Quantum Mechanics 2nd edition -PEARSON Education We work with leading authors to develop the strongest educational materials in physics, bringing cutting-edge thinking and best learning practice to a global market. Under a range of well-known imprints, including Prentice Hall, we craft high quality print and electronic publications which help readers to understand and apply their content, whether studying or at work. To find out more about the complete range of our publishing please visit us on the World Wide Web at: www.uk Quantum Mechanics 2nd edition B. Joachain PEARSON ------------ Prentice Hall Harlow, England • London • New York • Boston • San Francisco • Toronto • Sydney • Singapore • Hong Kong Tokyo • Seoul • Taipei • New Delhi • Cape Town • Madrid • Mexico City • Amsterdam • Munich.
Milan Pearson Education Limited Edinburgh Gate Harlow Essex CM20 2JE England and Associated Companies throughout the world Visit us on the World Wide Web at: www.uk First published under the Longman Scientific & Technical imprint 1989 Second edition 2000 © Pearson Education Limited 1989, 2000 The rights of B. Joachain to be identified as the authors of this Work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any fonn or by any means, electronic, mechanical, photocopying, recording or otherwise, without either the prior written pennission of the publisher or a licence pennitting restricted copying in the United Kingdom issued by the Copyright Licensing Agency Ltd, 90 Tottenham Court Road, London WIT 4LP.
ISBN-IO: 0-582-35691-1 ISBN-13: 978-0-582-35691-7 British Library Cataloguing-in-Publication Data A catalogue record for this book can be obtained from the British Library Library of Congress Cataloging-in-Publication Data Bransden, B. of: Introduction to quantum mechanics. Includes bibliographical references and index., 1926- Introduction to quantum mechanics. I 2-dc2 I 99-055742 10 9 876 0706 Typeset in Times at 10pt by 56.
Produced by Pearson Education Asia Pte Ltd., Printed in Great Britain by Henry Ling Limited, at the Dorset Press, Dorchester, DTI IHD. Contents Preface to the Second Edition xi Preface to the First Edition xiii Acknowledgements xiv 1 The origins of quantum theory 1 1.1 Black body radiation 2 1.2 The photoelectric effect 12 1.3 The Compton effect 16 1.4 Atomic spectra and the Bohr model of the hydrogen atom 19 1.5 The Stern-Gerlach experiment. Angular momentum and spin 33 1.6 De Broglie's hypothesis. Wave properties of matter and the genesis of quantum mechanics 38 Problems 45 2 The wave function and the uncertainty principle 51 2.1 Wave-particle duality 52 2.2 The interpretation of the wave function 56 2.3 Wave functions for particles having a definite momentum 58 2.5 The Heisenberg uncertainty principle 69 Problems 76 3 The Schrodinger equation 81 3.1 The time-dependent Schrodinger equation 82 3.2 Conservation of probability 86 3.3 Expectation values and operators 90 3.4 Transition from quantum mechanics to classical mechanics.
The Ehrenfest theorem 97 3.5 The time-independent Schrodinger equation.7 Properties of the energy eigenfunctions 115 v vi II Contents 3.8 General solution of the time-dependent Schrodinger equation for a time-independent potential 120 3.9 The Schrodinger equation in momentum space 124 Problems 128 4 One-dimensional examples 133 4.2 The free particle 134 4.3 The potential step 141 4.4 The potential barrier 150 4.5 The infinite square well 156 4.6 The square well 163 4.7 The linear harmonic oscillator 170 4.8 The periodic potential 182 Problems 189 5 The formalism of quantum mechanics 193 5.1 The state of a system 193 5.2 Dynamical variables and operators 198 5.3 Expansions in eigenfunctions 203 5.4 Commuting observables, compatibility and the Heisenberg uncertainty relations 210 5.6 Matrix representations of wave functions and operators 220 5.7 The Schrodinger equation and the time evolution of a system 231 5.8 The Schrodinger and Heisenberg pictures 238 5.10 Symmetry principles and conservation laws 245 5.11 The classical limit 256 Problems 260 6 Angular momentum 265 6.1 Orbital angular momentum 266 6.2 Orbital angular momentum and spatial rotations 270 6.3 The eigenvalues and eigenfunctions of L2 and L z 275 6.4 Particle on a sphere and the rigid rotator 289 6.5 General angular momentum. The spectrum of J2 and Jz 292 Contents .6 Matrix representations of angular momentum operators 296 6.7 Spin angular momentum 299 6.8 Spin one-half 303 6.9 Total angular momentum 311 6.10 The addition of angular momenta 315 Problems 323 7 The Schrodinger equation in three dimensions 327 7.1 Separation of the Schrodinger equation in Cartesian coordinates 328 7. Separation of the Schrodinger equation in spherical polar coordinates 336 7.3 The free particle 341 7.4 The three-dimensional square well potential 347 7.5 The hydrogenic atom 351 7.6 The three-dimensional isotropic oscillator 367 Problems 370 8 Approximation methods for stationary problems 375 8.1 Time-independent perturbation theory for a non-degenerate energy level 375 8.2 Time-independent perturbation theory for a degenerate energy level 386 8.3 The variational method 399 8.4 The WKB approximation 408 Problems 427 9 Approximation methods for time-dependent problems 431 9.1 Time-dependent perturbation theory.2 Time-independent perturbation 435 9.4 The adiabatic approximation 447 9.5 The sudden approximation 458 Problems 466 10 Several- and many-particle systems 469 10.2 Systems of identical particles 472 10.3 Spin-l/2 particles in a box. The Fermi gas 478 viii • Contents 10.4 Two-electron atoms 485 10.5 Many-electron atoms 492 10.7 Nuclear systems 506 Problems 511 11 The interaction of quantum systems with radiation 515 11.1 The electromagnetic field and its interaction with one-electron atoms 516 11.2 Perturbation theory for harmonic perturbations and transition rates 522 11.4 Selection rules for electric dipole transitions 533 11.5 lifetimes, line intensities, widths and shapes 538 11.6 The spin of the photon and helicity 544 11.8 Photodisintegration 550 Problems 555 12 The interaction of quantum systems with external electric and magnetic fields 557 12.1 The Stark effect 557 12.2 Interaction of particles with magnetic fields 563 12.3 One-electron atoms in external magnetic fields 574 12.4 Magnetic resonance 576 Problems 585 13 Quantum collision theory 587 13.1 Scattering experiments and cross-sections 588 13.3 The method of partial waves 595 13.4 Applications of the partial-wave method 599 13.5 The integral equation of potential scattering 608 13.6 The Born approximation 615 13.7 Collisions between identical particles 620 13.8 Collisions involving composite systems 627 Problems 635 Contents • ix 14 Quantum statistics 641 14.1 The density matrix 642 14.2 The density matrix for a spin-l/2 system.3 The equation of motion of the density matrix 654 14.4 Quantum mechanical ensembles 655 14.5 Applications to single-particle systems 661 14.6 Systems of non-interacting particles 663 14.7 The photon gas and Planck's law 667 14.8 The ideal gas 668 Problems 676 15 Relativistic quantum mechanics 679 15.1 The Klein-Gordon equation 679 15.2 The Dirac equation 684 15.3 Covariant formulation of the Dirac theory 690 15.4 Plane wave solutions of the Dirac equation 696 15.5 Solutions of the Dirac equation for a central potential 702 15.6 Non-relativistic limit of the Dirac equation 711 15.7 Negative-energy states.
Hole theory 715 Problems 717 16 Further applications of quantum mechanics 719 16.1 The van der Waals interaction 719 16.2 Electrons in solids 723 16.3 Masers and lasers 735 16.4 The decay of K-mesons 746 16.5 Positronium and charmonium 753 17 Measurement and interpretation 759 17.2 The Einstein-Podolsky-Rosen paradox 760 17.4 The problem of measurement 766 17.5 Time evolution of a system. Discrete or continuous? 772 A Fourier integrals and the Dirac delta function 775 A.1 Fourier series 775 x fI Contents A.2 Fourier transforms 776 B WKB connection formulae 783 References 787 Table of fundamental constants 789 Table of conversion factors 791 Index 793 Preface to the Second Edition The purpose of this book remains as outlined in the preface to the first edition: to provide a core text in quantum mechanics for students in physics at undergraduate level. It has not been found necessary to make major alterations to the contents of the book. However, we have taken advantage of the opportunity provided by the preparation of a new edition to make a number of minor improvements throughout the text, to introduce some new topics and to include a new chapter on relativistic quantum mechanics.
This inclusion stems from a reconsideration of our earlier decision to exclude this material. We believe that a significant number of core courses now include an introduction to relativistic quantum mechanics; this is the subject of the new chapter (Chapter 15). Among the other important changes are the inclusion of the Feynman path integral approach to quantum mechanics (Chapter 5), a discussion of the Berry phase (Chapter 9) with applications (Chapters 10 and 12), an account of the Aharonov-Bohm effect (Chapter 12) and a discussion of quantumjumps (Chapter 17). We have also included the integral equation of potential scattering in our treatment of quantum collision theory (Chapter 13) and have given a more extended discussion of Bose-Einstein condensation in Chapter 14.
It is a pleasure to acknowledge the many helpful comments made to us by colleagues who have used the first edition of this book. Their remarks have been of great benefit to us in preparing this new edition. One of us (CJJ) would like to thank Professor H. Walther for his hospitality at the Max-Planck-Institut fur Quantenoptik in Garching, where part of this work was carried out.
We also wish to thank Mrs R. Lareppe for her expert and careful typing of the manuscript. Joachain, Brussels August 1999 Preface to the First Edition The study of quantum mechanics and its applications pervades much of the modern undergraduate course in physics. Virtually all undergraduates are expected to become familiar with the principles of non-relativistic quantum mechanics, with a variety of approximation methods and with the application of these methods to simple systems occurring in atomic, nuclear and solid state physics.
This core material is the subject of this book. We have finnly in mind students of physics, rather than of mathematics, whose mathematical equipment is limited, particularly at the beginning of their studies. Relativistic quantum theory, the application of group theoretical methods and many-body techniques are usually taught in the fonn of optional courses and we have made no attempt to cover more advanced material of this nature. Although a fairly large number of examples drawn from atomic, nuclear and solid state physics are given in the text, we assume that the reader will be following separate systematic courses on those subjects, and only as much detail as necessary to illustrate the theory is given here.
Following an introductory chapter in which the evidence that led to the development of quantum theory is reviewed, we develop the concept of a wave function and its interpretation, and discuss Heisenberg's uncertainty relations. Chapter 3 is devoted to the Schrodinger equation and in the next chapter a variety of applications to one- dimensional problems is discussed. The next three chapters deal with the fonnal development of the theory, the properties of angular momenta and the application of Schrodinger's wave mechanics to simple three-dimensional systems. Chapters 8 and 9 deal with approximation methods for time-independent and time- dependent problems, respectively, and these are followed by six chapters in which the theory is illustrated through application to a range of specific systems of fundamental importance.
These include atoms, molecules, nuclei and their interaction with static and radiative electromagnetic fields, the elements of collision theory and quantum statistics. Finally, in Chapter 17, we discuss briefly some of the difficulties that arise in the interpretation of quantum theory. Problem sets are provided covering all the most important topics, which will help the student monitor his understanding of the theory. We wish to thank our colleagues and students for numerous helpful discussions and suggestions.
Particular thanks are due to Professor A. Castoldi and Dr J. It is also a pleasure to thank Miss P. Pean and Mrs M.
Leclercq for their patient and careful typing of the manuscript, and Mrs H. Joachain-Bukowinski and Mr C. Depraetere for preparing a large number of the diagrams.