MODERN QUANTUM MECHANICS S E C O N D E D I T I O N J. Sakurai • Jim Napolitano www.com MODERN QUANTUM MECHANICS Second Edition www.com MODERN QUANTUM MECHANICS Second Edition ·Addison.:wesle- y Boston Columbus Indianapolis New York San Francisco Upper Saddle River Amsterdam Cape Town Dubai London Madrid Milan Munich Paris Montreal Toronto Delhi Mexico City Sao Paulo Sydney Hong Kong Seoul Singapore Taipei Tokyo www.com Publisher: Jim Smith Director of Development: Michael Gillespie Editorial Manager: Laura Kenney Senior Project Editor: Katie Conley Editorial Assistant: Dyan Menezes Managing Editor: Corinne Benson Production Project Manager: Beth Collins Production Management, Composition, and Art Creation: Techsetters, Inc. Copyeditor: Connie Day Cover Designer: Blake Kim; Seventeenth Street Studios Photo Editor: Donna Kalal Manufacturing Buyer: Jeff Sargent Senior Marketing Manager: Kerry Chapman Cover Photo Illustration: Blake Kim Credits and acknowledgments borrowed from other sources and reproduced, with permission, in this textbook appear on the appropriate page within the text. Copyright© 1994, 201 1 Pearson Education, Inc., publishing as Addison-Wesley, 1 301 Sansome Street, San Francisco, CA 941 1 1.
All rights reserved. Manufactured in the United States of America. This publication is protected by Copyright and permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. To obtain permission(s) to use material from this work, please submit a written request to Pearson Education, Inc.
For information regarding permissions, call (847) 486-2635. Many of the designations used by manufacturers and sellers to distinguish their products are claimed as trademarks. Where those designations appear in this book, and the publisher was aware of a trademark claim, the designations have been printed in initial caps or all caps. Library of Congress Cataloging-in-Publication Data Sakurai, J.
Modern quantum mechanics. Sakurai, Jim Napolitano. Quantum theory-Textbooks. 12--dc22 2010022349 ISBN 10: 0-8053-829 1-7; ISBN 1 3: 978-0-8053-8291-4 1 2 3 4 5 6 7 8 9 10-CRK-14 13 12 1 1 10 Addison-Wesley is an imprint of I PEARSON www.com Contents Foreword to the First Edition.
IX Preface to the Revised Edition. XI Preface to the Second Edition. XIII In Memoriam. XVII 1 • Fundamental Concepts 1 1.1 The Stem-Gerlach Experiment 1 1 .2 Kets, Bras, and Operators 1 0 1 .3 Base Kets and Matrix Representations 1 7 1 .4 Measurements, Observables, and the Uncertainty Relations 23 1 .5 Change of Basis 35 1 .6 Position, Momentum, and Translation 40 1 .7 Wave Functions in Position and Momentum Space 50 2 • Quantum Dynamics 66 2.
1 Time-Evolution and the Schrodinger Equation 66 2.2 The Schrodinger Versus the Heisenberg Picture 80 2.3 Simple Harmonic Oscillator 89 2.4 SchrOdinger's Wave Equation 97 2.5 Elementary Solutions to SchrOdinger's Wave Equation 103 2.6 Propagators and Feynman Path Integrals 1 16 2.7 Potentials and Gauge Transformations 1 29 3 • Theory of Angular Momentum 157 3.1 Rotations and Angular-Momentum Commutation Relations 1 57 3.2 Spin � Systems and Finite Rotations 1 63 3.3 S0(3), SU(2), and Euler Rotations 172 v www.com VI Contents 3.4 Density Operators and Pure Versus Mixed Ensembles 178 3.5 Eigenvalues and Eigenstates of Angular Momentum 1 9 1 3.6 Orbital Angular Momentum 199 3.7 Schrodinger's Equation for Central Potentials 207 3.8 Addition of Angular Momenta 217 3.9 Schwinger's Oscillator Model of Angular Momentum 232 3.10 Spin Correlation Measurements and Bell's Inequality 238 3. 1 1 Tensor Operators 246 4 • Symmetry in Quantum Mechanics 262 4. 1 Symmetries, Conservation Laws, and Degeneracies 262 4.2 Discrete Symmetries, Parity, or Space Inversion 269 4.3 Lattice Translation as a Discrete Symmetry 280 4.4 The Time-Reversal Discrete Symmetry 284 5 • Approximation Methods 303 5.1 Time-Independent Perturbation Theory: Nondegenerate Case 303 5.2 Time-Independent Perturbation Theory: The Degenerate Case 3 16 5.3 Hydrogen-Like Atoms: Fine Structure and the Zeeman Effect 321 5.5 Time-Dependent Potentials: The Interaction Picture 336 5.6 Hamiltonians with Extreme Time Dependence 345 5.7 Time-Dependent Perturbation Theory 355 5.8 Applications to Interactions with the Classical Radiation Field 365 5.9 Energy Shift and Decay Width 37 1 6 • Scattering Theory 386 6.1 Scattering as a Time-Dependent Perturbation 386 6.2 The Scattering Amplitude 391 6.3 The Born Approximation 399 6.4 Phase Shifts and Partial Waves 404 6.6 Low-Energy Scattering and Bound States 423 6.8 Symmetry Considerations in Scattering 433 6.9 Inelastic Electron-Atom Scattering 436 7 • Identical Particles 446 7.2 Symmetrization Postulate 450 www.com Contents vii 7.3 Two-Electron System 452 7.4 The Helium Atom 455 7. 6 Quantization of the Electromagnetic Field 472 8 • Relativistic Quantum Mechanics 486 8.1 Paths to Relativistic Quantum Mechanics 486 8.2 The Dirac Equation 494 8 .3 Symmetries of the Dirac Equation 501 8.4 Solving with a Central Potential 506 8.5 Relativistic Quantum Field Theory 5 1 4 A • Electromagnetic Units 519 A.
1 Coulomb's Law, Charge, and Current 5 19 A.2 Converting Between Systems 520 B • Brief Summary of Elementary Solutions to Schrodinger's Wave Equation 523 B.2 Piecewise Constant Potentials in One Dimension 524 B.3 Transmission-Reflection Problems 525 B .4 Simple Harmonic Oscillator 526 B.5 The Central Force Problem [Spherically Symmetrical Potential V = V(r)] 527 B.6 Hydrogen Atom 5 3 1 C • Proof of the Angular-Momentum Addition Rule Given by Equation (3.38) 533 Bibliography 535 Index 537 www.com Forewo rd to the Fi rst Ed ition J. Sakurai was always a very welcome guest here at CERN, for he was one of those rare theorists to whom the experimental facts are even more interesting than the theoretical game itself. Nevertheless, he delighted in theoretical physics and in its teaching, a subject on which he held strong opinions. He thought that much theoretical physics teaching was both too narrow and too remote from application: ".
we see a number of sophisticated, yet uneducated, theoreticians who are con versant in the LSZ formalism of the Heisenberg field operators, but do not know why an excited atom radiates, or are ignorant of the quantum theoretic derivation of Rayleigh's law that accounts for the blueness of the sky." And he insisted that the student must be able to use what has been taught: "The reader who has read the book but cannot do the exercises has learned nothing." Advanced Quantum Mechanics He put these principles to work in his fine book Invariance Principles and Elementary Particles ( 1 967) and in ( 1 964), both of Modern which have been very much used in the CERN library. This new book, Quantum Mechanics, should be used even more, by a larger and less specialized group. The book combines breadth of interest with a thorough practicality. Its readers will find here what they need to know, with a sustained and successful effort to make it intelligible.
Sakurai's sudden death on November 1 , 1 982 left this book unfinished. Reinhold Bertlmann and I helped Mrs. Sakurai sort out her husband's papers at CERN. Among them we found a rough, handwritten version of most of the book and a large collection of exercises.
Though only three chapters had been com pletely finished, it was clear that the bulk of the creative work had been done. It was also clear that much work remained to fill in gaps, polish the writing, and put the manuscript in order. That the book is now finished is due to the determination of N oriko Sakurai and the dedication of San Fu Tuan. Upon her husband's death, Mrs.
Sakurai re solved immediately that his last effort should not go to waste. With great courage and dignity she became the driving force behind the project, overcoming all ob stacles and setting the high standards to be maintained. San Fu Tuan willingly gave his time and energy to the editing and completion of Sakurai's work. Per haps only others close to the hectic field of high-energy theoretical physics can fully appreciate the sacrifice involved.
For me personally, J. had long been far more than just a particularly dis tinguished colleague. It saddens me that we will never again laugh together at physics and physicists and life in general, and that he will not see the success of his last work. But I am happy that it has been brought to fruition.
Bell CERN, Geneva IX www.com Preface to the Revised Ed ition Since 1 989 the editor has enthusiastically pursued a revised edition of Modern Quantum Mechanics by his late great friend J. Sakurai, in order to extend this text's usefulness into the twenty-first century. Much consultation took place with the panel of Sakurai friends who helped with the original edition, but in particular with Professor Yasuo Hara of Tsukuba University and Professor Akio Sakurai of Kyoto Sangyo University in Japan. This book is intended for the first-year graduate student who has studied quan tum mechanics at the junior or senior level.
It does not provide an introduction to quantum mechanics for the beginner. The reader should have had some expe rience in solving time-dependent and time-independent wave equations. A famil iarity with the time evolution of the Gaussian wave packet in a force-free region is assumed, as is the ability to solve one-dimensional transmission-reflection prob lems. Some of the general properties of the energy eigenfunctions and the energy eigenvalues should also be known to the student who uses this text.
The major motivation for this project is to revise the main text. There are three important additions and/or changes to the revised edition, which otherwise pre serves the original version unchanged. These include a reworking of certain por tions of Section 5.2 on time-independent perturbation theory for the degenerate case, by Professor Kenneth Johnson of M., taking into account a subtle point that has not been properly treated by a number of texts on quantum mechanics in this country. Professor Roger Newton of lndiana University contributed refine ments on lifetime broadening in Stark effect and additional explanations of phase shifts at resonances, the optical theorem, and the non-normalizable state.
These appear as "remarks by the editor" or "editor's note" in the revised edition. Pro fessor Thomas Fulton of the Johns Hopkins University reworked his Coulomb scattering contribution (Section 7. 13); it now appears as a shorter text portion emphasizing the physics, with the mathematical details relegated to Appendix C. Though not a major part of the text, some additions were deemed necessary to take into account developments in quantum mechanics that have become promi nent since November 1 , 1 982.
To this end, two supplements are included at the end of the text. Supplement I is on adiabatic change and geometrical phase (pop ularized by M. Berry since 1 983) and is actually an English translation of the supplement on this subject written by Professor Akio Sakurai for the Japanese ver sion of Modern Quantum Mechanics (copyright© Yoshioka-Shoten Publishing of Kyoto). Supplement II on nonexponential decays was written by my colleague here, Professor Xerxes Tata, and read over by Professor E.
Sudarshan of the University of Texas at Austin. Although nonexponential decays have a long XI www.com xii Preface to the Revised Edition history theoretically, experimental work on transition rates that tests such decays indirectly was done only in 1 990. Introduction of additional material is of course a subjective decision on the part of the editor; readers can judge its appropriateness for themselves. Thanks to Professor Akio Sakurai, the revised edition has been diligently searched to correct misprint errors of the first ten printings of the origi nal edition.
My colleague Professor Sandip Pakvasa provided me overall guidance and encouragement throughout this process of revision. In addition to the acknowledgments above, my former students Li Ping, Shi Xiaohong, and Yasunaga Suzuki provided the sounding board for ideas on the revised edition when taking may graduate quantum mechanics course at the Uni versity of Hawaii during the spring of 1 992. Suzuki provided the initial translation from Japanese of Supplement I as a course term paper. Andy Acker provided me with computer graphics assistance.