Quantum Chaos (eBook)

G. Casati, I. Guarneri, U. Smilansky (Herausgeber)

eBook Download: PDF

2015 | 1. Auflage
437 Seiten
Elsevier Science (Verlag)
978-1-4832-9032-4 (ISBN)

Infos

The study of quantum systems which are chaotic in the classical limit (quantum chaos or quantum chaology) is a very new field of research. Not long ago, it was still considered as an esoteric subject, however this attitude changed radically when it was realized that this subject is relevant to many of the more mature branches of physics.This book presents the accumulated knowledge available up until now and at the same time introduces topics which are being intensively studied at present. Their relevance to other fields such as condensed matter, atomic and nuclear physics is also discussed. The lectures have been divided into two rough categories - background and advanced lectures.

Inhalt

Front Cover 1
Quantum Chaos 4
Copyright Page 5
Table of Contents 6
Preface 12
Chapter 1. Hyperbolic Structure in Classical Chaos 20
1. Introduction. 20
2. Transverse homoclinic orbits 27
3. Anti-integrable limit 37
4. Interlude 41
5. Anosov systems 43
6. Nonzero Lyapunov exponents 52
7. Flux 55
8. Summary 65
A guide to the literature 65
APPENDIX: Mathematical notation 66
Chapter 2. A New Paradigm in Quantum Chaos: Aubry's Theory of Equilibrium States for the Adiabatic Holstein Model 70
1. Introduction 70
2. The model 70
3. The anti-integrable limit t = 0 71
4. Small hopping 72
5. Explicit estimates 75
6. Properties 82
7. Extensions 86
APPENDIX A: Notation and basic mathematical results 88
APPENDIX B: Comments on the proof of [1] 89
APPENDIX C: Solution of a recurrence inequality 91
APPENDIX D: Variation of the electronic energy with configuration u 92
REFERENCES 94
Chapter 3. Periodic-Orbit Theory 96
1. Introduction 96
2. Semi-classical periodic-orbit theory 98
3. Organizing chaos 103
4. Symmetries 111
5. The three-disk system 116
6. Classical periodic-orbit theory 122
7. Matrix elements 125
8. Final remarks 127
REFERENCES 128
Chapter 4. The Semi-Classical Helium Atom. 132
1. Introduction 132
2. Classical motion in helium 134
3. Semi-classical quantization 143
4. Adiabatic vs. chaotic motion 157
5. Summary and conclusions 160
REFERENCES 161
Chapter 5. The Riemann Zeta-Function and Quantum Chaology 164
1. The Riemann zeta-function 166
2. The functional equation 168
3. The staircase of zeros 170
4. The quantum chaology connection 172
5. Convergence properties of periodic-orbit formulae 177
6. Riemann-Siegel resummation 183
7. The pair correlation of the zeros 192
APPENDIX: Probabilistic number theory and the pairwise distribution of the primes 200
REFERENCES 203
Chapter 6. Quantum Localization 206
1. Introduction 206
2. The kicked rotor 207
3. Anderson localization 211
4. The mapping of the kicked-rotor problem on the Anderson model 216
5. Adiabatic localization 222
6. Measures and manifestations of localization 226
7. Summary 233
8. Related problems 233
REFERENCES 234
Chapter 7. Dynamical Localization in the Hydrogen Atom 240
1. Introduction 240
2. Classical dynamics 242
3. One-dimensional model 244
4. Kepler map 245
5. Photonic localization 248
6. Derealization 251
7. Quantization of the Kepler map and the scattering problem 254
9. Conclusion 255
REFERENCES 256
Chapter 8. Dynamical Localization, Dissipation and Noise 260
1. Introduction 260
2. Dissipative quantum dynamics 263
3. Dynamical localization in the dissipative kicked-rotor model 269
4. Rydberg atoms in a noisy waveguide 277
REFERENCES 283
Chapter 9. Statistics of Quasi-Energy Spectrum. 284
1. Introduction 284
2. Some time-dependent models with classical chaos 287
3. The kicked-rotator model 290
4. General properties of the quasi-energy spectrum and quantum resonance 295
5. Uncorrelated statistics of quasi-energy 300
6. Maximal statistical properties of quasi-energy spectra 303
7. Intermediate statistics caused by the localization 309
REFERENCES 323
Chapter 10. Scattering and Resonances: Classical and Quantum Dynamics. 326
1. Introduction 326
2. Classical scattering and chaotic repellers 327
3. Semi-classical quantum scattering 364
4. Conclusions 393
REFERENCES 399
Chapter 11. Exotic Fractals and Atomic Decay. 404
1. Introduction 404
2. What is the one-dimensional kicked hydrogen atom (1DKH)? 405
3. Why consider the 1DKH? 405
4. Equations of motion 406
5. The mapping 408
6. The never-come-back property 409
7. The Jacobian 410
8. Eigenvalues 410
9. Hyperbolicity 411
10. The tent map, a model for ionization 414
11. Power law decay of the 1DKH 415
12. Exotic (scale-broken) fractals 415
13. Summary 416
REFERENCES 416
Chapter 12. Quantum Chaotic Scattering and Microwave Experiments. 418
1. Introduction 418
2. The model 420
3. Semi-classical theory 423
4. S-matrix element energy correlations 423
5. S-matrix correlations under general perturbations 424
6. The Wigner time delay 427
7. Absorption 428
8. Conclusions 431
REFERENCES 431

Produkt-Details

• Erscheint lt. Verlag: 11.8.2015
• Sprache: englisch
• Themenwelt: Naturwissenschaften ► Physik / Astronomie ► Quantenphysik
• Themenwelt: Technik
• ISBN-10: 1-4832-9032-8 / 1483290328
• ISBN-13: 978-1-4832-9032-4 / 9781483290324