Quantum Mechanics: Symmetries

Quantum Mechanics (Symmetries) deals with a particularly appealing and successful concept in advanced quantum mechanics. After a brief introduction to symmetries in classical mechanics, the text turns to their relevance in quantum mechanics, the consequences of rotation symmetry, and the general theory of Lie groups. The isospin group, hypercharge, SU(3) and their applications are all dealt with in depth before chapters on charm, SU(4), and dynamical symmetries lead to the frontiers of research in particle physics. This unique text comprises more than 120 detailed, worked examples and problems. As the third reprint of the second edition, this book has been revised to bring the text up to date.

Table of 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 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.

Review :
From the reviews:

"a ] A student who works through the examples will be in an excellent position to calculate the consequences of symmetry in atomic, nuclear and particles systems." American Journal of Physics

"? A student who works through the examples will be in an excellent position to calculate the consequences of symmetry in atomic, nuclear and particles systems." American Journal of Physics