QUANTUM MECHANICS SYMMETRIESPDF电子书下载

1．Symmetries in Quantum Mechanics 1

1.1 Symmetries in Classical Physics 1

1.2 Spatial Translations in Quantum Mechanics 18

1.3 The Unitary Translation Operator 19

1.4 The Equation of Motion for States Shifted in Space 20

1.5 Symmetry and Degeneracy of States 22

1.6 Time Displacements in Quantum Mechanics 30

1.7 Mathematical Supplement:Definition of a Group 32

1.8 Mathematical Supplement:Rotations and their Group Theoretical Properties 35

1.9 An Isomorphism of the Rotation Group 37

1.9.1 Infinitesimal and Finite Rotations 39

1.9.2 Isotropy of Space 41

1.10 The Rotation Operator for Many-Particle States 50

1.11 Biographical Notes 51

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

2.1．Irreducible Representations of the Rotation Group 53

2.2 Matrix Representations of Angular Momentum Operators 57

2.3 Addition of Two Angular Momenta 66

2.4 Evaluation of Clebsch-Gordan Coefficients 70

2.5 Recursion Relations for Clebsch-Gordan Coefficients 71

2.6 Explicit Calculation of Clebsch-Gordan Coefficients 72

2.7 Biographical Notes 79

3．Mathematical Supplement:Fundamental Properties of Lie Groups 81

3.1 General Structure of Lie Groups 81

3.2 Interpretation of Commutators as Generalized Vector Products,Lie's Theorem,Rank of Lie Group 91

3.3 Invariant Subgroups,Simple and Semisimple Lie Groups,Ideals 93

3.4 Compact Lie Groups and Lie Algebras 101

3.5 Invariant Operators(Casimir Operators) 101

3.6 Theorem of Racah 102

3.7 Comments on Multiplets 102

3.8 Invariance Under a Symmetry Group 104

3.9 Construction of the Invariant Operators 108

3.10 Remark on Casimir Operators of Abelian Lie Groups 110

3.11 Completeness Relation for Casimir Operators 110

3.12 Review of Some Groups and Their Properties 112

3.13 The Connection Between Coordianate Transformations and Transformations of Functions 113

3.14 Biographical Notes 126

4．Symmetry Groups and Their Physical Meaning-General Considerations 127

4.1 Biographical Notes 132

5．The Isospin Group(Isobaric Spin) 133

5.1 Isospin Operators for a Multi-Nucleon System 139

5.2 General Properties of Representations of a Lie Algebra 146

5.3 Regular(or Adjoint)Representation of a Lie Algebra 148

5.4 Transformation Law for Isospin Vectors 152

5.5 Experimental Test of Isospin Invariance 159

5.6 Biographical Notes 174

6．The Hypercharge 175

6.1 Biographical Notes 181

7．The SU(3)Symmetry 183

7.1 The 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 Operators 198

7.5 Coupling of T-,U-and V-Multiplets 201

7.6 Quantitative Analysis of Our Reasoning 202

7.7 Further Remarks About the Geometric Form of an SU(3)Multiplet 204

7.8 The Number of States on Mesh Points on Inner Shells 205

8．Quarks and SU(3) 217

8.1 Searching for Quarks 219

8.2 The Transformation Properties of Quark States 220

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

8.4 Construction of the Representation D(p,q)from Quarks and Antiquarks 228

8.4.1．The Smallest SU(3)Representations 231

8.5 Meson Multiplets 240

8.6 Rules for the Reduction of Direct Products of SU(3)Multiplets 244

8.7 U-spin Invariance 248

8.8 Test of U-spin Invariance 250

8.9 The Gell-Mann-Okubo Mass Formula 252

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

8.11 Quark Models with Inner Degrees of Freedom 257

8.12 The Mass Formula in SU(6) 283

8.13 Magnetic Moments in the Quark Model 284

8.14 Excited Meson and Baryon States 286

8.14.1 Combinations of More Than Three Quarks 286

8.15 Excited States with Orbital Angular Momentum 288

9．Representations of the Permutation Group and Young Tableaux 291

9.1 The Permutation Group and Identical Particles 291

9.2 The Standard Form of Young Diagrams 295

9.3 Standard Form and Dimension of Irreducible Representations of the Permutation Group SN 297

9.4 The Connection Between SU(2)and S2 307

9.5 The Irreducible Representations of SU(n) 310

9.6 Determination of the Dimension 316

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

9.8 Decomposition of the Tensor Product of Two Multiplets 322

10．Mathematical Excursion.Group Characters 327

10.1 Definition of Group Characters 327

10.2 Schur's Lemmas 328

10.2.1 Schur's First Lemma 328

10.2.2 Schur's Second Lemma 328

10.3 Orthogonality Relations of Representations and Discrete Groups 329

10.4 Equivalence Classes 331

10.5 Orthogonality Relations of the Group Characters for Discrete Groups and Other Relations 334

10.6 Orthogonality Relations of the Group Characters for the Example of the Group D3 334

10.7 Reduction of a Representation 336

10.8 Criterion for Irreducibility 337

10.9 Direct Product of Representations 337

10.10 Extension to Continuous,Compact Groups 338

10.11 Mathematical Excursion:Group Integration 339

10.12 Unitary Groups 340

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

10.14 Integration over Unitary Groups 344

10.15 Group Characters of Unitary Groups 347

11．Charm and SU(4) 365

11.1 Particles 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

11.5 Advanced Considerations 385

11.5.1 Decay of Mesons with Hidden Charm 385

11.5.2 Decay of Mesons with Open Charm 386

11.5.3 Baryon Multiplets 387

11.6 The Potential Model of Charmonium 398

11.7 The SU(4)[SU(8)]Mass Formula 406

11.8 The γResonances 409

12．Mathematical Supplement 413

12.1 Introduction 413

12.2 Root Vectors and Classical Lie Algebras 417

12.3 Scalar Products of Eigenvalues 421

12.4 Cartan-Wevl Normalization 424

12.5 Graphic Representation of the Root Vectors 424

12.6 Lie Algebra of Rank 1 425

12.7 Lie Algebras of Rank 2 426

12.8 Lie Algebras of Rank 1＞2 426

12.9 The Exceptional Lie Algebras 427

12.10 Simple Roots and Dynkin Diagrams 428

12.11 Dynkin's Prescription 430

12.12 The Cartan Matrix 432

12.13 Determination of all Roots from the Simple Roots 433

12.14 Two Simple Lie Algebras 435

12.15 Representations of the Classical Lie Algebras 436

13．Special Discrete Symmetries 441

13.1 Space Reflection(Parity Transformation) 441

13.2 Reflected States and Operators 443

13.3 Time Reversal 444

13.4 Antiunitary Operators 445

13.5 Many-Particle Systems 450

13.6 Real Eigenfunctions 451

14．Dynamical Symmetries 453

14.1 The Hydrogen Atom 453

14.2 The Group SO(4) 455

14.3 The Energy Levels of the Hydrogen Atom 456

14.4 The Classical Isotropic Oscillator 458

14.4.1 The Quantum Mechanical Isotropic Oscillator 458

15．Mathematical Excursion:Non-compact Lie Groups 473

15.1 Definition and Examples of Non-compact Lie Groups 473

15.2 The Lie Group SO(2,1) 480

15.3 Application to Scattering Problems 484

Subject Index 489