Group Theory in Quantum Mechanics

An Introduction to Its Present Usage

1st Edition - January 1, 1960
Author: Volker Heine
Editor: D. Ter Haar
Language: English
eBook ISBN:
9 7 8 - 1 - 4 8 3 1 - 5 2 0 0 - 4

Group Theory in Quantum Mechanics: An Introduction to its Present Usage introduces the reader to the three main uses of group theory in quantum mechanics: to label energy levels… Read more

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Group Theory in Quantum Mechanics: An Introduction to its Present Usage introduces the reader to the three main uses of group theory in quantum mechanics: to label energy levels and the corresponding eigenstates; to discuss qualitatively the splitting of energy levels as one starts from an approximate Hamiltonian and adds correction terms; and to aid in the evaluation of matrix elements of all kinds, and in particular to provide general selection rules for the non-zero ones. The theme is to show how all this is achieved by considering the symmetry properties of the Hamiltonian and the way in which these symmetries are reflected in the wave functions. This book is comprised of eight chapters and begins with an overview of the necessary mathematical concepts, including representations and vector spaces and their relevance to quantum mechanics. The uses of symmetry properties and mathematical expression of symmetry operations are also outlined, along with symmetry transformations of the Hamiltonian. The next chapter describes the three uses of group theory, with particular reference to the theory of atomic energy levels and transitions. The following chapters deal with the theory of free atoms and ions; representations of finite groups; the electronic structure and vibrations of molecules; solid state physics; and relativistic quantum mechanics. Nuclear physics is also discussed, with emphasis on the isotopic spin formalism, nuclear forces, and the reactions that arise when the nuclei take part in time-dependent processes. This monograph will be of interest to physicists and mathematicians.

Preface

Notation

I. Symmetry Transformations

1. The Uses of Symmetry Properties

2. Expressing Symmetry Operations Mathematically

3. Symmetry Transformations of the Hamiltonian

4. Groups of Symmetry Transformations

5. Group Representations

6. Applications to Quantum Mechanics

II. The Quantum Theory of a Free Atom

7. Some Simple Groups and Representations

8. The Irreducible Representations of the Full Rotation Group

9. Reduction of the Product Representation D(j) X D(j')

10. Quantum Mechanics of a Free Atom; Orbital Degeneracy

11. Quantum Mechanics of a Free Atom including Spin

12. The Effect of the Exclusion Principle

13. Calculating Matrix Elements and Selection Rules

III. The Representations of Finite Groups

14. Group Characters

15. Product Groups

16. Point-Groups

17. the Relationship between Group Theory and the Dirac Method

IV. Further Aspects of the Theory of Free Atoms and Ions

18. Paramagnetic Ions in Crystalline Fields

19. Time-Reversal and Kramers' Theorem

20. Wigner and Racah Coefficients

21. Hyperfine Structure

V. The Structure and Vibrations of Molecules

22. Valence Bond Orbitals and Molecular Orbitals

23. Molecular Vibrations

24. Infra-Red and Raman Spectra

VI. Solid State Physics

25. Brillouin Zone Theory of Simple Structures

26. Further Aspects of Brillouin Zone Theory

27. Tensor Properties of Crystals

VII. Nuclear Physics

28. The Isotopic Spin Formalism

29. Nuclear Forces

30. Reactions

VIII. Relativistic Quantum Mechanics

31. The Representations of the Lorentz Group

32. The Dirac Equation

33. Beta Decay

34. Positronium

Appendices

A. Matrix Algebra

B. Homomorphism and Isomorphism

C. Theorems on Vector Spaces and Group Representation

D. Sohur's Lemma

E. Irreducible Representations of Abelian Groups

F. Momenta and Infinitesimal Transformations

G. The Simple Harmonic Oscillator

H. the Irreducible Representations of the Complete Lorentz Group

I. Table of Wigner Coefficients (jj' mm'