Quantum Mechanics www.com Quantum Mechanics Fourth edition Alastair I. Rae Department of Physics University of Birmingham UK Institute of Physics Publishing Bristol and Philadelphia www.com c IOP Publishing Ltd 2002 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 permission of the publisher. Multiple copying is permitted in accordance with the terms of licences issued by the Copyright Licensing Agency under the terms of its agreement with Universities UK (UUK).
British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. ISBN 0 7503 0839 7 Library of Congress Cataloging-in-Publication Data are available First edition 1980 Second edition 1986 Reprinted 1987 Reprinted with corrections 1990 Reprinted 1991 Third edition 1992 Reprinted 1993 Reprinted with corrections 1996 Reprinted 1998, 2001 Fourth edition 2002 Commissioning Editor: James Revill Production Editor: Simon Laurenson Production Control: Sarah Plenty Cover Design: Frédérique Swist Published by Institute of Physics Publishing, wholly owned by The Institute of Physics, London Institute of Physics Publishing, Dirac House, Temple Back, Bristol BS1 6BE, UK US Office: Institute of Physics Publishing, The Public Ledger Building, Suite 1035, 150 South Independence Mall West, Philadelphia, PA 19106, USA Typeset in the UK by Text 2 Text, Torquay, Devon Printed in the UK by MPG Books Ltd, Bodmin, Cornwall www.com To Angus and Gavin www.com Contents Preface to Fourth Edition xi Preface to Third Edition xiii Preface to Second Edition xv Preface to First Edition xvii 1 Introduction 1 1.1 The photoelectric effect 2 1.2 The Compton effect 3 1.3 Line spectra and atomic structure 5 1.4 de Broglie waves 6 1.5 Wave–particle duality 7 1.6 The rest of this book 12 Problems 13 2 The one-dimensional Schrödinger equations 14 2.1 The time-dependent Schrödinger equation 14 2.2 The time-independent Schrödinger equation 18 2.5 Quantum mechanical tunnelling 27 2.6 The harmonic oscillator 33 Problems 38 3 The three-dimensional Schrödinger equations 39 3.1 The wave equations 39 3.2 Separation in Cartesian coordinates 41 3.3 Separation in spherical polar coordinates 45 3.4 The hydrogenic atom 53 Problems 59 www.com viii Contents 4 The basic postulates of quantum mechanics 60 4.2 The dynamical variables 62 4.5 The uncertainty principle 76 4.6 The time dependence of the wavefunction 81 4.8 The harmonic oscillator again 86 4.9 The measurement of momentum by Compton scattering 88 Problems 92 5 Angular momentum I 94 5.1 The angular-momentum operators 95 5.2 The eigenvalues and eigenfunctions 96 5.3 The experimental measurement of angular momentum 100 5.4 General solution to the eigenvalue problem 103 Problems 108 6 Angular momentum II 109 6.2 Pauli spin matrices 112 6.3 Spin and the quantum theory of measurement 114 6.5 Spin–orbit coupling and the Zeeman effect 119 6.1 The strong-field Zeeman effect 121 6.2 Spin–orbit coupling 122 6.3 The weak-field Zeeman effect 124 6.6 A more general treatment of the coupling of angular momenta 126 Problems 132 7 Time-independent perturbation theory and the variational principle 134 7.1 Perturbation theory for non-degenerate energy levels 135 7.2 Perturbation theory for degenerate levels 141 7.1 Nearly degenerate systems 143 7.3 The variational principle 151 Problems 155 8 Time dependence 157 8.1 Time-independent Hamiltonians 158 8.2 The sudden approximation 163 8.3 Time-dependent perturbation theory 165 8.5 The Ehrenfest theorem 174 8.6 The ammonia maser 176 Problems 179 www.com Contents ix 9 Scattering 181 9.1 Scattering in one dimension 181 9.2 Scattering in three dimensions 186 9.3 The Born approximation 189 9.4 Partial wave analysis 193 Problems 203 10 Many-particle systems 205 10.3 Non-interacting particles 208 10.5 Many-particle systems 212 10.6 The helium atom 216 10.7 Scattering of identical particles 223 Problems 224 11 Relativity and quantum mechanics 226 11.1 Basic results in special relativity 226 11.2 The Dirac equation 227 11.4 Other wave equations 235 11.5 Quantum field theory and the spin-statistics theorem 235 Problems 239 12 Quantum information 241 12.4 Quantum computing 249 Problems 252 13 The conceptual problems of quantum mechanics 253 13.1 The conceptual problems 253 13.2 Hidden-variable theories 255 13.4 The quantum-mechanical measurement problem 273 13.5 The ontological problem 287 Problems 288 Hints to solution of problems 290 Index 296 www.com Preface to Fourth Edition When I told a friend that I was working on a new edition, he asked me what had changed in quantum physics during the last ten years. In one sense very little: quantum mechanics is a very well established theory and the basic ideas and concepts are little changed from what they were ten, twenty or more years ago. However, new applications have been developed and some of these have revealed aspects of the subject that were previously unknown or largely ignored.
Much of this development has been in the field of information processing, where quantum effects have come to the fore. In particular, quantum techniques appear to have great potential in the field of cryptography, both in the coding and possible de-coding of messages, and I have included a chapter aimed at introducing this topic. I have also added a short chapter on relativistic quantum mechanics and introductory quantum field theory. This is a little more advanced than many of the other topics treated, but I hope it will be accessible to the interested reader.
It aims to open the door to the understanding of a number of points that were previously stated without justification. Once again, I have largely re-written the last chapter on the conceptual foundations of the subject. The twenty years since the publication of the first edition do not seem to have brought scientists and philosophers significantly closer to a consensus on these problems. However, many issues have been considerably clarified and the strengths and weaknesses of some of the explanations are more apparent.
My own understanding continues to grow, not least because of what I have learned from formal and informal discussions at the annual UK Conferences on the Foundations of Physics. Other changes include a more detailed treatment of tunnelling in chapter 2, a more gentle transition from the Born postulate to quantum measurement theory in chapter 4, the introduction of Dirac notation in chapter 6 and a discussion of the Bose–Einstein condensate in chapter 10. I am grateful to a number of people who have helped me with this edition. Glenn Cox shared his expertise on relativistic quantum mechanics when he read a draft of chapter 11; Harvey Brown corrected my understanding of the de Broglie–Bohm hidden variable theory discussed in the first part of chapter 13; Demetris Charalambous read a late draft of the whole book and suggested several www.com xi xii Preface to Fourth Edition improvements and corrections.
Of course, I bear full responsibility for the final version and any remaining errors. Modern technology means that the publishers are able to support the book at the web site http://bookmarkphysics. This is where you will find references to the wider literature, colour illustrations, links to other relevant web sites, etc. If any mistakes are identified, corrections will also be listed there.
Readers are also invited to contribute suggestions on what would be useful content. The most convenient form of communication is by e-mail to 0750308397@bookmarkphysics. Finally I should like to pay tribute to Ann for encouraging me to return to writing after some time. Her support has been invaluable.com Preface to Third Edition In preparing this edition, I have again gone right through the text identifying points where I thought the clarity could be improved.
As a result, numerous minor changes have been made. More major alterations include a discussion of the impressive modern experiments that demonstrate neutron diffraction by macroscopic sized slits in chapter 1, a revised treatment of Clebsch–Gordan coefficients in chapter 6 and a fuller discussion of spontaneous emission in chapter 8. I have also largely rewritten the last chapter on the conceptual problems of quantum mechanics in the light of recent developments in the field as well as of improvements in my understanding of the issues involved and changes in my own viewpoint. This chapter also includes an introduction to the de Broglie–Bohm hidden variable theory and I am grateful to Chris Dewdney for a critical reading of this section.com xiii Preface to Second Edition I have not introduced any major changes to the structure or content of the book, but I have concentrated on clarifying and extending the discussion at a number of points.
Thus the discussion of the application of the uncertainty principle to the Heisenberg microscope has been revised in chapter 1 and is referred to again in chapter 4 as one of the examples of the application of the generalized uncertainty principle; I have rewritten much of the section on spin–orbit coupling and the Zeeman effect and I have tried to improve the introduction to degenerate perturbation theory which many students seem to find difficult. The last chapter has been brought up to date in the light of recent experimental and theoretical work on the conceptual basis of the subject and, in response to a number of requests from students, I have provided hints to the solution of the problems at the ends of the chapters. I should like to thank everyone who drew my attention to errors or suggested improvements, I believe nearly every one of these suggestions has been incorporated in one way or another into this new edition.com xv Preface to First Edition Over the years the emphasis of undergraduate physics courses has moved away from the study of classical macroscopic phenomena towards the discussion of the microscopic properties of atomic and subatomic systems. As a result, students now have to study quantum mechanics at an earlier stage in their course without the benefit of a detailed knowledge of much of classical physics and, in particular, with little or no acquaintance with the formal aspects of classical mechanics.
This book has been written with the needs of such students in mind. It is based on a course of about thirty lectures given to physics students at the University of Birmingham towards the beginning of their second year—although, perhaps inevitably, the coverage of the book is a little greater than I was able to achieve in the lecture course. I have tried to develop the subject in a reasonably rigorous way, covering the topics needed for further study in atomic, nuclear, and solid state physics, but relying only on the physical and mathematical concepts usually taught in the first year of an undergraduate course. On the other hand, by the end of their first undergraduate year most students have heard about the basic ideas of atomic physics, including the experimental evidence pointing to the need for a quantum theory, so I have confined my treatment of these topics to a brief introductory chapter.
While discussing these aspects of quantum mechanics required for further study, I have laid considerable emphasis on the understanding of the basic ideas and concepts behind the subject, culminating in the last chapter which contains an introduction to quantum measurement theory. Recent research, particularly the theoretical and experimental work inspired by Bell’s theorem, has greatly clarified many of the conceptual problems in this area. However, most of the existing literature is at a research level and concentrates more on a rigorous presentation of results to other workers in the field than on making them accessible to a wider audience. I have found that many physics undergraduates are particularly interested in this aspect of the subject and there is therefore a need for a treatment suitable for this level.
The last chapter of this book is an attempt to meet this need. I should like to acknowledge the help I have received from my friends and colleagues while writing this book. I am particularly grateful to Robert Whitworth, who read an early draft of the complete book, and to Goronwy Jones and George Morrison, who read parts of it. They all offered many valuable and www.