimensions 35
N Newton 26 non-commutability 196 NSrlund 12, 13, 16, 18, 111,212, 216, 219, 262, 268
V van der Waerden 27 vector potentials 3 vector potential Ae for Ax ~ Ac 148
O one-dimensional coordinate system 40-42
W Wallenberg 212, 216 Weyl 28
P
Z
physical law 30
Zeno of Elea 26
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ERRATA
Advances in Imaging and Electron Physics Volume 137, Chapter 4, page 170. Equations (44) and (45) are missing on page 170. These equations are as follows.
pc = (1/47r2)(Ac/cT) 2, AI = eCTAmo~/hN,
Ae = 2AcAe(~)/e
G2(~,0,~) = -pcN2fi.~(~){(1 + 2iA1)~O,2,0(~,Or,) - [ 4 - 6 A ~ - 2i(2A1 + A~)lg'o,-1,o(q,0n)+ (1 - 2iAa)~I'o,-z,o(r G2-((, 0)= -pcN2Ae(r
(44)
+ 2iA1)~I'o,~,-1(~, 0n)
- 4[-6A 2 + 2i(2A1 + h~)]~o,1,-l(C,0n) + (6 - 12A 2 - A~)~o,o,-1(r 0n) - [4 - 6 ~ - 2 ~ ( 2 ~ + ~ ) ] ~ o , - ~ , - ~ ( r
+ (1 - ~ ) ~ o , - 2 , - ~ ( r
(45)
On page xxi, "Henning F. Harmuth" should read "Henning F. Harmuth and Beate Meffert."
ADVANCES IN IMAGING AND ELECTRON PHYSICS, VOLUME 137 ISBN: 0-12-014779-3
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