Peter Hertel

Quantum Theory and Statistical Thermodynamics Principles and Worked Examples

Springer

Contents

1 B a s i c s '••••••' • • • • • 1 1.1 Introduction 2

1.1.1 The Quantum of Light 2 1.1.2 Electron Diffraction. . 3 1.1.3 Heisenberg Uncertainty Principle . 4 1.1.4 Does God Play Dice? 5 1.1.5 Summary. 6

1.2 Classical Framework. 6 1.2.1 Phase Space . . . . . . 7 1.2.2 Observables . . . . . . . . . . . . . . . . 8 1.2.3 Dynamics . . . . . . . . . . 8 1.2.4 States . . . . 10 1.2.5 Properties of Poisson Brackets . . . . . 10 1.2.6 Canonical Relations. 11 1.2.7 Pure and Mixed States . . . . . . . . 11 1.2.8 Summary 12

1.3 Quantum Framework 12 1.3.1 q-Numbers 13 1.3.2 Hubert Space . . 13 1.3.3 Linear Operators 14 1.3.4 Projectors .'. 16 1.3.5 Normal Linear Operators. 17 1.3.6 Trace of an Operator. 19 1.3.7 Expectation Values 20 1.3.8 Summary .. 21

1.4 Time and Space . . . . . . . . . . . . . . . . 22 1.4.1 Measurement and Experiment 23 1.4.2 Time Translation . . . . . . . 24 1.4.3 Space Translation 25

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1.4.4 Location 25 1.4.5 Rotation 26 1.4.6 Orbital Angular Momentum and Spin 27 1.4.7 Schrödinger Picture 28 1.4.8 Summary 29

2 Simple Examples 31 2.1 Ammonia Molecule 31

2.1.1 Hilbert Space and Energy Observable.". 33 2.1.2 Ammonia Molecule in an External Electric Field 34 2.1.3 Dipole Moment Expectation Value 35 2.1.4 Ammonium Maser 36 2.1.5 Summary 37

2.2 Quasi-Particles '::..' . . . . . 38 2.2.1 Hilbert Spaces C" and f 38 2.2.2 Hopping Model 40 2.2.3 Wave Packets . . . 41 2.2.4 Group Velocity and Effective Mass. ...... 42 2.2.5 Scattering at Defects. ,...'./ 43 2.2.6 Trapping by Defects . . . . . . . . . . . . . 45 2.2.7 Summary 46

2.3 Neutron Scattering on Molecules 46 2.3.1 Feynman's Approach , 47 2.3.2 Spherical and Piain Waves 48 2.3.3 Neutron Scattering on a Diatomic Molecule 48 2.3.4 Cross Section . . . . 49 2.3.5 Orientation Averaged Gross Section 50 2.3.6 Neutron Diffraction 52 2.3.7 Summary 53

2.4 Free Particles 53 2.4.1 Square Integrable Functions 54 2.4.2 Location 55 2.4.3 Linear Momentum .'. . . . . . . ' 55 2.4.4 Wave Packets 57 2.4.5 Motion of a Free Particle • 58 2.4.6 Spreading of a Free Particle 59 2.4.7 Summary . 60

2.5 Small Oscillations 60 2.5.1 The Hamitonian 61 2.5.2 Ladder Operators 62 2.5.3 Eigenstate Wave Functions 63 2.5.4 Summary 64

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3 Atoms and Molecules 65 3.1 Radial Schrödinger Equation . . . . . . . . . . . . . . . . 66

3.1.1 Spherical Coordinates 66 3.1.2 TheLaplacian 67 3.1.3 Spherical Harmonics 67 3.1.4 Spherical Symmetrie Potential. 69 3.1.5 Behavior at Origin and Infinity .'.., 70 3.1.6 Alternative Form . . . . . . . . . ; . . . . . 71 3.1.7 Summary 71

3.2 Hydrogen Atom 72 3.2.1 Atomic Units 72 3.2.2 Non-relativistic Hydrogen Atom 74 3.2.3 Orbitals 75 3.2.4 Relativistic Hydrogen Atom 77 3.2.5 Classical Hydrogen Atom 80 3.2.6 Summary , 81

3.3 Helium Atom 82 3.3.1 Wave Functions . 82 3.3.2 Minimal Ground State Energy 84 3.3.3 Sample Calculation 86 3.3.4 The Negative Hydrogen Ion 88 3.3.5 Summary , 88

3.4 Hydrogen Molecule . . . 89 3.4.1 Wave Functions and Hamiltonian 89 3.4.2 Born-Oppenheimer Approximation 91 3.4.3 The Molecular Potential ...,. 91 3.4.4 Molecular Vibrations 93 3.4.5 Molecular Rotations 95 3.4.6 Summary 95

3.5 More on Approximations 96 3.5.1 The Minimax Theorem 97 3.5.2 Remarks 98 3.5.3 Stationary Perturbations. . . . . . . . . . . . . 100 3.5.4 Coping with Degeneracies 101 3.5.5 Summary 101

3.6 Stark and Zeeman Effect 102 3.6.1 Multipoles 102 3.6.2 Electric Dipole Moment 105 3.6.3 Stark Effect. 106 3.6.4 Magnetic Dipole Moment 108 3.6.5 Zeeman Effect . . 109 3.6.6 Summary 110

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4 Decay and Scattering 111 4.1 Forced Transitions 112

4.1.1 Time Dependent External Field. . . . . . . . 112 4.1.2 Detailed Balance. 115 4.1.3 Incoherent Radiation 115 4.1.4 Summary 119

4.2 Spontaneous Transitions 119 4.2.1 Einstein's Argument 119 4.2.2 Lifetime of an Excited State 121 4.2.3 Comment on Einstein's Reasoning . . . . . . . . . . . . . . . . . . . . . 121 4.2.4 Classical Limit 122 4.2.5 Summary ,,. 123

4.3 Scattering Amplitude 123 4.3.1 Cross Section 124 4.3.2 Scattering Amplitude 125 4.3.3 Center of Mass and Laboratory Frame . . . . . . . . . . . . . . . . 126 4.3.4 Relativistic Description 128 4.3.5 Summary. 129

4.4 Coulomb Scattering 129 4.4.1 Scattering Schrödinger Equation 130 4.4.2 Born Approximation.. . . . . . . . . . . 131 4.4.3 Scattering of Point Charges. 132 4.4.4 Electron-Hydrogen Atom Scattering 133 4.4.5 Form Factor and Structure 134 4.4.6 Summary 136

5 Thermal Equilibrium 137 5.1 Entropy and the Gibbs State ." 138

5.1.1 Observables and States , ' . . . . . . 139 5.1.2 First Main Law.. 141 5.1.3 Entropy . ..;'.., . . 142 5.1.4 Second Main Law . 145 5.1.5 The Gibbs State 146 5.1.6 Free Energy and Temperature 147 5.1.7 Chemical Potentials. . .. . . . .'.' . . . ... 148 5.1.8 Minimal Free Energy 149 5.1.9 Summary 150

5.2 Thermodynamics 150 5.2.1 Reversible and Irreversible Processes 151 5.2.2 Free Energy as Thermodynamic Potential 152 5.2.3 More Thermodynamic Potentials. 153 5.2.4 Heat Capacity and Compressibility.....,". 154 5.2.5 Chemical Potential . . . . . 156 5.2.6 Chemical Reactions 158

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5.2.7 Particle Number as an Extemal Parameter 158 5.2.8 Summary... . . . . . . . 160

5.3 Continuum Physics. 160 5.3.1 Material Points 161 5.3.2 Balance Equat ions . . . . . . . 162 5.3.3 Particles, Mass and Electric Charge 163 5.3.4 Conduction and Covariance 165 5.3.5 Momentum, Energy, and the First Main Law 167 5.3.6 Entropy and the Second Main L a w . . . . . . . . . . . . . . . . . . . 170 5.3.7 Summary 172

5.4 Second Quantization 173 5.4.1 Number Operators 173 5.4.2 Plane Waves . 175 5.4.3 Local Quantum Field 175 5.4.4 Fermions 176 5.4.5 Some Observables . . . . . 178 5.4.6 Time 179 5.4.7 Summary 180

5.5 G a s e s . . . . . . . . . . . . . . 180 5.5.1 Fermi Gas. 181 5.5.2 Böse Gas . . 184 5.5.3 Black-Body Radiation. . . 186 5.5.4 Boltzmann Gas . . . 187 5.5.5 Rotating and Vibrating Molecules. 190 5.5.6 Cluster Expansion 192 5.5.7 Joule-Thomson Effect. . . 194 5.5.8 Summary. . . . . . . . 196

5.6 Crystal Lattice Vibrations 198 5.6.1 Phenomenological Description 198 5.6.2 Phonons . . . . 200 5.6.3 Summary... .'.'.. 207

5.7 Electronic Band Structure. '.'..'• • ... 207 5.7.1 Hopping Model . . 208 5.7.2 Fermi-Energy . 209 5.7.3 TheColdSol id . . . . . . : . . . . . . . . . . . . . . . . . . . . . 2 1 1 5.7.4 Metals, Dielectrics and Semiconductors 212 5.7.5 Summary . 213

5.8 Extemal Fields . . . .... ', 214 5.8.1 Matter and Electromagnetic Fields 214 5.8.2 Alignment of Electric Dipoles 217 5.8.3 Alignment of Magnetic Dipoles 218 5.8.4 Heisenberg Ferromagnet 220 5.8.5 Summary 223

Contents

Fluctuations and DLssipation. 225 6.1 Fluctuations 226

6.1.1 An Example 227 6.1.2 Density Fluctuations 228 6.1.3 Correlations and Khinchin's Theorem 230 6.1.4 Thermal Noise of a Resistor ......•• 232 6.1.5 Langevin Equation 232 6.1.6 Nyquist Formula 234 6.1.7 Remarks 236 6.1.8 Summary 236

6.2 Brownian Motion 237 6.2.1 Einstein's Explanation .-.,. ' . 237 6.2.2 The Diffusion Coefficient 241 6.2.3 Langevin's Approach 242 6.2.4 Summary. . . 244

6.3 Linear Response Theory 244 6.3.1 Perturbations 245 6.3.2 Dispersion Relations 250 6.3.3 Summary 252

6.4 Dissipation . 253 6.4.1 Wiener-Khinchin Theorem 253 6.4.2 Kubo-Martin-Schwinger Formula.. 254 6.4.3 Callen-Welton Theorem 255 6.4.4 Interaction with an Electromagnetic Field 257 6.4.5 Summary 259

Mathematical Aspects . . . 261 7.1 Topological Spaces. . . 261

7.1.1 Abstract Topological Space. 262 7.1.2 Metrie Space 263 7.1.3 Linear Space with Norm 264 7.1.4 Linear Space with Scalar Product 265 7.1.5 Convergent Sequences 265 7.1.6 Continuity. 266 7.1.7 Cauchy Sequences and Completeness 267

7.2 The Lebesgue Integral • . . . . . ' 268 7.2.1 Measure Spaces. 268 7.2.2 Measurable Functions . . . . . V 269 7.2.3 The Lebesgue Integral. . . . . . , . . . . . . . . . . . 270 7.2.4 Function Spaces 271

7.3 On Probabilities 272 7.3.1 Probability Spaces . . . 272 7.3.2 Random Variables. , 273 7.3.3 Law of Large Numbers and Central Limit Theorem 276

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7.4 Generalized Functions 277 7.4.1 Test Functions . . . 278 7.4.2 Distributions . . . . . . . 278 7.4.3 Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279 7.4.4 Fourier Transforms . . . . . . . . . 280

7.5 Linear Spaces . . . . . . . . . 283 7.5.1 Sealars . 284 7.5.2 Vectors . . . . . 284 7.5.3 Linear Subspaces 284 7.5.4 Dimension . . 285 7.5.5 Linear Mappings 285 7.5.6 Ring of Linear Operators 285

7.6 Hilbert Spaces 286 7.6.1 Operator Norm 288 7.6.2 Adjoint Operator. 288

7.7 Projection Operators 289 7.7.1 Projectors 289 7.7.2 Decomposition of Unity 291

7.8 Normal O p e r a t o r s . . . . . . . . . . . . . . . . . . . . . . . . . . . 291 7.8.1 Spectral Decomposition 291 7.8.2 Unitary Operators . 293 7.8.3 Self-Adjoint Operators 294 7.8.4 Positive Operators 295 7.8.5 Probability Operators 295

7.9 Operator Functions 296 7.9.1 Power Series 296 7.9.2 Normal Operator 297 7.9.3 Comparison. 297 7;9.4 Example ...'.•" 297

7.10 Translations 298 7.10.1 Periodic Boundary Conditions 299 7.10.2 Domain of Definition 299 7.10.3 Selfadjointness 300 7.10.4 Spectral Decomposition 301

7.11 Fourier Transform 302 7.11.1 Fourier Series 302 7.11.2 Fourier Expansion 303 7.11.3 Fourier Integral.. . . 303 7.11.4 Convolution Theorem 304

7.12 Position and Momentum. 306 7.12.1 Test Functions 306 7.12.2 Canonical Commutation Rules 307

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7.12.3 Uncertainty Relation . 308 7.12.4 Quasi-Eigenfunctions , 308

7.13 Ladder Operators 309 7.13.1 Raising and Lowering Operators 310 7.13.2 Ground State and Excited States. 310 7.13.3 Harmonie Oscillator . 311 7.13.4 Quantum Fields 312

7.14 Transformation Groups 314 7.14.1 Group 314 7.14.2 Finite Groups 314 7.14.3 Topological Groups 315 7.14.4 Angular Momentum 318

Glossary 321

Index 361