Quantum Theory Of Angular Momemtum

528 Pages·1988·131.08 MB·English

by  D.A. Varshalovich

Quantum Theory
of
Angular Momentum
This page is intentionally left blank
Quantum Theory
of
Angular Momentum
Irreducible Tensors, Spherical Harmonics,
Vector Coupling Coefficients, 3nj Symbols
D. A. Varshalovich
A. F. loffe Physical-Te chnica! Institute
A. N. Moskalev
B. P. Konstantinov Institute of Nuclear Physics
V. K. Khersonskii
Special Astrophysical Observatory
\\h World Scientific
• Singapore • New Jersey • Hong Kong
Published by
World Scientific Publishing Co. Pte. Ltd.,
P 0 Box 128, Farrer Road, Singapore 9128
USA office: 687 Hartwell Street, Teaneck, NJ 07666
UK office: 73 Lynton Mead, Totteridge, London N20 8DH
Library of Congress Cataloging-in-Publication Data
Varshalovich, D. A. (Dimitrll Aleksandrovich)
Quantum theory of angular momentum.
,...
Translation of: Kvantovaia teoriia uglovogo
momenta.
I. Angular momentum (Nuclear physics) 2. Quantum
theory. I. Moskalev, A. N. II. Khersonskii, V. K.
(Valerll Kel'manovich) Ill. Title.
QC793.3.A5V3713 1988 530.1'2 86-9279
ISBN 9971-50-107-4
9971-50-996-2 pbk
First published 1988
First reprint 1989
QUANTUM THEORY OF ANGULAR MOMENTUM
Copyright ©1988 by World Scientific Publishing Co Pte Ltd.
All rights reserved. This book, or parts thereof, may not be reproduced
in any form or by any means, electronic or mechan~al, including photo
copying, recording or any information storage and retrieval system now
known or to be invented, without written permission from the Publisher.
Printed in Singapore by Chong Moh Offset Printing Pte Ltd.
PREFACE
This book deals with one of the basic topics of quantum mechanics: the theory of angular momentum and
irreducible tensors. Being rather versatile, the mathematical apparatus of this theory is widely used in atomic
and molecular physics, in nuclear physics and elementary particle theory. It enables one to calculate atomic,
molecular and nuclear structures, energies of ground and excited states, fine and hyperfine splittings, etc. The
apparatus is also very handy for evaluating the probabilities of radiative transitions, cross sections of various
processes such as elastic and nonelastic scattering, different decays and reactions (both chemical and nuclear)
and for studying angular distributions and polarisations of particles.
Today this apparatus is finding ever increasing use in solving practical problems relating to quantum
chemistry, kinetics, plasma physics, quantum optics, radiophysics and astrophysics.
The basic ideas of the theory of angular momentum were first put forward by M. Born, P. Dirac, W. Heisen
berg and W. Pauli. However, the modern version of its mathematical apparatus was developed mainly in the
works of E. Wigner, J. Racah, L. Biedenharn and others who applied group theoretical methods to problems
in quantum mechanics.
At present a number of good books on the theory of angular momentum have been already published.
The general principles and results of the theory may be found in the books by M. Rose 131], A. Edmonds
ll8J,
[16], U. Fano and G. Racah A. P. Yutsis, I. B. Levinson and V. V. Vanagas 144], A. P. Yutsis and
A. A. Bandzaitis [45J, D. Brink and G. Satcher [9]. Nevertheless, many formulas and relationships essential
for practical calculations have escaped these books and are either scattered in various editions, or included as
appendices in papers discussing somewhat disparate topics, making them generally inaccessible. Even greater
difficulties arise when one tries to use the results, as each author employs his own phase conventions, initial
definitions and symbols.
The authors of this book aimed at collecting and compiling ample material on the quantum theory of
angular momentum within the framework of a single system of phases and definitions. This is why, in addition
to the basic theoretical results, the book also includes a great number of formulas and relationships essential
for practical applications.
This edition is the translated version of our book published in the USSR in 1975. In the course of its
preparation we have tried to comply with a number of suggestions from our readers. For instance, each
chapter opens with a comprehensive listing of its contents to ease the search for information needed. We
also included some new results relating to different aspects of angular momentum theory which have recently
appeared in journals. Unfortunately the limited volume of the present book prevented us from covering all the
aforementioned results. We offer sincere apologies to the authors whose results we failed to include.
The monograph is a kind of handbook. Consequently the material is presented in concise form. Most of
the formulas and relationships are given without proof. Their full derivation may be found in the literature
v
vi Preface
listed at the end of the text. Some results which have become generally known are given without references,
and for this we also apologize.
The sequence adopted is as follows: chapter, section and. subsection. Many chapters are self-contained and
can be read independently of the others. Sections have double numbering: the first figure denotes the number
of the chapter, the second, the number of the section. Equations are numbered within the confines of the
section they are included in. When referring to an equation from the same section only the number of the
equation is given, e.g., (3), (27); when reference is made to an equation from another section the numbers of
the chapter, section and equation~ given, e.g., Eq. 4.2.(17). A similar system is adopted when referring to
individual subsections, e.g., Sec. 1.2.5. For convenience the book also contains a glossary of all symbols used
in the text with references to the pages where their coITesponding definitions are given. The list of 11eferences
is divided into parts: the first part lists books and reviews; the second, papers on different subjects; the third,
tables; the fourth, references added during translation.
The authors hope that many specialists will find in the book some fresh and interesting information. The
material is prepared and aITanged so as to make it useful to those less familiar with theory and for students of
physics. These readers can effectively use the monograph as a supplementary text to their main coUJ!Ses.
For those who wish to thoroughly familiarize themselves with the fundamentals of angular m~mentum
theory we recommend the excellent new book by L. Biedenharn and J. Louck [132] Angular Mom~ntum in
Quantum Ph.g1ic1. Th.eorg and Applkationa.
The authors wish to express their deep appreciation to D. G.Yakovlev who took the trouble ofreading the
English translation of the book and gave some valuable, suggestions on its preparation.
Leningrad D. A. Varshalovich
A. N. Moskalev
V. K. Khersonskii
CONTENTS
Preface ..••.••••..•..•••••••••.••••.••••• v
Introduction: Basic Concepts ..................... 1
Chapter 1. Elements of Vector and Tensor Theory 3
1.1. Coordinate Systems. Basis Vectors 3
1.2. Vectors. Tensors . . . . . . . . 11
1.3. Differential Operations . . . . . 17
1.4. Rotations of Coordinate System . 21
Chapter 2. Angular Momentum Operators . . . 36
2.1. Total Angular Momentum Operator . 36
2.2. Orbital Angular Momentum Operator . 39
2.3. Spin Angular Momentum Operator . 42
2.4. Polarization Operators . . . . . . . . 44
= ! . . . . . . .
2.5. Spin Matrices for S 47
2.6. Spin Matrices and Polarization Operators for S = 1 . 51
Chapter 8. Irreducible Tensors . . . . . . . . . . . . . . . . 61
3.1. Definition and Properties of Irreducible Tensors . . . 61
3.2. Relation Between the Irreducible Tensor Algebra and Vector and Tensor Theory . 65
3.3. Recoupling in Irreducible Tensor Products . 69
Chapter 4. Wigner D-Functions . . . . . . . . . . . 72
4.1. Definition of DLu1(et,/3,'Y) . . . . . . 72
4.2. Differential Equations for DLu•(a,/3,1) . 74
4.3. Explicit Forms of the Wigner D-Functions . 76
4.4. Symmetries of d'ku1(/3) and DLu•(a,/3 1) . 79
1
4.5. Rotation Matrix Uf.tu• in Terms of Angles w, e, • . 80
4.6. Sums Involving D-Functions . . 84
4. 7. Addition of Rotations . . . . 87
4.8. Recursion Relations for Dku• . 90
vii
viii Contents
4.9. Differential Relations for Dftu• (a,P,'l') . . . . . . 94
4.10. Orthogonality and Completeness of the D-Fnnctions . 94
4.11. Integrals Involving the D-Functiou . . . . . . . . . 96
4.12. Invariant Summation of Integrals Involving DLu1(a,P,'l') . 97
4.13. Generating Fnnctiou for dLu•(P) . . . . . . . . . . . 98
4.14. Characters x1 (R) of Irreducible Representations of Rotation Group . 99
xf
4.15. Generalised Characters, (R), of Irreducible Representations of the Rotation Group 106
4.16. DL.u•(a,P,1) for Particular Valu• of tlle Arguments 112
4.17. Special Cases of DL.u1(a,P,1) for Particular Mor M' 113
4.18. Asymptotics of Df.t.u,(a,P,1) • . . • • . • . . . 115
4.19. Definitiou of DLu•(a,P,1) by Other Authors 117
4.20. Special Cases of dLu1(P) for Particular J,M and M' 117
4.21. Tables of df.t .Al' (P) for P = 'Ir /2 • 117
4.22. Special Cases of Uk.u•(w;e,~) 117
Chapter &. Spherical Barmonlc:a . • • . • . . • 130
5.1. Definition . . . . . . . . . . . 130
5.2. Explicit Fol'llll of the Spherical H~oI nica and Their Relatiom tO Other Functions 133
5.3. Integral Repraentatiom of the Spherical Harmonics [4, 22, 27) • • . • . • • ·• . 139
5.4. Symmetry Properties • • • • • • • • • • • • • • • • • • • • • 140
5.5. Behaviour of Yim(f,p) under Tranaformatiom of Coordinate Systems •..... 141
5.6. Expamions in Series of the Spherical Harmonica 143
5. 7. Recursion Relatiom • • • • • • • • • • • • • • • • 145
5.8. Differential Relations • • • . • , • . • • . • 146
5.9. Some Integrals Involving Spherical Harmonica 148
5.10. Suma Involving Spherical Harmonica • , • • • • 150
5.11. Generating Functiom for Yim(I, p) . 151
5.12. Asymptotic Expressions for Yim(f,p) ......••••.......... 152
5.13. Yim(f, p) for Special Valu• of l and m • • • • • • . . . • • . . . . . . . . 155
5.14. Yim(f,p) and (a/al)Yim(f,p) for Special I 158
5.15. Zeros of Yim(f, p) and (8/Bf)Yim(f, p) . . . . • . . . 158
5.16. Bipolar and Tripolar Spherical HU'Jllonica • • • • • • • 160
5.17. Exp&ll8iom of Functio:u Which Depend on Two Vectors 163
Chapter 8. Spin Punc:tlom . . . . . . . . . . . . . . . . . 170
6.1. Spin Functiou of Particles with Arbitrary Spin 170
= i . . . .
6.2. Spin Functions for S 178
=
6.3. Spin Functiom for S 1 185
Chapter f. Tenaor Spherical Harmonic• . • . . 196
7.1. General Properties of Tensor Spherical Harmonics 196
7.2. Spinor Spherical Harmonics • • • . . • • . . 202
7.3. Vector Spherical Harmonica ••...•... 208
7 .•. Other Notatiom for Tensor Spherical Harmonics 234
Contents ix
Chapter 8. Clebsch-Gordan Coefllclents and 3jm Symbols 235
8.1. Definition . . . . . . . . . . . . . . . . . . . • • 235
8.2. Explicit Forms of the Clebach-Gordan Coefficients and
Their Relations to Other Functions 237
8.3. Integral Representations 243
8.4. Symmetry Properties . . . . . . . . . . . . . . 244
8.5. Explicit Forms of the Clebach-Gordan Coefficients for
Special Values of the Arguments . . . . . . . . . 248
8.6. Recunion Relations for the Clebsch-Gordan Coefficients 252
8.7. Sums of Products of the Clebsch-Gordan Coefficients . 259
8.8. Generating Functions . . . . . . . . . . . . . . . 263
8.9. Classical Limit and Asymptotic Expressions for the Clebach-Gordan Coefficients 264
8.10. Zeros of the Vector-Addition Coefficients ...............•.. 268
8.11. Connection of the Clebach-Gordan Coefficients and
the 3jm Symbols with Analogous Functions of Other Authors 268
8.12. Algebraic Tables of the Clebsch-Gordan Coefficients 270
8.13. Numerical Tables of the Clebsch-Gordan Coefficients 270
Chapter 9. 6j Symbols and the Racah Coefficients . 290
9.1. Definition . . . . . . . . . . . . . 290
9.2. General Expressions for the 6j Symbols.
Relations Between the 6j Symbols and Other Functions 29S
9.3. Integral Representations of the 6j Symbols . . . . . 297
9.4. Symmetries of the 6j Symbols and the Racah Coefficients 298
9.5. Explicit Forms of the 6:j Symbols for Certain Arguments 299
sos
9.6. Recunion Relations
9.7. Generating Function .... S05
9.8. Sums Involving the 6j Symbols 305
9.9. Asymptotics of the 6j Symbols for Large Angular Momenta 306
9.10. Relations Between the Wigner 6j Symbols and Analogous Functions of
Other Authors . . . . . . . . . . . . . . . . 310
9.11. Tables of Algebraic Expressions for the 6j Symbols 310
9.12. Numerical Values of the 6j Symbols ..... . S10
Chapter 10. 9j and 12:j Symbols 333
10.1. Definition of the 9j Symbols ................... . 333
10.2. Explicit Forms of the 9:j Symbols and Their Relations to Other Functions 3S6
10.3. Integral Representations of the 9j Symbols 3'0
10.4. Symmetry Properties of the 9j Symbols 3'2
10.5. Recunion Relations for the 9:j Symbols 345
10.6. Generating Function of the 9:j Symbols 351
10. 7. Asymptotic Expression for a 9j Symbol 351
10.8. Explicit Forms of the 9:j Symbols at Some Relations Between Arguments 352
10.9. Explicit Forms of the 9j Symbols for Special Values of the Arguments . 357