Theory of Groups and Its Application to Physical Problems

Theory of Groups and Its Application to Physical Problems

1st Edition - January 1, 1969

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  • Authors: S. Bhagavantam, T. Venkatarayudu
  • eBook ISBN: 9781483275987

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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

Product details

  • No. of pages: 294
  • Language: English
  • Copyright: © Academic Press 1969
  • Published: January 1, 1969
  • Imprint: Academic Press
  • eBook ISBN: 9781483275987

About the Authors

S. Bhagavantam

T. Venkatarayudu

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