Group Theory and Quantum Mechanics

Description

This graduate-level text develops the aspects of group theory most relevant to physics and chemistry (such as the theory of representations) and illustrates their applications to quantum mechanics. The first five chapters focus chiefly on the introduction of methods, illustrated by physical examples, and the final three chapters offer a systematic treatment of the quantum theory of atoms, molecules, and solids. The formal theory of finite groups and their representation is developed in Chapters 1 through 4 and illustrated by examples from the crystallographic point groups basic to solid-state and molecular theory. Chapter 5 is devoted to the theory of systems with full rotational symmetry, Chapter 6 to the systematic presentation of atomic structure, and Chapter 7 to molecular quantum mechanics. Chapter 8, which deals with solid-state physics, treats electronic energy band theory and magnetic crystal symmetry. A compact and worthwhile compilation of the scattered material on standard methods, this volume presumes a basic understanding of quantum theory. PREFACE......Page 4 3 Theory of Group Representations......Page 8 5 Full Rotation Group and Angular Momentum......Page 9 7 Molecular Quantum Mechanics......Page 10 Index......Page 11 1-1 The Nature of the Problem......Page 12 1-2 The Role of Symmetry......Page 14 2-1 Definitions and Nomenclature......Page 17 2-2 Illustrative Examples......Page 18 2-3 Rearrangement Theorem......Page 19 2.......Page 21 ii......Page 22 2-7 Conjugate Elements and Class Structure......Page 23 CMi,......Page 24 Isomorphy and homomorphy.......Page 25 W = 5(2-2)......Page 26 2-4......Page 27 REFERENCES......Page 28 3-1 Definitions......Page 29 Reducible and irreducible representations.......Page 30 3-2 Proof of the Orthogonality Theorem......Page 31 3-3 The Character of a Representation......Page 36 3-4 Construction of Character Tables......Page 39 3-5 Decomposition of Reducible Representations......Page 40 3-6 Application of Representation Theory in Quantum Mechanics Transformation operators.......Page 42 3-7 Illustrative Representations of Abelian Groups......Page 48 3-8 Basis Functions for Irreducible Representations......Page 50 3-9 Direct-product Groups......Page 54 3-10 Direct-product Representations within a Group......Page 57 PHYSICAL APPLICATIONS OF GROUP THEORY......Page 61 Successive operations.......Page 62 Fig. 4-1.......Page 63 2b.......Page 64 o.......Page 65 x......Page 68 Fig. 4-5.......Page 70 Classification of functions by transformation properties.......Page 73 4-4 Elementary Representations of the Three-dimensional Rotation Group......Page 76 4-5 Crystal-field Splitting of Atomic Energy Levels......Page 78 of lower symmetry.......Page 82 J......Page 83 the......Page 84 Physical picture for splittings.......Page 85 4-7 Weak-crystal-field Case and Crystal Double Groups......Page 86 Crystal double groups.......Page 87 Application of double groups to term splittings.......Page 88 4-8 Introduction of Spin Effects in the Medium-field Case......Page 89 l» ~ r2 + r4 + r5.......Page 90 Generalized Unsold theorem.......Page 91 of......Page 92 4-10 Selection Rules and Parity......Page 93 p.......Page 94 Inversion and parity.......Page 95 r,x F......Page 96 x......Page 97 11......Page 98 Vx.-i.-x = K-s......Page 99 4-12 Application of Group Theory to Directed Valence......Page 100 147,......Page 104 V,......Page 105 60x-^o L n......Page 106 Fig. 5-1.......Page 108 The Hurwitz invariant integral.......Page 109 5-3 Representation of Rotations through Eulerian Angles......Page 112 x......Page 114 o......Page 115 I (5-27)......Page 116 a......Page 117 u(......Page 120 2......Page 121 5-7 Application of the Rotation-representation Matrices......Page 122 c'......Page 123 Fig. 5-4.......Page 125 |>8 Vector Model for Addition of Angular Momenta......Page 126 Wei......Page 129 u)......Page 130 46......Page 131 9......Page 137 7>(......Page 142 13 The Racah Coefficients......Page 144 1......Page 145 5-14 Application of Racah Coefficients......Page 148 5-15 The Rotation-Inversion Group......Page 150 5-16 Time-reversal Symmetry......Page 152 Determination of the time-inversion operator.......Page 153 p......Page 158 EXERCISES......Page 162 REFERENCES......Page 164 QUANTUM MECHANICS OF ATOMS......Page 165 6-1 Review of Elementary Atomic Structure and Nomenclature......Page 166 6-3 Approximate Eigenfunctions Simple approximate eigenfunctions.......Page 168 6-4 Calculation of Matrix Elements between Determinantal Wavefunctions......Page 173 6-5 Hartree-Fock Method......Page 178 6-6 Calculation of......Page 181 6-7 Evaluation of Matrix Elements of the Energy......Page 184 6-8 Eigenfunctions and Angular-momentum Operations......Page 189 6-9 Calculation of Fine Structure......Page 192 6-10 Zeeman Effect......Page 199 6-11 Magnetic Hyperfine Structure......Page 204 6-12 Electric Hyperfine Structure......Page 212 EXERCISES......Page 213 REFERENCES......Page 219 7-1 Born-Oppenheimer Approximation......Page 221 7-2 Simple Electronic Eigenfunctions......Page 224 7-3 Irreducible Representations for Linear Molecules......Page 227 7-4 The Hydrogen Molecule......Page 230 7-5 Molecular Orbitals......Page 231 7-6 Heitler-London Method......Page 234 7-7 Orthogonal Atomic Orbitals......Page 237 7-8 Group Theory and Molecular Orbitals......Page 239 7-9 Selection Rules for Electronic Transitions......Page 244 7-10 Vibration of Diatomic Molecules......Page 245 7-11 Normal Modes in Polyatomic Molecules......Page 249 7-12 Group Theory and Normal Modes......Page 254 7-13 Selection Rules for Vibrational Transitions......Page 259 7-14 Molecular Rotation......Page 261 7-15 Effect of Nuclear Statistics on Molecular Rotation......Page 264 7-16 Asymmetric Rotor......Page 266 7-17 Vibration-Rotation Interaction......Page 268 7-18 Rotation-Electronic Coupling......Page 271 EXERCISES......Page 275 REFERENCES......Page 277 8-1 Symmetry Properties in Solids......Page 278 8-2 The Reciprocal Lattice and Brillouin Zones......Page 281 8-3 Form of Energy-band Wavefunctions......Page 286 8-4 Crystal Symmetry and the Group of the k Vector......Page 290 8-5 Pictorial Consideration of Eigenfunctions......Page 292 8-6 Formal Consideration of Degeneracy and Compatibility......Page 295 8-7 Group Theory and the Plane-wave Approximation......Page 301 8-8 Connection between Tightand Loose-binding Approximations......Page 304 8-9 Spin-orbit Coupling in Band Theory......Page 306 8-10 Time Reversal in Band Theory......Page 308 8-11 Magnetic Crystal Groups......Page 310 8-12 Symmetries of Magnetic Structures......Page 314 8-13 The Landau Theory of Second-order Phase Transitions......Page 320 8-14 Irreducible Representations of Magnetic Groups......Page 323 EXERCISES......Page 324 REFERENCES......Page 326 Vectors......Page 328 Matrices and Transformations......Page 329 Generalizations......Page 330 Diagonalization of Matrices......Page 331 Nomenclature and Properties of Special Matrices......Page 332 REFERENCES......Page 333 APPENDIX B CHARACTER TABLES FOR POINT-SYMMETRY GROUPS......Page 334 REFERENCES......Page 341 INDEX......Page 346 255......Page 351