Quantum Mechanics

1. Symmetries in Quantum Mechanics.- 1.1 Symmetries in Classical Physics.- 1.2 Spatial Translations in Quantum Mechanics.- 1.3 The Unitary Translation Operator.- 1.4 The Equation of Motion for States Shifted in Space.- 1.5 Symmetry and Degeneracy of States.- 1.6 Time Displacements in Quantum Mechanics.- 1.7 Mathematical Supplement: Definition of a Group.- 1.8 Mathematical Supplement: Rotations and their Group Theoretical Properties.- 1.9 An Isomorphism of the Rotation Group.- 1.9.1 Infinitesimal and Finite Rotations.- 1.9.2 Isotropy of Space.- 1.10 The Rotation Operator for Many-Particle States.- 1.11 Biographical Notes.- 2. Angular Momentum Algebra Representation of Angular Momentum Operators - Generators of SO(3).- 2.1 Irreducible Representations of the Rotation Group.- 2.2 Matrix Representations of Angular Momentum Operators.- 2.3 Addition of Two Angular Momenta.- 2.4 Evaluation of Clebsch-Gordan Coefficients.- 2.5 Recursion Relations for Clebsch-Gordan Coefficients.- 2.6 Explicit Calculation of Clebsch-Gordan Coefficients.- 2.7 Biographical Notes.- 3. Mathematical Supplement: Fundamental Properties of Lie Groups.- 3.1 General Structure of Lie Groups.- 3.2 Interpretation of Commutators as Generalized Vector Products, Lie's Theorem, Rank of Lie Group.- 3.3 Invariant Subgroups, Simple and Semisimple Lie Groups, Ideals.- 3.4 Compact Lie Groups and Lie Algebras.- 3.5 Invariant Operators (Casimir Operators).- 3.6 Theorem of Racah.- 3.7 Comments on Multiplets.- 3.8 Invariance Under a Symmetry Group.- 3.9 Construction of the Invariant Operators.- 3.10 Remark on Casimir Operators of Abelian Lie Groups.- 3.11 Completeness Relation for Casimir Operators.- 3.12 Review of Some Groups and Their Properties.- 3.13 The Connection Between Coordianate Transformations and Transformations of Functions.- 3.14 Biographical Notes.- 4. Symmetry Groups and Their Physical Meaning -General Considerations.- 4.1 Biographical Notes.- 5. The Isospin Group (Isobaric Spin).- 5.1 Isospin Operators for a Multi-Nucleon System.- 5.2 General Properties of Representations of a Lie Algebra.- 5.3 Regular (or Adjoint) Representation of a Lie Algebra.- 5.4 Transformation Law for Isospin Vectors.- 5.5 Experimental Test of Isospin Invariance.- 5.6 Biographical Notes.- 6. The Hypercharge.- 6.1 Biographical Notes.- 7. The SU(3) Symmetry.- 7.1 The Groups U(n) and SU(n).- 7.1.1. The Generators of U(n) and SU(n).- 7.2 The Generators of SU(3).- 7.3 The Lie Algebra of SU(3).- 7.4 The Subalgebras of the SU(3)-Lie Algebra and the Shift Operators.- 7.5 Coupling of T-, U- and V-Multiplets.- 7.6 Quantitative Analysis of Our Reasoning.- 7.7 Further Remarks About the Geometric Form of an SU(3) Multiplet.- 7.8 The Number of States on Mesh Points on Inner Shells.- 8. Quarks and SU(3).- 8.1 Searching for Quarks.- 8.2 The Transformation Properties of Quark States.- 8.3 Construction of all SU(3) Multiplets from the Elementary Representations [3] and 3.- 8.4 Construction of the Representation D(p, q) from Quarks and Antiquarks.- 8.4.1. The Smallest SU(3) Representations.- 8.5 Meson Multiplets.- 8.6 Rules for the Reduction of Direct Product of SU(3) Multiplets.- 8.7 U-spin Invariance.- 8.8 Test of U-spin Invariance.- 8.9 The Gell-Mann-Okubo Mass Formula.- 8.10 The Clebsch-Gordan Coefficients of the SU(3).- 8.11 Quark Models with Inner Degrees of Freedom.- 8.12 The Mass Formula in SU(6).- 8.13 Magnetic Moments in the Quark Model.- 8.14 Excited Meson and Baryon States.- 8.14.1 Combinations of More Than Three Quarks.- 8.15 Excited States with Orbital Angular Momentum.- 9. Representations of the Permutation Group and Young Tableaux.- 9.1 The Permutation Group and Identical Particles.- 9.2 The Standard Form of Young Diagrams.- 9.3 Standard Form and Dimension of Irreducible Representations of the Permutation Group SN.- 9.4 The Connection Between SU(2) and S2.- 9.5 The Irreducible Representations of SU(n).- 9.6 Determination of the Dimension.- 9.7 The SU(n - 1) Subgroups of SU(n).- 9.8 Decomposition of the Tensor Product of Two Multiplets.- 10. Mathematical Excursion. Group Characters.- 10.1 Definition of Group Characters.- 10.2 Schur's Lemmas.- 10.2.1 Schur's First Lemma.- 10.2.2 Schur's Second Lemma.- 10.3 Orthogonality Relations of Representations and Discrete Groups.- 10.4 Equivalence Classes.- 10.5 Orthogonality Relations of the Group Characters for Discrete Groups and Other Relations.- 10.6 Orthogonality Relations of the Group Characters for the Example of the Group D3.- 10.7 Reduction of a Representation.- 10.8 Criterion for Irreducibility.- 10.9 Direct Product of Representations.- 10.10 Extension to Continuous, Compact Groups.- 10.11 Mathematical Excursion: Group Integration.- 10.12 Unitary Groups.- 10.13 The Transition from U(N) to SU(N) for the Example SU(3).- 10.14 Integration over Unitary Groups.- 10.15 Group Characters of Unitary Groups.- 11. Charm and SU(4).- 11.1 Particles with Charm and the SU(4).- 11.2 The Group Properties of SU(4).- 11.3 Tables of the Structure Constants fijk and the Coefficients dijk for SU(4).- 11.4 Multiplet Structure of SU(4).- 11.5 Advanced Considerations.- 11.5.1 Decay of Mesons with Hidden Charm.- 11.5.2 Decay of Mesons with Open Charm.- 11.5.3 Baryon Multiplets.- 11.6 The Potential Model of Charmonium.- 11.7 The SU(4) [SU(8)] Mass Formula.- 11.8 The ? Resonances.- 12. Mathematical Supplement.- 12.1 Introduction.- 12.2 Root Vectors and Classical Lie Algebras.- 12.3 Scalar Products of Eigenvalues.- 12.4 Cartan-Weyl Normalization.- 12.5 Graphic Representation of the Root Vectors.- 12.6 Lie Algebra of Rank 1.- 12.7 Lie Algebras of Rank 2.- 12.8 Lie Algebras of Rank l > 2.- 12.9 The Exceptional Lie Algebras.- 12.10 Simple Roots and Dynkin Diagrams.- 12.11 Dynkin's Prescription.- 12.12 The Cartan Matrix.- 12.13 Determination of all Roots from the Simple Roots.- 12.14 Two Simple Lie Algebras.- 12.15 Representations of the Classical Lie Algebras.- 13. Special Discrete Symmetries.- 13.1 Space Reflection (Parity Transformation).- 13.2 Reflected States and Operators.- 13.3 Time Reversal.- 13.4 Antiunitary Operators.- 13.5 Many-Particle Systems.- 13.6 Real Eigenfunctions.- 14. Dynamical Symmetries.- 14.1 The Hydrogen Atom.- 14.2 The Group SO(4).- 14.3 The Energy Levels of the Hydrogen Atom.- 14.4 The Classical Isotropic Oscillator.- 14.4.1 The Quantum Mechanical Isotropic Oscillator.- 15. Mathematical Excursion: Non-compact Lie Groups.- 15.1 Definition and Examples of Non-compact Lie Groups.- 15.2 The Lie Group SO(2,l).- 15.3 Application to Scattering Problems.