Advanced Quantum Theory, Third Edition

^•^H

QJJANTUM THEORY THIRD EDITION

Michael D S c a d r o n

ADVANCED- 'q QJJANTUM- ^ THEORY*^

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University of Arizona, USA

N E W JERSEY • L O N D O N • S I N G A P O R E

• BEIJING

• SHANGHAI

• HONG KONG • T A I P E I •

CHENNAI

Published by Imperial College Press 57 Shelton Street Covent Garden London WC2H9HE and World Scientific Publishing Co. Pte. Ltd. 5 TohTuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

We thankfully acknowledge Springer-Verlag for releasing the copyright of the original editions. Professor Michael Scadron has made corrections to the text and included a new Chapter 17. We regret that Professor Scadron's health condition was not ideal at the time of publication to finish the Preface to this new edition.

ADVANCED QUANTUM THEORY (3rd Edition) Copyright © 2007 by Imperial College Press and World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

ISBN 981-270-050-1

Printed in Singapore by World Scientific Printers (S) Pte Ltd

To my wife and daughters,

Arlene, Kari, and Lisa

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Preface to the Second Edition

For the past five years, my editor at Springer-Verlag has asked me to write a second edition of this text that would incorporate new material on the quark model. Because this is a subject at the forefront of modern physics, whose central ideas are perpetually in flux, such an addition is not a simple task. Nevertheless, I have tried to discuss quark model topics that should stand the test of time and be of interest to introductory advanced quantum mechanics students as examples of the Feynman diagram technique. I have also tried to eliminate errors made in the first edition. I appreciate the work of R. Miller, who graciously typed the additional material. My colleagues V. Elias, T. Hakioglu, S. Kocic, N. Paver, and R. Thews helped me formulate the quark model chapter. Tucson, Arizona May 1990

M. D. Scadron

vu

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Preface to the First Edition

The fundamental goal of physics is an understanding of the forces of nature in their simplest and most general terms. Yet the scientific method inadvertently steers us away from that course by requiring an ever finer subdivision of the problem into constituent components, so that the overall objective is often obscured, even to the experts. The situation is most frustrating and acute for today's graduate students, who must try to absorb as much general knowledge as is possible and also try to digest only a small fraction of the ever increasing morass of observational data or detailed theories to write a dissertation. This book is based on the premise that to study a subject in depth is only half the battle; the remaining struggle is to put the pieces together in a broad but comprehensive manner. Accordingly, the primary purpose of this text is to cut across the barriers existing between the various fields of modern physics (elementary particles; nuclear, atomic, and solid state physics; gravitation) and present a unified description of the quantum nature of forces encountered in each field at the level of the second-year physics graduate student. This unification is based on one-body perturbation techniques, covariantly generalized to what are now called "Feynman diagrams," and is formulated as a simple (but nontrivial) extension of ordinary nonrelativistic, one-particle quantum theory. No attempt will be made, however, to cover all or even the major topics in each field. Only those subjects will be discussed which, in the author's opinion, serve best to illustrate the utility of the Feynman-diagram approach and have a direct bearing on the fundamental forces. IX

x Preface to the First Edition

In presenting the two themes of scattering diagrams and the fundamental forces, this book is divided into three parts. Part I is devoted to transformation theory, which, while admittedly somewhat formal, is the most streamlined way to review the concepts of nonrelativistic quantum mechanics and to formulate the relativistic Klein-Gordon, Maxwell, and Dirac wave equations for relativistic spin-0, massless spin-l, and spin4 particles, respectively. The language of group theory is used to write relativistic Lorentz transformations in a form similar to ordinary rotations and to describe the important discrete symmetries of C, P, and T. In Part II, quantum mechanics is reformulated in the language of scattering theory, with the momentum-space S-matrix replacing the coordinate-space hamiltonian as the central dynamical operator. Nonrelativistic perturbation scattering diagrams are then developed, and simple applications given for nuclear, atomic, and solid-state scattering problems. Part III deals with relativistic scattering diagrams built up from covariant Feynman propagators and vertices in a manner consistent with the CPT theorem. The theory is systematically applied to the lowest-order fundamental electromagnetic, strong, weak, and gravitational interactions. Finally, the last chapter surveys the use of higherorder Feynman diagrams to explain more detailed aspects of quantum electrodynamics (QED) and strong-interaction elementary-particle physics. Throughout the text the notion of currents is used to exploit the underlying symmetries and dynamical interactions of the various quantum forces. The material in Parts I and II is usually described as advanced quantum mechanics, while the topics treated in Part III often appear in texts on field theory or elementary-particle physics. It is the author's opinion, however, that every graduate student in physics should know most of the subjects covered in this book and that these topics can best be understood in proper context when presented as an integral whole. This material can be reasonably absorbed when introduced as a second-year course in quantum mechanics, with problems such as those given at the conclusion of this book thoroughly integrated into the dialogue. It is intended that this survey of advanced, one-particle scattering quantum mechanics and Feynman diagrams will provide adequate preparation for further studies in relativistic quantum field theory or nonrelativistic many-body theory. For readers with no intention of pursuing modern quantum theory in any greater depth, it is hoped that a mastery of the material presented in this book will give them an intuitive feel for the meaning and power of the method as well as an understanding of the quantum nature of forces. At the very least, serious readers should be able to calculate simple diagrams, lifetimes, and cross sections correctly—because, according to the teaching of a famous physicist, "If you do not get the factors of 2n straight, you do not understand anything." The author wishes to acknowledge the kind indulgence of students in past years at the University of Arizona and Northwestern University, where much of this material was presented in somewhat unpolished form. In particular, D. Jones, T. MacMullen, B. Wilde, J. Mich, and D. Eimerol have

Preface to the First Edition xi

detected many misprints in the preliminary notes. Moreover, this work could not have been satisfactorily completed without the careful reading of the entire manuscript by Professor R. Delbourgo, much of it by Dr. H. F. Jones and various chapters by Drs. P. McNamee, P. Herczeg, C. Isham, S. Coon, K. Stelle, R. Thews, R. Young, and Professors J. D. Garcia, T. Kibble, M. Moravcsik, R. Parmenter, and P. Winternitz, all of whom offered many useful suggestions and improvements. Last but not least the author is grateful for the excellent typing and drafting support given by Barbara Bickel, Ann Williams, and Diane Hartman, the latter two of the University of Arizona, Sarah Cowen of Imperial College and W. Beiglbock and the staff at Springer-Verlag for the outstanding technical skill and speed with which they transformed the manuscript into the reality of this book. Tucson, Arizona August 1979

M. D. Scadron

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Contents

1

Preface to the Second Edition

vii

Preface to the First Edition

ix

PART I

1

Introduction

l.A l.B l.C l.D

2

Principles of Quantum Mechanics 3 Angular-Momentum Expansions 5 Unitary Operators and Transformation Theory Translations in Time 11

3

9

Transformations in Space

2.A 2.B 2.C 2.D 2.E

3

Transformation Theory

Continuous Symmetry Transformations and Group Theory 15 Translations in Space 17 Rotations in Space 18 The Rotation Group 0(3) 20 Tensor Operators and the Wigner-Eckart Theorem 25

Transformations in Space-Time

3.A 3.B 3.C 3.D

15

29

Lorentz Velocity Transformations 29 Homogeneous Lorentz Group if 33 Inhomogeneous Lorentz Group 9 38 Helicity Formalism 41 xm

xiv Contents 4

Boson Wave Equations

4.A 4.B 4.C 4.D

5

Derivations of the Dirac Equation 64 Covariant Formulation 69 Free-Particle Solutions of the Dirac Equation 74 Dirac Equation in an External Field 81 Wave Equations for Other Fermi Particles 92

Discrete Symmetries

6.A 6.B 6.C 6.D

Charge-Conjugation Transformation 96 Space-Reflection Transformations 103 Time-Reversal Transformation 108 CPT Invariance 112

PART II 7

Scattering Theory

Formal Theory of Scattering

7. A 7.B 7.C 7.D 7.E 7.F 7.G 7.H

8

61

Spin-i Dirac Equation

5.A 5.B 5.C 5.D 5.E

6

Spin-0 Klein-Gordon Equation 48 Spin-1 Wave Equation 55 Spin-1 Maxwell Equation 57 Second Quantization: Photons and Phonons

Formulation of the Scattering Problem 119 Time-Independent Potential Scattering 121 Time-Dependent Scattering in the Schrodinger Picture Time-Dependent Scattering in the Interaction Picture The S-Matrix 133 Transition Probabilities 136 Unitarity and Probability Conservation 139 Bound States in Scattering Language 140

Simple Scattering Dynamics

8.A 8.B 8.C 8.D 8.E 8.F

Partial Waves and Phase Shifts 142 Low-Energy Scattering and Bound States 145 Resonance Scattering, Formation, and Decay 149 Born Approximation in Potential Scattering 153 Form Factors 155 High-Energy Scattering 156

Contents xv Nonrelativistic Perturbation Theory

9.A 9.B 9.C 9.D

Time-Independent Perturbation Theory 159 Time-Dependent Perturbation Theory 161 Electron-Photon Interactions in Atoms 164 Electron-Phonon Interactions in Solids 172

PART III 10

13

202

228

Yukawa Force 228 Isospin 230 One-Pion-Exchange Nucleon-Nucleon Force 234 Low-Energy nN Scattering 240 Hadronic Vector Currents 246 Hadronic Axial-Vector Currents 251

Lowest-Order Weak Interactions

13.A 13.B 13.C 13.D 13.E 13.F 13.G

183

Coulomb Scattering 202 Moller Scattering 208 Bhabha Scattering 211 Compton Scattering 214 Bremsstrahlung and Pair Production 219 Electromagnetic Interactions of Hadrons 223 Static Electromagnetic Potentials 226

Low-Energy Strong Interactions

12.A 12.B 12.C 12.D 12.E 12.F

181

Covariant Kinematics 183 Covariant S-Matrix 188 Covariant Vertices 190 Covariant Feynman Propagators 193 Feynman Rules in Momentum Space 199

Lowest-Order Electromagnetic Interactions

I LA ll.B II .C ll.D ll.E 11.F ll.G

12

Covariant Feynman Diagrams

Covariant Feynman Rules

10.A 10.B 10.C 10.D 10.E 11

159

Phenomenology of Weak Decays 257 Current-Current Hypothesis 259 Muon Decay 263 Neutron 0-Decay 266 Charged-Pion Decay 269 Cabibbo Universality 271 Nonleptonic Decays 275

257

xvi Contents 14

Lowest-Order Gravitational Interactions

14.A 14.B 14.C 14.D 14.E 14.F

15

16

17.F 17.G 17.H 17.1 17.J

347

Flavor, Spin, and Color Symmetry of Quarks 348 Magnetic Moments and Mass Formulae 351 Chiral Symmetry and Nambu-Goldstone Pions 354 Chiral Symmetry and Quark Loops 357 Linear a Model to One-Loop Order 363

Chiral Symmetry Breaking and Pions

17.A 17.B 17.C 17.D 17.E

297

Closed-Loop Diagrams 297 Electron Anomalous Magnetic Moment 302 Self-Energy Loop Diagrams 308 Free-Electron Charge Form Factor 313 Bound-State Lamb Shift 317 Renormalization in Field Theory 323 Dispersion Theory and QED 326 Dispersion Theory and Strong Interactions 337

Quark Model at Low Energies

16.A 16.B 16.C 16.D 16.E

17

Graviton Wave Function and Propagator 281 Graviton Vertices 285 Graviton Spontaneous Emission 287 Quantum Corrections to the Newtonian Force Law 289 Gravitational Light Bending 291 Connection between Quantum Theory of Gravity and General Relativity 293

Higher-Order Covariant Feynman Diagrams

15.A 15.B 15.C 15.D 15.E 15.F 15.G 15.H

281

Nonstrange Constituent Quark Masses 369 ChSB qq Pion Mass 371 ChSB Kaon Masses 371 B.W.Lee's Null Tadpole Sum 372 Dimensional Regularization Lemma/(R. Delbourgo, M.D. Scadron 1995) 373 Dynamically Generating the Scalar Squared Mass 374 Cutoff Version of the LDGE 374 ChSB nNa Term 376 U(3) Scalar Nonet 377 Quark Triangle Graphs 379

368

Contents xvii Problems

383

Appendices I

Units and Conventions

1. 2: 3. 4. 5.

Units 405 Metric and y-Matrices 406 Normalizations 406 Decay Rates and Cross Sections 406 Covariant Integrals and Feynman Parametrization

405

406

II

Rotation-Group Formulae

407

III

Elementary-Particle Zoo

408

Bibliography

409

Books

411

Research Articles

415

Index

421

PART I

TRANSFORMATION THEORY

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CHAPTER 1

Introduction

The fundamentals of nonrelativistic quantum mechanics will be summarized to lay a foundation for understanding more advanced concepts in relativistic quantum mechanics and scattering theory. Dirac's elegant notation is used to highlight the orthogonality and completeness properties of eigenstates of physical observables and to stress the importance of transformation theory through the use of Dirac transformation coefficients and unitary operators. The Schrodinger equation is embedded in a unitary time translation operator, and the Schrodinger, Heisenberg, and Dirac-interaction pictures are reviewed. Notation conventions and units are also discussed in this introductory chapter.

l.A Principles of Quantum Mechanics To begin let us briefly review the principles of nonrelativistic quantum mechanics stated in Dirac notation: i. A physical system is characterized by a state vector | if/} in a normed linear vector space (Hilbert space) with the corresponding coordinate-space wave function t/f(x) = <x|i/r>. The probability density |i/f(x)|2 = ip*(x)\j/(x) > 0 of finding the system at x in state |i/f> has physical significance. In rigorous terms, it can be shown that the absolute square of the norm \tp \2 = = \dx \\JJ{X)\2

(1.1)

and can be chosen as unity for a stable system. 3

4 Introduction

ii. Every physical observable is represented by a linear hermitian operator which, under suitable mathematical restrictions, has real eigenvalues. The hamiltonian operator H with real energy eigenvalues E„ is a case in point: H | tfO = £„!*„>, (1.2) where the eigenvectors |^„> are orthogonal: can govern the state of the system, then the superposition principle guarantees that |i/^> = cv l ^ ) + c2|iA2> does also. In terms of eigenstates, the superposition principle states that | \p} can be represented by the eigenstate expansion

l = Ic n |.A n >,

(1.3)

n

where c„ = . The discrete eigenstates | i/f„> are sometimes written as | n> in analogy with the continuous basis | x>. Since these bases are in no way fundamental, expansion theorems like (1.1) and (1.3) can be used to transform the system from one representation to another. iv. The time development of the state vectors | •/'(f )> is governed by the Schrodinger equation (Schrodinger 1926)

s ih ±m=„|Wl)>.

(u)

An alternative but equivalent picture is the time development of the operators O, given by the Heisenberg equation of motion (Heisenberg 1925) ihd^

= [0, H] = OH- HO.

(1.5)

Unless otherwise specified, we shall assume that operators 0 and H do not contain any explicit time dependence. v. There is an hermitian probability current corresponding to a particle velocity p/m with density j(x, t) = - ^ [>A*(x, tftiHx, t) - (V^*(x, tMx,

t)],

(1.6)

which can be combined with the probability density | i/^(x, t) \2 in a continuity equation j t |^(x,t)| 2 + V - j = 0.

(1.7)

This insures that the physical requirement of probability conservation is satisfied. vi. Spin-j fermions (e.g., electrons) must obey the exclusion principle (Pauli 1925). This corresponds to a wave function describing the combined state of two electrons which is antisymmetric under interchange of the electrons, ^nuni(XU

X2)=

-lAn,,,,^,*!)-

(1-8)

Angular-Momentum Expansions 5

The extension of (1.8) to all pairs of half-integer fermions along with the analogous symmetric property for pairs of identical integer-spin bosons (e.g., photons) under interchange is referred to as "the connection between spin and statistics". Its proof goes beyond the bounds of nonrelativistic quantum mechanics (Pauli 1940), and so we assume it as a basic postulate. Connection with Classical Physics. The foregoing six postulates are sufficient to develop a complete quantum mechanics, the significance of which is clarified by the following principles: a. The correspondence principle (Bohr 1923) notes that the predictions of quantum mechanics approach the classical limit as either h -> 0 or the quantum numbers of bound systems become large, n -* oo. b. The uncertainty principle (Heisenberg 1927) relates the inherent spread of finite wave packets with respect to position or time and in the Fourier-transform variable, Ax A/c > 1, At AOJ > 1 (valid for Maxwell waves as well as Schrodinger waves), with the scales set by h via the de Broglie and Planck relations, p = hk,E = ha. This results in the uncertainty products Ax Ap > h, Af A£ > h. c. The complementarity principle (Bohr 1928) states that any given experiment can probe either the wave or particle nature of radiation or matter, but not both together. These principles are linked to the fundamental postulates via the probability interpretation of \\ji |2. The significance of \\i itself, however, as a material "matter wave" as opposed to a statistical measure of the behavior of an ensemble of particles, is a debated question. Nevertheless it is generally accepted that the physical consequence of the theory is to add to classical physics the four purely quantum effects: uncertainty-principle "jittering", quantum-mechanical "tunneling", discrete energy levels of bound systems, and "exclusion-principle repulsion" for identical fermions or "symmetrization-induced attraction" for identical bosons. Units. We shall henceforth adhere to the convention (unless otherwise stated) that h = c = 1. This reduces the three fundamental units of length, time, and mass to only one, e.g., mass. To convert from one form of this basic unit to another, it will be convenient to use h = 0.658 x 10~ 15 eV-sec, or he = 1973 eV-A = 197.3 MeV-fm, where c = 3 x 1010 cm/sec. We shall also use rationalized electromagnetic units, so that the dimensionless finestructure constant is a = e2/4n = 1/137. See Appendix I.

1.B Angular-Momentum Expansions Transformation Coefficients. Given the postulates of Section LA, it is useful to condense many of the results of nonrelativistic quantum mechanics in terms of the transformation coefficients c„ in (1.3). Consider, for example, the hermitian orbital-angular-momentum operator L = r x p, with eigenstates

6 Introduction | /m> satisfying L2|/m> = /(/+l)|/m>,

(1.9)

L3|/m> = m|/m>

(1.10)

for nonnegative integers /, where the 2/ + 1 integral values of m range from — / to /. These eigenstates obey the formal completeness relation £|/m>