Jj sakurai modern quantum mechanics pdf free download

Modern Quantum Mechanics J. J. Sakurai Wa Z Y WY _ N N oS Revised EditionModern Quantum Mechanics Revised Edition J. J. Sakurai Late, University of California, Los Angeles San Fu Tuan, Editor University of Hawaii, Manoa Addison-Wesley Publishing Company Reading, Massachusetts * Menlo Park, Califomia * New York Don Mills, Ontario * Wokingham, England « Amsterdam + Bonn Sydney + Singapore + Tokyo * Madrid + San Juan « Milan « ParisSponsoring Editor: Stuart W. Johnson Assistant Editor: Jennifer Duggan Senior Production Coordinator: Amy Willcutt Manufacturing Manager: Roy Logan Library of Congress Cataloging-in-Publication Data Sakurai, J. J. (Jun John), Modern quantum mechanics / J. J. Sakurai ; San Fu Tuan, editor.— Rev. ed. p. cm. Includes bibliographical references and index. ISBN 1. Quantum theory. I. Tuan, San Fu, . IL. Title. QC '2—de20 , cIP Copyright © by Addison-Wesley Publishing Company, Inc. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher. Printed in the United States of America. 5 6 7 8 9 MAForeword J. 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 sophisti- cated, yet uneducated, theoreticians who are conversant in the LSZ for- malism 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.” He put these principles to work in his fine book Advanced Quantum Mechanics () and in Invariance Principles and Elementary Particles (), 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, 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 completely 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 Noriko Sakurai and the dedication of San Fu Tuan. Upon her husband’s death, Mrs. Sakurai resolved 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 obstacles 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. Perhaps only others close to the hectic field of high-energy theoretical physics can fully appreciate the sacrifice involved. For me personally, J. J. had long been far more than just a particu- larly distinguished 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Preface to the Revised Edition Since the Editor has enthusiastically pursued a revised edition of Modern Quantum Mechanics by his late great friend J. J. Sakurai, in order to extend this text’s usefulness into the twenty-first century. Much con- sultation 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. 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 preserves the original version unchanged. These include a reworking of certain portions of Section on time-independent per- turbation theory for the degenerate case by Professor Kenneth Johnson of M.L.T., 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 Indiana University contributed refinements on lifetime broadening in Stark effect, additional explanations of phase shifts at res- onances, the optical theorem, and on 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 Cou- lomb Scattering contribution (Section ) so that 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 prominent since November 1, To this end, two sup- plements are included at the end of the text. Supplement I is on adiabatic change and geometrical phase (popularized by M. V. Berry since ) and is actually an English translation of the supplement on this subject written by Professor Akio Sakurai for the Japanese version of Modern Quantum Mechanics (copyright © Yoshioka-Shoten Publishing of Kyoto). Supplement II is on non-exponential decays written by my colleague here, Professor Xerxes Tata, and read over by Professor E. C. G. Sudarshan of the University of Texas at Austin. Though non-exponential decays have a long history theoretically, experimental work on transition rates that tests indirectly such decays was done only in Introduction of additional material is of course a subjective matter on the part of the Editor; the readers will evaluate for themselves its appropriateness. Thanks to Pro- fessor Akio Sakurai, the revised edition has been “finely toothcombed” for misprint errors of the first ten printings of the original edition. My colleague, Professor Sandip Pakvasa, provided overall guidance and en- couragement to me throughout this process of revision.Preface to the Revised Edition v 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 my graduate quantum me- chanics course at the University of Hawaii during the spring of Suzuki provided the initial translation from Japanese of Supplement I as a course term paper. Dr. Andy Acker provided me with computer graphic assis- tance. The Department of Physics and Astronomy and particularly the High Energy Physics Group of the University of Hawaii at Manoa provided again both the facilities and a conducive atmosphere for me to carry out my editorial task. Finally I wish to express my gratitude to Physics (and sponsoring) Senior Editor, Stuart Johnson, and his Editorial Assistant, Jennifer Duggan, as well as Senior Production Coordinator Amy Willcutt, of Addison-Wesley for their encouragement and optimism that the revised edition will indeed materialize. San Fu TUAN Honolulu, HawaiiJ. J. Sakurai In Memoriam Jun John Sakurai was born in in Tokyo and came to the United States as a high school student in He studied at Harvard and at Cornell, where he received his Ph.D. in He was then appointed assistant professor of Physics at the University of Chicago, and became a full professor in He stayed at Chicago until when he moved to the University of California at Los Angeles, where he remained until his death. During his lifetime he wrote articles in theoretical physics of elementary particles as well as several books and monographs on both quantum and particle theory. The discipline of theoretical physics has as its principal aim the formulation of theoretical descriptions of the physical world that are at once concise and comprehensive. Because nature is subtle and complex, the pursuit of theoretical physics requires bold and enthusiastic ventures to the frontiers of newly discovered phenomena. This is an area in which Sakurai reigned supreme with his uncanny physical insight and intuition and also his ability to explain these phenomena in illuminating physical terms to the unsophisticated. One has but to read his very lucid textbooks on Invariance Principles and Elementary Particles and Advanced Quantum Mechanics as well as his reviews and summer school lectures to appreciate this. Without exaggeration I could say that much of what I did understand in particle physics came from these and from his articles and private tutoring. When Sakurai was still a graduate student, he proposed what is now known as the V-A theory of weak interactions, independently of (and simultaneously with) Richard Feynman, Murray Gell-Mann, Robert Marshak, and George Sudarshan. In he published in Annals of Physics a prophetic paper, probably his single most important one. It was concerned with the first serious attempt to construct a theory of strong interactions based on Abelian and non-Abelian (Yang-Mills) gauge invariance. This seminal work induced theorists to attempt an understanding of the mecha- nisms of mass generation for gauge (vector) fields, now realized as the Higgs mechanism. Above all it stimulated the search for a realistic unification of forces under the gauge principle, now crowned with success in the cel- ebrated Glashow-Weinberg-Salam unification of weak and electromagnetic forces. On the phenomenological side, Sakurai pursued and vigorously advocated the vector mesons dominance model of hadron dynamics. He was the first to discuss the mixing of w and ¢ meson states. Indeed, he made numerous important contributions to particle physics phenomenology in aviii In Memoriam much more general sense, as his heart was always close to experimental activities, I knew Jun John for more than 25 years, and I had the greatest admiration not only for his immense powers as a theoretical physicist but also for the warmth and generosity of his spirit. Though a graduate student himself at Cornell during , he took time from his own pioneering research in K-nucleon dispersion relations to help me (via extensive corre- spondence) with my Ph.D. thesis on the same subject at Berkeley. Both Sandip Pakvasa and I were privileged to be associated with one of his last papers on weak couplings of heavy quarks, which displayed once more his infectious and intuitive style of doing physics. It is of course gratifying to us in retrospect that Jun John counted this paper among the score of his published works that he particularly enjoyed. The physics community suffered a great loss at Jun John Sakurai’s death. The personal sense of loss is a severe one for me. Hence J am profoundly thankful for the opportunity to edit and complete his manuscript on Modern Quantum Mechanics for publication. In my faith no greater gift can be given me than an opportunity to show my respect and love for Jun John through meaningful service. San Fu TuanContents Foreword Preface In Memoriam FUNDAMENTAL CONCEPTS The Stern-Gerlach Experiment Kets, Bras, and Operators Base Kets and Matrix Representations Measurements, Observables, and the Uncertainty Relations Change of Basis Position, Momentum, and Translation Wave Functions in Position and Momentum Space Problems n QUANTUM DYNAMICS, Time Evolution and the Schrédinger Equation The Schrédinger Versus the Heisenberg Picture Simple Harmonic Oscillator Schrédinger’s Wave Equation Propagators and Feynman Path Integrals Potentials and Gauge Transformations Problems o THEORY OF ANGULAR MOMENTUM Rotations and Angular Momentum Commutation Relations Spin 1/2 Systems and Finite Rotations $O(3), SU(2), and Euler Rotations Density Operators and Pure Versus Mixed Ensembles Eigenvalues and Eigenstates of Angular Momentum Orbital Angular Momentum Addition of Angular Momenta Schwinger’s Oscillator Model of Angular Momentum Spin Correlation Measurements and Bell’s Inequality Tensor Operators Problems 4 SYMMETRY IN QUANTUM MECHANICS Symmetries, Conservation Laws, and Degeneracies Discrete Symmetries, Parity, or Space Inversion 43 Lattice Translation as a Discrete Symmetry 44 The Time-Reversal Discrete Symmetry Problems 5 APPROXIMATION METHODS Time-Independent Perturbation Theory: Nondegenerate Case Time-Independent Perturbation Theory: The Degenerate Case Hydrogenlike Atoms: Fine Structure and the Zeeman Effect Variational Methods Time-Dependent Potentials: The Interaction Picture Time-Dependent Perturbation Theory Applications to Interactions with the Classical Radiation Field Energy Shift and Decay Width Problems 6 IDENTICAL PARTICLES Permutation Symmetry Symmetrization Postulate Two-Electron System The Helium Atom Permutation Symmetry and Young Tableaux Problems 7 SCATTERING THEORY TW 72 73 74 75 17 78 79 TAL The Lippmann-Schwinger Equation The Born Approximation Optical Theorem Eikonal Approximation Free-Particle States: Plane Waves Versus Spherical Waves Method of Partial Waves Low-Energy Scattering and Bound States Resonance Scattering Identical Particles and Scattering Symmetry Considerations in Scattering Time-Dependent Formulation of Scattering Inelastic Electron-Atom Scattering Coulomb Scattering Problems Appendix A Appendix B Appendix C Supplement I Adiabatic Change and Geometrical Phase Supplement II Non-Exponential Decays Bibliography Index Contents Modern Quantum MechanicsCHAPTER 1 Fundamental Concepts The revolutionary change in our understanding of microscopic phenomena that took place during the first 27 years of the twentieth century is unprecedented in the history of natural sciences. Not only did we witness severe limitations in the validity of classical physics, but we found the alternative theory that replaced the classical physical theories to be far richer in scope and far richer in its range of applicability. The most traditional way to begin a study of quantum mechanics is to follow the historical developments—Planck’s radiation law, the Einstein- Debye theory of specific heats, the Bohr atom, de Broglie’s matter waves, and so forth—together with careful analyses of some key experiments such as the Compton effect, the Franck-Hertz experiment, and the Davisson- Germer-Thompson experiment. In that way we may come to appreciate how the physicists in the first quarter of the twentieth century were forced to abandon, little by little, the cherished concepts of classical physics and how, despite earlier false starts and wrong turns, the great masters— Heisenberg, Schrédinger, and Dirac, among others—finally succeeded in formulating quantum mechanics as we know it today. However, we do not follow the historical approach in this book. Instead, we start with an example that illustrates, perhaps more than any other example, the inadequacy of classical concepts in a fundamental way. We hope that by exposing the reader to a “shock treatment” at the onset, he 12 Fundamental Concepts or she may be attuned to what we might call the “quantum-mechanical way of thinking” at a very early stage. THE STERN-GERLACH EXPERIMENT The example we concentrate on in this section is the Stern-Gerlach experi- ment, originally conceived by O. Stern in and carried out in Frankfurt by him in collaboration with W. Gerlach in This experiment illustrates in a dramatic manner the necessity for a radical departure from the concepts of classical mechanics. In the subsequent sections the basic for- malism of quantum mechanics is presented in a somewhat axiomatic manner but always with the example of the Stern-Gerlach experiment in the back of our minds. In a certain sense, a two-state system of the Stern-Gerlach type is the least classical, most quantum-mechanical system. A solid understand- ing of problems involving two-state systems will turn out to be rewarding to any serious student of quantum mechanics. It is for this reason that we refer repeatedly to two-state problems throughout this book. Description of the Experiment We now present a brief discussion of the Stern-Gerlach experiment, which is discussed in almost any book on modern physics. First, silver (Ag) atoms are heated in an oven. The oven has a small hole through which some of the silver atoms escape. As shown in Figure , the beam goes through a collimator and is then subjected to an inhomogeneous magnetic field produced by a pair of pole pieces, one of which has a very sharp edge. We must now work out the effect of the magnetic field on the silver atoms. For our purpose the following oversimplified model of the silver atom suffices. The silver atom is made up of a nucleus and 47 electrons, where 46 out of the 47 electrons can be visualized as forming a spherically symmetrical electron cloud with no net angular momentum. If we ignore the nuclear spin, which is irrelevant to our discussion, we see that the atom as a whole does have an angular momentum, which is due solely to the spin— intrinsic as opposed to orbital—angular momentum of the single 47th (Ss) electron. The 47 electrons are attached to the nucleus, which is ~ % times heavier than the electron; as a result, the heavy atom as a whole possesses a magnetic moment equal to the spin magnetic moment of the 47th electron. In other words, the magnetic moment p of the atom is *For an elementary but enlightening discussion of the Stern-Gerlach experiment, see French and Taylor (, ). The Stern-Gerlach Experiment 3 z-axis Beam direction B -field Sea Cotimating magnet " (pole pieces) FIGURE The Stern-Gerlach experiment. proportional to the electron spin S, pas, () where the precise proportionality factor turns out to be e/m,c (e