McWeeny / Jones Symmetry

An Introduction to Group Theory and Its Applications
1. Auflage 2013
ISBN: 978-1-4832-2624-8
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 262 Seiten, Web PDF

Symmetry: An Introduction to Group Theory and its Application is an eight-chapter text that covers the fundamental bases, the development of the theoretical and experimental aspects of the group theory. Chapter 1 deals with the elementary concepts and definitions, while Chapter 2 provides the necessary theory of vector spaces. Chapters 3 and 4 are devoted to an opportunity of actually working with groups and representations until the ideas already introduced are fully assimilated. Chapter 5 looks into the more formal theory of irreducible representations, while Chapter 6 is concerned largely with quadratic forms, illustrated by applications to crystal properties and to molecular vibrations. Chapter 7 surveys the symmetry properties of functions, with special emphasis on the eigenvalue equation in quantum mechanics. Chapter 8 covers more advanced applications, including the detailed analysis of tensor properties and tensor operators. This book is of great value to mathematicians, and math teachers and students.

Autoren/Hrsg.

McWeeny, R.

Jones, H.

Inhaltsverzeichnis

1;Front Cover;1
2;Symmetry: An Introduction to Group Theory and its Applications;4
3;Copyright Page;5
4;Table of Contents;10
5;THE INTERNATIONAL ENCYCLOPEDIA OF PHYSICAL CHEMISTRY AND CHEMICAL PHYSICS;7
6;INTRODUCTION;8
7;PREFACE;14
8;CHAPTER 1. GROUPS;16
8.1;1.1 Symbols and the group property;16
8.2;1.2 Definition of a group;21
8.3;1.3 The multiplication table;22
8.4;1.4 Powers, products, generators;24
8.5;1.5 Subgroups, cosets, classes;26
8.6;1.6 Invariant subgroups. The factor group;28
8.7;1.7 Homomorphisms and isomorphisms;29
8.8;1.8 Elementary concept of a representation;31
8.9;1.9 The direct product;33
8.10;1.10 The algebra of a group;34
9;CHAPTER 2. LATTICES AND VECTOR SPACES;37
9.1;2.1 Lattices. One dimension;37
9.2;2.2 Lattices. Two and three dimensions;40
9.3;2.3 Vector spaces;42
9.4;2.4 n-Dimensional space. Basis vectors;43
9.5;2.5 Components and basis changes;46
9.6;2.6 Mappings and similarity transformation;48
9.7;2.7 Representations. Equivalence;53
9.8;2.8 Length and angle. The metric;56
9.9;2.9 Unitary transformations;62
9.10;2.10 Matrix elements as scalar products;64
10;CHAPTER 3. POINT AND SPACE GROUPS;69
10.1;3.1 Symmetry operations as orthogonal transformations;69
10.2;3.2 The axial point groups;74
10.3;3.3 The tetrahedral and octahedral point groups;84
10.4;3.4 Compatibility of symmetry operations;90
10.5;3.5 Symmetry of crystal lattices;93
10.6;3.6 Derivation of space groups;100
11;CHAPTER 4. REPRESENTATIONS OF POINT AND TRANSLATION GROUPS;106
11.1;4.1 Matrices for point group operations;106
11.2;4.2 Nomenclature. Representations;110
11.3;4.3 Translation groups. Representations and reciprocal space;120
12;CHAPTER 5. IRREDUCIBLE REPRESENTATIONS;124
12.1;5.1 Reducibility. Nature of the problem;124
12.2;5.2 Reduction and complete reduction. Basic theorems;125
12.3;5.3 The orthogonality relation;131
12.4;5.4 Group characters;136
12.5;5.5 The regular representation;139
12.6;5.6 The number of distinct irreducible representations;140
12.7;5.7 Reduction of representations;141
12.8;5.8 Idempotents and projection operators;146
12.9;5.9 The direct product;148
13;CHAPTER 6. APPLICATIONS INVOLVING ALGEBRAIC FORMS;155
13.1;6.1 Nature of applications;155
13.2;6.2 Invariant forms. Symmetry restrictions;156
13.3;6.3 Principal axes. The eigenvalue problem;162
13.4;6.4 Symmetry considerations;165
13.5;6.5 Symmetry classification of molecular vibrations;166
13.6;6.6 Symmetry coordinates in vibration theory;174
14;CHAPTER 7. APPLICATIONS INVOLVING FUNCTIONS AND OPERATORS;181
14.1;7.1 Transformation of functions;181
14.2;7.2 Functions of Cartesian coordinates;185
14.3;7.3 Operator equations. Invariance;189
14.4;7.4 Symmetry and the eigenvalue problem;196
14.5;7.5 Approximation methods. Symmetry functions;202
14.6;7.6 Symmetry functions by projection;205
14.7;7.7 Symmetry functions and equivalent functions;210
14.8;7.8 Determination of equivalent functions;212
15;CHAPTER 8. APPLICATIONS INVOLVING TENSORS AND TENSOR OPERATORS;218
15.1;8.1 Scalar, vector and tensor properties;218
15.2;8.2 Significance of the metric;221
15.3;8.3 Tensor properties. Symmetry restrictions;223
15.4;8.4 Symmetric and antisymmetric tensors;226
15.5;8.5 Tensor fields. Tensor operators;233
15.6;8.6 Matrix elements of tensor operators;239
15.7;8.7 Detennination of coupling coefficients;246
16;APPENDIX 1: Representations carried by harmonic functions;250
17;APPENDIX 2: Alternative bases for cubic groups;256
18;INDEX;260