An Introduction to Groups and Their Matrices for Science Students

339 Pages·2022·27.229 MB·English

by  Robert Kolenkow

AN INTRODUCTION TO
GROUPS
AND THEIR
MATRICES
FOR SCIENCE STUDENTS
T0; = £
1=1
ROBERT KOLENKOW
An Introduction to Groups and Their Matrices
for Science Students
Group theory, originating from algebraic structures in mathematics, has long been a
powerful tool in many areas of physics, chemistry, and other applied sciences, but it
has seldom been covered in a manner accessible to undergraduates. This book from
renowned educator Robert Kolenkow introduces group theory and its applications
starting with simple ideas of symmetry, through quantum numbers, and working up to
particle physics. It features clear explanations, accompanying problems and exercises,
and numerous worked examples from experimental research in the physical sciences.
Beginning with key concepts and necessary theorems, topics are introduced systemat
ically, including molecular vibrations and lattice symmetries; matrix mechanics; wave
mechanics; rotation and quantum angular momentum; atomic structure; and finally
particle physics. This comprehensive primer on group theory is ideal for advanced
undergraduate topics courses, reading groups, or self-study, and it will help prepare
graduate students for higher-level courses.
Robert Kolenkow was formerly Associate Professor of Physics at Massachusetts
Institute of Technology and was awarded the Everett Moore Baker Award for
Outstanding Teaching. He was the lead author of Physical Geography Today and
coauthor, with Daniel Kleppner, of An Introduction to Mechanics (also published by
Cambridge University Press).

AN INTRODUCTION TO
GROUPS AND THEIR
MATRICES FOR SCIENCE
STUDENTS
ROBERT KOLENKOW
Formerly Massachusetts Institute of Technology
НЯ Cambridge
UNIVERSITY PRESS
Cambridge
UNIVERSITY PRESS
University Printing House, Cambridge CB2 8BS, United Kingdom
One Liberty Plaza, 20th Floor, New York, NY 10006, USA
477 Williamstown Road, Port Melbourne, VIC 3207, Australia
314 321, 3rd Floor, Plot 3, Splendor Forum, Jasola District Centre,
New Delhi 110025, India
103 Penang Road, #05-06/07, Visioncrest Commercial, Singapore 238467
Cambridge University Press is part of the University of Cambridge.
It furthers the University’s mission by disseminating knowledge in the pursuit of
education, learning, and research at the highest international levels of excellence.
www.cambridge.org
Information on this title: www.cambridge.org/9781108831086
DOI: 10.1017/9781108923217
© Robert Kolenkow 2022
This publication is in copyright. Subject to statutory exception
and to the provisions of relevant collective licensing agreements,
no reproduction of any part may take place without the written
permission of Cambridge University Press.
First published 2022
A catalogue record for this publication is available from the British Library.
Library of Congress Cataloging-in-Publication Data
Names: Kolenkow, Robert J , author.
Title: An introduction to groups and their matrices for science students /
Robert Kolenkow.
Description: Cambridge ; New York, NY : Cambridge University Press, 2022. |
Includes bibliographical references and index.
Identifiers: LCCN 2021051448 (print) | LCCN 2021051449 (ebook) | ISBN
9781108831086 (hardback) | ISBN 9781108923217 (ebook)
Subjects: LCSH: Group theory. | Matrices. | Science-Mathematics. | BISAC:
SCIENCE / Physics / Mathematical & Computational
Classification: LCC QA174.2 .K64 2022 (print) |LCC QA174.2 (ebook) | DDC
512/.2-dc23/eng/20220128
LC record available at https://lccn.loc.gov/2021051448
LC ebook record available at https://lccn.loc.gov/2021051449
ISBN 978-1-108-83108-6 Hardback
Cambridge University Press has no responsibility for the persistence or accuracy of
URLs for external or third-party internet websites referred to in this publication
and does not guarantee that any content on such websites is, or will remain,
accurate or appropriate.
To Marcia
my help and support

Contents
Preface xi
1 Fundamental Concepts 1
1.1 Introduction ............................................................................................ 1
1.2 Operations ................................................................................................ 2
1.3 What Is a Group? ................................................................................... 8
1.4 Examples of Groups ................................................................................ 9
1.5 Matrix Representations of Groups ...................................................... 12
1.6 Matrix Algebra ...................................................................................... 15
1.7 Special Matrices ...................................................................................... 18
1.8 A Brief History of Group Theory ......................................................... 20
1.9 Brief Bios ............................................................................................... 21
Summary of Chapter 1 ...................................................................................... 21
Problems and Exercises ...................................................................................... 22
2 Matrix Representations of Discrete Groups 26
2.1 Introduction ............................................................................................. 26
2.2 Basis Functions and Representations ................................................... 26
2.3 Similarity Transformations ................................................................... 30
2.4 Equivalent Representations ................................................................... 31
2.5 Similarity Transformations and Unitary Matrices ............................. 33
2.6 Character and Its Invariance under Similarity Transformations . . . 34
2.7 Irreducible Representations ................................................................... 36
2.8 Kronecker (Direct) Product ................................................................... 48
2.9 Kronecker Sum ....................................................................................... 51
Summary of Chapter 2 ...................................................................................... 51
Stated Theorems in Chapter 2 ............................................................................ 52
Prob lems and Exercises .................................................................................. 53
3 Molecular Vibrations 59
3.1 Introduction ............................................................................................. 59
vii
viii Contents
3.2 Oscillating Systems and Newton’s Laws ............................................. 59
3.3 Normal Modes and Group Theory ....................................................... 63
3.4 Normal Modes of a Water Molecule ................................................... 65
3.5 Visualizing Normal Modes .................................................................... 69
3.6 Infrared (IR) Spectroscopy .................................................................... 74
3.7 Raman Spectroscopy ............................................................................. 78
3.8 Brief Bios ................................................................................................ 81
Summary of Chapter 3 ...................................................................................... 81
Pro blems and Exercises ................................................................................ 83
4 Crystalline Solids 87
4.1 Introduction ............................................................................................. 87
4.2 Bravais Lattices ....................................................................................... 87
4.3 X Ray Crystallography .......................................................................... 90
4.4 Fourier Transform ................................................................................... 92
4.5 Reciprocal Lattice ................................................................................... 94
4.6 Lattice Translation Group ....................................................................... 98
4.7 Crystallographic Point Groups and Rotation Symmetry .................... 102
4.8 Crystallographic Space Groups and the Seitz Operator ....................... 104
4.9 Crystal Symmetry Operations ................................................................ 105
4.10 Lattice Vibrations ................................................................................... 108
4.11 Brief Bios ................................................................................................ 114
Summary of Chapter 4 ...................................................................................... 115
Problems and Exercises ...................................................................................... 117
5 Bohr’s Quantum Theory and Matrix Mechanics 121
5.1 Introduction ............................................................................................. 121
5.2 Bohr’s Model .......................................................................................... 121
5.3 Matrix Mechanics ................................................................................... 125
5.4 Matrix Mechanics Quantization ............................................................. 128
5.5 Consequences of Matrix Mechanics .................................................... 131
5.6 Heisenberg Uncertainty Relation .......................................................... 134
5.7 Brief Bios ................................................................................................ 136
Summary of Chapter 5 ...................................................................................... 137
Problems and Exercises ...................................................................................... 138
6 Wave Mechanics, Measurement, and Entanglement 140
6.1 Introduction ............................................................................................. 140
6.2 Schrodinger’s Wave Mechanics................................................................140
6.3 The Wave Equation ................................................................................ 142
6.4 Quantization Conditions in Wave Mechanics ....................................... 147
6.5 Matrix Diagonalization .......................................................................... 154
6.6 Quantum Measurement .......................................................................... 156
6.7 The EPR Paradox and Entanglement .................................................... 157
Contents ix
6.8 Brief Bio ................................................................................................ 163
Summary of Chapter 6 ...................................................................................... 163
Problems and Exercises ...................................................................................... 164
7 Rotation 168
7.1 Introduction ............................................................................................. 168
7.2 Two Ways of Looking at Rotation ....................................................... 168
7.3 Rotation of a Function .......................................................................... 172
7.4 The Axial Rotation Group ................................................................... 173
7.5 The U(1) and SU(2) Groups ................................................................ 176
7.6 Pauli Matrices, SU(2), and Rotation ................................................... 179
7.7 Euler Angles ............................................................................................. 182
7.8 Finite Rotations Don’t Commute .......................................................... 184
7.9 . . . But Rotations Do Commute to First Order .................................... 185
7.10 Brief Bios ................................................................................................ 185
Summary of Chapter 7 ...................................................................................... 186
Problems and Exercises ...................................................................................... 187
8 Quantum Angular Momentum 189
8.1 Introduction ............................................................................................. 189
8.2 Stern and Gerlach: An Important Experiment (1922) .......................... 189
8.3 Rotation and Angular Momentum Operators ....................................... 193
8.4 Commutation Relations .......................................................................... 196
8.5 The Axial Rotation Group Again .......................................................... 198
8.6 Raising and Lowering (Ladder) Operators .......................................... 199
8.7 Angular Momentum Operators and Representations of the Rotation
Group ......................................................................................... 202
8.8 The ujm Are Spherical Harmonics ....................................................... 206
8.9 Spin Basis Functions and Pauli Matrices ............................................. 208
8.10 Coupling (Adding) Angular Momenta ................................................ 209
8.11 Wigner-Eckart Theorem........................................................................215
8.12 Selection Rules ....................................................................................... 217
8.13 Brief Bios ................................................................................................ 221
Summary of Chapter 8 ...................................................................................... 222
Problems and Exercises ...................................................................................... 223
9 The Structure of Atoms 226
9.1 Introduction ............................................................................................. 226
9.2 Zeeman: An Important Experiment (1897) .......................................... 226
9.3 Quantum Theory of the Zeeman Effect ................................................ 228
9.4 Fine Structure .......................................................................................... 233
9.5 Example: Intermediate Field Zeeman .................................................... 235
9.6 Nuclear Spin and Hyperfine Structure ................................................ 239
9.7 Multi-electron Atoms ............................................................................. 243