Table Of Content

Useful Constants and Conversion Factors Quoted to a useful number of significant figures. Speed of light in vacuum Electron charge magnitude Planck's constant Boltzmann's constant Avogadro's number Coulomb's law constant Electron rest mass Proton rest mass Neutron rest mass Atomic mass unit (C 12 = 12) c = 2.998 x 108 m/sec e = 1.602 x 10 = 19 coul h = 6.626 x 10-34 joule-sec h = h/27c = 1.055 x 10 -34 joule-sec = 0.6582 x 10 -15 eV-sec k = 1.381 x 10 -23 joule/°K = 8.617 x 10-5 eV/°K No = 6.023 x 1023/mole 1/47rE0 = 8.988 x 109 nt-m2/coul2 me = 9.109 x 10 -31 kg = 0.5110 MeV/c2 m p = 1.672 x 10 -27 kg = 938.3 MeV/c2 m„ = 1.675 x 10 -Z7 kg = 939.6 MeV/c 2 u = 1.661 x 10 -27 kg = 931.5 MeV/c 2 Bohr magneton Nuclear magneton Bohr radius Bohr energy Electron Compton wavelength Fine-structure constant kT at room temperature ub = eh/2me = 9.27 x 10 -24 amp-m2 (or joule/tesla) µn = eh/2m, = 5.05 x 10 -27 amp-m2 (or joule/tesla) ao = 47c€0h2/mee2 = 5.29 x 10 -11 m = 0.529 A E1 = — mee4/(4rcE0)22h2 = —2.17 x 10 -18 joule = —13.6 eV Ac = h/mec = 2.43 x 10-12 m = 0.0243 A a = e2/4nE0hc = 7.30 x 10 -3 1/137 k300°K = 0.0258 eV ^ 1/40 eV 1eV= 1.602 x 10 -19 joule 1 A=10 -10 m i joule = 6.242 x 1018 eV 1F=10 -15 m � l barn (bn)= 10-28m2 QUANTUM PHYSICS Assisted by yid O CaIgweal Univer^^#y^qf^#^rni^ ^^ arbara United'•°Stalês C^^t^^ ^,;^^ Odemy The figure on the cover is frori ; èction „ 9-4, where it is used to show the tendency for two identical spin 1/2 particles (such as electrons) to avoid each other if their spins are essentially parallel. This tendency, or its inverse for the antiparallel case, is one of the recurring themes in quantum physics explanations of the properties of atoms, molecules, solids, nuclei, and particles. QUANTUM PHYSICS of Atoms, Molecules, Solids, Nuclei, and Particles Second Edition ROBERT EISBERG University of California, Santa Barbara JOHN WILEY & SONS New York Chichester Brisbane Toronto Singapore Copyright © 1974, 1985, by John Wiley & Sons, Inc. All rights reserved. Published simultaneously in Canada. Reproduction or translation of any part of this work beyond that permitted by Sections 107 and 108 of the 1976 United States Copyright Act without the permission of the copyright owner is unlawful. Requests for permission or further information should be addressed to the Permissions Department, John Wiley & Sons. Library of Congress Cataloging in Publication Data: Eisberg, Robert Martin. Quantum physics of atoms, molecules, solids, nuclei, and particles. Includes index. 1. Quantum theory. I. Resnick, Robert, 1923— II. Title, QC174.12.E34 1985 � 530.1'2 � 84-10444 ISBN 0-471-87373-X Printed in the United States of America Printed and bound by the Hamilton Printing Company. 30 29 28 27 26 25 24 23 PREFACE TO THE SECOND EDITION The many developments that have occurred in the physics of quantum systems since the publication of the first edition of this book—particularly in the field of elementary particles—have made apparent the need for a second edition. In preparing it, we solicited suggestions from the instructors that we knew to be using the book in their courses (and also from some that we knew were not, in order to determine their objections to the book). The wide acceptance of the first edition made it possible for us to obtain a broad sampling of thought concerning ways to make the second edition more useful. We were not able to act on all the suggestions that were re- ceived, because some were in conflict with others or were impossible to carry out for technical reasons. But we certainly did respond to the general consensus of these suggestions. Many users of the first edition felt that new topics, typically more sophisticated aspects of quantum mechanics such as perturbation theory, should be added to the book. Yet others said that the level of the first edition was well suited to the course they teach and that it should not be changed. We decided to try to satisfy both groups by adding material to the new edition in the form of new appendices, but to do it in such a way as to maintain the decoupling of the appendices and the text that characterized the original edition. The more advanced appendices are well integrated in the text but it is a one-way, not two-way, integration. A student reading one of these appendices will find numerous references to places in the text where the development is motivated and where its results are used. On the other hand, a student who does not read the appendix because he is in a lower level course will not be frustrated by many references in the text to material contained in an appendix he does not use. Instead, he will find only one or two brief parenthetical statements in the text advising him of the existence of an optional appendix that has a bearing on the subject dealt with in the text. The appendices in the second edition that are new or are significantly changed are: Appendix A, The Special Theory of Relativity (a number of worked-out examples added and an important calculation simplified); Appendix D, Fourier Integral De- scription of a Wave Group (new); Appendix G, Numerical Solution of the Time- Independent Schroedinger Equation for a Square Well Potential (completely rewritten to include a universal program in BASIC for solving second-order differential equations on microcomputers); Appendix J, Time-Independent Perturbation Theory (new); Appendix K, Time-Dependent Perturbation Theory (new); Appendix L, The Born Approximation (new); Appendix N, Series Solutions of the Angular and Radial Equations for a One-Electron Atom (new); Appendix Q, Crystallography (new); Appendix R, Gauge Invariance in Classical and Quantum Mechanical Electromag- netism (new). Problem sets have been added to the ends of many of the appendices, both old and new. In particular, Appendix A now contains a brief but comprehensive set of problems for use by instructors who begin their "modern physics" course with a treatment of relativity. v PREFACE TO THE S ECOND EDITIO N A large number of small changes and additions have been made to the text to improve and update it. There are also several quite substantial pieces of new material, including: the new Section 13-8 on electron-positron annihilation in solids; the additions to Section 16-6 on the Mössbauer effect; the extensive modernization of the last half of the introduction to elementary particles in Chapter 17; and the en- tirely new Chapter 18 treating the developments that have occurred in particle physics since the first edition was written. We were very fortunate to have secured the services of Professor David Caldwell of the University of California, Santa Barbara, to write the new material in Chapters 17 and 18, as well as Appendix R. Only a person who has been totally immersed in research in particle physics could have done what had to be done to produce a brief but understandable treatment of what has happened in that field in recent years. Furthermore, since Caldwell is a colleague of the senior author, it was easy to have the interaction required to be sure that this new material was closely integrated into the earlier parts of the book, both in style and in content. Prepublication reviews have made it clear that Caldwell's material is a very strong addition to the book. Professor Richard Christman, of the U.S. Coast Guard Academy, wrote the new material in Section 13-8, Section 16-6, and Appendix Q, receiving significant input from the authors. We are very pleased with the results. The answers to selected problems, found in Appendix S, were prepared by Profes- sor Edward Derringh, of the Wentworth Institute of Technology. He also edited the new additions to the problem sets and prepared a manual giving detailed solutions to most of the problems. The solutions manual is available to instructors from the publisher. It is a pleasure to express our deep appreciation to the people mentioned above. We also thank Frank T. Avignone, III, University of South Carolina; Edward Cecil, Colorado School of Mines; L. Edward Millet, California State University, Chico; and James T. Tough, The Ohio State University, for their very useful prepublication reviews. The following people offered suggestions or comments which helped in the development of the second edition: Alan H. Barrett, Massachusetts Institute of Technology; Richard H. Behrman, Swarthmore College; George F. Bertsch, Michigan State Uni- versity; Richard N. Boyd, The Ohio State University; Philip A. Casabella, Rensselaer Polytechnic Institute; C. Dewey Cooper, University of Georgia; James E. Draper, University of California at Davis; Arnold Engler, Carnegie-Mellon University; A. T. Fromhold, Jr., Auburn University; Ross Garrett, University of Auckland; Russell Hobbie, University of Minnesota; Bei-Lok Hu, University of Maryland; Hillard Hun- tington, Rensselaer Polytechnic Institute; Mario Iona, University of Denver; Ronald G. Johnson, Trent University; A. L. Laskar, Clemson University; Charles W. Leming, Henderson State University; Luc Leplae, University of Wisconsin-Milwaukee; Ralph D. Meeker, Illinois Benedictine College; Roger N. Metz, Colby College; Ichiro Miya- gawa, University of Alabama; J. A. Moore, Brock University; John J. O'Dwyer, State University of New York at Oswego; Douglas M. Potter, Rutgers State University; Russell A. Schaffer, Lehigh University; John W. Watson, Kent State University; and Robert White, University of Auckland. We appreciate their contribution. Santa Barbara, California � Robert Eisberg Troy, New York � Robert Resnick PREFACE TO THE FIRST EDITION The basic purpose of this book is to present clear and valid treatments of the properties of almost all of the important quantum systems from the point of view of elementary quantum mechanics. Only as much quantum mechanics is developed as is required to accomplish the purpose. Thus we have chosen to emphasize the applications of the theory more than the theory itself. In so doing we hope that the book will be well adapted to the attitudes of contemporary students in a terminal course on the phenomena of quantum physics. As students obtain an insight into the tre- mendous explanatory power of quantum mechanics, they should be motivated to learn more about the theory. Hence we hope that the book will be equally well adapted to a course that is to be followed by a more advanced course in formal quantum mechanics. The book is intended primarily to be used in a one year course for students who have been through substantial treatments of elementary differential and integral calculus and of calculus level elementary classical physics. But it can also be used in shorter courses. Chapters 1 through 4 introduce the various phenomena of early quantum physics and develop the essential ideas of the old quantum theory. These chapters can be gone through fairly rapidly, particularly for students who have had some prior exposure to quantum physics. The basic core of quantum mechanics, and its application to one- and two-electron atoms, is contained in Chapters 5 through 8 and the first four sections of Chapter 9. This core can be covered well in appre- ciably less than half a year. Thus the instructor can construct a variety of shorter courses by adding to the core material from the chapters covering the essentially independent topics: multielectron atoms and molecules, quantum statistics and solids, nuclei and particles. Instructors who require a similar but more extensive and higher level treatment of quantum mechanics, and who can accept a much more restricted coverage of the applications of the theory, may want to use Fundamentals of Modern Physics by Robert Eisberg (John Wiley & Sons, 1961), instead of this book. For instructors requir- ing a more comprehensive treatment of special relativity than is given in Appendix A, but similar in level and pedagogic style to this book, we recommend using in addition Introduction to Special Relativity by Robert Resnick (John Wiley & Sons, 1968). Successive preliminary editions of this book were developed by us through a pro- cedure involving intensive classroom testing in our home institutions and four other schools. Robert Eisberg then completed the writing by significantly revising and extending the last preliminary edition. He is consequently the senior author of this book. Robert Resnick has taken the lead in developing and revising the last preliminary edition so as to prepare the manuscript for a modern physics counterpart at a somewhat lower level. He will consequently be that book's senior author. The pedagogic features of the book, some of which are not usually found in books at this level, were proven in the classroom testing to be very suçcessful. These features are: detailed outlines at the beginning of each chapter, numerous worked out vii PREFACE TO THE FIRST EDITION examples in each chapter, optional sections in the chapters and optional appendices, summary sections and tables, sets of questions at the end of each chapter, and long and varied sets of thoroughly tested problems at the end of each chapter, with subsets of answers at the end of the book. The writing is careful and expansive. Hence we believe that the book is well suited to self-learning and to self-paced courses. We have employed the MKS (or SI) system of units, but not slavishly so. Where general practice in a particular field involves the use of alternative units, they are used here. It is a pleasure to express our appreciation to Drs. Harriet Forster, Russell Hobbie, Stuart Meyer, Gerhard Salinger, and Paul Yergin for constructive reviews, to Dr. David Swedlow for assistance with the evaluation and solutions of the problems, to Dr. Benjamin Chi for assistance with the figures, to Mr. Donald Deneck for editorial and other assistance, and to Mrs. Cassie Young and Mrs. Carolyn Clemente for typing and other secretarial services. Santa Barbara, California � Robert Eisberg Troy, New York � Robert Resnick CONTENTS 1 THERMAL RADIATION AND PLANCK'S POSTULATE � 1 1-1 Introduction � 2 1-2 Thermal Radiation � 2 1-3 Classical Theory of Cavity Radiation � 6 1-4 Planck's Theory of Cavity Radiation � 13 1-5 The Use of Planck's Radiation Law in Thermometry � 19 1-6 Planck's Postulate and Its Implications � 20 1-7 A Bit of Quantum History � 21 2 PHOTONS—PARTICLELIKE PROPERTIES OF RADIATION � 26 2-1 Introduction � 27 2-2 The Photoelectric Effect � 27 2-3 Einstein's Quantum Theory of the Photoelectric Effect � 29 2-4 The Compton Effect � 34 2-5 The Dual Nature of Electromagnetic Radiation � 40 2-6 Photons and X-Ray Production � 40 2-7 Pair Production and Pair Annihilation � 43 2-8 Cross Sections for Photon Absorption and Scattering � 48 3 DE BROGLIE'S POSTULATE—WAVELIKE PROPERTIES OF PARTICLES � 55 3-1 Matter Waves � 56 3-2 The Wave-Particle Duality � 62 3-3 The Uncertainty Principle � 65 3-4 Properties of Matter Waves � 69 3-5 Some Consequences of the Uncertainty Principle � 77 3-6 The Philosophy of Quantum Theory � 79 4 BOHR'S MODEL OF THE ATOM � 85 4-1 Thomson's Model � 86 4-2 Rutherford's Model � 90 4-3 The Stability of the Nuclear Atom � 95 4-4 Atomic Spectra � 96 4-5 Bohr's Postulates � 98 4-6 Bohr's Model � 100 4-7 Correction for Finite Nuclear Mass � 105 4-8 Atomic Energy States � 107 4-9 Interpretation of the Quantization Rules � 110 4-10 Sommerfeld's Model � 114 4-11 The Correspondence Principle � 117 4-12 A Critique of the Old Quantum Theory � 118 ix CONTENTS 5 SCHROEDINGER'S THEORY OF QUANTUM MECHANICS � 124 5-1 Introduction � 125 5-2 Plausibility Argument Leading to Schroedinger's Equation � 128 5-3 Born's Interpretation of Wave Functions � 134 5-4 Expectation Values � 141 5-5 The Time-Independent Schroedinger Equation � 150 5-6 Required Properties of Eigenfunctions � 155 5-7 Energy Quantization in the Schroedinger Theory � 157 5-8 Summary � 165 6 SOLUTIONS OF TIME-INDEPENDENT SCHROEDINGER EQUATIONS � 176 6-1 Introduction � 177 6-2 The Zero Potential � 178 6-3 The Step Potential (Energy Less Than Step Height) � 184 6-4 The Step Potential (Energy Greater Than Step Height) � 193 6-5 The Barrier Potential � 199 6-6 Examples of Barrier Penetration by Particles � 205 6-7 The Square Well Potential � 209 6-8 The Infinite Square Well Potential � 214 6-9 The Simple Harmonic Oscillator Potential � 221 6-10 Summary � 225 7 ONE-ELECTRON ATOMS � 232 7-1 Introduction � 233 7-2 Development of the Schroedinger Equation � 234 7-3 Separation of the Time-Independent Equation � 235 7-4 Solution of the Equations � 237 7-5 Eigenvalues, Quantum Numbers, and Degeneracy � 239 7-6 Eigenfunctions � 242 7-7 Probability Densities � 244 7-8 Orbital Angular Momentum � 254 7-9 Eigenvalue Equations � 259 8 MAGNETIC DIPOLE MOMENTS, SPIN, AND TRANSITION RATES � 266 8-1 Introduction � 267 8-2 Orbital Magnetic Dipole Moments � 267 8-3 The Stern-Gerlach Experiment and Electron Spin � 272 8-4 The Spin-Orbit Interaction � 278 8-5 Total Angular Momentum � 281 8-6 Spin-Orbit Interaction Energy and the Hydrogen Energy Levels � 284 8-7 Transition Rates and Selection Rules � 288 8-8 A Comparison of the Modern and Old Quantum Theories � 295 9 MULTIELECTRON ATOMS—GROUND STATES AND X-RAY EXCITATIONS � 300 9-1 Introduction � 301 9-2 Identical Particles � 302 9-3 The Exclusion Principle � 308 9-4 Exchange Forces and the Helium Atom � 310 9-5 The Hartree Theory � 319 x 9-6 Results of the Hartree Theory � 322 9-7 Ground States of Multielectron Atoms and the Periodic Table � 331 9-8 X-Ray Line Spectra � 337 S1N3lNO0 10 MULTIELECTRON ATOMS—OPTICAL EXCITATIONS � 347 10-1 Introduction � 348 10-2 Alkali Atoms � 349 10-3 Atoms with Several Optically Active Electrons � 352 10-4 LS Coupling � 356 10-5 Energy Levels of the Carbon Atom � 361 10-6 The Zeeman Effect � 364 10-7 Summary � 370 11 QUANTUM STATISTICS � 375 11-1 Introduction � 376 11-2 Indistinguishability and Quantum Statistics � 377 11-3 The Quantum Distribution Functions � 380 11-4 Comparison of the Distribution Functions � 384 11-5 The Specific Heat of a Crystalline Solid � 388 11-6 The Boltzmann Distributions as an Approximation to Quantum Distributions � 391 11-7 The Laser � 392 11-8 The Photon Gas � 398 11-9 The Phonon Gas � 399 11-10 Bose Condensation and Liquid Helium � 399 11-11 The Free Electron Gas � 404 11-12 Contact Potential and Thermionic Emission � 407 11-13 Classical and Quantum Descriptions of the State of a System � 409 12 MOLECULES � 415 12-1 Introduction � 416 12-2 Ionic Bonds � 416 12-3 Covalent Bonds � 418 12-4 Molecular Spectra � 422 12-5 Rotational Spectra � 423 12-6 Vibration-Rotation Spectra � 426 12-7 Electronic Spectra � 429 12-8 The Raman Effect � 432 12-9 Determination of Nuclear Spin and Symmetry Character � 434 13 SOLIDS—CONDUCTORS AND SEMICONDUCTORS � 442 13-1 Introduction � 443 13-2 Types of Solids � 443 13-3 Band Theory of Solids � 445 13-4 Electrical Conduction in Metals � 450 13-5 The Quantum Free-Electron Model � 452 13-6 The Motion of Electrons in a Periodic Lattice � 456 13-7 Effective Mass � 460 13-8 Electron-Positron Annihilation in Solids � 464 13-9 Semiconductors � 467 13-10 Semiconductor Devices � 472 CONTENTS 14 SOLIDS—SUPERCONDUCTORS AND MAGNETIC PROPERTIES �483 14-1 Superconductivity � 484 14-2 Magnetic Properties of Solids � 492 14-3 Paramagnetism � 493 14-4 Ferromagnetism � 497 14-5 Antiferromagnetism and Ferrimagnetism � 503 15 NUCLEAR MODELS � 508 15-1 Introduction � 509 15-2 A Survey of Some Nuclear Properties � 510 15-3 Nuclear Sizes and Densities � 515 15-4 Nuclear Masses and Abundances � 519 15-5 The Liquid Drop Model � 526 15-6 Magic Numbers � 530 15-7 The Fermi Gas Model � 531 15-8 The Shell Model � 534 15-9 Predictions of the Shell Model � 540 15-10 The Collective Model � 545 15-11 Summary � 549 16 NUCLEAR DECAY AND NUCLEAR REACTIONS � 554 16-1 Introduction � 555 16-2 Alpha Decay � 555 16-3 Beta Decay � 562 16-4 The Beta-Decay Interaction � 572 16-5 Gamma Decay � 578 16-6 The Mössbauer Effect � 584 16-7 Nuclear Reactions � 588 16-8 Excited States of Nuclei � 598 16-9 Fission and Reactors � 602 16-10 Fusion and the Origin of the Elements � 607 17 INTRODUCTION TO ELEMENTARY PARTICLES � 617 17-1 Introduction � 618 17-2 Nucleon Forces � 618 17-3 Isospin � 631 17-4 Pions � 634 17-5 Leptons � 641 17-6 Strangeness � 643 17-7 Families of Elementary Particles � 649 17-8 Observed Interactions and Conservation Laws � 653 18 MORE ELEMENTARY PARTICLES � 666 18-1 Introduction � 667 18-2 Evidence for Partons � 667 18-3 Unitary Symmetry and Quarks � 673 18-4 Extensions of SU(3)—More Quarks � 678 18-5 Color and the Color Interaction � 683 18-6 Introduction to Gauge Theories � 688 18-7 Quantum Chromodynamics � 691 18-8 Electroweak Theory � 699 18-9 Grand Unification and the Fundamental Interactions � 706 Appendix A The Special Theory of Relativity Appendix B Radiation from an Accelerated Charge Appendix C The Boltzmann Distribution Appendix D Fourier Integral Description of a Wave Group Appendix E Rutherford Scattering Trajectories Appendix F Complex Quantities Appendix G Numerical Solution of the Time-Independent Schroedinger Equation for a Square Well Potential Appendix H Analytical Solution of the Time-Independent Schroedinger Equation for a Square Well Potential Appendix I � Series Solution of the Time-Independent Schroedinger Equation for a Simple Harmonic Oscillator Potential Appendix J Time-Independent Perturbation Theory Appendix K Time-Dependent Perturbation Theory Appendix L The Born Approximation Appendix M The Laplacian and Angular Momentum Operators in Spherical Polar Coordinates Appendix N Series Solutions of the Angular and Radial Equations for a One-Electron Atom Appendix O The Thomas Precession Appendix P The Exclusion Principle in LS Coupling Appendix Q Crystallography Appendix R Gauge Invariance in Classical and Quantum Mechanical Electromagnetism Appendix S Answers to Selected Problems Index S1N3L NOJ