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Group Theory - 1st Edition - ISBN: 9780127505503, 9780323152785

Group Theory

1st Edition

And its Application to the Quantum Mechanics of Atomic Spectra

Author: Eugene Wigner
eBook ISBN: 9780323152785
Imprint: Academic Press
Published Date: 1st January 1959
Page Count: 386
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Group Theory: And Its Application To The Quantum Mechanics Of Atomic Spectra aims to describe the application of group theoretical methods to problems of quantum mechanics with specific reference to atomic spectra. Chapters 1 to 3 discuss the elements of linear vector theory, while Chapters 4 to 6 deal more specifically with the rudiments of quantum mechanics itself. Chapters 7 to 16 discuss the abstract group theory, invariant subgroups, and the general theory of representations. These chapters are mathematical, although much of the material covered should be familiar from an elementary course in quantum theory. Chapters 17 to 23 are specifically concerned with atomic spectra, as is Chapter 25. The remaining chapters discuss topics such as the recoupling (Racah) coefficients, the time inversion operation, and the classical interpretations of the coefficients. The text is recommended for physicists and mathematicians who are interested in the application of group theory to quantum mechanics. Those who are only interested in mathematics can choose to focus on the parts more devoted to that particular area of the subject.

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 to 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 Unitarity of 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 Lande g-Formula

The Interval Rule

24. Racah Coefficients

Conjugate Complex Representations

Symmetric Form of the Vector Coupling Coefficients

Co variant and Contra variant 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
1st January 1959
Academic Press
eBook ISBN:

About the Author

Eugene Wigner

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