Group Theory in Quantum Mechanics - 1st Edition - ISBN: 9780080092423, 9781483152004

Group Theory in Quantum Mechanics

1st Edition

An Introduction to Its Present Usage

Authors: Volker Heine
Editors: D. Ter Haar
eBook ISBN: 9781483152004
Imprint: Pergamon
Published Date: 1st January 1960
Page Count: 478
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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.

Table of Contents

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. Moment


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© Pergamon 1960
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About the Author

Volker Heine

About the Editor

D. Ter Haar

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