Symmetries in Quantum Physics

Transcription

1 Symmetries in Quantum Physics U. Fano Department of Physics and James Franck Institute University of Chicago Chicago, Illinois A. R. P. Rau Department of Physics and Astronomy louisiana State University Baton Rouge, louisiana Academic Press San Diego New York Boston London Sydney Tokyo Toronto

2 Contents Preface xiii 1 Introduction 1 Transformation Theories: Klein's and Dirac's Symmetry and the Selection of Variables Examples of tensorial equations Algebraic Elements Vectors, tensors, and related quantities Addition and direct product of tensorial sets Linear transformation Reduction Procedure and Irreducible Tensorial Sets An analytical example: Reduction of tensors Further Aspects of Reduction Reduction procedures Labeling of set elements Block diagonalization of the reduction Phase normalization Group theory Reduction as an expansion into eigenfunctions Structure of the Book Alternative sets of commuting invariant operators Quaternions 28 Problems 29 v

3 vi PART A STATE REPRESENTATIVES AND r-transformations: THEIR CONSTRUCTION AND PROPERTIES 2 Infinitesimal Rotations and Angular Momentum Basic Relations Analytical Example: Infinitesimal Transformation of Cartesian Coordinates The Angular Momentum Matrices of Quantum Mechanics Phase normalization Definition of a standard base The Fundamental Representation Significance of half-integer j 47 Problems 49 3 Frame Reversal and Complex Conjugation Analytical Representation and Implications of Frame Reversal Explicit form of the matrix U Properties of the matrix U Contragredience and the Construction of Invariants Contragredient tensorial sets Invariant products Notation Cartesian Base for Integer j > Cartesian-to-standard transformation Phase normalization of spherical harmonics 72 Problems 74 4 Standard r-transformation Matrices and Their Applications Explicit Form and Properties Spinor method 78

4 Vll Algebraic approach First order differential system Second order differential equation Symmetries of the standard r-transformations Integrals r-transformations in the Cartesian frame Macroscopic Applications Applications to Quantum Physics Particle transmission through a Stern-Gerlach magnet Angular distribution of a particle in orbital motion Rotational eigenfunctions and eigenvalues for symmetric-top polyatomic molecules and heteronuclear diatomics Spinor and vector harmonics Coordinate Inversion and Parity Eigenfunctions 93 Problems 97 5 Reduction of Direct Products (Addition of Angular Momenta) Structure and Properties of the Reducing Matrix Spinor approach Normalization Recurrence relations Symmetries Reduction in the Cartesian frame Reduction of r-transformation Products Irreducible Product Sets Special cases Symmetry Products of contragredient sets Wave-mechanical examples Multiple products Coupling diagrams Symmetrization of Wigner Coefficients: Invariant Triple Product and 3-j Coefficients 119 Problems. 123

5 Vlll PART В TENSORIAL ASPECTS OF QUANTUM PHYSICS 6 Tensorial Sets of Quantum Operators The Liouville Representation of Quantum Mechanics Quantum Mechanics of Particles with Spin Base sets of matrices and operators Two-Level Systems Atom in a radiation field Light polarization and Stokes parameters Further applications of two-level systems: Occupation, creation, and annihilation operators Particles with Spin j> : Wigner-Eckart Theorem Density matrix Multipole expansion of operators G Physical implications of the triangular relation к < 2j Systems with 2j + l Levels Transfer of Angular Momentum Calculation of Matrix Elements 150 Problems Recoupling Transformations: 6-j and 9-j Coefficients Transformation Matrices and Their Analysis Diagrams Group properties Factorization of transformations Symmetrized Recoupling: 6-j and 9-j Coefficients j coefficients j coefficients Alternative perspectives Products of Operators Unit operators General operator Commutators Schrödinger equation for a (2j + l)-level system Combining Operators of Different Systems 183

6 IX 7.5 Illustrations Interaction matrix elements Projection of operators Correlations 191 Problems Partially Filled Shells of Atoms or Nuclei Qualitative Discussion Two-particle states States of three or more equivalent particles Quantum numbers for many-particle states Shell-wide Treatment Triple tensors and their matrices Coefficients of fractional parentage Algebra of Triple Tensors and Its Applications Interpretation of Х(*«*.*г) Quasi-spin and seniority Quasiparticles for the / shell Determination of fractional parentage Operator matrices 224 PART С SYMMETRIES OF HIGHER DIMENSIONS 9 Discrete Transformations of Coordinates Point Symmetry Operations and Their Groups Characters of Group Representations and Their Applications Abelian groups Non-Abelian groups Characters of the rotation group 50(3) Reduction of representations Reduction of set products Symmetries of Molecules and Crystals 243

7 X Symmetry combinations Vibrational motions Molecular rotations Stability analysis of nuclear positions 246 Problems Rotation Groups in Higher Dimensions: Multiparticle Problems Four-Dimensional Rotations: The Coulomb-Kepler Problem Spherical and parabolic representations Rotations in four dimensions Hydrogen atom in momentum space Alternative subgroups of 50(4): The hydrogen atom in external fields Clebsch-Gordan coefficients for products of 50(4) Orthogonal Groups in Higher Dimensions Hypersphere in D dimensions Hyperspherical coordinates for multiparticle systems Transformation between alternative schemes Further Developments Invariance and noninvariance groups "Dynamical symmetries" for atoms and nuclei Adjoining an extra degree of freedom Alternative reduction schemes for multiparticle systems Lorentz Transformations and the Lorentz and Poincare Groups Lorentz Transformations Generators and Representations of the Lorentz Group Four-vectors and the Lorentz metric Generators of the proper Lorentz group Lorentz transformations to r-transformations Spinor representations Neutrino and electron spinor states Electromagnetism and its quantum The Inhomogeneous Lorentz (Poincare) Group Generators and commutation relationships 296

8 11.4 Field Representations Massive systems Representations of massless entities Symmetries of the Scattering Continuum Symmetries of Radial Eigenfunctions The Full Noninvariance Group of Hydrogen Alternative decompositions of the noninvariance group Dynamics and Symmetry Transformations 310 Bibliography 313 Index 317