Heine / Ter Haar Group Theory in Quantum Mechanics

An Introduction to Its Present Usage
1. Auflage 2014
ISBN: 978-1-4831-5200-4
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 478 Seiten, Web PDF

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.

Autoren/Hrsg.

Heine, Volker

Ter Haar, D.

Inhaltsverzeichnis

1;Front Cover;1
2;Group Theory in Quantum Mechanics: An Introduction to
its Present Usage;4
3;Copyright Page;5
4;Table of Contens;6
5;PREFACE;8
6;Chapter I. SYMMETRY TRANSFORMATIONS IN QUANTUM MECHANICS;12
6.1;1. The Uses of Symmetry Properties;12
6.2;2. Expressing Symmetry Operations Mathematically;14
6.3;3. Symmetry Transformations of the Hamiltonian;17
6.4;4. Groups of Symmetry Transformations;23
6.5;5. Group Representations;35
6.6;6. Application to Quantum Mechanics;52
7;Chapter II. THE QUANTUM THEORY OF A FREE ATOM
;59
7.1;7. Some Simple Groups and Representations;59
7.2;8. The Irreducible Representations of the Full Rotation Group
;63
7.3;9. Reduction of the Product Representation D(1) X D(1');78
7.4;10. Quantum Mechanics of a Free Atom; Orbital Degeneracy
;84
7.5;11. Quantum Mechanics of a Free Atom including Spin;89
7.6;12. The Effect of the Exclusion Principle;100
7.7;13. Calculating Matrix Elements and Selection Rules;110
8;Chapter III. THE REPRESENTATIONS OF FINITE GROUPS
;124
8.1;14. Group Characters;124
8.2;15· Product Groups;136
8.3;16. Point-groups;139
8.4;17. The Relationship Between Group Theory and the Dirac Method;154
9;Chapter IV. FURTHER ASPECTS OF THE THEORY OF FREE ATOMS AND IONS;159
9.1;18. Paramagnetic Ions in Crystalline Fields;159
9.2;19. Time-Reversal and Kramers' Theorem;175
9.3;20. Wigner and Racah Coefficients;187
9.4;21, Hyper fine Structure;200
10;Chapter V. THE STRUCTURE AND VIBRATIONS OF MOLECULES;217
10.1;22. Valence Bond Orbitals and Molecular Orbitals;217
10.2;23. Molecular Vibrations;240
11;Chapter VI. SOLID STATE PHYSICS;276
11.1;25. Brillouin Zone Theory of Simple Structures;276
11.2;26. Further Aspects of Brillouin Zone Theory;295
11.3;27. Tensor Properties of Crystals;315
11.4;Chapter VII. NUCLEAR PHYSICS
;324
11.5;28. The Isotopic Spin Formalism;324
11.6;29. Nuclear Forces;332
11.7;30. Reactions;345
12;Chapter VIII. RELATIVISTIC QUANTUM MECHANICS
;362
12.1;31. The Representations of the Lorentz Group;362
12.2;32. The Dirac Equation;374
12.3;33. Beta Decay;395
12.4;34. Positronium;408
13;Appendix;415
13.1;Appendix A: Matrix Algebra
;415
13.2;Appendix B: Homomorphism and Isomorphism;421
13.3;Appendix C: Theorems on Vector Spaces and Group Representations;423
13.4;Appendix D: Schur's Lemma;429
13.5;Appendix E: Irreducible Representations of Abelian Groups;431
13.6;Appendix F: Momenta and Infinitesimal Transformations;433
13.7;Appendix G: The Simple Harmonie Oscillator
;435
13.8;Appendix H: The Irreducible Representations of the Complete Lorentz Group
;439
13.9;Appendix I: Table of Wigner Coefficients (jj'mm'\JM);443
13.10;Appendix J: Notation for the Thirty-two Crystal Point-groups;457
13.11;Appendix K: Character Tables for the Crystal Point-groups;459
13.12;Appendix L: Character Tables for the Axial Rotation Group and Derived Groups
;466
14;GENERAL REFERENCES;468
15;BIBLIOGRAPHY;470
16;INDEX;475