Theory of Groups and Its Application to Physical Problems - 1st Edition - ISBN: 9781483233130, 9781483275987

Theory of Groups and Its Application to Physical Problems

1st Edition

Authors: S. Bhagavantam T. Venkatarayudu
eBook ISBN: 9781483275987
Imprint: Academic Press
Published Date: 1st January 1969
Page Count: 294
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Description

Theory of Groups and Its Application to Physical Problems is an introductory study of the theory of groups for persons with no easy access to an orthodox mathematical treatise on the subject. The aim is to provide an understanding of the method of applying group theory to various problems and appreciate the advantages thereof. It is hoped that this account of the theory of groups will serve a real need for physicists interested in the subject. The book opens with a discussion of the concept of groups. This is followed by separate chapters on the one-dimensional and two-dimensional lattices, some properties of groups, matrix groups, and the wave equation and its properties. Subsequent chapters deal with vibrations of a dynamical system, vibrational Raman effect and infrared absorption, molecular structure and normal modes, three-dimensional lattices, Raman and infrared spectra of crystals, crystal symmetry and physical properties, rotation groups, and applications to problems of atomic spectra.

Table of Contents


Preface to American Edition

Preface to First Edition

I. Groups

Group Postulates

Displacement of a Rigid Body

Symmetry Operations

Point Groups

Space Groups

II. One-Dimensional Lattice

Symmetry of the Lattice

One-Dimensional Motives

Two-Dimensional Motives

Three-Dimensional Motives

III. Lattices in Two Dimensions

Symmetry of the Lattices

Two-Dimensional Motives

Three-Dimensional Motives

IV. Some Properties of Groups

Abstract Groups

Subgroups

Classes of Conjugate Elements

Self-Conjugate Subgroups

Factor Groups

Permutation Groups

Isomorphous Groups

Direct Product Groups

V. Matrix Groups

Matrices

Linear Transformations

Equivalent Matrices

Reducible and Irreducible Matrix Representations of Groups

Kronecker Square and Symmetrized Kronecker Square Representations

Kronecker Direct Product of Two Representations

VI. The Wave Equation and Its Properties

Vibrations of a String

The Wave Equation

Eigenvalues and Eigenfunctions

Linear Operators and Manifolds

Invariant Manifolds

Physical Quantities as Operators

Harmonic Oscillator

Eigenfunctions of Hydrogen-like Atoms

The Rigid Rotator

VII. Vibrations of a Dynamical System

Kinetic and Potential Energies of a Dynamical System

Lagrangian Equations of Motion

Normal Modes of Oscillation

Normal Frequencies

Orthogonality Relation between the Normal Co-ordinates

Symmetry Properties of Normal Modes

Representation Defined by the Cartesian Co-ordinates

Determination of the Normal Co-ordinates

Splitting of the Secular Equation

F and G Matrices

VIII. Vibrational Raman Effect and Infra-Red Absorption

The Molecule as a Dynamical System

Raman Scattering by a Diatomic Molecule

Infra-Red Absorption and Electric Moment

Selection Rules for Fundamentals

Overtone and Combination Lines

Selection Rules in Some Special Cases

IX. Molecular Structure and Normal Modes

Triatomic Molecules. Pyramidal Molecules

The Nitrate and the Carbonate Ions

Diatomic and Other Linear Molecules

Sulfur

X. Molecular Structure and Normal Frequencies

Interatomic Forces

Water

Phosphorus

XI. Lattices in Three Dimensions

Space Lattices

Crystal Classes

Space Groups

XII. Raman and Infra-Red Spectra of Crystals

The Internal Structure of a Crystal

Application of Group Theory

Lattice Oscillations in Calcite and Sodium Nitrate

Some Special Cases

Lattice Oscillations in Some Organic Crystals

Raman Spectra and Different Crystalline Modifications

Splitting of Degenerate Modes in Crystals of Lower Symmetry

Special Case of Diamond

XIII. Crystal Symmetry and Physical Properties

General Considerations

Crystal Optics

Elasticity and Photoelasticity

Description of the General Method

Results

Enantiomorphism and Optical Activity

Isotropic Solids

XIV. Rotation Groups

The Rotation Groups in Two and Three Dimensions

Unitary Substitutions of Two Variables

Irreducible Manifolds with Respect to U2

Irreducible Representations of U2

Characters of the Group U2

The Irreducible Components of iD X λD

Isomorphism between the Rotation and the Unitary Groups

XV. Application to Problems of Atomic Spectra

Solutions of the Wave Equation

Angular Momentum Operators

Quantization of Angular Momentum and Its Components

Vector Addition of Angular Momenta

Reduction of the Product Manifold

Selection Rules and Intensities of Spectral Lines

Pauli Theory

Pauli Exclusion Principle

XVI. Other Applications

The Hydrogen Molecule

Rotational Specific Heat of Hydrogen

Nuclear Spin

Intensities of Rotational Raman Lines

Appendices

I. Representations of Finite Groups

II. Transformation of Matrices

III. Kramers-Heisenberg Dispersion Formula

IV. Evaluation of Group Characters

V. Properties of Some Polynomial Functions

VI. Laplacian Operator

VII. Parameter Groups

VIII. Character Tables and Irreducible Representations in Respect of Various Point Groups

Index

Details

No. of pages:
294
Language:
English
Copyright:
© Academic Press 1969
Published:
Imprint:
Academic Press
eBook ISBN:
9781483275987

About the Author

S. Bhagavantam

T. Venkatarayudu