Groifi* • ••««•• a nd pufiiii i*«i M il l i « a « «l i i n i . i Mi T h is H i / w l i mh It • - I h -.I it t ip i relevant In |iliv»»i» • • • I !•> . n M I a I1 (n IS) mi.I Hill li n|i | >: I I l li I I The lit si (|v< rluilil' I •» (• • •••« • hi' II M systemal l( MIIMMMIMI II and solids T he formal • < IIMlMpH Mill I Mil | I I I HUM>I \ • oped in ( liaph i I Mllllll I Mini lllllHll ill li l || M : crystallography • M l| Ml I III ill Ih| I • M il I ory. Chapter f> I? i ll V i ll h M li MM symmetry, Chaplm IIMIIII M y i i li ( in i i l l . on ture, and Chapter / * l l l l ll 1111M111 M i ll t ii | which deals with solid nl.ih pli\ • I• in a hIM I t h e o ry and magnetic < i V M M M\ niiin I I A c o m p a ct and worthwhile • < MI NO . .11. standard m e t h o d s, this •I H II III VOIIIIIM I»M IIIIH q u a n t um theory. Dover (2003) unabridged icpnmn muni «• i in* • . Ml llllH||ll t he McGraw-Hill Book Company, Nrw mil ill* 1 lll|i|l 1 ures. 33 tables. Appendix. Index. S'/m h M / r A 6 ^ 4, ~ 4 ALSO AVAILAIil I Paul M , LECTURES ON QUANTUM MECHANICS A, M 41713-1 THE THEORY OF GROUPS AND QUAN TUM MECHANICS, Wi'vl 3 5 /s x 872. 60269-9 Albert Messiah. 1,152pp. !">'/« \ '/j I QUANTUM MECHANICS, M HMi I For current price information write to Dover P • l| IllM Mil I" www.doverpublications.com—and see every Dovei Free Dover Mathematics and Science Catalog (f»!MX. M) ,» Ml MM upon request. ISBN II • i|l< i HI) I < LU - IN U SA $ 3 7 . SD IN C A N A DA I W "T W W f ' • " If • 53A \ k f5 ~ T \ \J GROUP THEORY AND QUANTUM MECHANICS Michael Tinkham Rum ford Professor of Physics and Cordon McKay Professor of Applied Physics Harvard University © ^ Si 2 30 DOVER PUBLICATIONS, INC. Mineola, New York Copyright Copyright © 1964 by McGraw-Hill, Inc. Copyright © renewed 1992 by Michael Tinkham All rights reserved. Bibliographical Note This Dover edition, first published in 2003, is an unabridged republication of the work originally published by McGraw-Hill Book Company, New York, in 1964. Library of Congress Cataloging-in-Publication Data Tinkham, Michael Group theory and quantum mechanics / Michael Tinkham. p. cm. Originally published: New York : McGraw-Hill, cl964, in series: International series in pure and applied physics. Includes bibliographical references and index. ISBN 0-486-43247-5 (pbk.) 1. Quantum theory. 2. Group theory. I. Title. QC174.17.G7T54 2003 530.12—dc22 2003061780 Manufactured in the United States of America Dover Publications, Inc., 31 East 2nd Street, Mineola, N.Y. 11501 PREFACE This book has developed from a set of mimeographed notes, written to accompany a series of 44 lectures which comprise a semester course on group theory and the quantum mechanics of atoms, molecules, and solids. The course has been given by the author to graduate students at the University of California since 1957. Since this course is followed by another dealing entirely with solid-state theory, the applications to solid- state problems in the book are largely limited to areas in which the apparatus of group theory is particularly essential. Because of the resulting emphasis on atomic and molecular problems, many chemists take the course, typically about a third of the class. Since students are assumed to have had a year of quantum theory as a prerequisite to the course, such standard elementary examples as the hydrogen atom and the harmonic oscillator have been dispensed with in the interest of brevity. However, a real attempt has been made throughout to keep the treatment as simple as possible and under- standable to students with a variety of backgrounds. In particular, the group theory is developed on the assumption that the student has no pre- vious knowledge of it, and elegant formal elaborations have been avoided in favor of presenting only a useful core of ideas as simply as possible. The treatment throughout is definitely intended to be useful to experimentalists who desire a better understanding of the current theoretical picture of the structure of atoms, molecules, and solids. The book is intended for use as an introductory text rather than as a definitive treatise. Relatively little completely original work is presented. Rather, an attempt has been made to collect in a compact and unified form some of the scattered material on standard methods which is needed for an understanding of current research work. vi PREFACE The book divides logically into two parts. The first five chapters are primarily devoted to the introduction of methods, illustrated by physical examples, whereas the last three chapters present a systematic treatment of the quantum theory of atoms, molecules, and solids, additional methods being introduced as needed to handle specific problems as they arise. After a brief introductory chapter, the formal theory of finite groups and their representation is developed in Chapters 2 and 3 in a manner which draws heavily on Wigner's classic "Group Theory." Chapter 4 is devoted to the illustration of the group theory. This is done by developing the crystallographic point groups which are basic for both solid-state and molecular theory. The classic work of Bethe on the splitting of atomic terms in crystalline fields of the various symmetries is then presented, in- cluding the introduction of the crystal double groups. This beautiful and illuminating work has not been readily accessible before, though basic for the theory of magnetic materials. As a final illustration of the point groups, we consider the directed-valence theory of hybrid orbitals. Chapter 5 is devoted to the theory of systems with full rotational sym- metry. First we develop the relation between angular momentum and trans- formation properties under rotation. Then, closely following Wigner, the homomorphism between the full rotation group and the unimodular unitary group is used to develop ail the representations of the rotation group. From these, the Wigner or Clebsch-Gordan coefficients are derived in general form and applied through the Wigner-Eckart theorem to matrix elements of tensor operators. Finally, the Racah coefficients are introduced to handle more general problems arising in the application of tensor operators. This latter treatment is based on the monograph of Rose, "Elementary Theory of Angular Momentum." The chapter concludes with a discussion of space- and time-inversion symmetries and their consequences in parity conservation and Kramers' degeneracy. The systematic presentation of atomic structure in Chapter 6 draws heavily on the monumental treatise by Condon and Shortley, "The Theory of Atomic Structure." In particular, the determinantal method developed by Slater is used for calculation of L-S-term energies and multiplet structure. However, the group-theoretical viewpoint is emphasized where it provides helpful insight, rather than being minimized. The Hartree-Fock method is treated, including Slater's modified form, and some reference is made to recent work on the unrestricted Hartree-Fock method. The rigorous group- theoretical derivation of the Lande g-factor formula is given, as is a dis- cussion of the Paschen-Back transition in the Zeeman effect. The discussion of magnetic hyperfine structure follows that of Ramsey ("Nuclear Moments") but again makes more use of group-theoretical arguments. The electric hyperfine structure is treated as an application of the Racah coefficients, although the connection with the usual elementary approach is indicated. PREFACE vii Chapter 7 on molecular structure makes considerable use of material from "Quantum Chemistry" by Eyring, Walter, and Kimball. It opens with a discussion of the Born-Oppenheimer approximation. The rest of the chapter provides a systematic following through of the approximation. First the electronic energy levels for the problem of "clamped nuclei" are attacked, the method of molecular orbitals being emphasized. The group theory is employed in classifying and constructing these. With the elec- tronic energy used as an effective potential, vibrational motion is discussed, together with the symmetry properties of combination and overtone levels in polyatomic molecules. The rotational energy of a rigid molecule is then considered, and a discussion of centrifugal distortion effects is given. At the end of the chapter a variety of cross terms which involve a higher-order Born-Oppenheimer approximation are indicated. Chapter 8, on solid-state physics, treats two important topics of solid- state theory in which group-theoretical symmetry considerations are par- ticularly useful, namely, electronic energy band theory and magnetic crystal symmetry. The general periodic-potential problem is set up, and some standard approximate approaches are discussed briefly, though the problems of actual quantitative calculations are not dealt with. The principal em- phasis is on the group-theoretical discussion of the symmetry classification of wave functions in crystals, as initiated by Bouckaert, Smoluchowski, and Wigner and extended by many others. The introduction of spin-orbit coupling and the implications of time-reversal symmetry in band theory are indicated. This leads to a consideration of magnetic materials, in which time-reversal symmetry in the usual sense is not effective. The Shubnikov groups are developed, following Tavger and Zaitsev, and illustrated with examples. Finally, the Landau theory of second-order phase transitions is presented and applied to treat the change in symmetry of a magnetic material upon ordering. As this summary indicates, a wide range of material, which introduces many techniques, is covered. An emphasis on the application of group theory to give insight into physical problems serves as a unifying principle. Of necessity, the treatment is rather brief, since with some few omissions the material is intended for a one-semester course. Accordingly, little material has been included merely for completeness. Rather, an attempt has been made to pick out some of the main features of the picture, things which should be part of the working understanding of nature for physicists and chemists. It is hoped that this approach will help the reader to acquire the new concepts and advanced methods and give him the sort of perspective which seems to be so easy to miss. He should then be able to continue with confidence to further independent study of particular topics required in his research work. For the benefit of those who are familiar with the mimeographed version viii PREFACE of this book, which has been widely circulated since 1957, we might note that the text has been completely rewritten in an effort to improve its clarity and completeness. New material added includes the entire final chapter on band theory and magnetic-crystal symmetry groups, a major expansion of the section on time-reversal symmetry, a major expansion of the treatment of molecular vibrations, a short discussion of the 3-j coefficients, a treatment of the Zeeman effect in atomic spectra, and problems at the end of each chapter. The new material, together with the general expansion of the old upon rewriting, accounts for the increase by about one-half in the length of the present volume compared with the notes. The author wishes to acknowledge the contributions of Professors C. Kittel and R. Karplus, the notes from whose previous lecture courses pro- vided the starting point for this book. Thanks are also due to many persons, particularly Professors L. I. Schiff, J. H. Van Vleck, and L. C. Allen, for their encouragement when the work was progressing slowly. The support of a John Simon Guggenheim Memorial Foundation grant and the hospitality of the Massachusetts Institute of Technology provided the author the oppor- tunity to go over the proofs of the book with the care necessary to eliminate errors. Finally, it is a pleasure to thank Dr. V. Heine for his very thorough and helpful scrutiny of the entire manuscript, which has eliminated many obscurities and some errors. The author is grateful for this help, and that of other reviewers of the manuscript, in improving the presentation, but he, of course, accepts all responsibility for the final result. M. Tinkham CONTENTS Preface V 1 Introduction I 1-1 The Nature of the Problem 1 1-2 The Role of Symmetry 3 2 Abstract Group Theory 6 2-1 Definitions and Nomenclature 6 2-2 Illustrative Examples 7 2-3 Rearrangement Theorem 8 2-4 Cyclic Groups 9 2-5 Subgroups and Cosets 9 2-6 Example Groups of Finite Order 10 2-7 Conjugate Elements and Class Structure 12 2-8 Normal Divisors and Factor Groups 13 2-9 Class Multiplication 15 Exercises 16 References 17 3 Theory of Group Representations 18 3-1 Definitions 18 3-2 Proof of the Orthogonality Theorem 20 3-3 The Character of a Representation 25 X CONTENTS 3-4 Construction of Character Tables 28 3-5 Decomposition of Reducible Representations 29 3-6 Application of Representation Theory in Quantum Mechanics 31 3-7 Illustrative Representations of Abelian Groups 37 3-8 Basis Functions for Irreducible Representations 39 3-9 Direct-product Groups 43 3-10 Direct-product Representations within a Group 46 Exercises 47 References 48 4 Physical Applications of Group Theory SO 4-1 Crystal-symmetry Operators 51 4-2 The Crystallographic Point Groups 54 4-3 Irreducible Representations of the Point Groups 62 4-4 Elementary Representations of the Three-dimensional Rotation Group 4-5 Crystal-field Splitting of Atomic Energy Levels 67 4-6 Intermediate Crystal-field-splitting Case 69 4-7 Weak-crystal-field Case and Crystal Double Groups 75 4-8 Introduction of Spin Effects in the Medium-field Case 78 4-9 Group-theoretical Matrix-element Theorems 80 4-10 Selection Rules and Parity 82 4-11 Directed Valence 87 4-12 Application of Group Theory to Directed Valence 89 Exercises 92 References 93 5 Full Rotation Group and Angular Momentum 94 5-1 Rotational Transformation Properties and Angular Momentum 94 5-2 Continuous Groups 98 5-3 Representation of Rotations through Eulerian Angles 101 5-4 Homomorphism with the Unitary Group 103 5-5 Representations of the Unitary Group 106 5-6 Representation of the Rotation Group by Representations of the Unitary Group 109 5-7 Application of the Rotation-representation Matrices 111 5-8 Vector Model for Addition of Angular Momenta 115 5-9 The Wigner or Clebsch-Gordan Coefficients 117 5-10 Notation, Tabulations, and Symmetry Properties of the Wigner Coefficients 121 5-11 Tensor Operators 124 CONTENTS 5-12 The Wigner-Eckart Theorem 131 5-13 The Racah Coefficients 133 5-14 Application of Racah Coefficients 137 5-15 The Rotation-Inversion Group 139 5-16 Time-reversal Symmetry 141 5-17 More General Invariances 147 Exercises 151 References 153 6 Quantum Mechanics of Atoms 154 6-1 Review of Elementary Atomic Structure and Nomenclature 155 6-2 The Hamiltonian 157 6-3 Approximate Eigenfunctions 157 6-4 Calculation of Matrix Elements between Determinantal Wavefunctions 162 6-5 Hartree-Fock Method 167 6-6 Calculation of L-S-term Energies 170 6-7 Evaluation of Matrix Elements of the Energy 173 6-8 Eigenfunctions and Angular-momentum Operations 178 6-9 Calculation of Fine Structure 181 6-10 Zeeman Effect 188 6-11 Magnetic Hyperfine Structure 193 6-12 Electric Hyperfine Structure 201 Exercises 206 References 208 7 Molecular Quantum Mechanics 210 7-1 Born-Oppenheimer Approximation 210 7-2 Simple Electronic Eigenfunctions 213 7-3 Irreducible Representations for Linear Molecules 216 7-4 The Hydrogen Molecule 219 7-5 Molecular Orbitals 220 7-6 Heitler-London Method 223 7-7 Orthogonal Atomic Orbitals 226 7-8 Group Theory and Molecular Orbitals 227 7-9 Selection Rules for Electronic Transitions 233 7-10 Vibration of Diatomic Molecules 234 7-11 Normal Modes in Polyatomic Molecules 238 7-12 Group Theory and Normal Modes 242 7-13 Selection Rules for Vibrational Transitions 248