Symmetries in Quantum Physics

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

Contents Preface xiii 1 Introduction 1 Transformation Theories: Klein's and Dirac's 3 1.1 Symmetry and the Selection of Variables 5 1.1.1 Examples of tensorial equations 5 1.2 Algebraic Elements 11 1.2.1 Vectors, tensors, and related quantities 11 1.2.2 Addition and direct product of tensorial sets... 14 1.2.3 Linear transformation 14 1.3 Reduction Procedure and Irreducible Tensorial Sets 17 1.3.1 An analytical example: Reduction of tensors... 18 1.4 Further Aspects of Reduction 20 1.4.1 Reduction procedures 21 1.4.2 Labeling of set elements 22 1.4.3 Block diagonalization of the reduction 23 1.4.4 Phase normalization 23 1.4.5 Group theory 24 1.4.6 Reduction as an expansion into eigenfunctions... 25 1.5 Structure of the Book 26 1.5.1 Alternative sets of commuting invariant operators 27 1.6 Quaternions 28 Problems 29 v

vi PART A STATE REPRESENTATIVES AND r-transformations: THEIR CONSTRUCTION AND PROPERTIES 2 Infinitesimal Rotations and Angular Momentum 33 2.1 Basic Relations 34 2.2 Analytical Example: Infinitesimal Transformation of Cartesian Coordinates. 38 2.3 The Angular Momentum Matrices of Quantum Mechanics 41 2.3.1 Phase normalization 43 2.3.2 Definition of a standard base 44 2.4 The Fundamental Representation 45 2.4.1 Significance of half-integer j 47 Problems 49 3 Frame Reversal and Complex Conjugation 51 3.1 Analytical Representation and Implications of Frame Reversal 54 3.1.1 Explicit form of the matrix U 58 3.1.2 Properties of the matrix U 59 3.2 Contragredience and the Construction of Invariants 62 3.2.1 Contragredient tensorial sets 64 3.2.2 Invariant products 64 3.2.3 Notation 67 3.3 Cartesian Base for Integer j > 1 68 3.3.1 Cartesian-to-standard transformation 70 3.3.2 Phase normalization of spherical harmonics 72 Problems 74 4 Standard r-transformation Matrices and Their Applications 75 4.1 Explicit Form and Properties 76 4.1.1 Spinor method 78

Vll 4.1.2 Algebraic approach 80 4.1.3 First order differential system 81 4.1.4 Second order differential equation 82 4.1.5 Symmetries of the standard r-transformations.... 82 4.1.6 Integrals 83 4.1.7 r-transformations in the Cartesian frame 85 4.2 Macroscopic Applications 85 4.3 Applications to Quantum Physics 88 4.3.1 Particle transmission through a Stern-Gerlach magnet 88 4.3.2 Angular distribution of a particle in orbital motion 89 4.3.3 Rotational eigenfunctions and eigenvalues for symmetric-top polyatomic molecules and heteronuclear diatomics 90 4.3.4 Spinor and vector harmonics 92 4.4 Coordinate Inversion and Parity Eigenfunctions 93 Problems 97 5 Reduction of Direct Products (Addition of Angular Momenta) 99 5.1 Structure and Properties of the Reducing Matrix 100 5.1.1 Spinor approach 102 5.1.2 Normalization 104 5.1.3 Recurrence relations 107 5.1.4 Symmetries 107 5.1.5 Reduction in the Cartesian frame 109 5.2 Reduction of r-transformation Products 110 5.3 Irreducible Product Sets 112 5.3.1 Special cases 112 5.3.2 Symmetry 113 5.3.3 Products of contragredient sets 113 5.3.4 Wave-mechanical examples 114 5.3.5 Multiple products 115 5.3.6 Coupling diagrams 116 5.4 Symmetrization of Wigner Coefficients: Invariant Triple Product and 3-j Coefficients 119 Problems. 123

Vlll PART В TENSORIAL ASPECTS OF QUANTUM PHYSICS 6 Tensorial Sets of Quantum Operators 127 6.1 The Liouville Representation of Quantum Mechanics... 128 6.2 Quantum Mechanics of Particles with Spin 130 6.2.1 Base sets of matrices and operators 132 6.3 Two-Level Systems 135 6.3.1 Atom in a radiation field 136 6.3.2 Light polarization and Stokes parameters 137 6.3.3 Further applications of two-level systems: Occupation, creation, and annihilation operators.. 138 6.4 Particles with Spin j> : Wigner-Eckart Theorem 140 6.4.1 Density matrix 140 6.4.2 Multipole expansion of operators G 143 6.4.3 Physical implications of the triangular relation к < 2j 145 6.5 Systems with 2j + l Levels 146 6.6 Transfer of Angular Momentum 148 6.7 Calculation of Matrix Elements 150 Problems 154 7 Recoupling Transformations: 6-j and 9-j Coefficients 157 7.1 Transformation Matrices and Their Analysis 160 7.1.1 Diagrams 161 7.1.2 Group properties 162 7.1.3 Factorization of transformations 163 7.2 Symmetrized Recoupling: 6-j and 9-j Coefficients 168 7.2.1 6-j coefficients 170 7.2.2 9-j coefficients 173 7.2.3 Alternative perspectives 175 7.3 Products of Operators 176 7.3.1 Unit operators 176 7.3.2 General operator 179 7.3.3 Commutators 180 7.3.4 Schrödinger equation for a (2j + l)-level system... 181 7.4 Combining Operators of Different Systems 183

IX 7.5 Illustrations 186 7.5.1 Interaction matrix elements 187 7.5.2 Projection of operators 189 7.5.3 Correlations 191 Problems 195 8 Partially Filled Shells of Atoms or Nuclei 199 8.1 Qualitative Discussion 200 8.1.1 Two-particle states 202 8.1.2 States of three or more equivalent particles 202 8.1.3 Quantum numbers for many-particle states 204 8.2 Shell-wide Treatment 208 8.2.1 Triple tensors and their matrices 210 8.2.2 Coefficients of fractional parentage 213 8.3 Algebra of Triple Tensors and Its Applications 214 8.3.1 Interpretation of Х(*«*.*г) 215 8.3.2 Quasi-spin and seniority 217 8.3.3 Quasiparticles for the / shell 218 8.3.4 Determination of fractional parentage 220 8.3.5 Operator matrices 224 PART С SYMMETRIES OF HIGHER DIMENSIONS 9 Discrete Transformations of Coordinates 229 9.1 Point Symmetry Operations and Their Groups 230 9.2 Characters of Group Representations and Their Applications 235 9.2.1 Abelian groups 235 9.2.2 Non-Abelian groups 237 9.2.3 Characters of the rotation group 50(3) 239 9.2.4 Reduction of representations 240 9.2.5 Reduction of set products 242 9.3 Symmetries of Molecules and Crystals 243

X 9.3.1 Symmetry combinations 244 9.3.2 Vibrational motions 245 9.3.3 Molecular rotations 245 9.3.4 Stability analysis of nuclear positions 246 Problems 250 10 Rotation Groups in Higher Dimensions: Multiparticle Problems 251 10.1 Four-Dimensional Rotations: The Coulomb-Kepler Problem 252 10.1.1 Spherical and parabolic representations 253 10.1.2 Rotations in four dimensions 255 10.1.3 Hydrogen atom in momentum space 258 10.1.4 Alternative subgroups of 50(4): The hydrogen atom in external fields 259 10.1.5 Clebsch-Gordan coefficients for products of 50(4). 261 10.2 Orthogonal Groups in Higher Dimensions 263 10.2.1 Hypersphere in D dimensions 264 10.2.2 Hyperspherical coordinates for multiparticle systems 267 10.2.3 Transformation between alternative schemes 271 10.3 Further Developments 272 10.3.1 Invariance and noninvariance groups 272 10.3.2 "Dynamical symmetries" for atoms and nuclei... 274 10.3.3 Adjoining an extra degree of freedom 276 10.3.4 Alternative reduction schemes for multiparticle systems 277 11 Lorentz Transformations and the Lorentz and Poincare Groups 279 11.1 Lorentz Transformations 281 11.2 Generators and Representations of the Lorentz Group... 283 11.2.1 Four-vectors and the Lorentz metric 284 11.2.2 Generators of the proper Lorentz group 285 11.2.3 Lorentz transformations to r-transformations... 287 11.2.4 Spinor representations 289 11.2.5 Neutrino and electron spinor states 291 11.2.6 Electromagnetism and its quantum 294 11.3 The Inhomogeneous Lorentz (Poincare) Group 295 11.3.1 Generators and commutation relationships 296

11.4 Field Representations 297 11.4.1 Massive systems 298 11.4.2 Representations of massless entities 299 12 Symmetries of the Scattering Continuum 301 12.1 Symmetries of Radial Eigenfunctions 302 12.2 The Full Noninvariance Group of Hydrogen 305 12.2.1 Alternative decompositions of the noninvariance group 308 12.3 Dynamics and Symmetry Transformations 310 Bibliography 313 Index 317