THEORY OF GROUPS AND ITS APPLICATION TO PHYSICAL PROBLEMS
CONTENTS INTRODUCTION A BRIEF HISTORICAL OVERVIEW SYMMETRIES SYMMETRIES OF BODIES SYMMETRIES IN CLASSICAL PHYSICS SYMMETRIES IN QUANTUM MECHANICS REFERENCE GROUPS: DE?NITIONS AND PROPERTIES DE?NITION OF GROUPS AND SIMPLE PROPERTIES EXAMPLES THE GROUP UPPER D 2 THE CYCLIC GROUP UPPER C 4 THE GROUP UPPER D 3 CLASSES OF CONJUGATE ELEMENTS, SUBSETS, AND COSETS THE REARRANGEMENT THEOREM DE?NITION AND PROPERTIES OF CLASSES CLASS MULTIPLICATION SUB-GROUPS AND COSETS NORMAL SUB-GROUPS FACTOR GROUPS PRODUCT GROUPS POINT GROUPS DE?NITION OF POINT GROUPS THE POINT GROUPS OF THE FIRST KIND THE GROUPS UPPER C SUBSCRIPT N THE GROUPS UPPER D SUBSCRIPT N THE TETRAHEDRAL GROUP UPPER T THE CUBIC GROUP UPPER O ICOSAHEDRON GROUP UPPER Y POINT GROUPS IN SOLIDS THE POINT GROUPS OF THE SECOND KIND IMPROPER POINT GROUPS WITHOUT THE INVERSION IMPROPER POINT GROUPS WHICH INCLUDE THE INVERSION THE 32 POINT GROUPS IN SOLIDS THE SEVEN CRYSTAL SYSTEMS REPRESENTATIONS AND CHARACTERS MATRIX GROUPS EQUIVALENT AND IRREDUCIBLE MATRIX GROUPS SCHUR’S LEMMA REPRESENTATIONS ORTHOGONALITY THEOREMS THE FUNDAMENTAL THEOREM IN THE THEORY OF REPRESENTATIONS CONSEQUENCES THEOREM 4: ORTHOGONALITY OF THE CHARACTERS PROOF OF THEOREMS 1–4 CLEAR CRITERION FOR THE IRREDUCIBILITY OF A REPRESENTATION QUANTUM MECHANICS AND GROUP THEORY REPRESENTATION SPACES DE?NITION OF REPRESENTATION SPACES REPRESENTATION FUNCTIONS OF IRREDUCIBLE REPRESENTATIONS REPRESENTATION SPACES AND INVARIANT SUB-SPACES IRREDUCIBILITY OF REPRESENTATION SPACES THE EXPANSION THEOREM PROJECTION OPERATORS THEOREM ON THE ORTHOGONALITY OF REPRESENTATION SPACES HAMILTONIANS WITH SYMMETRIES REMINDER: DEGENERACIES IN QUANTUM MECHANICS GROUP THEORETICAL TREATMENT IRREDUCIBILITY POSTULATE EXAMPLE: A PARTICLE IN A ONE-DIMENSIONAL POTENTIAL DIAGONALIZATION OF HAMILTONIANS IRREDUCIBLE REPRESENTATIONS OF THE POINT GROUPS IN SOLIDS CHARACTER TABLE WITH REPRESENTATION FUNCTIONS EXAMPLE: A PARTICLE IN A CUBIC BOX GROUP THEORY IN STATIONARY PERTURBATION THEORY CALCULATIONS REMINDER: RAYLEIGH-SCHRÖDINGER PERTURBATION THEORY SUBDUCED REPRESENTATIONS DEGENERATE PERTURBATION THEORY APPLICATION: SPLITTING OF ATOMIC ORBITALS IN CRYSTAL FIELDS THE ATOMIC PROBLEM SPLITTING OF ORBITAL ENERGIES IN CRYSTAL FIELDS MATRIX ELEMENTS IN PERTURBATION THEORY MATERIAL TENSORS AND TENSOR OPERATORS MATERIAL TENSORS PHYSICAL MOTIVATION TRANSFORMATION OF TENSORS PRODUCT REPRESENTATIONS INDEPENDENT TENSOR COMPONENTS TENSOR OPERATORS DE?NITION OF TENSOR OPERATORS IRREDUCIBLE TENSOR COMPONENTS MATRIX ELEMENTS OF TENSOR OPERATORS: THE WIGNER-ECKART THEOREM CLEBSCH-GORDAN COEF?CIENTS AND THE WIGNER-ECKART THEOREM FOR ANGULAR MOMENTA CLEBSCH-GORDAN COEF?CIENTS THE WIGNER-ECKART THEOREM FOR ANGULAR MOMENTA MATRIX ELEMENTS IN THE TIME-DEPENDENT PERTURBATION THEORY COUPLING OR CLEBSCH-GORDAN COEF?CIENTS THE WIGNER-ECKHART THEOREM DOUBLE GROUPS AND THEIR REPRESENTATIONS PARTICLES WITH SPIN 1/2 DE?NITION OF DOUBLE GROUPS THE ALGEBRA OF THE DOUBLE GROUPS THE CLASSES OF THE DOUBLE GROUPS THE IRREDUCIBLE REPRESENTATIONS OF THE DOUBLE GROUPS SYMMETRIC REPRESENTATIONS EXTRA REPRESENTATIONS SPACE GROUPS DE?NITIONS THE REAL AF?NE GROUP SYMMORPHIC AND NON-SYMMORPHIC SPACE GROUPS NON-PRIMITIVE TRANSLATIONS DIFFERENCE BETWEEN SYMMORPHIC AND NON-SYMMORPHIC SPACE GROUPS INEQUIVALENT SPACE GROUPS MATRIX SPACE GROUPS THE 14 INEQUIVALENT BRAVAIS LATTICES CLASSI?CATION OF SPACE GROUPS REPRESENTATIONS OF SPACE GROUPS IRREDUCIBLE REPRESENTATIONS OF THE TRANSLATION GROUP THE IRREDUCIBLE REPRESENTATIONS OF SYMMORPHIC SPACE GROUPS IRREDUCIBLE REPRESENTATIONS OF A STAR IN GENERAL POSITION IRREDUCIBLE REPRESENTATIONS OF A STAR IN NON-GENERAL POSITION SPECTRUM OF A HAMILTONIAN WITH SPACE GROUP SYMMETRY PARTICLES IN PERIODIC POTENTIALS SCHRÖDINGER EQUATION, BLOCH THEOREM IRREDUCIBLE PART OF THE BRILLOUIN ZONE COMPATIBILITY CONDITIONS SOLUTION OF THE EIGENVALUE PROBLEM WITH PLANE WAVES TIGHT-BINDING MODELS DERIVATION OF TIGHT-BINDING MODELS THE SLATER KOSTER PARAMETERS REFERENCES