Group Theory
And its Application to the Quantum Mechanics of Atomic Spectra
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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
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
Product details
- No. of pages: 386
- Language: English
- Copyright: © Academic Press 1959
- Published: January 1, 1959
- Imprint: Academic Press
- eBook ISBN: 9780323152785