Basic quantum mechanics

216 Pages·1982·14.772 MB·English

by  J. M. Cassels FRS (auth.)

Basic quantum mechanics
Basic quantum
mechanics
J. M. Cassels, FRS
Lyon Jones Professor of Physics
University of Liverpool, Liverpool, UK
Second edition
M
© J. M. Cassels 1982
Softcover reprint of the hardcover 1st edition 1982
All rights reserved. No part of this publication may be reproduced or transmitted, in
any form or by any means, without permission
First edition published 1970 by
McGraw-Hill Ltd
Second edition published 1982 by
THE MACMILLAN PRESS LTD
London and Basingstoke
Companies and representatives throughout the world
ISBN 978-0-333-31768-6 ISBN 978-1-349-16671-8 (eBook)
DOI 10.1007/978-1-349-16671-8
The paperback edition of the book is sold subject to the condition that it shall not, by
way of trade or otherwise, be lent, resold, hired out, or otherwise circulated without
the publisher's prior consent in any form of binding or cover other than that in which
it is published and without a similar condition including this condition being imposed
on the subsequent purchaser.
Contents
Preface to the second edition IX
Preface to the first edition XI
List of abbreviations X Ill
1 Fundamentals of particle quantum mechanics 1
§ 1 Introduction 1
§ 2 Basic assumptions 8
Problems 14
2 Mathematical and physical development of the basic
assumptions 16
§ 3 Solutions to Schrodinger's equation 16
§ 4 Mathematical properties of wave functions and
operators 19
§ 5 General properties of quantum mechanics 30
Problems 35
3 One-dimension applications 38
§ 6 Free particles: momentum probability density 38
§ 7 Solutions of the TISE 46
§ 8 The linear harmonic oscillator 59
Problems 66
4 Three-dimensional applications 71
§ 9 Free particles 71
§ 10 Orbital angular momentum 74
§ 11 Central potentials 87
§ 12 The hydrogen atom 88
v
vi CONTENTS
§ 13 The three-dimensional harmonic oscillator 94
§ 14 The deuteron 97
Problems 101
S Approximate methods for stationary states 103
§ 15 Perturbation theory 103
§ 16 Hydrogen atom in an electric field 105
§ 17 The variational method 107
§ 18 The WKB approximation 109
Problems 115
6 Magnetic fields and spin angular momentum 116
§ 19 Review of classical mechanics: general rules for
quantisation 116
§ 20 Magnetic fields 118
§ 21 Spin angular momentum 122
§ 22 Combination of orbital and spin angular momenta 124
§ 23 Spin-orbit coupling and fine structure: Zeeman
effects 128
Problems 134
7 Identical particles and the Pauli principle 135
§ 24 Identical particles 135
Problems 141
8 Transitions 142
§ 25 Spin precession and magnetic resonance 142
§ 26 Transitions caused by a perturbation independent of
time 147
§ 27 Nuclear beta decay 155
§ 28 Radioactivity 158
Problems 162
9 Scattering 163
§ 29 Introduction: analysis into partial waves of definite
angular momentum 163
§ 30 Neutron-proton scattering at low energies 170
§ 31 Resonant scattering 17 4
§ 32 The Born approximation 179
Problems 183
CONTENTS vii
10 Radiation 185
§ 33 Quantisation of the radiation field 185
§ 34 The 2p-1 s radiative transition in the hydrogen atom 192
Problems 200
Appendix 201
Index 203
Preface to the second edition
I am glad to have the opportunity to write a preface to a second
edition of Basic Quantum Mechanics. The first edition has been out
of print for a few years but, with the permission of the publishers, it
has been circulating vigorously among students in photocopied
form. This has encouraged me to believe that it would be worth
while to produce an improved version of the book.
There are two major changes that I have made. In chapter 3 there
is now a much more thorough account of the relationship between
potential energy curves, energy levels, and wave functions. A diffi
culty here is that the discussion rests on the WKB approximation, an
analysis of which would hold up progress at a point at which speedy
movement is vital. My solution is to state and use the important
features of the WKB approximation without proofs, which come
later-in the same place as in the first edition, chapter 5. Since the
statements required are few and succinct I believe this procedure
will be acceptable.
The second major change is to the treatments of radioactive states
and the related resonant scattering, in chapters 8 and 9. It has
always surprised me to notice how many physicists spend careers
measuring widths and inferring lifetimes without asking for proofs of
the formulae required. Certainly few elementary books on quantum
mechanics are of any use to them in this respect. In the first edition I
worked out decay and scattering for the same particular potential,
pointed out the relationship of interest, and stated the general
extension. That was better than nothing, but not really satisfactory.
Now in this second edition I have given a more general treatment,
based on the S-matrix, which owes much to research papers by Ning
Hu and by Peierls, published thirty-three and twenty years ago,
ix
X PREFACE TO THE SECOND EDITION
respectively. I intend the version in this book to be comprehensible
at the transition between British undergraduate and postgraduate
studies, like the treatments of other advanced topics.
I have also made many minor changes that are intended to
improve the balance of the argument and to smooth the reader's
path. Referring to the preface to the first edition, I cannot yet claim
that the book is as easy to read as a novel, but perhaps it is now
more nearly so.
I have changed over to SI units throughout. This seems to be
mandatory now, but I regret the clumsiness that has been intro
duced into many of the formulae thereby.
Finally I should thank Professor Sir Rudolf Peierls, and Drs
Allcock and Huby for valuable advice, Dr Wormald for computa
tional help, Misses Calland and Owen for typing the manuscript,
and Mrs Cheyne for tracing the diagrams. I thank also Dr Carroll
for help with the proofs.
Liverpool, 1981 J. M. CASSELS
Preface to the first edition
When I first learnt about quantum mechanics, I thought it was
horrible. The trouble was that the course was taught in a semi
historical spirit, with a conscious attempt to bring in new ideas only
very gradually. Every calculation involved further tinkering with the
rules, until a point was reached when I had lost all confidence that a
stable new view of physics would be achieved. It was a great relief in
the following year to attend the lectures of Professor Dirac, which
followed essentially the plan of his evergreen book, The Principles of
Quantum Mechanics (Oxford University Press). I learnt at last that
quantum mechanics had an extremely clear and logical structure,
and that classical mechanics could be seen as a limiting case, in a
very beautiful way.
My lectures to third year Honours Physics students at Liverpool
have therefore followed the logical approach, albeit at a more
elementary level than Professor Dirac's. This book follows the same
plan at the same level; all the new concepts are in chapter 1 and the
rest of the book simply follows the consequences.
A commitment to the logical approach also seems to me to make
it advisable to give a full account of the mathematics. I have
therefore tried very hard to avoid simply quoting 'well-known'
mathematical results which in fact might well be encountered for the
first time.
I should make it clear that the book is about mechanics, and
makes no claim to be a connected account of elementary atomic,
nuclear, or particle physics. I have provided examples from these
fields, however, to illustrate every important point in the discussion.
I am aware that my emphasis on logic and basic structure, on full
presentation of mathematical argument, and on overall brevity has
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