QUANTUM rae • napoltano PHYSICS QUANTUM MECHANICS SIXTH MECHANICS EDITION “This is a very versatile textbook, which could be used in a variety of courses ranging from an ‘honors’ introductory course to a challenging undergraduate upper-class course. … The book is divided into parts, making it easy for an instructor to choose the relevant material based on the level of the class.” —Robert Pelcovits, Professor of Physics, Brown University “The new sixth edition of this well-known textbook should be thought of as one of the best SIXTH EDITION options available for undergraduate quantum mechanics courses, among a very large class of QUANTUM MECHANICS introductory books. … Detailed worked examples and asides on associated applications of the principles discussed enhance the educational aspects of this book.” —Aaron Lindenberg, Associate Professor, Department of Materials Science and Engineering/Photon Science, Stanford University/SLAC National Accelerator Laboratory “This book by Rae and Napolitano distinguishes itself with a unique approach by including more material on practical applications of the theoretical concepts … a great choice of textbook for upper-class undergraduate students in physics or students entering graduate studies in engineering schools.” —Professor Chunhui Chen, Iowa State University “This text provides an updated treatment of quantum mechanics, suitable for the standard senior- level undergraduate course at U. colleges and universities.
Pete Markowitz, Professor, Department of Physics, Florida International University Quantum Mechanics, Sixth Edition builds on its highly praised predecessors to make the text even more accessible to a wider audience. This sixth edition contains three new chapters that review prerequisite physics and mathematics, laying out the notation, formalism, and physical basis necessary for the rest of the book. Along with more problems, this edition also presents short descriptions of sixth edition numerous applications relevant to the physics discussed, offering a brief look at what quantum mechanics has made possible industrially and scientifically. Rae 6000 Broken Sound Parkway, NW Suite 300, Boca Raton, FL 33487 K24456 ISBN: 978-1-4822-9918-2 90000 Jim Napolitano 711 Third Avenue New York, NY 10017 an informa business 2 Park Square, Milton Park Abingdon, Oxon OX14 4RN, UK 9 78 1 482 299 1 82 w w w.com QUANTUM MECHANICS SIXTH EDITION www.com QUANTUM MECHANICS SIXTH EDITION Alastair I.
Rae University of Birmingham, U. Jim Napolitano Temple University, U. Boca Raton London New York CRC Press is an imprint of the Taylor & Francis Group, an informa business www.com CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2016 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U. Government works Version Date: 20151105 International Standard Book Number-13: 978-1-4822-9921-2 (eBook - PDF) This book contains information obtained from authentic and highly regarded sources.
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For organizations that have been granted a photo- copy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Visit the Taylor & Francis Web site at http://www.com and the CRC Press Web site at http://www.com To Angus and Gavin To Julia www.com Contents PART I Waves, Electromagnetism, and the Limits of Classical Physics CHAPTER 1 The Physics and Mathematics of Waves 3 1.1 A REVIEW OF SIMPLE HARMONIC MOTION 4 1.2 THE STRETCHED STRING EQUATION OF MOTION 7 1.3 STANDING WAVES, AND FOURIER SERIES 11 1.4 THE FOURIER TRANSFORM 16 1.5 PROBLEMS 20 CHAPTER 2 Maxwell’s Equations and Electromagnetic Waves 23 2.1 MAXWELL’S EQUATIONS AS INTEGRALS 25 2.2 SURFACE THEOREMS IN VECTOR CALCULUS 29 2.3 MAXWELL’S EQUATIONS AS DERIVATIVES 32 2.6 PROBLEMS 41 CHAPTER 3 Particle Mechanics, Relativity, and Photons 45 3.1 NEWTON, MAXWELL, AND EINSTEIN 46 3.2 SPACETIME IN SPECIAL RELATIVITY 49 3.3 VELOCITY, MOMENTUM, AND ENERGY 52 3.4 PROBLEMS 58 CHAPTER 4 The Early Development of Quantum Mechanics 61 4.1 THE PHOTOELECTRIC EFFECT 62 4.2 THE COMPTON EFFECT 65 vii www.com viii Contents 4.3 LINE SPECTRA AND ATOMIC STRUCTURE 67 4.4 DE BROGLIE WAVES 69 4.5 WAVE-PARTICLE DUALITY 72 4.6 THE REST OF THIS BOOK 76 4.7 PROBLEMS 77 PART II Elementary Wave Mechanics CHAPTER 5 The One-dimensional Schrödinger Equations 81 5.1 THE TIME-DEPENDENT SCHRÖDINGER EQUATION 82 5.2 THE TIME-INDEPENDENT SCHRÖDINGER EQUATION 85 5.4 THE INFINITE SQUARE WELL 87 5.5 THE FINITE SQUARE WELL 91 5.6 QUANTUM MECHANICAL TUNNELLING 95 5.7 THE HARMONIC OSCILLATOR 102 5.8 PROBLEMS 106 CHAPTER 6 The Three-dimensional Schrödinger Equations 109 6.1 THE WAVE EQUATIONS 110 6.2 SEPARATION IN CARTESIAN COORDINATES 111 6.3 SEPARATION IN SPHERICAL POLAR COORDINATES 115 6.4 THE HYDROGENIC ATOM 124 6.5 PROBLEMS 130 PART III Formal Foundations CHAPTER 7 The Basic Postulates of Quantum Mechanics 135 7.1 THE WAVE FUNCTION 137 7.2 THE DYNAMICAL VARIABLES 137 7.5 THE UNCERTAINTY PRINCIPLE 153 7.6 THE TIME DEPENDENCE OF THE WAVE FUNCTION 158 7.com Contents ix 7.8 THE HARMONIC OSCILLATOR AGAIN 163 7.9 THE MEASUREMENT OF MOMENTUM BY COMPTON SCATTERING 165 7.10 PROBLEMS 169 CHAPTER 8 Angular Momentum I 173 8.1 THE ANGULAR-MOMENTUM OPERATORS 174 8.2 THE ANGULAR MOMENTUM EIGENVALUES AND EIGENFUNCTIONS 176 8.3 THE EXPERIMENTAL MEASUREMENT OF ANGULAR MOMENTUM 179 8.4 A GENERAL SOLUTION TO THE ANGULAR- MOMENTUM EIGENVALUE PROBLEM 182 8.5 PROBLEMS 188 CHAPTER 9 Angular Momentum II 191 9.2 PAULI SPIN MATRICES 195 9.3 SPIN AND THE QUANTUM THEORY OF MEASUREMENT 197 9.5 SPIN-ORBIT COUPLING AND THE ZEEMAN EFFECT 202 9.6 A MORE GENERAL TREATMENT OF THE COUPLING OF ANGULAR MOMENTA 209 9.7 PROBLEMS 216 PART IV Extensions and Approximation Schemes CHAPTER 10 Time-independent Perturbation Theory and the Variational Principle 221 10.1 PERTURBATION THEORY FOR NONDEGENERATE ENERGY LEVELS 222 10.2 PERTURBATION THEORY FOR DEGENERATE ENERGY LEVELS 228 10.3 THE VARIATIONAL PRINCIPLE 238 10.com x Contents CHAPTER 11 Time Dependence 245 11.1 TIME-INDEPENDENT HAMILTONIANS 247 11.2 THE SUDDEN APPROXIMATION 252 11.3 TIME-DEPENDENT PERTURBATION THEORY 254 11.4 TRANSITIONS BETWEEN ATOMIC ENERGY LEVELS 260 11.5 THE EHRENFEST THEOREM 266 11.6 THE AMMONIA MASER 267 11.7 PROBLEMS 270 CHAPTER 12 Scattering 273 12.1 SCATTERING IN ONE DIMENSION 274 12.2 SCATTERING IN THREE DIMENSIONS 279 12.3 THE BORN APPROXIMATION 281 12.4 PARTIAL WAVE ANALYSIS 286 12.5 PROBLEMS 297 CHAPTER 13 Many-particle Systems 299 13.5 MANY-PARTICLE SYSTEMS 306 13.6 THE HELIUM ATOM 314 13.7 SCATTERING OF IDENTICAL PARTICLES 320 13.8 PROBLEMS 321 PART V Advanced Topics CHAPTER 14 Relativity and Quantum Mechanics 325 14.1 BASIC RESULTS IN SPECIAL RELATIVITY 326 14.2 THE DIRAC EQUATION 326 14.4 OTHER WAVE EQUATIONS 337 14.5 QUANTUM FIELD THEORY AND THE SPIN- STATISTICS THEOREM 337 14.com Contents xi CHAPTER 15 Quantum Information 343 15.3 CLONING AND TELEPORTATION 349 15.5 PROBLEMS 364 CHAPTER 16 The Conceptual Problems of Quantum Mechanics 365 16.1 THE CONCEPTUAL PROBLEMS 366 16.2 HIDDEN-VARIABLE THEORIES 368 16.4 THE QUANTUM MEASUREMENT PROBLEM 385 16.5 THE ONTOLOGICAL PROBLEM 400 16.6 PROBLEMS 401 B IBLIOGRAPHY 403 I NDEX 407 www.com Preface to Sixth Edition Quantum mechanics is an indispensable part of any undergraduate physics curricu- lum. Nevertheless, the level at which quantum mechanics is taught varies widely from one college or university to another, as do the mathematics and physics prerequisites for the course.
Our goal in producing this new edition is to make the textbook more accessible to a wider readership. At the same time, we do not want the book to appear unfamiliar to those that have come to know it over the years. As part of an attempt to reach these goals, the book is now divided into five “parts” that separately cover broad topics some or all of which might make up portions of any general course on quantum mechanics. Part 1 is comprised of three new chapters that review prerequisite physics and mathematics, along with a slightly revised version of the first chapter from the fifth edition.
Our purpose here is not to cover classical waves, electromagnetism and special relativity, along with the attendant mathematics, at a level of detail that would suitable for a full course in these. Rather, we lay out the notation, formalism, and physical basis that we feel is necessary to set the stage for the rest of the book. That is, the chapters should be considered to be review material and a reference source for the later topics. Also, many of the end-of-chapter problems in these chapters are designed to extend the discussion in the text, so one approach to covering this material could be to assign those problems after a review of the these chapters.
Part 2 consists of elementary wave mechanics based on the Schrödinger equations and their solutions in one and three dimensions, culminating in the energy levels and wavefunctions for the hydrogen atom. Part 3 introduces formal quantum mechanics including the theory of angular momentum and spin-orbit coupling. Part 4 extends the basic theory to include approximation schemes, scattering and many-body systems. Part 5 contains a selection of advanced topics, including the Dirac equation, quantum information and a discussion of conceptual problems.
We hope that this division will assist teachers to select those topics that best fit the background of their students. Thus, an instructor might choose not to cover all or some of Part I, especially if that material is part of one or more prerequisite courses. (This approach would be closest to using the fifth edition of the book.) For example, xiii www.com xiv Preface to Sixth Edition many curricula in the US include a prerequisite course in “Modern Physics,” covering all of Part I in great detail, and possibly elements of Part II. In this case, the instructor could start at Part 2, while having students revise these topics by reading through, and working some problems from, the first four chapters.
On the other hand, many curricula will see the material in those first four chapters covered in different courses in different years. For example, the vector calculus formulation of electromagnetism may not be seen until the same time students are taking a quantum mechanics course. In this case, the instructor might spend some class time at the start of the term, working through the material in Sections 2.3 before moving on to Part 2. Instructors and students need to appreciate the breadth versus depth balance in the new chapters.
A further innovation in this edition is the inclusion of short descriptions of a number of applications relevant to the physics being discussed at various points through the book. These are not meant to be complete descriptions by any means, but rather a brief look at what quantum mechanics has made possible, both industrially and scientifically. It is our hope that these short sections will inspire the interested reader to learn more about the subjects they describe.