Quantum Mechanics and the particles of nature: An outline for mathematicians

369 Pages·1986·44.931 MB·English

by  Anthony Sudberry

I
Quantum mechanics and
the particles of nature
AN OUTLINE FOR MATHEMATICIANS
ANTHONY SUDBERY
Department of Mathematics, University of York
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CAMBRIDGE UNIVERSITY PRESS
Cambridge
London New York New Rochelle
Melbourne Sydney
Published by the Press Syndicate of the University of Cambridge
The Pitt Building, Trumpington Street, Cambridge CB2 I RP
32 East 57th Street, New York, NY 10022, USA
10 Stamford Road, Oakleigh, Melbourne 3166, Australia
0 Cambridge University Press 1986
First published 1986
Printed in Great Britain at the University Press, Cambridge
British Library cataloguing in publication data
Sudbery, Anthony
Quantum mechanics and the particles of nature.
I. Quantum theory
I. Title
530.1'2 QC174.12
Library of Congress cataloging-in-publication data
Sudbery, Anthony.
Quantum mechanics and the particles of nature.
Bibliography: p.
Includes index.
I. Quantum theory. 2. Particles (Nuclear physics)
I. Title.
QC174.12.S89 1986 530.1'2 85-29124
ISBN 0 521 25891 X hard covers
ISBN 0 521 27765 5 paperback
MP
Contents
Preface xi
1 Particles and forces 1
A. THE ANALYSIS OF MATIER
1.1 Molecules and atoms 1
1.2 Electrons, protons and neutrons 3
1.3 Neutrinos 6
1.4 Antiparticles: baryons and leptons 8
Summary: the first four particles 10
1.5 Quarks and leptons 11
1.6 Observation of particles 16
B. THE ANALYSIS OF FORCE 17
I. 7 Kinds of force 1 7
1.8 Particles of force 19
Further reading 35
Problems 35
2 Quantum statics 36
2.1 Some examples 36
A note on probability 41
2.2 State space 43
2.3 The results of experiments 47
2.4 Observables 52
Operators and matrices 53; Hermitian conjugation 54; Statistical properties of
observables 56; Compatibility of observables 57
2.5 Observables of a particle moving in space 59
A note on Hilbert space 67
2.6 Combined systems 68
Identical particles 70
Appendix: properties of hermitian and unitary operators 74
Bones of Chapter 2 75
Further reading 75
Problems 75
\'Ill Contents
3 Quantum dynamics 78
3.1 The equations of motion 78
I
Rate of change of expectation value 82; The probability current 85; Boundary conditions
for tire Sclrrodinger equation 85
3.2 lnvariances and constants of the motion 88
Parity 95; Time reversal 96; Combined systems 98; Transformation of observables 99;
Actir;e and passh·e transformations /02
3.3 Groups of operations 103
3.4 The Heisenberg picture 109
3.5 Time-dependent perturbation theory Ill
First-order theory 112; The phase-space factor in decay rates 116; Second-order theory
117; Interpretation: transition amplitudes 117: Dyson's form of the perturbation series
119: Exponential decay 120
3.6 Feynman's formulation of quantt,~m mechanics 123
Bones of Chapter 3 127
Further reading 117
Problems 127
4 Some quantum systems 133
4.1 Angular momentum 133
Orbital angular momentum 137: Spin 138; Spin-! particles 139; Rotation operators 139;
The rotation group and SU(2) 141; Intrinsic parity 142: Massless particles 143
4.2 Addition of angular momentum 144
The Wigner-Eckart theorem 147
4.3 Two-particle systems 149
4.4 The hydrogen atom 153
Tire periodic table 157
4.5 The harmonic oscillator 158
The three-dimensional harmonic oscillator 160
4.6 Annihilation and creation operators 162
Bosonic systems 162: Fermionic systems 164; Several types of particle 166: Charge
conjugation 167; Particle-changing interactions 169
Bones of Chapter 4 173
Further reading 174
Problems 17 4
5 Quantum metaphysics 178
5.1 Statistical formulations of classical and quantum mechanics 178
(a) Classical mechanics 178
(b) Quantum mechanics: the statistical operator 180
Combined systems 185
5.2 Quantum theory of measurement 185
Schrodinger's cat paradox 187; Properties of macroscopic apparatus 189; Continuous
observation 191
5.3 Hidden variables and locality 194
The Einstein-Podolsk~Rosen paradox 194; Thede Broglie/Bohm pilot wave theory 196;
Belrs inequalities 198
Contents ix
5.4 Alternative formulations of quantum mechanics 201
Algebraic formulations 201: Quantum logic 203; Superse/ection rules 208; Three-valued
logic 209
5.5 Interpretations of quantum mechanics 210
Some meta-scientific vocabulary 2 10; The peculiarities of quantum mechanics 210; Nine
interprerations 212: Conclusion 224
Bones of Chapter 5 224
Further Reading 225
Problems 225
6 Quantum numbers The properties of particles 227
6.1 Isospin 227
Isospin and charge conjugation 232; Irreducible isospin operators 234; lsospin in
electromagnetic and weak interactions 236
6.2 Strangeness 238
Newral K-mesons 241: CP non-conservation 244
6.3 The eightfold way 244
Quark strucwre 250: SU(4, 5 and 6) 255
6.4 Hadron spectroscopy 256
Charm, beauty and truth 261
6.5 The colour force 265
Colour 265: Gluons 268: Asymptotic freedom and confinement 269: The Zweig rule 271
6.6 The electroweak force 273
Non-conservation of parity 273; The Sa/am-Weinberg Hamiltonian 276: Experimental
confirmation of the Sa/am-Weinberg theory 280
6.7 Further speculations 281
Grand unification 281: Supersymmetry 283: Substructure 285
Further reading 286
Problems 286
7 Quantum fields 290
7.1 Field operators 290
(a) The electromagnetic field 290
Relativistic formulation 294
(b) Second quantisation 297
The Klein-Gordon equation 300
7.2 The Dirac equation 301
Parity 305: Spinor bilinears 305; Second quantisation 309
7.3 Field dynamics 311
In variances and conserved quantities 314; quantum electrodynamics 317: renormalisation
319
7.4 Gauge theories 320
Quantum chromodynamics 324
7.5 Hidden symmetries 325
Self-interacting fields 325; Global symmetry: Goldstone bosons 326; Hidden local
symmetry: the Higgs mechanism 329
X Contents
7.6 Quantum navourdynamics 331
Bones of Chapter 7 334
Further reading 335
Problems 335
Appendix 1: 3-vector and 4-vector algebra 337
Appendix II: Particle properties 340
Appendix Ill: Clebsch--Gordan coefficients 342
Answers to all(/ help with problems 346
Bibliography 350
Index 354
Preface
This book is addressed to the reader who wants, as an educated person, to have
an outline of the present state of knowledge of the constituents of the material
world; who has a logical cast of mind and will follow a mathematical
argument; but who may have little knowledge of physics and no intention of
becoming deeply involved in the subject. In practice I have imagined this
reader as a mathematics student taking a third-year undergraduate course in
quantum mechanics such as is commonly offered as a part of the mathematics
degree course in British universities. Only a minority of such students will be
intending to pursue the subject further, and it seems more appropriate to aim
for a wide survey of the interesting bits than to try to provide a sound basis for
a training as a quantum mechanic.
The emphasis in the book, therefore, is on providing a coherent account of
the basic theoretical concepts of quantum mechanics and particle physics.
Experimental detail, mathematical rigour and calculational facility are all
given lower priority than conceptual coherence. However, I hope that I have
given sufficient experimental reason for every major statement of theory; that
the mathematics is honest, with gaps acknowledged and without the
inconsistencies which can puzzle and dishearten (or arouse the scorn of)
mathematics students; and that there are enough problems at the end of
chapters to enable readers to test their grasp of the concepts.
This approach to the subject has led me to omit several topics which would
normally be included in a quantum mechanics course; for example, there is no
scattering theory and little discussion of the Schrodinger equation as a
differential equation. These topics may be indispensable to anyone who wants
to work in the area, but they are not actually needed in explaining the results of
the research in which they were tools. On the other hand, there are conceptual
problems which can be (and often have been) ignored by the working physicist,
but which seem much more important to the spectator who wants to
understand more of the game. These metaphysical problems often arouse
great interest in students, who find that it is poorly catered for in quantum
mechanics textbooks (perhaps because the discussion is likely to be either
Xll Preface
inconclusive or unconvincing, and quite possibly both). I have devoted a
chapter to such problems; it is indeed inconclusive, and may well be
unconvmcmg.
The reading of mathematics and physics books is hopefully embarked upon
more often than it is successfully completed. This fact of human nature is
allowed for in the structure of this book, which has several points at which a
reader can feel that they have reached the end of a journey. Thereafter the
journey is started again, but in a different craft and at a different level. The first
chapter is a general description of the structure of matter, leading to the
introduction of quarks and leptons, and an account of the first ideas of
quantum mechanics. This material will be familiar to many students, but not
to all; it is included to make the book accessible to mathematics students who
may have studied no physics, or have forgotten what they have studied, and to
meet the complaint that books on particle physics always assume that you
a!r eady know about particles. At the end of this chapter the reader will know
what the particles of matter are, and what forces act between them.
The next two chapters contain the theoretical development of quantum
mechanics, in the state-vector formalism with its standard interpretation. At
the end of Chapter 3 the reader will know the basic assumptions and
theoretical apparatus of quantum theory. Chapter 4 continues the study of
quantum mechanics, but should perhaps be regarded as a prelude to the
remaining chapters, being largely concerned with constructing more
apparatus for later use (angular momentum theory, annihilation and creation
operators), though it also contains the theory of the hydrogen atom as being of
intrinsic interest.
The last three chapters provide three independent journeys, which can be
taken in any order. Chapter 5 goes over the ground of Chapters 2 and 3 again,
examining the concepts of quantum mechanics more critically. This journey
ends in a muddy river delta, the mainstream having split into nine mouths.
Chapter 6 goes over the ground of Chapter 1, the language of quantum
mechanics now being available for a more detailed description of particles.
Annihilation and creation operators are used to give a simplified treatment of
the forces between particles- a kind of quantum field theory without space
and time. Finally, the ideas of quantum field theory proper are described in
Chapter 7. The first half of this chapter is a continuation of the formal
development of quantum mechanics, and carries on directly from Chapter 4.
The second half describes how quantum field theory is applied to particle
physics in quantum chromodynamics and quantum flavourdynamics, and
constitutes a third passage over the ground of particle physics.
Although the development is not formally axiomatic, the book does have a
logical skeleton consisting of postulates (stated as such) and a chain of
propositi-ons (marked by the symbol e) deduced from them. The bones are
listed at the end of each chapter (except Chapters 1 and 6, which are
cartilaginous). The. mathematical arguments are not as rigorous as they might
Preface XIII
be, but the physical arguments are slightly more rigorous than they often are.
The mathematical style is mainly algebraic, being based on commutation
relations: the notion of a wave function is not needed in this logical skeleton.
(The spectrum of the hydrogen atom is found by Pauli's original method,
which predates the Schrodinger equation.} However, since it would be an
impoverished idea of quantum mechanics that did not include wave functions,
and since students arc likely to have met them elsewhere, they are included
from the beginning as an example of a type of state vector, and the usual
assumptions about them (e.g. boundary conditions for the Schrodinger
equation) are justified.
Teachers of quantum mechanics arc divided, and no doubt always will be,
about the suitability of the state-vector formalism for a first course in quantum
mechanics. Those who, like myself, first learnt quantum mechanics by reading
Dirac's immortal Pri11ciples have no doubt that the state-vector formalism is
the best introduction to the subject.ln deference to the other half of the world I
have to say that this book mighr be found difficult by students who have not
taken a first course on quantum mechanics based on wave functions.
Formally, however, it requires no knowledge of any physics. Formally, also,
the only mathematics required is vector algebra and vector calculus; but the
reader with no knowledge of linear algebra will probably find it heavy going,
and an acquaintance with the idea of a group and the elements of analytical
mechanics will be helpful in places. Until Chapter 7 the only fact from special
relativity that is used is the energy-momentum relation ( 1.5); for Chapter 7 the
reader will need the 4-vector formalism and a knowledge of Maxwell's
equations.
Bold type is used to indicate that a word or phrase is being defined, and the
reader is not expected to know what it means. The symbol • denotes the end
of a proof (or a proposition whose proof has already appeared). Complex
conjugation is denoted by an overbar (not by an asterisk).
I would like to thank Mark Lawson, Chris Clarke, Richard Crossley, Peter
Landshoff, Ian Drummond, Jeremy Rogers, Stephen McGahan, Alison
Ramsay. Clifford Bishop, Denis Cronin, Roland Hall, Anne Thompson and
Steve Roberts, all of whom read parts of the manuscript and made useful
suggestions. I am grateful to the Scientific Information Service of CERN,
Geneva, for supplying me with photographs and for their permission to use
them. Finally, I would like to record my appreciation of the sensitive and
patient editorship of Simon Capel in, and the care and forbearance of Sheila
Shepherd and the other staff of Cambridge University Press.
Tony Sudbery
York, July 1985