Group Theory

Group Theory and its Application to the Quantum Mechanics of Atomic Spectra describes the applications of group theoretical methods to problems of quantum mechanics with particular reference to atomic spectra.
The manuscript first takes a look at vectors and matrices, generalizations, and principal axis transformation. Topics include principal axis transformation for unitary and Hermitian matrices; unitary matrices and the scalar product; linear independence of vectors; and real orthogonal and symmetric matrices. The publication also ponders on the elements of quantum mechanics, perturbation theory, and transformation theory and the bases for the statistical interpretation of quantum mechanics.
The book discusses abstract group theory and invariant subgroups, including theorems of finite groups, factor group, and isomorphism and homomorphism. The text also reviews the algebra of representation theory, rotation groups, three-dimensional pure rotation group, and characteristics of atomic spectra. Discussions focus on eigenvalues and quantum numbers, spherical harmonics, and representations of the unitary group.
The manuscript is a valuable reference for readers interested in the applications of group theoretical methods.

Inhalt

Contents
Author's Preface
Translator's Preface

1. Vectors and Matrices
   Linear Transformations
   Linear Independence of Vectors
2. Generalizations
3. The Principal Axis Transformation
   Unitary Matrices and the Scalar Product
   The Principal Axis Transformation for Unitary and Hermitian Matrices
   Real Orthogonal and Symmetric Matrices
4. The Elements of Quantum Mechanics
5. Perturbation Theory
6. Transformation Theory and the Bases for the Statistical Interpretation of Quantum Mechanics
7. Abstract Group Theory
   Theorems for Finite Groups
   Examples of Groups
   Conjugate Elements and Classes
8. Invariant Subgroups
   The Factor Group
   Isomorphism and Homomorphism
9. The General Theory of Representations
10. Continuous Groups
11. Representations and Eigenfunctions
12. The Algebra of Representation Theory
13. The Symmetric Group
    Appendix to Chapter 13. A Lemma related to the Symmetric Group
14. The Rotation Groups
15. The Three-Dimensional Pure Rotation Group
    Spherical Harmonics
    The Homomorphism of the Two-Dimensional Unitary Group onto the Rotation Group
    The Representations of the Unitary Group
    The Representations of the Three-Dimensional Pure Rotation Group
16. The Representations of the Direct Product
17. The Characteristics of Atomic Spectra
    Eigenvalues and Quantum Numbers
    The Vector Addition Model
    Appendix t o Chapter 17. A Relationship Among Binomial Coefficients
18. Selection Rules and the Splitting of Spectral Lines
19. Partial Determination of Eigenfunctions from Their Transformation Properties
20. Electron Spin
    The Physical Basis for the Pauli Theory
    Invariance of the Description under Spatial Rotation
    Connection with Representation Theory
    Appendix to Chapter 20. Linearity and unitary Rotation Operators
21. The Total Angular Momentum Quantum Number
22. The Fine Structure of Spectral Lines
23. Selection and Intensity Rules with Spin
    The Hönl-Kronig Intensity Formulas
    The Landé G-Formula
    The Interval Rule
24. Racah Coefficients
    Conjugate Complex Representations
    Symmetric Form of the Vector Coupling Coefficients
    Covariant and Contravariant Vector Coupling Coefficients
    Racah Coefficients
    Matrix Elements of Spin-Free Tensor Operators
    General Two-Sided Tensor Operators
25. The Building-Up Principle
26. Time Inversion
    Time Inversion and Antiunitary Operators
    Determination of the Time Inversion Operator
    Transformation of the Eigenfunctions under Antiunitary Operators
    Reduction of Corepresentations
    Determination of the Irreducible Corepresentations
    Consequences of Invariance under Time Inversion
27. Physical Interpretation and Classical Limits of Representation Coefficients, Three- and Six-j Symbols
    Representation Coefficients
    Vector Coupling Coefficients
    Racah Coefficients
    Appendix A. Conventions
    Appendix B. Summary of Formulas
    Subject Index