Group Theory - 1st Edition - ISBN: 9781483227535, 9781483275765

Group Theory

1st Edition

And Its Application to the Quantum Mechanics of Atomic Spectra

Authors: Eugene P. Wigner
Editors: H. S. W. Massey
eBook ISBN: 9781483275765
Imprint: Academic Press
Published Date: 1st January 1959
Page Count: 384
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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.

Table of 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


No. of pages:
© Academic Press 1959
Academic Press
eBook ISBN:

About the Author

Eugene P. Wigner

About the Editor

H. S. W. Massey