《GROUP THEORY AND ITS APPLICATION TO THE QUANTUM MECHANICS OF ATOMIC SPECTRA EXPANDED AND IMPROVED ED》

1．Vectors and Matrices1

Linear Transformations1

Linear Independence of Vectors10

2．Generalizations13

3．The Principal Axis Transformation20

Unitary Matrices and the Scalar Product24

The Principal Axis Transformation for Unitary and Hermitian Matrices26

Real Orthogonal and Symmetric Matrices29

4．The Elements of Quantum Mechanics31

5．Perturbation Theory40

6．Transformation Theory and the Bases for the Statistical Interpretation of Quantum Mechanics47

7．Abstract Group Theory58

Theorems for Finite Groups59

Examples of Groups61

Conjugate Elements and Classes65

8．Invariant Subgroups67

The Factor Group68

Isomorphism and Homomorphism69

9．The General Theory of Representations72

10．Continuous Groups88

11．Representations and Eigenfunctions102

12．The Algebra of Representation Theory112

13．The Symmetric Group124

Appendix to Chapter 13．A Lemma Related to the Symmetric Group140

14．The Rotation Groups142

15．The Three-Dimensional Pure Rotation Group163

Spherical Harmonics153

The Homomorphism of the Two-Dimensionai Unitary Group onto the Rotation Group157

The Representations of the Unitary Group161

The Representations of the Three-Dimensional Pure Rotation Group166

16．The Representations of the Direct Product171

17．The Characteristics of Atomic Spectra177

Eigenvalues and Quantum Numbers177

The Vector Addition Model184

Appendix to Chapter 17．A Relationship Among Binomial Coefficients194

18．Selection Rules and the Splitting of Spectral Lines195

19．Partial Determination of Eigenfunctions from Their Transformation Properties210

20．Electron Spin220

The Physical Basis for the Pauli Theory220

Invariance of the Description under Spatial Rotation223

Connection with Representation Theory227

Appendix to Chapter 20．Linearity and Unitarity of Rotation Operators233

21．The Total Angular Momentum Quantum Number237

22．The Fine Structure of Spectral Lines251

23．Selection and Intensity Rules with Spin266

The Honl-Kronig Intensity Formulas275

The Landé g-Formula278

The Interval Rule280

24．Racah Coefficients284

Conjugate Complex Representations285

Symmetric Form of the Vector Coupling Coefficients289

Covariant and Contravariant Vector Coupling Coefficients292

Racah Coefficients296

Matrix Elements of Spin-Free Tensor Operators303

General Two-Sided Tensor Operators305

25．The Building-Up Principle309

26．Time Inversion325

Time Inversion and Antiunitary Operators325

Determination of the Time Inversion Operator329

Transformation of the Eigenfunctions under Antiunitary Operators333

Reduction of Corepresentations335

Determination of the Irreducible Corepresentations340

Consequences of In variance under Time Inversion344

27．Physical Interpretation and Classical Limits of Representation Coefficients，Three- and Six-j Symbols349

Representation Coefficients350

Vector Coupling Coefficients351

Racah Coefficients355

Appendix A．Conventions357

Appendix B．Summary of Formulas361

Subject Index365