M O D E R N QUANTUM MECHANICS

1 MODERNQUANTUMMECHANICSSECOND Sakurai Jim NapolitanoMODERN QUANTUM MECHANICS Second Edition 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 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.

2 Copyright 1994, 2011 Pearson Education, Inc., publishing as Addison-Wesley, 1301 Sansome Street, San Francisco, CA 941 11. 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., Permissions Department, 1900 E. Lake Ave., Glenview, IL 60025. 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.

3 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. J. (Jun John), 1933-1982. Modern QUANTUM MECHANICS . - 2nd ed. I Sakurai, Jim Napolitano. p. cm. ISBN 978-0-8053-8291-4 (alk. paper) 1. QUANTUM theory-Textbooks. I. Napolitano, Jim. II. Title. 2011 ISBN 10: 0-8053-8291-7; ISBN 13: 978-0-8053-8291-4 1 2 3 4 5 6 7 8 9 10-CRK-14 13 12 11 10 Addison-Wesley is an imprint of I PEARSON 2010022349 Contents Foreword to the First Edition Preface to the Revised Edition Preface to the Second Edition In Memoriam 1 Fundamental Concepts The Stem-Gerlach Experiment Kets, Bras, and Operators 10 1 Base Kets and Matrix Representations 17 Measurements, Observables, and the Uncertainty Relations 23 Change of Basis 35 Position, Momentum.

4 And Translation 40 Wave Functions in Position and Momentum Space 50 2 QUANTUM Dynamics Time-Evolution and the Schrodinger Equation 66 The Schrodinger Versus the Heisenberg Picture 80 Simple Harmonic Oscillator 89 SchrOdinger's Wave Equation 97 Elementary Solutions to SchrOdinger's Wave Equation 103 Propagators and Feynman Path Integrals 116 2. 7 Potentials and Gauge Transformations 129 . IX . XI .. XIII .. XVII 1 66 3 Theory of Angular Momentum 157 Rotations and Angular-Momentum Commutation Relations 157 Spin Systems and Finite Rotations 163 S0(3), SU(2), and Euler Rotations 172 v VI Contents Density Operators and Pure Versus Mixed Ensembles 178 Eigenvalues and Eigenstates of Angular Momentum 191 Orbital Angular Momentum 199 Schrodinger's Equation for Central Potentials 207 Addition of Angular Momenta 217 Schwinger's Oscillator Model of Angular Momentum 232 Spin Correlation Measurements and Bell's Inequality 238 Tensor Operators 246 4 Symmetry in QUANTUM MECHANICS 4.

5 1 Symmetries, Conservation Laws, and Degeneracies 262 Discrete Symmetries, Parity, or Space Inversion 269 Lattice Translation as a Discrete Symmetry 280 The Time-Reversal Discrete Symmetry 284 262 5 Approximation Methods 303 Time-IndependentPerturbation Theory: NondegenerateCase 303 Time-Independent Perturbation Theory: The Degenerate Case 316 Hydrogen-Like Atoms: Fine Structure and the Zeeman Effect 321 Variational Methods 332 Time-Dependent Potentials: The Interaction Picture 336 Hamiltonians with Extreme Time Dependence 345 Time-Dependent Perturbation Theory 355 Applications to Interactions with the Classical Radiation Field 365 Energy Shift and Decay Width 37 1 6 Scattering Theory Scattering as a Time-Dependent Perturbation 386 The Scattering Amplitude 391 The Born Approximation 399 Phase Shifts and Pa rtial Waves 404 Eikonal Approximation 417 Low-Energy Scattering and Bound States 423 6.

6 7 Resonance Scattering 430 Symmetry Considerations in Scattering 433 Inelastic Electron-Atom Scattering 436 7 Identical Particles Permutation Symmetry 446 Symmetrization Postulate 450 386 446 Contents Two-Electron System 452 The Helium Atom 455 Multiparticle States 459 7. 6 Quantization of the Electromagnetic Field 4 72 8 relativistic QUANTUM MECHANICS Paths to relativistic QUANTUM MECHANICS 486 The Dirac Equation 494 Symmetries of the Dirac Equation 501 Solving with a Central Potential 506 relativistic QUANTUM Field Theory 514 vii 486 A Electromagnetic Units 519 A. 1 Coulomb's Law, Charge, and Current 519 Converting Between Systems 520 B Brief Summary of Elementary Solutions to Schrodinger's Wave Equation 523 Free Particles (V = 0) 523 Piecewise Constant Potentials in One Dimension 524 Transmission-Reflection Problems 525 Simple Harmonic Oscillator 526 The Central Force Problem [Spherically Symmetrical Potential V = V(r)] 527 Hydrogen Atom 531 C Proof of the Angular-Momentum Addition Rule Given by Equation ( ) 533 Bibliography 535 Index 537 Foreword to the First Edition J.

7 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.

8 " He put these principles to work in his fine book Advanced QUANTUM MECHANICS (1967) and in Invariance Principles and Elementary Particles (1964), both of which have been very much used in the CERN library. This new book, Modern 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. J. J. Sakurai's sudden death on November 1, 1982 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.

9 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 Tu an. 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 Tu an 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.

10 For me personally, J. 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. John S. Bell CERN, Geneva IX Preface to the Revised Edition Since 1989 the editor has enthusiastically pursued a revised edition of Modern QUANTUM MECHANICS by his late great fri end J. 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 Ya suo Hara of Tsukuba University and Professor Akio Sakurai of Kyoto Sangyo University in Japan.