Group Theory and Quantum Mechanics Book

	Preface	V
1	Introduction	1
1-1	The Nature of the Problem	1
1-2	The Role of Symmetry	3
2	Abstract Group Theory	6
2-1	Definitions and Nomenclature	6
2-2	Illustrative Examples	7
2-3	Rearrangement Theorem	8
2-4	Cyclic Groups	9
2-5	Subgroups and Cosets	9
2-6	Example Groups of Finite Order	10
2-7	Conjugate Elements and Class Structure	12
2-8	Normal Divisors and Factor Groups	13
2-9	Class Multiplication	15
	Exercises	16
	References	17
3	Theory of Group Representations	18
3-1	Definitions	18
3-2	Proof of the Orthogonality Theorem	20
3-3	The Character of a Representation	25
3-4	Construction of Character Tables	28
3-5	Decomposition of Reducible Representations	29
3-6	Application of Representation Theory in Quantum Mechanics	31
3-7	Illustrative Representations of Abelian Groups	37
3-8	Basis Functions for Irreducible Representations	39
3-9	Direct-product Groups	43
3-10	Direct-product Representations within a Group	46
	Exercises	47
	References	48
4	Physical Applications of Group Theory	50
4-1	Crystal-symmetry Operators	51
4-2	The Crystallographic Point Groups	54
4-3	Irreducible Representations of the Point Groups	62
4-4	Elementary Representations of the Three-dimensional Rotation Group	65
4-5	Crystal-field Splitting of Atomic Energy Levels	67
4-6	Intermediate Crystal-field-splitting Case	69
4-7	Weak-crystal-field Case and Crystal Double Groups	75
4-8	Introduction of Spin Effects in the Medium-field Case	78
4-9	Group-theoretical Matrix-element Theorems	80
4-10	Selection Rules and Parity	82
4-11	Directed Valence	87
4-12	Application of Group Theory to Directed Valence	89
	Exercises	92
	References	93
5	Full Rotation Group and Angular Momentum	94
5-1	Rotational Transformation Properties and Angular Momentum	94
5-2	Continuous Groups	98
5-3	Representation of Rotations through Eulerian Angles	101
5-4	Homomorphism with the Unitary Group	103
5-5	Representations of the Unitary Group	106
5-6	Representation of the Rotation Group by Representations of the Unitary Group	109
5-7	Application of the Rotation-representation Matrices	111
5-8	Vector Model for Addition of Angular Momenta	115
5-9	The Wigner or Clebsch-Gordan Coefficients	117
5-10	Notation, Tabulations, and Symmetry Properties of the Wigner Coefficients	121
5-11	Tensor Operators	124
5-12	The Wigner-Eckart Theorem	131
5-13	The Racah Coefficients	133
5-14	Application of Racah Coefficients	137
5-15	The Rotation-Inversion Group	139
5-16	Time-reversal Symmetry	141
5-17	More General Invariances	147
	Exercises	151
	References	153
6	Quantum Mechanics of Atoms	154
6-1	Review of Elementary Atomic Structure and Nomenclature	155
6-2	The Hamiltonian	157
6-3	Approximate Eigenfunctions	157
6-4	Calculation of Matrix Elements between Determinantal Wavefunctions	162
6-5	Hartree-Fock Method	167
6-6	Calculation of L-S-term Energies	170
6-7	Evaluation of Matrix Elements of the Energy	173
6-8	Eigenfunctions and Angular-momentum Operations	178
6-9	Calculation of Fine Structure	181
6-10	Zeeman Effect	188
6-11	Magnetic Hyperfine Structure	193
6-12	Electric Hyperfine Structure	201
	Exercises	206
	References	208
7	Molecular Quantum Mechanics	210
7-1	Born-Oppenheimer Approximation	210
7-2	Simple Electronic Eigenfunctions	213
7-3	Irreducible Representations for Linear Molecules	216
7-4	The Hydrogen Molecule	219
7-5	Molecular Orbitals	220
7-6	Heitler-London Method	223
7-7	Orthogonal Atomic Orbitals	226
7-8	Group Theory and Molecular Orbitals	227
7-9	Selection Rules for Electronic Transitions	233
7-10	Vibration of Diatomic Molecules	234
7-11	Normal Modes in Polyatomic Molecules	238
7-12	Group Theory and Normal Modes	242
7-13	Selection Rules for Vibrational Transitions	248
7-14	Molecular Rotation	250
7-15	Effect of Nuclear Statistics on Molecular Rotation	252
7-16	Asymmetric Rotor	255
7-17	Vibration-Rotation Interaction	257
7-18	Rotation-Electronic Coupling	260
	Exercises	264
	References	266
8	Solid-state Theory	267
8-1	Symmetry Properties in Solids	267
8-2	The Reciprocal Lattice and Brillouin Zones	270
8-3	Form of Energy-band Wavefunctions	275
8-4	Crystal Symmetry and the Group of the k Vector	279
8-5	Pictorial Consideration of Eigenfunctions	281
8-6	Formal Consideration of Degeneracy and Compatibility	284
8-7	Group Theory and the Plane-wave Approximation	290
8-8	Connection between Tight- and Loose-binding Approximations	293
8-9	Spin-orbit Coupling in Band Theory	295
8-10	Time Reversal in Band Theory	297
8-11	Magnetic Crystal Groups	299
8-12	Symmetries of Magnetic Structures	303
8-13	The Landau Theory of Second-order Phase Transitions	309
8-14	Irreducible Representations of Magnetic Groups	311
	Exercises	313
	References	315
	Appendix
A	Review of Vectors, Vector Spaces, and Matrices	317
B	Character Tables for Point-symmetry Groups	323
C	Tables of c[superscript k] and a[superscript k] Coefficients for s, p, and d Electrons	331
	Index	335