Symmetries in Quantum Physics

Symmetries in Quantum Physics

1st Edition - May 30, 1996

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  • Authors: U. Fano, A. Rau
  • eBook ISBN: 9780080542171

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This text focuses on the physics of symmetries, developing symmetries and transformations through concrete physical examples and contexts rather than presenting the information axiomatically, mathematically, and abstractly. Readers are introduced gradually to advanced mathematical procedures, including the Wigner and Racah algebras and their applications to various symmetry groups. The book also includes some of the latest research on the use of non-invariance and non-compact groups in the consideration of relativistic and many-particle problems of atoms and nuclei.This book is an updated replacement for the text Irreducible Tensorial Sets (Academic Press, 1959). Parts A and B of the present book grew out of occasional lectures in the intervening decades at the University of Chicago, where it became neccessary to update or elaborate upon certain points. Part C has been built more recently to deal with innovations and new information in the field of mathematical physics. The book as a whole develops the subject of symmetry from a physical point of view, allowing students and researchers to gain new insight on their subject. This book can be used both as a text and as a reference by students and scientists in the field.

Key Features

Adapts and extends the earlier Irreducible Tensor Sets (Academic Press, 1959) to classroom use
Extends to multi-particle systems and relativity
Includes problems in each chapter for homework assignments
Embraces the latest research on non-invariance groups


Graduate students in physics and chemistry, physicists, quantum physicists, physical chemists, mathematicians.

Table of Contents

  • Introduction: Symmetry and the Selection of Variables. Algebraic Elements. Reduction Procedure and Irreducible Tensorial Sets. Further Aspects of Reduction. Structure of the Book. Quaternions. Part A: State Representatives and r<$>-Transformations: Their Construction and Properties. Infinitesimal Rotations and Angular Momentum:<$> Basic Relations. Analytical Example: Infinitesimal Transformation of Cartesian Coordinates. The Angular Momentum Matrices of Quantum Mechanics. The Fundamental Representation. Frame Reversal and Complex Conjugation:<$> Analytical Representation and Implications of Frame Reversal. Contragradience and the Construction of Invariants. Cartesian Base for Integer j<$> [greater than or equal to sign here]1. Standard r<$>-Transformation Matrices and Their Applications:<$> Explicit Form and Properties. Macroscopic Applications. Applications to Quantum Physics. Coordinate Inversion and Parity Eigenfunctions. Reduction of Direct Products (Addition of Angular Momenta):<$> Structure and Properties of the Reducing Matrix. Reduction of r<$>-Transformation Products. Irreducible Product Sets. Symmetrization of Wigner Coefficients: Invariant Triple Product and 3-j<$> Coefficients. Part B: Tensorial Aspects of Quantum Physics.<$> Tensorial Sets of Quantum Operators:<$> The Liouville Representation of Quantum Mechanics. Quantum Mechanics of Particles with Spin 1/2. Two-Level Systems. Particles With Spin j<$>>1/2: Wigner-Eckart Theorem. Systems with 2j<$>+1 Levels. Transfer of Angular Momentum. Calculation of Matrix Elements. Recoupling Transformations: 6-j<$> and 9-j<$> Coefficients:<$> Transformation Matrices and Their Analysis. Symmetrized Recoupling: 6-j<$> and9-j<$> Coefficients. Products of Operators. Combining Operators of Different Systems. Illustrations. Partially Filled Shells of Atoms or Nuclei:<$> Qualitative Discussion. Shell-wide Treatment. Algebra of Triple Tensors and Its Applications. PartC: Symmetries of Higher Dimensions.<$> Discrete Transformations of Coordinates:<$> Point Symmetry Operations and Their Groups. Characters of Group Representations and Their Applications. Symmetries of Molecules and Crystals. Rotation Groups in Higher Dimensions: Multiparticle Problems:<$> Four-Dimensional Rotations: the Coulomb-Kepler Problem. Orthogonal Groups in Higher Dimensions. Further Developments. Lorentz Transformations and the Lorentz and Poincare Groups:<$> Lorentz Transformations. Generators and Representations of the Lorentz Group. The Inhomogenous Lorentz (Poincare) Group. Field Representations. Symmetries of the Scattering Continuum:<$> Symmetries of Radial Eigenfunctions. The Full Noninvariance Group of Hydrogen. Dynamics as Symmetry Transformations. Bibliography. Index.

Product details

  • No. of pages: 333
  • Language: English
  • Copyright: © Academic Press 1996
  • Published: May 30, 1996
  • Imprint: Academic Press
  • eBook ISBN: 9780080542171

About the Authors

U. Fano

Affiliations and Expertise

University of Chicago

A. Rau

Affiliations and Expertise

Louisiana State University

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