《QUANTUM MECHANICS SYMMETRIES》

1．Symmetries in Quantum Mechanics1

1.1Symmetries in Classical Physics1

1.2 Spatial Translations in Quantum Mechanics18

1.3 The Unitary Translation Operator19

1.4 The Equation of Motion for States Shifted in Space20

1.5 Symmetry and Degeneracy of States22

1.6 Time Displacements in Quantum Mechanics30

1.7 Mathematical Supplement:Definition of a Group32

1.8 Mathematical Supplement:Rotations and their Group Theoretical Properties35

1.9An Isomorphism of the Rotation Group37

1.9.1 Infinitesimal and Finite Rotations39

1.9.2 Isotropy of Space41

1.10 The Rotation Operator for Many-Particle States50

1.11 Biographical Notes51

2．Angular Momentum Algebra Representation of Angular Momentum Operators—Generators of SO(3)53

2.1．Irreducible Representations of the Rotation Group53

2.2Matrix Representations of Angular Momentum Operators57

2.3 Addition of Two Angular Momenta66

2.4 Evaluation of Clebsch-Gordan Coefficients70

2.5 Recursion Relations for Clebsch-Gordan Coefficients71

2.6 Explicit Calculation of Clebsch-Gordan Coefficients72

2.7 Biographical Notes79

3．Mathematical Supplement:Fundamental Properties of Lie Groups81

3.1General Structure of Lie Groups81

3.2 Interpretation of Commutators as Generalized Vector Products,Lie's Theorem,Rank of Lie Group91

3.3 Invariant Subgroups,Simple and Semisimple Lie Groups,Ideals93

3.4 Compact Lie Groups and Lie Algebras101

3.5 Invariant Operators(Casimir Operators)101

3.6 Theorem of Racah102

3.7 Comments on Multiplets102

3.8 Invariance Under a Symmetry Group104

3.9 Construction of the Invariant Operators108

3.10 Remark on Casimir Operators of Abelian Lie Groups110

3.11 Completeness Relation for Casimir Operators110

3.12 Review of Some Groups and Their Properties112

3.13 The Connection Between Coordianate Transformations and Transformations of Functions113

3.14 Biographical Notes126

4．Symmetry Groups and Their Physical Meaning-General Considerations127

4.1Biographical Notes132

5．The Isospin Group(Isobaric Spin)133

5.1Isospin Operators for a Multi-Nucleon System139

5.2 General Properties of Representations of a Lie Algebra146

5.3 Regular(or Adjoint)Representation of a Lie Algebra148

5.4 Transformation Law for Isospin Vectors152

5.5 Experimental Test of Isospin Invariance159

5.6 Biographical Notes174

6．The Hypercharge175

6.1Biographical Notes181

7．The SU(3)Symmetry183

7.1The Groups U(n)and SU(n)183

7.1.1．The Generators of U(n)and SU(n)185

7.2 The Generators of SU(3)187

7.3 The Lie Algebra of SU(3)190

7.4 The Subalgebras of the SU(3)-Lie Algebra and the Shift Operators198

7.5 Coupling of T-,U-and V-Multiplets201

7.6 Quantitative Analysis of Our Reasoning202

7.7 Further Remarks About the Geometric Form of an SU(3)Multiplet204

7.8 The Number of States on Mesh Points on Inner Shells205

8．Quarks and SU(3)217

8.1Searching for Quarks219

8.2 The Transformation Properties of Quark States220

8.3 Construction of all SU(3)Multiplets from the Elementary Representations[3]and[？]226

8.4Construction of the Representation D(p,q)from Quarks and Antiquarks228

8.4.1．The Smallest SU(3)Representations231

8.5 Meson Multiplets240

8.6 Rules for the Reduction of Direct Products of SU(3)Multiplets244

8.7 U-spin Invariance248

8.8 Test of U-spin Invariance250

8.9 The Gell-Mann-Okubo Mass Formula252

8.10 The Clebsch-Gordan Coefficients of the SU(3)254

8.11 Quark Models with Inner Degrees of Freedom257

8.12 The Mass Formula in SU(6)283

8.13 Magnetic Moments in the Quark Model284

8.14 Excited Meson and Baryon States286

8.14.1 Combinations of More Than Three Quarks286

8.15 Excited States with Orbital Angular Momentum288

9．Representations of the Permutation Group and Young Tableaux291

9.1The Permutation Group and Identical Particles291

9.2 The Standard Form of Young Diagrams295

9.3 Standard Form and Dimension of Irreducible Representations of the Permutation Group SN297

9.4 The Connection Between SU(2)and S2307

9.5 The Irreducible Representations of SU(n)310

9.6 Determination of the Dimension316

9.7 The SU(n-1)Subgroups of SU(n)320

9.8 Decomposition of the Tensor Product of Two Multiplets322

10．Mathematical Excursion.Group Characters327

10.1Definition of Group Characters327

10.2Schur's Lemmas328

10.2.1 Schur's First Lemma328

10.2.2 Schur's Second Lemma328

10.3 Orthogonality Relations of Representations and Discrete Groups329

10.4 Equivalence Classes331

10.5 Orthogonality Relations of the Group Characters for Discrete Groups and Other Relations334

10.6 Orthogonality Relations of the Group Characters for the Example of the Group D3334

10.7 Reduction of a Representation336

10.8 Criterion for Irreducibility337

10.9 Direct Product of Representations337

10.10 Extension to Continuous,Compact Groups338

10.11 Mathematical Excursion:Group Integration339

10.12 Unitary Groups340

10.13 The Transition from U(N)to SU(N)for the Example SU(3)342

10.14 Integration over Unitary Groups344

10.15 Group Characters of Unitary Groups347

11．Charm and SU(4)365

11.1Particles with Charm and the SU(4)367

11.2 The Group Properties of SU(4)367

11.3 Tables of the Structure Constants fijk and the Coefficients dijk for SU(4)376

11.4 Multiplet Structure of SU(4)378

11.5Advanced Considerations385

11.5.1 Decay of Mesons with Hidden Charm385

11.5.2 Decay of Mesons with Open Charm386

11.5.3 Baryon Multiplets387

11.6 The Potential Model of Charmonium398

11.7 The SU(4)[SU(8)]Mass Formula406

11.8 The γResonances409

12．Mathematical Supplement413

12.1Introduction413

12.2 Root Vectors and Classical Lie Algebras417

12.3 Scalar Products of Eigenvalues421

12.4 Cartan-Wevl Normalization424

12.5 Graphic Representation of the Root Vectors424

12.6 Lie Algebra of Rank 1425

12.7 Lie Algebras of Rank 2426

12.8 Lie Algebras of Rank 1＞2426

12.9 The Exceptional Lie Algebras427

12.10 Simple Roots and Dynkin Diagrams428

12.11 Dynkin's Prescription430

12.12 The Cartan Matrix432

12.13 Determination of all Roots from the Simple Roots433

12.14 Two Simple Lie Algebras435

12.15 Representations of the Classical Lie Algebras436

13．Special Discrete Symmetries441

13.1Space Reflection(Parity Transformation)441

13.2 Reflected States and Operators443

13.3 Time Reversal444

13.4 Antiunitary Operators445

13.5 Many-Particle Systems450

13.6 Real Eigenfunctions451

14．Dynamical Symmetries453

14.1The Hydrogen Atom453

14.2 The Group SO(4)455

14.3 The Energy Levels of the Hydrogen Atom456

14.4The Classical Isotropic Oscillator458

14.4.1 The Quantum Mechanical Isotropic Oscillator458

15．Mathematical Excursion:Non-compact Lie Groups473

15.1Definition and Examples of Non-compact Lie Groups473

15.2 The Lie Group SO(2,1)480

15.3 Application to Scattering Problems484

Subject Index489