Quantum mechanics/ Jean-Louis Basdevant

1. Quantum Phenomena 1
1.1 The Franck and Hertz Experiment 3
1.2 Interference of Matter Waves 5
1.2.1 The Young Double-Slit Experiment 6
1.2.2 Interference of Atoms in a Double-Slit Experiment.... 7
1.2.3 Probabilistic Aspect of Quantum Interference 8
1.3 The Experiment of Davisson and Germer 10
1.3.1 Diffraction of X Rays by a Crystal 10
1.3.2 Electron Diffraction 12
1.4 Summary of a Few Important Ideas 15
Further Reading 15
Exercises 16
2. The Wave Function and the Schrodinger Equation 17
2.1 The Wave Function 18
2.1.1 Description of the State of a Particle 18
2.1.2 Position Measurement of the Particle 19
2.2 Interference and the Superposition Principle 20
2.2.1 De Broglie Waves 20
2.2.2 The Superposition Principle 21
2.2.3 The Wave Equation in Vacuum 22
2.3 Free Wave Packets 24
2.3.1 * Definition of a Wave Packet 24
2.3.2 Fourier Transformation 24
2.3.3 Structure of the Wave Packet 25
2.3.4 Propagation of a Wave Packet: the Group Velocity ... 26
2.3.5 Propagation of a Wave Packet:
Average Position and Spreading 27
2.4 Momentum Measurements and Uncertainty Relations 28
2.4.1 The Momentum Probability Distribution 29
2.4.2 Heisenberg Uncertainty Relations 30
2.5 The Schrodinger Equation 31XII Contents
2.5.1 Equation of Motion 32
2.5.2 Particle in a Potential: Uncertainty Relations 32
2.5.3 Stability of Matter 33
2.6 Momentum Measurement in a Time-of-Flight Experiment ... 34
Further Reading 36
Exercises
3. Physical Quantities and Measurements • • 39
3.1 Measurements in Quantum Mechanics 40
3.1.1 The Measurement Procedure 40
3.1.2 Experimental Facts 41
3.1.3 Reinterpretation of Position
and Momentum Measurements 41
3.2 Physical Quantities and Observables 42
3.2.1 Expectation Value of a Physical Quantity 42
3.2.2 Position and Momentum Observables 43
3.2.3 Other Observables: the Correspondence Principle 44
3.2.4 Commutation of Observables 44
3.3 Possible Results of a Measurement 45
3.3.1 Eigenfunctions and Eigenvalues of an Observable 45
3.3.2 Results of a Measurement
and Reduction of the Wave Packet 46
3.3.3 Individual Versus Multiple Measurements 47
3.3.4 Relation to Heisenberg Uncertainty Relations 47
3.3.5 Measurement and Coherence of Quantum Mechanics.. 48
3.4 Energy Eigenfunctions and Stationary States 48
3.4.1 Isolated Systems: Stationary States 49
3.4.2 Energy Eigenstates and Time Evolution oO
3.5 The Probability Current
3.6 Crossing Potential Barriers ^2
3.6.1 The Eigenstates of the Hamiltonian 52
3.6.2 Boundary Conditions at the Discontinuities
of the Potential
3.6.3 Reflection and Transmission on a Potential Step 54
3.6.4 Potential Barrier and Tunnel Effect 56
3.7 Summary of Chapters 2 and 3
Further Reading
. 60
Exercises
4. Quantization of Energy in Simple Systems 53
4.1 Bound States and Scattering States 53
4.1.1 Stationary States of the Schrodinger Equation 54
4.1.2 Bound States
4.1.3 Scattering States
4.2 The One Dimensional Harmonic Oscillator 56Contents XIII
4.2.1 Definition and Classical Motion 66
4.2.2 The Quantum Harmonic Oscillator 67
4.2.3 Examples 69
4.3 Square-Well Potentials 70
4.3.1 Relevance of Square Potentials 70
4.3.2 Bound States in a One-Dimensional
Square-Well Potential 71
4.3.3 Infinite Square Well 73
4.3.4 Particle in a Three-Dimensional Box 74
4.4 Periodic Boundary Conditions 75
4.4.1 A One-Dimensional Example 75
4.4.2 Extension to Three Dimensions 77
4.4.3 Introduction of Phase Space 78
4.5 The Double Well Problem and the Ammonia Molecule 78
4.5.1 Model of the NH3 Molecule 79
4.5.2 Wave Functions 79
4.5.3 Energy Levels 81
4.5.4 The Tunnel Effect and the Inversion Phenomenon .... 82
4.6 Other Applications of the Double Well 84
Further Reading 86
Exercises 87
5. Principles of Quantum Mechanics 89
5.1 Hilbert Space 90
5.1.1 The State Vector 90
5.1.2 Scalar Products and the Dirac Notations 90
5.1.3 Examples 91
5.1.4 Bras and Kets, Brackets 92
5.2 Operators in Hilbert Space 92
5.2.1 Matrix Elements of an Operator 92
5.2.2 Adjoint Operators and Hermitian Operators 93
5.2.3 Eigenvectors and Eigenvalues 94
5.2.4 Summary: Syntax Rules in Dirac's Formalism 95
5.3 The Spectral Theorem 95
5.3.1 Hilbertian Bases 95
5.3.2 Projectors and Closure Relation 96
5.3.3 The Spectral Decomposition of an Operator 96
5.3.4 Matrix Representations 97
5.4 Measurement of Physical Quantities 99
5.5 The Principles of Quantum Mechanics 100
5.6 Structure of Hilbert Space 104
5.6.1 Tensor Products of Spaces 104
5.6.2 The Appropriate Hilbert Space 105
5.6.3 Properties of Tensor Products 105
5.6.4 Operators in a Tensor Product Space 106XIV Contents
5.6.5 Simple Examples 106
5.7 Reversible Evolution and the Measurement Proce.ss 107
Further Reading 110
Exercises Ill
6. Two-State Systems, Principle of the Maser 115
6.1 Two-Dimensional Hilbert Space 115
6.2 A Familiar Example: the Polarization of Light 116
6.2.1 Polarization States of a Photon 116
6.2.2 Measurement of Photon Polarizations 118
6.2.3 Successive Measurements and "Quantum Logic" 119
6.3 The Model of the Ammonia Molecule 120
6.3.1 Restriction to a Two-Dimensional Hilbert Space 120
6.3.2 The Basis {|^s),|Vm>} 121
6.3.3 The Basis {|V-'r)> \^l)} 123
6.4 The Ammonia Molecule in an Electric Field 123
6.4.1 The Coupling of NH3 to an Electric Field 124
6.4.2 Energy Levels in a Fixed Electric Field 125
6.4.3 Force Exerted on the Molecule
by an Inhomogeneous Field 127
6.5 Oscillating Fields and Stimulated Emission 129
6.6 Principle and Applications of Masers 131
6.6.1 Amplifier 131
6.6.2 Oscillator 132
6.6.3 Atomic Clocks 132
Further Reading 132
Exercises 133
7. Commutation of Observables 135
7.1 Commutation Relations 136
7.2 Uncertainty Relations 137
7.3 Ehrenfest's Theorem 138
7.3.1 Evolution of the Expectation Value of an Observable.. 138
7.3.2 Particle in a Potential V(r) 139
7.3.3 Constants of Motion 140
7.4 Commuting Observables 142
7.4.1 Existence of a Common Eigenbasis
for Commuting Observables 142
7.4.2 Complete Set of Commuting Observables(CSCO).... 142
7.4.3 Completely Prepared Quantum State 143
7.4.4 Symmetries of the Hamiltonian
and Search of Its Eigenstates 145
7.5 Algebraic Solution of the Harmonic-Oscillator Problem 148
7.5.1 Reduced Variables 148
7.5.2 Annihilation and Creation Operators a and 148Contents XV
7.5.3 Eigenvalues of the Number Operator N 149
7.5.4 Eigenstates 150
Further Reading 151
Exercis&s 152
8. The Stern-Gerlach Experiment 157
8.1 Principle of the Experiment 157
8.1.1 Classical Analysis 157
8.1.2 Experimental Results 159
8.2 The Quantum Description of the Problem 161
8.3 The Observables and find 163
8.4 Discussion 165
8.4.1 Incompatibility of Measurements Along Different Axes 165
8.4.2 Classical Versus Quantum Analysis 166
8.4.3 Measurement Along an Arbitrary Axis 167
8.5 Complete Description of the Atom 168
8.5.1 Hilbert Space 168
8.5.2 Representation of States and Observables 169
8.5.3 Energy of the Atom in a Magnetic Field 170
8.6 Evolution of the Atom in a Magnetic Field 170
8.6.1 Schrodinger Equation 170
8.6.2 Evolution in a Uniform Magnetic Field 171
8.6.3 Explanation of the Stern-Gerlach Experiment 173
8.7 Conclusion 175
Further Reading 175
Exercises 176
9. Approximation Methods «
9.1 Perturbation Theory 177
9.1.1 Definition of the Problem 177
9.1.2 Power Expansion of Energies and Eigenstates 178
9.1.3 First-Order Perturbation in the Nondegenerate Case .. 179
9.1.4 First-Order Perturbation in the Degenerate Case 179
9.1.5 First-Order Perturbation to the Eigenstates 180
9.1.6 Second-Order Perturbation to the Energy Levels 181
9.1.7 Examples 181
9.1.8 Remarks on the Convergence of Perturbation Theory ..182
9.2 The Variational Method 183
9.2.1 The Ground State 183
9.2.2 Other Levels 184
9.2.3 Examples of Applications of the Variational Method .. 185
Exercises 187XVI Contents
10. Angular Momentum 189
10.1 Orbital Angular Momentum and the Coniiniitation Relations 190
10.2 Eigenvalues of Angular Monicntuni 190
10.2.1 The Observables and J. and the Basis States |J, ni) 191
10.2.2 The Operators .J± 192
10.2.3 Action of J± on the States \j. ni) 192
10.2.4 Quantization of and m 193
10.2.5 Measurement of and Jy .. 195
10.3 Orbital Angular Momentum 196
10.3.1 The Quantum Numbers in and(are Integers 196
10.3.2 Spherical Coordinates 197
10.3.3 Eigenfunctions of and L.: the Spherical Harmonics 198
10.3.4 Examples of Spherical Harmonics 199
10.3.5 Example: Rotational Energy of a Diatomic Molecule .. 200
10.4 Angular Momentum and Magnetic Moment 201
10.4.1 Orbital Angular Momentum and Magnetic Moment... 202
10.4.2 Generalization to Other Angular Momenta 203
10.4.3 What Should we Think
about Half-Integer Values ofjand 7n ? 204
Further Reading 204
Exercises 205
11. Initial Description of Atoms 207
11.1 The Two-Body Problem; Relative Motion 208
11.2 Motion in a Central Potential 210
11.2.1 Spherical Coordinates 210
11.2.2 Eigenfunctions Common to H, and Lz 211
11.3 The Hydrogen Atom 215
11.3.1 Orders of Magnitude:
Appropriate Units in Atomic Physics 215
11.3.2 The Dimensionless Radial Equation 216
11.3.3 Spectrum of Hydrogen 219
11.3.4 Stationary States of the Hydrogen Atom 220
-11.3.5 Dimensions and Orders of Magnitude 221
11.3.6 Time Evolution of States of Low Energies 223
11.4 Hydrogen-Like Atoms 224
11.5 Muonic Atoms 224
11.6 Spectra of Alkali Atoms 226
Further Reading 227
Exercises 228Contents XVII
12. Spin 1/2 and Magnetic Resonance 231
12.1 The Hilbert Space of Spin 1/2 232
12.1.1 Spin Observables 233
12.1.2 Representation in a Particular Basis 233
12.1.3 Matrix Representation 234
12.1.4 Arbitrary Spin State 234
12.2 Complete Description of a Spin-1/2 Particle 235
12.2.1 Hilbert Space 235
12.2.2 Representation of States and Observables 235
12.3 Spin Magnetic Moment 236
12.3.1 The Stern-Gerlach Experiment 236
12.3.2 Anomalous Zeeman Effect 237
12.3.3 Magnetic Moment of Elementary Particles 237
12.4 Uncorrelated Space and Spin Variables 238
12.5 Magnetic Resonance 239
12.5.1 Larmor Precession in a Fixed Magnetic Field Rq 239
12.5.2 Superposition of a Fixed Field and a Rotating Field .. 240
12.5.3 Rabi's Experiment 242
12.5.4 Applications of Magnetic Resonance 244
12.5.5 Rotation of a Spin 1/2 Particle by 27r 245
Further Reading 246
Exercises 247
13. Addition of Angular Momenta,
Fine and Hyperfine Structure of Atomic Spectra 249
13.1 Addition of Angular Momenta 249
13.1.1 The Total-Angular Momentum Operator 249
13.1.2 Factorized and Coupled Bases 250
13.1.3 A Simple Case: the Addition of Two Spins of 1/2 251
13.1.4 Addition of Two Arbitrary Angular Momenta 254
13.1.5 One-Electron Atoms, Spectroscopic Notations 258
13.2 Fine Structure of Monovalent Atoms 258
13.3 Hyperfine Structure; the 21cm Line of Hydrogen 261
13.3.1 Interaction Energy 261
13.3.2 Perturbation Theory 262
13.3.3 Diagonalization of 263
13.3.4 The Effect of an External Magnetic Field 265
13.3.5 The 21cm Line in Astrophysics 265
Further Reading 268
Exercises 269XVIII Contents
14. Entangled States, EPR Paradox and Bell's Inequality 2^3
Written in collaboration with Philippe Grangier
14.1 The EPR Paradox and Bell's Inequality 274
14.1.1 "God Does Not Play Dice" 274
14.1.2 The EPR Argument 275
14.1.3 Bell's Inequality 278
14.1.4 Experimental Tests 281
14.2 Quantum Cryptography • • • 282
14.2.1 The Communication Between Alice and Bob 282
14.2.2 The Quantum No cloning Theorem 285
14.2.3 Present Experimental Setups 286
14.3 The Quantum Computer 287
14.3.1 The Quantum Bits, or "'Q-Bits" 287
14.3.2 The Algorithm of Peter Shor 288
14.3.3 Principle of a Quantum Computer 289
14.3.4 Decoherence 290
Further Reading 290
901
Exercises
15. The Lagrangian and Hamiltonian Formalisms,
Lorentz Force in Quantum Mechanics 293
15.1 Lagrangian Formalism and the Least-Action Principle 294
15.1.1 Least Action Principle 294
15.1.2 Lagrange Equations 295
15.1.3 Energy 297
15.2 Canonical Formalism of Hamilton 297
15.2.1 Conjugate Momenta 297
15.2.2 Canonical Equations 298
15.2.3 Poisson Brackets 299
15.3 Analytical Mechanics and Quantum Mechanics 300
15.4 Classical Charged Particles in an Electromagnetic Field 301
15.5 Lorentz Force in Quantum Mechanics 302
15.5.1 Hamiltonian ^^2
1*5.5.2 Gauge Invariance 303
15.5.3 The Hydrogen Atom Without Spin
in a Uniform Magnetic Field • • 304
15.5.4 Spin-1/2 Particle in an Electromagnetic Field 305
Further Reading
Exercises
16. Identical Particles and the Pauli Principle 309
16.1 Indistinguishability of Two Identical Particles 310
16.1.1 Identical Particles in Classical Physics 310
16.1.2 The Quantum Problem 310
16.2 Two-Particle Systems; the Exchange Operator 312Contents XIX
16.2.1 The Hilbert Space for the Two Particle System 312
16.2.2 The Exchange Operator
Between Two Identical Particles 312
16.2.3 Symmetry of the States 313
16.3 The Pauli Principle 314
16.3.1 The Case of Two Particles 314
16.3.2 Independent Fermions and Exclusion Principle 315
16.3.3 The Case of N Identical Particles 316
16.3.4 Time Evolution 317
16.4 Physical Consequences of the Pauli Principle 317
16.4.1 Exchange Force Between Two Fermions 318
16.4.2 The Ground State
of N Identical Independent Particles 318
16.4.3 Behavior of Fermion and Boson Sj'stems
at Low Temperature 320
16.4.4 Stimulated Emission and the Laser Effect 322
16.4.5 Uncertainty Relations for a System of N Fermions.... 323
16.4.6 Complex Atoms and Atomic Shells 324
Further Reading 326
Exercises 327
17. The Evolution of Systems 331
Written in collaboration with Gilbert Grynberg
17.1 Time-Dependent Perturbation Theory 332
17.1.1 Transition Probabilities 332
17.1.2 Evolution Equations 332
17.1.3 Perturbative Solution 333
17.1.4 First-Order Solution: the Born Approximation 334
17.1.5 Particular Cases 334
17.1.6 Perturbative and Exact Solutions 335
17.2 Interaction of an Atom with an Electromagnetic Wave 336
17.2.1 The Electric-Dipole Approximation 336
17.2.2 Justification of the Electric Dipole Interaction 337
17.2.3 Absorption of Energy by an Atom 338
17.2.4 Selection Rules 339
17.2.5 Spontaneous Emission 339
17.2.6 Control of Atomic Motion by Light 341
17.3 Decay of a System 343
17.3.1 The Radioactivity of ^'^Fe 343
17.3.2 The Fermi Golden Rule 345
17.3.3 Orders of Magnitude 346
17.3.4 Behavior for Long Times 347
17.4 The Time-Energy Uncertainty Relation 350
17.4.1 Isolated Systems and Intrinsic Interpretations 350
17.4.2 Interpretation of Landau and Peierls 351XX Contents
17.4.3 The Einstein Bohr Controver.sy 352
Further Reading 353
Exercises 353
18. Scattering Processes 357
18.1 Concept of Cross Section 358
18.1.1 Definition of Cross Section 358
18.1.2 Classical Calculation 359
18.1.3 Examples 360
18.2 Quantum Calculation in the Born Approximation 361
18.2.1 Asymptotic States 361
18.2.2 Transition Probability 362
18.2.3 Scattering Cross Section 363
18.2.4 Validity of the Born Approximation 364
18.2.5 Example: the Yukawa Potential 365
18.2.6 Range of a Potential in Quantum Mechanics 366
18.3 Exploration of Composite Systems 367
18.3.1 Scattering Off a Bound State and the Form Factor ... 367
18.3.2 Scattering by a Charge Distribution 368
18.4 General Scattering Theory 372
18.4.1 Scattering States 372
18.4.2 The Scattering Amplitude 373
18.4.3 The Integral Equation for Scattering 374
18.5 Scattering at Low Energy 375
18.5.1 The Scattering Length 375
18.5.2 Explicit Calculation of a Scattering Length 376
18.5.3 The Case of Identical Particles 377
Further Reading 378
Exercises 378
19. Qualitative Physics on a Macroscopic Scale 381
Written in collaboration with Alfred Vidal-Madjar
19.1 Confined Particles and Ground State Energy 382
19.1.1 The Quantum Pressure 382
19.1.2 Hydrogen Atom 383
19.1.3 7V-Fermion Systems and Complex Atoms 383
19.1.4 Molecules, Liquids and Solids .. 384
19.1.5 Hardness of a Solid 385
19.2 Gravitational Versus Electrostatic Forces 386
19.2.1 Screening of Electrostatic Interactions 386
19.2.2 Additivity of Gravitational Interactions 387
19.2.3 Ground State of a Gravity-Dominated Object 388
19.2.4 Liquefaction of a Solid and the Height of Mountains .. 390
19.3 White Dwarfs, Neutron Stars
and the Gravitational Catastrophe 392Contents XXI
19.3.1 White Dwarfs and the Chandrasekhar* Mass 392
19.3.2 Neutron Stars 394
Further Reading 396
20. Early History of Quantum Mechanics 397
20.1 The Origin of Quantum Concepts 397
20.1.1 Planck's Radiation Law 397
20.1.2 Photons 398
20.2 The Atomic Spectrum 398
20.2.1 Empirical Regularities of Atomic Spectra 398
20.2.2 The Structure of Atoms 399
20.2.3 The Bohr Atom 399
20.2.4 The Old Theory of Quanta 400
20.3 Spin 400
20.4 Heisenberg's Matrices 401
20.5 Wave Mechanics 403
20.6 The Mathematical Formalization 404
20.7 Some Important Steps in More Recent Years 405
Further Reading 406
Appendix A. Concepts of Probability Theory 407
1 Fundamental Concepts 407
2 Examples of Probability Laws 408
2.1 Discrete Laws 408
2.2 Continuous Probability Laws
in One or Several Variables 408
3 Random Variables 409
3.1 Definition 409
3.2 Conditional Probabilities 410
3.3 Independent Random Variables 411
3.4 Binomial Law and the Gaussian Approximation 411
4 Moments of Probability Distributions 412
4.1 Mean Value or Expectation Value 412
4.2 Variance and Mean Square Deviation 412
4.3 ^ Bienayme-Tchebycheff Inequality 413
4.4 Experimental Verification of a Probability Law 413
Exercises 414
Appendix B. Dirac Distribution, Fourier Transformation .... 417
1 Dirac Distribution, or <5 "Function" 417
1.1 Definition of <5(.t) 417
1.2 Examples of Functions Which Tend to (5(a:) 418
1.3 Properties of S{x) 419
2 Distributions 420
2.1 The Space S 420XXII Contents
2.2 Linear Fimctionals 420
2.3 Derivative of a Di.stributioii 421
2.4 Convolution Product 422
3 Fourier Transformation 422
3.1 Definition 422
3.2 Fourier Transform of a Gaussian 423
3.3 Inversion of the Fourier Transformation 423
3.4 Parseval-Plancherel Theorem 424
3.5 Fourier Transform of a Distribution 425
3.6 Uncertainty Relation 426
Exercises 427
Appendix C. Operators in Infinite-Dimensional Spaces 429
1 Matrix Elements of an Operator 429
2 Continuous Bases 430
Appendix D. The Density Operator 435
1 Pure States 436
1.1 A Mathematical Tool: the Tiace of an Operator 436
1.2 The Density Operator of Pure States 437
1.3 Alternative Formulation of Quantum Mechanics
for Pure States 438
2 Statistical Mixtures 439
2.1 A Particular Case: an Unpolarized Spin-1/2 System .. 439
2.2 The Density Operator for Statistical Mixtures 440
3 Examples of Density Operators 441
3.1 The Micro-Canonical and Canonical Ensembles 441
3.2 The Wigner Distribution of a Spinless Point Particle.. 442
4 Entangled Systems 444
4.1 Reduced Density Operator 444
4.2 Evolution of a Reduced Density Operator 444
4.3 Entanglement and Measurement 445
Further Reading 446
Exerdses 446