Group Theory in Quantum Mechanics

Group Theory in Quantum Mechanics

An Introduction to Its Present Usage

1st Edition - January 1, 1960

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  • Author: Volker Heine
  • eBook ISBN: 9781483152004

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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


    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


    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'| JM)

    J. Notation For the Thirty-Two Crystal Point Groups

    K. Character Tables for the Crystal Point-Groups

    L. Character Tables for the Axial Rotation Group and Derived Groups

    List of General References, with Reviews


    Subject Index

Product details

  • No. of pages: 478
  • Language: English
  • Copyright: © Pergamon 1960
  • Published: January 1, 1960
  • Imprint: Pergamon
  • eBook ISBN: 9781483152004

About the Author

Volker Heine

About the Editor

D. Ter Haar

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