An Introduction to Group Theory and Its Applications

1st Edition - January 1, 1963

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  • Author: R. McWeeny
  • eBook ISBN: 9781483226248

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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.

Table of Contents

  • Preface

    Chapter 1 Groups

    1.1 Symbols and the Group Property

    1.2 Definition of a Group

    1.3 The Multiplication Table

    1.4 Powers, Products, Generators

    1.5 Subgroups, Cosets, Classes

    1.6 Invariant Subgroups. The Factor Group

    1.7 Homomorphisms and Isomorphisms

    1.8 Elementary Concept of a Representation

    1.9 The Direct Product

    1.10 The Algebra of a Group

    Chapter 2 Lattices and Vector Spaces

    2.1 Lattices. One Dimension

    2.2 Lattices. Two and Three Dimensions

    2.3 Vector Spaces

    2.4 n-Dimensional Space. Basis Vectors

    2.5 Components and Basis Changes

    2.6 Mappings and Similarity Transformations

    2.7 Representations. Equivalence

    2.8 Length and Angle. The Metric

    2.9 Unitary Transformations

    2.10 Matrix Elements as Scalar Products

    2.11 The Eigenvalue Problem

    Chapter 3 Point and Space Groups

    3.1 Symmetry Operations as Orthogonal Transformations

    3.2 The Axial Point Groups

    3.3 The Tetrahedral and Octahedral Point Groups

    3.4 Compatibility of Symmetry Operations

    3.5 Symmetry of Crystal Lattices

    3.6 Derivation of Space Groups

    Chapter 4 Representations of Point and Translation Groups

    4.1 Matrices for Point Group Operations

    4.2 Nomenclature. Representations

    4.3 Translation Groups. Representations and Reciprocal Space

    Chapter 5 Irreducible Representations

    5.1 Reducibility. Nature of the Problem

    5.2 Reduction and Complete Reduction. Basic Theorems

    5.3 The Orthogonality Relations

    5.4 Group Characters

    5.5 The Regular Representation

    5.6 The Number of Distinct Irreducible Representations

    5.7 Reduction of Representations

    5.8 Idempotents and Projection Operators

    5.9 The Direct Product

    Chapter 6 Applications Involving Algebraic Forms

    6.1 Nature of Applications

    6.2 Invariant Forms. Symmetry Restrictions

    6.3 Principal Axes. The Eigenvalue Problem

    6.4 Symmetry Considerations

    6.5 Symmetry Classification of Molecular Vibrations

    6.6 Symmetry Coordinates in Vibration Theory

    Chapter 7 Applications Involving Functions and Operators

    7.1 Transformation of Functions

    7.2 Functions of Cartesian Coordinates

    7.3 Operator Equations. Invariance

    7.4 Symmetry and the Eigenvalue Problem

    7.5 Approximation Methods. Symmetry Functions

    7.6 Symmetry Functions by Projection

    7.7 Symmetry Functions and Equivalent Functions

    7.8 Determination of Equivalent Functions

    Chapter 8 Applications Involving Tensors and Tensor Operators

    8.1 Scalar, Vector and Tensor Properties

    8.2 Significance of the Metric

    8.3 Tensor Properties. Symmetry Restrictions

    8.4 Symmetric and Antisymmetric Tensors

    8.5 Tensor Fields. Tensor Operators

    8.6 Matrix Elements of Tensor Operators

    8.7 Determination of Coupling Coefficients

    Appendix 1 Representations Carried by Harmonic Functions

    Appendix 2 Alternative Bases for Cubic Groups


Product details

  • No. of pages: 262
  • Language: English
  • Copyright: © Pergamon 1963
  • Published: January 1, 1963
  • Imprint: Pergamon
  • eBook ISBN: 9781483226248

About the Author

R. McWeeny

Affiliations and Expertise

Universita di pisa

About the Editor

H. Jones

Affiliations and Expertise

University of Sheffield, Sheffield, UK

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