Quantum Mechanics - Symmetries

Cover page
Contents
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 Equation of Motion
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
1.10 The Rotation Operator for Many-Particle States
2 Angular Momentum Algebra
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
3 Basics on 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, Ideal
3.4 Compact Lie Groups and Lie Algebras
3.5 Casimir Operators
3.6 Tbeorem of Racab
3.7 Comments on Multiplets
3.8 Invariance Under a Symmetry Group
3.9 Construction of the Invariant Operators
3.10 Remark on Casimir Opera torts of Abelian Lie Groups
3.11 Completeness Relation for Casimir Operators
4 Symmetry Groups and their Significance in Physics
5 Isospin Group
5.1 Isospin Operators for a Multi-Nucleon System
5.2 General Properties of Representations of a Lie Algebra
5.4 Transformation Law for Isospin Vectors
5.5 Experimental Test of Isospin Invariance
6 Hypercharge
6.1 Hypercharge of Nuclei
6.2 Hypercharge of Delta Resonances
6.3 The Baryons
6.4 Antibaryons
6.5 Isospin and Hypercharge of Baryon Resonances
7 SU(3) Symmetry
7.1 The Groups 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 [3J and [3J
8.4 Construction of the Representation D(p,q) from Quarks
and Antiquarks
8.5 Meson Multiplets
8.6 Rules for the Reduction of Direct Products
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
8.15 Excited States with Orbital Angular Momentum
9 Permutation Group, SU(n) 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
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 Group Characters
10.1 Definition of Group Characters
10.2 Schur's Lemmas
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 Relation
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
for SU(4)
11.4 Multiplet Structure of SU(4)
11.5 Advanced Considerations
11.6 The Potential Model of Charmonium
11.7 The SU(4) [SU(8)] Mass Formula
11.8 The Y-Resonances
12 Basics on Lie Algebras
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 >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
15 Non-compact Lie Groups
15.1 Definition and Examples of Non-compact Lie Groups
15.2 The Lie Group SO (2, 1)
15.3 Application to Scattering Problems
Subject Index