Symmetries in Quantum Mechanics: From Angular Momentum to Supersymmetry (PBK)

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ISBN: 9780750304085 | 0750304081
Cover: Nonspecific Binding
Copyright: 1/1/1997

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This paperback provides a thorough, didactic exposition of the role of symmetry, particularly rotational symmetry, in quantum mechanics. The bulk of the book covers the description of rotations (geometrically and group-theoretically) and their representations, and the quantum theory of angular momentum. Later chapters introduce more advanced topics such as relativistic theory, supersymmetry, anyons, fractional spin, and statistics. With clear, in-depth explanations, the book is ideal for use as a course text in physics and theoretical physics. It can also serve as an accessible introduction to this important area of quantum theory.

Preface

xiii

1 Introduction

(15)

1.1 Notation

(3)

1.2 Some basic concepts in quantum mechanics

(3)

1.3 Some basic objects of group theory

(7)

1.3.1 Groups: finite, infinite, continuous, Abelian, non-Abelian; subgroup of a group, cosets

(1)

1.3.2 Isomorphism, automorphism, homomorphism

(1)

1.3.3 Lie groups and Lie algebras

(1)

1.3.4 Representations: faithful, irreducible, reducible, completely reducible (decomposable), indecomposable, adjoint, fundamental

(1)

1.3.5 Relation between Lie algebras and Lie groups, Casimir operators, rank of a group

(1)

1.3.6 Schur's lemmas

(1)

1.3.7 Semidirect sum of Lie algebras and semidirect product of Lie groups (inhomogeneous Lie algebras and groups)

(1)

1.3.8 The Haar measure

(1)

1.4 Remark about the introduction of angular momentum

(2)

2 Symmetry in Quantum Mechanics

(37)

2.1 Definition of symmetry

(5)

2.1.1 General considerations

(4)

2.1.2 Formal definition of symmetry; ray correspondence

(1)

2.2 Wigner's theorem: the existence of unitary or anti-unitary representations

(7)

2.3 Continuous matrix groups and their generators

(14)

2.3.1 General considerations

(1)

2.3.2 Continuous matrix groups; decomposition into pieces

(1)

2.3.3 The Lie algebra (Lie ring, infinitesimal ring)

(2)

2.3.4 Canonical coordinates

(4)

2.3.5 The structure of the group and its infinitesimal ring

(2)

2.3.6 Summary: continuous matrix groups and their Lie algebra

(1)

2.3.7 Group representations

(3)

2.4 The physical significance of symmetries

(11)

2.4.1 Continuous groups connected to the identity; Noether's theorem

(3)

2.4.2 Pieces not connected to the identity; discrete groups

(1)

2.4.3 Super-selection rules

(4)

2.4.4 Complete symmetry group, complete sets of commuting observables, complete sets of states

(3)

2.4.5 Summary of the chapter

(1)

3 Rotations in Three-Dimensional Space

(19)

3.1 General remarks on rotations

(3)

3.1.1 Interpretation

(1)

3.1.2 Parameters describing a rotation

(1)

3.1.3 Representation of a rotation

(1)

3.2 Sequences of rotations

(7)

3.2.1 Considering the `abstract' rotations R

(2)

3.2.2 Considering the 3 x 3 rotation matrices M (R)

(4)

3.3 The Lie algebra and the local group

(6)

3.3.1 The rotation matrix M p(n)

(2)

3.3.2 The generators of the rotation group

(2)

3.3.3 The local group

(1)

3.3.4 Canonical parameters of the first and the second kind

(1)

3.4 The unitary representation U (R) induced by the three-dimensional rotation R

(3)

4 Angular Momentum Operators and Eigenstates

(49)

4.1 The operators of angular momentum J(1), J(2) and J(3)

(3)

4.1.1 The physical significance of J

(3)

4.1.2 The angular momentum component in a direction n

(1)

4.2 Commutation relations for angular momenta

(5)

4.3 Direct sum and direct product

(5)

4.4 Angular momenta of interacting systems

(2)

4.5 Irreducible representations; Schur's lemma

(4)

4.6 Eigenstates of angular momentum

(8)

4.7 Orbital angular momentum

(12)

4.7.1 Angular momentum operators in polar coordinates

100

(2)

4.7.2 Construction of the eigenfunctions

102

(2)

4.7.3 Orbital angular momenta have only integer eigenvalues

104

(1)

4.7.4 Spherical harmonics

105

(3)

4.7.5 The phase convention

108

(1)

4.7.6 Parity

108

(1)

4.7.7 Particular cases

108

(3)

4.7.8 Further formulae

111

(1)

4.8 Spin-1/2 eigenstates and operators

111

(3)

4.9 Double-valued representations; the covering group SU(2)

114

(2)

4.10 Construction of the general j, m-state from spin-1/2 states

116

(5)

5 Addition of Angular Momenta

121

(49)

5.1 The general problem

121

(1)

5.2 Complete sets of mutually commuting (angular momentum) observables

122

(6)

5.3 Combining two angular momenta; Clebsch-Gordan (Wigner) coefficients

128

(31)

5.3.1 Notation

128

(3)

5.3.2 Definition and some properties of the Clebsch-Gordan coefficients

131

(2)

5.3.3 Orthogonality of the Clebsch-Gordan coefficients

133

(1)

5.3.4 Sketch of the calculation of the Clebsch-Gordan coefficients; phase convention and reality

134

(4)

5.3.5 Calculation of [j(1)m(1)j(2)j -- m(1)|jj]

138

(2)

5.3.6 Obvious symmetry relations for CGCs

140

(7)

5.3.7 Wigner's 3j-symbol and Racah's V (j(1)j(2)j(3)|m(1)m(2)m(3))-symbol

147

(2)

5.3.8 Racah's formula for the CGCs

149

(5)

5.3.9 Regge's symmetry of CGCs

154

(2)

5.3.10 Collection of formulae for the CGCs; a table of special values

156

(3)

5.4 Combining three angular momenta; recoupling coefficients

159

(9)

5.4.1 General remarks; statement of the problem

159

(3)

5.4.2 The 6j-symbol and the Racah coefficients

162

(2)

5.4.3 Collection of formulae for recoupling coefficients

164

(4)

5.5 Combining more than three angular momenta

168

(1)

5.6 Numerical tables and important references on addition of angular momenta

168

(2)

6 Representations of the Rotation Group

170

(26)

6.1 Active and passive interpretation; definition of D^(j)(m'm); the invariant subspaces Hj

170

(3)

6.2 The explicit form of D^(j)(m'm) (Alpha, Beta, Gamma)

173

(4)

6.2.1 The spin-1/2 case

173

(2)

6.2.2 The general case

175

(2)

6.3 General properties of D(j)

177

(19)

6.3.1 Relation to the Clebsch-Gordan coefficients

177

(2)

6.3.2 Significance of the relation to the CGCs

179

(4)

6.3.3 Relation to the eigenfunctions of angular momentum

183

(5)

6.3.4 Orthogonality relations and integrals over D-matrices

188

(5)

6.3.5 A projection formula

190

(1)

6.3.6 Completeness relation for the D-matrices

191

(3)

6.3.7 Symmetry properties of the D-matrices

194

(2)

7 The Jordan-Schwinger Construction and Representations

196

(11)

7.1 Bosonic operators

196

(3)

7.2 Realization of su(2) Lie algebra and the rotation matrix in terms of bosonic operators

199

(5)

7.3 A short note about the new field of quantum groups

204

(3)

8 Irreducible Tensors and Tensor Operators

207

(20)

8.1 Introduction

207

(2)

8.2 Definition and properties

209

(2)

8.3 Tensor product; irreducible combination of irreducible tensors; scalar product

211

(2)

8.4 Invariants and covariant equations

213

(2)

8.5 Spinor and vector spherical harmonics

215

(4)

8.6 Angular momenta as spherical tensor operators

219

(1)

8.7 The Wigner-Eckart theorem

220

(2)

8.8 Examples of applications of the Wigner-Eckart theorem

222

(1)

8.8.1 The trace of T(kq)

222

(1)

8.8.2 Tensors of rank O (scalars, invariants)

223

(1)

8.8.3 The angular momentum operators

223

(1)

8.9 Projection theorem for irreducible tensor operators of rank 1

223

(4)

9 Peculiarities of Two-Dimensional Rotations: Anyons, Fractional Spin and Statistics

227

(12)

9.1 Introduction

227

(1)

9.2 Properties of rotations in two-dimensional space and fractional statistics

228

(4)

9.3 Particle-flux system: example of anyon

232

(5)

9.4 Possible role of anyons in physics

237

(2)

10 A Brief Glance at Relativistic Problems

239

(36)

10.1 Introduction

239

(1)

10.2 The generators of the inhomogeneous Lorentz group (Poincare group)

240

(8)

10.2.1 Translations; four-momentum

241

(1)

10.2.2 The homogeneous Lorentz group; angular momentum

242

(6)

10.3 The angular momentum operators

248

(1)

10.3.1 Commutation relations of the J^(mv) with each other

248

(1)

10.3.2 Commutation relations of the J^(mv) with P^p

249

(1)

10.4 A complete set of commuting observables

249

(14)

10.4.1 The spin four-vector w^m and the spin tensor S^(mv)

250

(2)

10.4.2 Commutation relations for w^m and S^(mv)

252

(3)

10.4.3 Construction of a complete set of commuting observables; helicity

255

(5)

10.4.4 Zero-mass particles

260

(3)

10.5 The use of helicity states in elementary particle physics

263

(12)

10.5.1 Construction of one-particle helicity states of arbitrary p

263

(2)

10.5.2 Two-particle helicity states

265

(1)

10.5.3 Eigenstates of the total angular momentum

265

(2)

10.5.4 The S-matrix; cross-sections

267

(2)

10.5.5 Evaluation of cross-section formulae

269

(1)

10.5.6 Discrete symmetry relations: parity, time reversal, identical particles

270

(5)

11 Supersymmetry in Quantum Mechanics and Particle Physics

275

(12)

11.1 What is supersymmetry?

275

(4)

11.2 SUSY quantum mechanics

279

(2)

11.3 Factorization and the hierarchy of Hamiltonians

281

(4)

11.4 Broken supersymmetry

285

(2)

Appendix A

287

(4)

Appendix B

291

(2)

Bibliography

293

(4)

Index

297

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Symmetries in Quantum Mechanics: From Angular Momentum to Supersymmetry (PBK)

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