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Quantum Theory for Mathematicians

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Table Of Content:

1 The Experimental Origins of Quantum Mechanics 1
1.1 Is Light a Wave or a Particle? ……………. 1
1.2 Is an Electron a Wave or a Particle? ………… 7
1.3 Schr¨odinger and Heisenberg . . . . . . . . . . . . . . . . . 13
1.4 A Matter of Interpretation . . . . . . . . . . . . . . . . . . 14
1.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2 A First Approach to Classical Mechanics 19
2.1 Motion in R1 . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2 Motion in Rn . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.3 Systems of Particles . . . . . . . . . . . . . . . . . . . . . . 26
2.4 Angular Momentum . . . . . . . . . . . . . . . . . . . . . . 31
2.5 Poisson Brackets and Hamiltonian Mechanics . . . . . . . 33
2.6 The Kepler Problem and the Runge–Lenz Vector . . . . . 41
2.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3 A First Approach to Quantum Mechanics 53
3.1 Waves, Particles, and Probabilities . . . . . . . . . . . . . 53
3.2 A Few Words About Operators and Their Adjoints . . . . 55
3.3 Position and the Position Operator . . . . . . . . . . . . . 58
3.4 Momentum and the Momentum Operator . . . . . . . . . 59
3.5 The Position and Momentum Operators . . . . . . . . . . 62
3.6 Axioms of Quantum Mechanics: Operators
and Measurements . . . . . . . . . . . . . . . . . . . . . . 64
xi
xii Contents
3.7 Time-Evolution in Quantum Theory . . . . . . . . . . . . 70
3.8 The Heisenberg Picture . . . . . . . . . . . . . . . . . . . . 78
3.9 Example: A Particle in a Box . . . . . . . . . . . . . . . . 80
3.10 Quantum Mechanics for a Particle in Rn . . . . . . . . . . 82
3.11 Systems of Multiple Particles . . . . . . . . . . . . . . . . 84
3.12 Physics Notation . . . . . . . . . . . . . . . . . . . . . . . 85
3.13 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
4 The Free Schr¨odinger Equation 91
4.1 Solution by Means of the Fourier Transform . . . . . . . . 92
4.2 Solution as a Convolution . . . . . . . . . . . . . . . . . . 94
4.3 Propagation of the Wave Packet: First Approach . . . . . 97
4.4 Propagation of the Wave Packet: Second Approach . . . . 100
4.5 Spread of the Wave Packet . . . . . . . . . . . . . . . . . 104
4.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
5 A Particle in a Square Well 109
5.1 The Time-Independent Schr¨odinger Equation . . . . . . . 109
5.2 Domain Questions and the Matching Conditions . . . . . . 111
5.3 Finding Square-Integrable Solutions . . . . . . . . . . . . . 112
5.4 Tunneling and the Classically Forbidden Region . . . . . 118
5.5 Discrete and Continuous Spectrum . . . . . . . . . . . . . 119
5.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
6 Perspectives on the Spectral Theorem 123
6.1 The Difficulties with the Infinite-Dimensional Case . . . . 123
6.2 The Goals of Spectral Theory . . . . . . . . . . . . . . . . 125
6.3 A Guide to Reading . . . . . . . . . . . . . . . . . . . . . . 126
6.4 The Position Operator . . . . . . . . . . . . . . . . . . . . 126
6.5 Multiplication Operators . . . . . . . . . . . . . . . . . . . 127
6.6 The Momentum Operator . . . . . . . . . . . . . . . . . . 127
7 The Spectral Theorem for Bounded Self-Adjoint
Operators: Statements 131
7.1 Elementary Properties of Bounded Operators . . . . . . . 131
7.2 Spectral Theorem for Bounded Self-Adjoint
Operators, I . . . . . . . . . . . . . . . . . . . . . . . . . . 137
7.3 Spectral Theorem for Bounded Self-Adjoint
Operators, II . . . . . . . . . . . . . . . . . . . . . . . . . . 144
7.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
8 The Spectral Theorem for Bounded Self-Adjoint
Operators: Proofs 153
8.1 Proof of the Spectral Theorem, First Version . . . . . . . . 153
Contents xiii
8.2 Proof of the Spectral Theorem, Second Version . . . . . . 162
8.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
9 Unbounded Self-Adjoint Operators 169
9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 169
9.2 Adjoint and Closure of an Unbounded Operator . . . . . . 170
9.3 Elementary Properties of Adjoints and Closed
Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
9.4 The Spectrum of an Unbounded Operator . . . . . . . . . 177
9.5 Conditions for Self-Adjointness and Essential
Self-Adjointness . . . . . . . . . . . . . . . . . . . . . . . . 179
9.6 A Counterexample . . . . . . . . . . . . . . . . . . . . . . 182
9.7 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . 184
9.8 The Basic Operators of Quantum Mechanics . . . . . . . . 185
9.9 Sums of Self-Adjoint Operators . . . . . . . . . . . . . . . 190
9.10 Another Counterexample . . . . . . . . . . . . . . . . . . . 193
9.11 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
10 The Spectral Theorem for Unbounded Self-Adjoint
Operators 201
10.1 Statements of the Spectral Theorem . . . . . . . . . . . . . 202
10.2 Stone’s Theorem and One-Parameter Unitary Groups . . . 207
10.3 The Spectral Theorem for Bounded Normal
Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
10.4 Proof of the Spectral Theorem for Unbounded
Self-Adjoint Operators . . . . . . . . . . . . . . . . . . . . 220
10.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224
11 The Harmonic Oscillator 227
11.1 The Role of the Harmonic Oscillator . . . . . . . . . . . . 227
11.2 The Algebraic Approach . . . . . . . . . . . . . . . . . . . 228
11.3 The Analytic Approach . . . . . . . . . . . . . . . . . . . . 232
11.4 Domain Conditions and Completeness . . . . . . . . . . . 233
11.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236
12 The Uncertainty Principle 239
12.1 Uncertainty Principle, First Version . . . . . . . . . . . . . 241
12.2 A Counterexample . . . . . . . . . . . . . . . . . . . . . . 245
12.3 Uncertainty Principle, Second Version . . . . . . . . . . . . 246
12.4 Minimum Uncertainty States . . . . . . . . . . . . . . . . . 249
12.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
13 Quantization Schemes for Euclidean Space 255
13.1 Ordering Ambiguities . . . . . . . . . . . . . . . . . . . . . 255
13.2 Some Common Quantization Schemes . . . . . . . . . . . . 256
xiv Contents
13.3 The Weyl Quantization for R2n . . . . . . . . . . . . . . . 261
13.4 The “No Go” Theorem of Groenewold . . . . . . . . . . . 271
13.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275
14 The Stone–von Neumann Theorem 279
14.1 A Heuristic Argument . . . . . . . . . . . . . . . . . . . . 279
14.2 The Exponentiated Commutation Relations . . . . . . . . 281
14.3 The Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 286
14.4 The Segal–Bargmann Space . . . . . . . . . . . . . . . . . 292
14.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301
15 The WKB Approximation 305
15.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 305
15.2 The Old Quantum Theory and the Bohr–Sommerfeld
Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . 306
15.3 Classical and Semiclassical Approximations . . . . . . . . . 308
15.4 The WKB Approximation Away from the Turning
Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311
15.5 The Airy Function and the Connection Formulas . . . . . 315
15.6 A Rigorous Error Estimate . . . . . . . . . . . . . . . . . . 320
15.7 Other Approaches . . . . . . . . . . . . . . . . . . . . . . . 328
15.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329
16 Lie Groups, Lie Algebras, and Representations 333
16.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334
16.2 Matrix Lie Groups . . . . . . . . . . . . . . . . . . . . . . 335
16.3 Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . 338
16.4 The Matrix Exponential . . . . . . . . . . . . . . . . . . . 339
16.5 The Lie Algebra of a Matrix Lie Group . . . . . . . . . . . 342
16.6 Relationships Between Lie Groups and Lie Algebras . . . . 344
16.7 Finite-Dimensional Representations of Lie Groups
and Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . 350
16.8 New Representations from Old . . . . . . . . . . . . . . . . 358
16.9 Infinite-Dimensional Unitary Representations . . . . . . . 360
16.10 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363
17 Angular Momentum and Spin 367
17.1 The Role of Angular Momentum
in Quantum Mechanics . . . . . . . . . . . . . . . . . . . . 367
17.2 The Angular Momentum Operators in R3 . . . . . . . . . 368
17.3 Angular Momentum from the Lie Algebra Point
of View . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369
17.4 The Irreducible Representations of so(3) . . . . . . . . . . 370
17.5 The Irreducible Representations of SO(3) . . . . . . . . . . 375
17.6 Realizing the Representations Inside L2(S2) . . . . . . . . 376
Contents xv
17.7 Realizing the Representations Inside L2(R3) . . . . . . . . 380
17.8 Spin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383
17.9 Tensor Products of Representations: “Addition of
Angular Momentum” . . . . . . . . . . . . . . . . . . . . . 384
17.10 Vectors and Vector Operators . . . . . . . . . . . . . . . . 387
17.11 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 390
18 Radial Potentials and the Hydrogen Atom 393
18.1 Radial Potentials . . . . . . . . . . . . . . . . . . . . . . . 393
18.2 The Hydrogen Atom: Preliminaries . . . . . . . . . . . . . 396
18.3 The Bound States of the Hydrogen Atom . . . . . . . . . . 397
18.4 The Runge–Lenz Vector in the Quantum Kepler
Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401
18.5 The Role of Spin . . . . . . . . . . . . . . . . . . . . . . . 409
18.6 Runge–Lenz Calculations . . . . . . . . . . . . . . . . . . . 410
18.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416
19 Systems and Subsystems, Multiple Particles 419
19.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 419
19.2 Trace-Class and Hilbert–Schmidt Operators . . . . . . . . 421
19.3 Density Matrices: The General Notion
of the State of a Quantum System . . . . . . . . . . . . . . 422
19.4 Modified Axioms for Quantum Mechanics . . . . . . . . . 427
19.5 Composite Systems and the Tensor Product . . . . . . . . 429
19.6 Multiple Particles: Bosons and Fermions . . . . . . . . . . 433
19.7 “Statistics” and the Pauli Exclusion Principle . . . . . . . 435
19.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 438
20 The Path Integral Formulation of Quantum Mechanics 441
20.1 Trotter Product Formula . . . . . . . . . . . . . . . . . . . 442
20.2 Formal Derivation of the Feynman Path Integral . . . . . . 444
20.3 The Imaginary-Time Calculation . . . . . . . . . . . . . . 447
20.4 The Wiener Measure . . . . . . . . . . . . . . . . . . . . . 448
20.5 The Feynman–Kac Formula . . . . . . . . . . . . . . . . . 449
20.6 Path Integrals in Quantum Field Theory . . . . . . . . . . 451
20.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453
21 Hamiltonian Mechanics on Manifolds 455
21.1 Calculus on Manifolds . . . . . . . . . . . . . . . . . . . . 455
21.2 Mechanics on Symplectic Manifolds . . . . . . . . . . . . . 459
21.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465
22 Geometric Quantization on Euclidean Space 467
22.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 467
22.2 Prequantization . . . . . . . . . . . . . . . . . . . . . . . . 468
xvi Contents
22.3 Problems with Prequantization . . . . . . . . . . . . . . . 472
22.4 Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . 474
22.5 Quantization of Observables . . . . . . . . . . . . . . . . . 478
22.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482
23 Geometric Quantization on Manifolds 483
23.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 483
23.2 Line Bundles and Connections . . . . . . . . . . . . . . . . 485
23.3 Prequantization . . . . . . . . . . . . . . . . . . . . . . . . 490
23.4 Polarizations . . . . . . . . . . . . . . . . . . . . . . . . . . 492
23.5 Quantization Without Half-Forms . . . . . . . . . . . . . . 495
23.6 Quantization with Half-Forms: The Real Case . . . . . . . 505
23.7 Quantization with Half-Forms: The Complex Case . . . . . 518
23.8 Pairing Maps . . . . . . . . . . . . . . . . . . . . . . . . . 521
23.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523
A Review of Basic Material 527
A.1 Tensor Products of Vector Spaces . . . . . . . . . . . . . . 527
A.2 Measure Theory . . . . . . . . . . . . . . . . . . . . . . . . 529
A.3 Elementary Functional Analysis . . . . . . . . . . . . . . . 530
A.4 Hilbert Spaces and Operators on Them . . . . . .

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