| Preface 1 Setting the Stage 1.1 Introduction 1.2 Fundamental Assumptions of Quantum Mechanics 1.3 The Hydrogen Atom 1.4 The Periodic Table 1.5 Preliminary Mathematics 1.6 Spherical Harmonics 1.7 Equivalence Classes 1.8 Exercises 2 Linear Algebra over the Complex Numbers 2.1 Complex Vector Spaces 2.2 Dimension 2.3 Linear Transformations 2.4 Kernels and Images of Linear Transformations 2.5 Linear Operators 2.6 Cartesian Sums and Tensor Products 2.7 Exercises 3 Complex Scalar Product Spaces (a.k.a. Hilbert Spaces) 3.1 Lebesgue Equivalence and L2(N3) 3.2 Complex Scalar Products 3.3 Euclidean-style Geometry in Complex Scalar Product Spaces 3.4 Norms and Approximations 3.5 Useful Spanning Subspaces 3.6 Exercises 4 Lie Groups and Lie Group Representations 4.1 Groups and Lie Groups 4.2 The Key Players: SO(3), SU(2) and SO(4) 4.3 The Spectral Theorem for SU(2) and the Double Cover of SO(3) 4.4 Representations: Definition and Examples 4.5 Representations in Quantum Mechanics 4.6 Homogeneous Polynomials in Two Variables 4.7 Characters of Representations 4.8 Exercises 5 New Representations from Old 5.1 Subrepresentations 5.2 Cartesian Sums of Representations 5.3 Tensor Products of Representations 5.4 Dual Representations 5.5 The Representation Horn 5.6 Pullback and Pushforward Representations 5.7 Exercises 6 Irreducible Representations and Invariant Integration 6.1 Definitions and Schur's Lemma 6.2 Elementary States of Quantum Mechanical Systems 6.3 Invariant Integration and Characters of Irreducible Representations 6.4 Isotypic Decompositions (Optional) 6.5 Classification of the Irreducible Representations of SU(2) 6.6 Classification of the Irreducible Representations of SO(3) 6.7 Exercises 7 Representations and the Hydrogen Atom 8 The Algebra so(4)Symmetry of the Hydrogen Atom 9 The Group So(4)Symmetry of the Hydrogen Atom 10 Projective Representations and Spin 11 Independent Events and Tensor Products A Spherical Harmonics B Proof of the Correspondence Between Irreducible Linear Representations of SU(2)and C Suggested Paper Topics Bibliography Glossary of Symbols and Notation Index |
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