Quantum Mechanics in Hilbert Space Second Edition

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ISBN: 9780486453279 | 0486453278
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Copyright: 12/1/2006

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A rigorous, critical presentation of the basic mathematics of nonrelativistic quantum mechanics, this text is suitable for courses in functional analysis at the advanced undergraduate and graduate levels. Its readable, self-contained form is accessible to students without an extensive mathematical background. Numerous exercises include hints and solutions. 1981 edition.

Preface to the Second Edition

Preface to the First Edition

xvii

List of Symbols

xix

Introduction

I. Basic Ideas of Hilbert Space Theory

1. Vector Spaces

1.1 Vector spaces over fields of scalars

1.2 Linear independence of vectors

1.3 Dimension of a vector space

1.4 Isomorphism of vector spaces

Exercises

2. Euclidean (Pre-Hilbert) Spaces

2.1 Inner products on vector spaces

2.2 The concept of norm

2.3 Orthogonal vectors and orthonormal bases

2.4 Isomorphism of Euclidean spaces

Exercises

3. Metric Spaces

3.1 Convergence in metric spaces

3.2 Complete metric spaces

3.3 Completion of a metric space

Exercises

4. Hilbert Space

4.1 Completion of a Euclidean space

4.2 Separable Hilbert spaces

4.3 l² spaces as examples of separable Hilbert spaces

4.4 Orthonormal bases in Hilbert space

4.5 Isomorphism of separable Hilbert spaces

Exercises

5. Wave Mechanics of a Single Particle Moving in One Dimension

5.1 The formalism and its partial physical interpretation

5.2 The wave mechanical initial-value problem

5.3 Bound states of the system

5.4 A particle moving in a square-well potential

Exercises

References for Further Study

II. Measure Theory and Hilbert Spaces of Functions

1. Measurable Spaces

1.1 Boolean algebras and σ algebras of sets

1.2 Boolean algebras of intervals

1.3 Borel sets in Rn

1.4 Monotone classes of sets

Exercises

2. Measures and Measure Spaces

2.1 The concept of measure

2.2 Basic properties of measures

2.3 Extensions of measures and outer measures

2.4 Cartesian products of measure spaces

Exercises

3. Measurable and Integrable Functions

3.1 The concept of a measurable function

3.2 Properties of measurable functions

3.3 Positive-definite integrable functions

3.4 Real and complex integrable functions

3.5 Infinite sequences and sums of integrals

3.6 Integration on Cartesian products of measure spaces

Exercises

4. Spaces of Square-Integrable Functions

4.1 Square-integrable functions

101

4.2 Hilbert spaces of square-integrable functions

103

4.3 The separability of L² spaces

109

4.4 Change of variables of integration

115

Exercises

118

5. The Hilbert Space of Systems of n Different Particles in Wave Mechanics

5.1 The Schroedinger equation of n-particle systems

119

5.2 The center-of-mass frame of reference

122

5.3 The bound states of n-particle systems

126

5.4 Properties of the n-particle Schroedinger operator

128

5.5 The initial-value problem

131

Exercises

132

6. Direct Sums and Tensor Products of Hilbert Spaces

6.1 Direct sums of Euclidean spaces

132

6.2 Separability and completeness of direct sums of Hilbert spaces

134

6.3 Bilinear forms on vector spaces

137

6.4 Algebraic tensor products of vector spaces

140

6.5 Hilbert tensor products of Hilbert spaces

144

Exercises

147

7. The Two-Body Bound-State Problem with a Spherically Symmetric Potential

7.1 Two particles interacting via a spherically symmetric potential

147

7.2 The equation of motion in spherical coordinates

149

7.3 Spherical harmonics on the unit sphere

151

7.4 The completeness of trigonometric functions

153

7.5 The completeness of Legendre polynomials

157

7.6 Completeness of the spherical harmonics

162

7.7 The two-body problem with a Coulomb potential

164

Exercises

169

References for Further Study

171

III. Theory of Linear Operators in Hilbert Spaces

1. Linear and Antilinear Operators on Euclidean Spaces

1.1 Linear and antilinear transformations

172

1.2 Algebraic operations with linear transformations

174

1.3 Continuous and bounded transformations

178

1.4 Examples of bounded and unbounded operators

179

Exercises

180

2. Linear Operators in Hilbert Spaces

2.1 Linear functionals on normed spaces

182

2.2 The dual of a Hilbert space

183

2.3 Adjoints of linear operators in Hilbert spaces

186

2.4 Bounded linear operators in Hilbert spaces

188

2.5 Dirac notation for linear operators

190

2.6 Closed operators and the graph of an operator

191

2.7 Nonexistence of unbounded everywhere-defined self-adjoint operators

193

Exercises

195

3. Orthogonal Projection Operators

3.1 Projectors onto closed subspaces of a Hilbert space

197

3.2 Algebraic properties of projectors

200

3.3 Partial ordering of projectors

202

3.4 Projectors onto intersections and orthogonal sums of subspaces

203

3.5 Appendix: Extensions and adjoints of closed linear operators

209

Exercises

211

4. Isometric and Unitary Transformations

4.1 Isometric transformations in between Hilbert spaces

212

4.2 Unitary operators and the change of orthonormal basis

214

4.3 The Fourier-Plancherel transform

216

4.4 Cayley transforms of symmetric operators

219

4.5 Self-adjointness of position and momentum operators in wave mechanics

224

Exercises

226

5. Spectral Measures

5.1 The point spectrum of a self-adjoint operator

226

5.2 Spectral resolution of self-adjoint operators with pure point spectrum

227

5.3 Weak, strong, and uniform operator limits

229

5.4 Spectral measures and complex measures

231

5.5 Spectral functions

235

5.6 Appendix: Signed measures

236

Exercises

240

6. The Spectral Theorem for Unitary and Self-Adjoint Operators

6.1 Spectral decomposition of a unitary operator

241

6.2 Monotonic sequences of linear operators

242

6.3 Construction of spectral families for unitary operators

243

6.4 Uniqueness of the spectral family of a unitary operator

247

6.5 Spectral decomposition of a self-adjoint operator

249

6.6 The spectral theorem for bounded self-adjoint operators

253

Exercises

255

References for Further Study

256

IV. The Axiomatic Structure of Quantum Mechanics

1. Basic Concepts in the Quantum Theory of Measurement

1.1 Observables and states in quantum mechanics

257

1.2 The concept of compatible observables

260

1.3 Born's correspondence rule for determinative measurements

261

1.4 Born's correspondence rule for preparatory measurements

264

1.5 The stochastic nature of the quantum theory of measurement

267

Exercises

268

2. Functions of Compatible Observables

2.1 Fundamental and nonfundamental observables

269

2.2 Bounded functions of commuting self-adjoint operators

270

2.3 Algebras of compatible observables

274

2.4 Unbounded functions of commuting self-adjoint operators

277

Exercises

284

3 The Schroedinger, Heisenberg, and Interaction Pictures

3.1 The general form of the Schroedinger equation

285

3.2 The evolution operator

286

3.3 The Schroedinger picture

291

3.4 The Heisenberg picture and physical equivalence of formalisms

293

3.5 The formalism of matrix mechanics

297

3.6 The interaction picture

298

Exercises

300

4. State Vectors and Observables of Compound Systems

4.1 Superselection rules and state vectors

301

4.2 The Hilbert space of compound systems

302

4.3 Tensor products of linear operators

303

4.4 The observables of a system of distinct particles

304

4.5 Symmetric and antisymmetric tensor products of Hilbert spaces

305

4.6 The connection between spin and statistics

306

4.7 Spin and statistics for the n-body problem

308

Exercises

310

5. Complete Sets of Observables

5.1 The concept of a complete set of operators

311

5.2 Cyclic vectors and complete sets of operators

315

5.3 The construction of spectral representation spaces

321

5.4 Cyclicity and maximality

324

Exercises

328

6. Canonical Commutation Relations

6.1 The empirical significance of commutation relations

329

6.2 Representations of canonical commutation relations

331

6.3 One-parameter Abelian groups of unitary operators

334

6.4 Representations of Weyl relations

339

6.5 Appendix: Proof of von Neumann's theorem

342

Exercises

347

7. The General Formalism of Wave Mechanics

7.1 A derivation of one-particle wave mechanics

348

7.2 Wave mechanics of n-particle systems

351

7.3 The Schroedinger operator

354

7.4 Closures of linear operators

355

7.5 The Schroedinger kinetic energy operator

357

7.6 The Schroedinger potential energy operator

360

7.7 The self-adjointness of the Schroedinger operator

366

7.8 The angular momentum operators

369

7.9 Time-dependent Hamiltonians

372

Exercises

373

8. Completely Continuous Operators and Statistical Operators

8.1 Completely continuous operators

375

8.2 The trace of a linear operator

380

8.3 Hilbert-Schmidt operators

383

8.4 The trace norm and the trace class

385

8.5 Statistical ensembles and the process of measurement

390

8.6 The quantum mechanical state of an ensemble

392

8.7 The von Neumann equation in Liouville space

396

8.8 Density matrices on spectral representation spaces

400

8.9 Appendix: Classical and quantum statistical mechanics in master Liouville space

404

Exercises

411

References for Further Study

412

V. Quantum Mechanical Scattering Theory

1. Basic Concepts in Scattering Theory of Two Particles

1.1 Scattering theory and the initial-value problem

414

1.2 Asymptotic states in classical mechanics

416

1.3 Asymptotic states and scattering states in the Schroedinger picture

418

1.4 Moller wave operators

421

1.5 The scattering operator

423

1.6 The differential scattering cross section

425

1.7 The transition operator

426

1.8 The T-matrix formula for the differential cross section

430

Exercises

436

2. General Time-Dependent Two-Body Scattering Theory

2.1 The intertwining property of wave operators

438

2.2 The partial isometry of wave operators

440

2.3 Properties of the S operator

442

2.4 Initial and final domains of wave operators

445

2.5 Dyson's perturbation expansion

448

2.6 Criteria for existence of strong asymptotic states

451

2.7 The physical asymptotic condition

454

Exercises

457

3. General Time-Independent Two-Body Scattering Theory

3.1 The relation of the time-independent to the time-dependent approach

458

3.2 Lippmann-Schwinger equations in Hilbert space

463

3.3 Spectral integral representations of wave operators

471

3.4 The transition amplitude

472

3.5 The resolvent of an operator

474

3.6 The resolvent method in scattering theory

477

3.7 Appendix: Integration of vector- and operator-valued functions

479

3.8 Appendix: Scattering theory in Liouville space

486

Exercises

489

4. Eigenfunction Expansions in Two-Body Potential Scattering Theory

4.1 Free plane waves in three dimensions

491

4.2 Distorted plane waves

494

4.3 Free and distorted spherical waves

496

4.4 Eigenfunction expansions for complete sets of operators

498

4.5 Green's operators and Green functions

501

4.6 Lippmann-Schwinger equations for eigenfunction expansions

503

4.7 The on-shell T-matrix

506

4.8 The off-shell T-matrix and operators

510

4.9 Appendix: Scattering theory for long-range potentials

513

4.10 Appendix: Eigenfunctions and transition density matrices in statistical mechanics

516

Exercises

519

5. Green Functions in Potential Scattering

5.1 The free Green function

520

5.2 Partial wave free Green functions

524

5.3 Fredholm integral equations with Hilbert-Schmidt kernels

525

5.4 The full Green function

528

5.5 Fredholm expansion of the full Green function

529

5.6 Symmetry properties of the full Green function

531

5.7 Appendix: The spectrum of the Schroedinger operator

533

5.8 Appendix: Relations between resolvents and spectral functions

539

Exercises

542

6. Distorted Plane Waves in Potential Scattering

6.1 Potentials of Rollnik class

543

6.2 Fredholm series expressions for distorted plane waves

545

6.3 Asymptotic completeness and the generalized Parseval's equality

550

6.4 The scattering amplitude

553

6.5 The Born series

557

6.6 Distorted plane waves as solutions of the Schroedinger equation

560

6.7 Appendix: Analytic operator-valued functions

564

Exercises

567

7. Wave and Scattering Operators in Potential Scattering

7.1 The existence of strong asymptotic states

569

7.2 The completeness of the Moller wave operators

576

7.3 Proof of asymptotic completeness

580

7.4 Phase shifts for scattering in central potentials

585

7.5 The general phase-shift formula for the scattering operator

588

7.6 Partial-wave analysis for spherically symmetric potentials

592

Exercises

594

8. Fundamental Concepts in Multichannel Scattering Theory

8.1 The concept of channel

596

8.2 Channel Hamiltonians and wave operators

598

8.3 The uniqueness of channel strong asymptotic states

603

8.4 Interchannel scattering operators

607

8.5 The existence of strong asymptotic states in n-particle potential scattering

609

8.6 Two-Hilbert space formulation of multichannel scattering theory

612

8.7 Multichannel eigenfunction expansions and T-matrices

615

8.8 Multichannel Born approximations and Faddeev equations

621

8.9 Appendix: von Neumann's mean ergodic theorem

626

Exercises

628

References for Further Study

630

Hints and Solutions to Exercises

631

References

669

Index

679

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Quantum Mechanics in Hilbert Space Second Edition

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