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For further information please contact Springer-Verlag, Physics Editorial Department V, Tiergarten- strasse 17, W-6900 Heidelberg, FRG W. Dieter Heiss (Ed.) Chaos and Quantum Chaos Proceedings of the Eighth Chris Engelbrecht Summer School on Theoretical Physics Held at Blydepoort, Eastern Transvaal South Africa, 13-24 January 1992 Springer-Verlag Berlin Heidelberg NewYork London Paris Tokyo Hong Kong Barcelona Budapest " Editor W. Dieter Heiss Department of Physics University of the Witwatersrand, Johannesburg Private Bag 3, Wits 2050, South Africa ISBN 3-540-56253-2 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-56253-2 Springer-Verlag New York Berlin Heidelberg This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1992 Printed in Germany Typesetting: Camera ready by author/editor 58/3140-543 210 - Printed on acid-free paper Christian Albertus Engelbrecht 8 October 1935 - 30 July 1991 Chris Engelbrecht was the founder of the series of South African Summer Schools in Theoretical Physics. He negotiated its structure and its funding, determined its specific form and by applying his personal attention, he ensured that each school was relevant and of a high standard. Born in Johannesburg where he received his school education, he studied at Pretoria University for a BSc and MSc degree before going to Caltech where he obtained a PhD in 1960. Back in South Africa he held appointments as theo- retical physicist at the Atomic Energy Board (1961-1978) and at Stellenbosch University (1978-1991). Apart from his research and excellence in teaching, he served physics and science on numerous bodies. He was elected Presider/t of the SA Institute of Physics for two terms - 1987 - 1991. It is a fitting memorial to him and a tribute to his selfless, excellent and dedicated service to the cause of physics and his fellow scientists, to henceforth name this series The Chris Engelbrecht Summer Schools in Theoretical Physics. Preface Chaos and the quantum mechanical behaviour of classically chaotic systems have been attracting increasing attention. Initially, there was perhaps more emphasis on the theoretical side, but this is now being backed up by experimental work to an increasing extent. The words 'Quantum Chaos' are often used these days, usually with an undertone of unease, the reason being that, in contrast to classical chaos, quantum chaos is ill defined; some authors say it is non-existent. So, why is it that an increasing number of physicists are devoting their efforts to a subject so fuzzily defined? Short pulse laser techniques make it possible nowadays to probe nature on the border line between classical and quantum mechanics. Such experimental back-up is direly needed, since, in the case of classically chaotic systems, the formal tools have so far turned out to be insufficient for an understanding of this border line. The fact that the conceptual foundations of quantum mechanics are being challenged - or, at least, subjected to a search for deeper understanding - is of course ample explanation for this new field being so attractive. We were fortunate that we could assemble seven leading experts who have made major contributions in the field. The emphasis of the school was on quantum chaos and random matrix theory. The material presented in this volume is a reflection of lucid and nicely coordinated presentations. What it cannot reflect is the friendly working atmosphere that prevailed throughout the course. The Organizing Committee is indebted to the Foundation for Research Development for its financial support, without which such high-level courses would be impossible. We also wish to express our thanks to the Editors of Lecture Notes in Physics and Springer-Verlag who readily agreed to publish and assisted in the preparation of these proceedings. Johannesburg South Africa September 1992 W D Heiss Contents The Problem of Quantum Chaos Boris V C"hirikov . . Introduction: The Theory of Dynamical Systems and Statistical Physics Asymptotic Statistical Properties of Classical Dynamical Chaos . 4. . The Correspondence Principle and Quantum Chaos The Uncertainty Principle and the Time Scales of Quantum Dynamics Finite-Time Statistical Relaxation in Discrete Spectrum . 7. 8. The Quantum Steady State Asymptotic Statistical Properties of Quantum Chaos Conclusion: The Quantum Chaos and Traditional Statistical Mechanics 9 17 20 26 32 40 49 Semi-Classical Quantization of Chaotic Billiards Uzy. SmiIansky I Introduction H Classical Billiards HI Quantization - The Semi, Quantal Secular Equation HI.a Quantization of Convex Billiards HI.b Quantization of Billiards with Arbitrary Shapes III.c Properties of the Semi.Quantal Secular Equation IV V The Semi-Classical Secular Function Spectral Densities V.a The Averaged Spectral Density V.b The Gutzwiller Trace Formulae for the Spectral Density 57 58 62 67 68 70 75 80 90 91 95 VI Spectral Correlations VX.a VI.b VI.c S Matrix Spectral Correlations Energy Spectral Correlations Composite Billiards VII Conclusions Appendix A 98 100 104 106 112 115 Stochastic Scattering Theory or Random-Matrix Models for Fluctuations in Microscopic and Mesoscopic Systems Hans A WeidenmfilIer 1. Motivation : The Phenomena 1.1 Microwave Scattering in Cavities 1.2 Compound-Nucleus Scattering in the Domains of Isolated and of Overlapping Resonances 1.3 Chaotic Motion in Molecules 1.4 Passage of Light Through a Medium with a Spatially Randomly Varying Index of Refraction 1.5 Universal Conductance Fluctuations 2. Stochastic Modelling 2.1 Chaotic and Compound-Nuclens Scattering 2.2 Conductance Fluctuations 3. Methods of Averaging 3.1 Monte-Carlo Simulation 3.2 Disorder Perturbation Theory 3.3 The Generating Functional 4. Chaotic Scattering and Compound-Nudens Reactions 5. Universal Conductance Fluctuations 6. Persistent Currents in Mesoscopic Rings 7. Conclusions 121 122 124 126 128 130 130 133 134 135 137 137 138 139 141 151 159 164 XI Atomic and Molecular Physics Experiments in Quantum Chsology Peter M Koch 1. Introduction 1.1 The Diamagnetic Kepler Problem 1.2 Spectroscopy of Highly Excited Polyatomic Molecules 1.3 The Helium Atom 1.4 Swift Ions Traversing Foils 1.5 What This Paper Covers and Does Not Cover 2. Apparatus and Experimental Method 2.1 Apparatus 2.2 Experimental Methods 3. The Hamiltonian and Scaled Variables 4. Regimes of Behavior 4.1 "Ionization" Curves 5. Static Field Ionization 6. Regime-I : The Dynamic Tnnneling Regime 7. Regime-H : The Low Frequency Regime 8. Regime-HI : The Semiclassical Regime 8.1 Classical Kepler Maps for ld Motion 9. Regime-IV : The Transition Regime 9.1 Nonclassical Local Stability and "Scars" 10. Regime-V : The High Frequency Regime 11. Conclusions 167 168 170 171 172 173 174 176 176 179 182 187 187 190 191 194 196 199 203 206 212 215 ×ll Topics in Quantum Chaos R E Prauge I. Introduction A. Philosophy B. Time Scales C. ~ The Quasiclassical Approximation D. Pseudorandom Matrix Theory E. Types of Chaotic Systems F. Summary and Outline II. Quantum Longtime Behavior and Localization The Kicked Rotor Tnnneling and KAM Torii Dynamic Localization HI. A° B. C. D. E. F. G. Connection of Anderson Localization to Quantum Chaos Pseudorandomness of Tm An Aside on Liouville Numbers Comparison of Pseudorandom and Truly Random Cases H. Numerical Solutions I. Relationship of the Localization Length to Classical Diffusion Transitions to Chaos A. B. C. D. E. F. G. Introduction The Logistics Map Period Doubling Sequence Hamiltonian Maps Last KAM Toms Other Relevant Variables Planck's Constant as a Relevant Variable 225 225 225 227 228 229 230 232 233 233 234 237 241 242 242 243 243 244 244 244 244 246 252 252 254 254 [...]... "more random", or "true random", processes remains, as yet, open 3 T h e correspondence principle and quantum chaos Absence of the claassical-like chaos in quantum mechanics apparently contradicts not only with the correspondence principle, as mentioned above, but also with the fudamentai statistical nature of quantum mechanics However, even though the random element in quantum mechanics ( "quantum. .. solution The peculiarity of this and similar examples is in that to achieve the true chaos not only the quantum motion must be unbounded and, hence, of a continuous spectrum but the momenta have to grow exponentially in time This is why most physicists reject the above definition of quantum chaos and adhere to another one which reads (see, e.g., Ref.[ll]): quantum chaos is the quantum dynamics of classically... the quantum chaos is deterministic randomness in quantum mechanics over and above that contained in wavefunction or the expansion postulate The latter refers to the quantum measurement as mentioned above Some mathematicians implicitly accepted the same definition, and "succesfully" constructed the quantum analogue to the classical KS-entropy (see, e.g., second Ref.[39]) For bounded in phase space quantum. .. initial state, and only than the normal diffusion is restored An example of the time reversal in classical and qu0axtum standard map is shown in Fig.7 [50] The stability of quantun chaos on relaxation time scale is comprehensible as the random time scale (4.1) is 23 0 50 100 150 200 250 300 'l" Figure 7: The effect of time reversal at ~" = 150 in classical (1) and quantum (2) chaos for the standard map... regime of quantum motion depending on the quasiclassical parameter q as outlined in Fig.8 In this picture the asymptotic classical chaos is but a limitin# pattern to compare with the true (quantum) dynamics The real quantum chaos, nevertheless, is called sometimes pseudochaos or transient chaos to distinguish an "ugly" reality from the perfect ideal Of the two characteristic time scales of quantum motion... indeed, it can be singled out and separated from the proper quantum processes Namely, the fundamental randomness in quantum mechanics is related only to a very specific event - the quantum measurement which, in a sense, is foreign to the proper quantum system itself This allows to divide the whole problem of quantum dynamics into two qualitatively different parts: (i) the proper quantum dynamics as described... understanding of quantum chaos has been achieved, particularly, due to the above philosophy of separating the dynamical part of quantum mechanics accepted, explicitly or more often implicitly, by most researchers in this field Currently, there are several approaches to the definition of quantum chaos The first natural move was to extend onto the quantum mechanics the classical definition of dynamical chaos. .. Modulated Standing Light Wave 323 Dynamical Localization in Josephson Junctions 325 e The Problem of Quantum Chaos Boris V Chirikov Budker Institute of Nuclear Physics 630090 Nov0sibirsk, RUSSIA Abstract: The new phenomenon of quantum chaos has revealed the intrinsic complexity and richness of the dynamical motion with discrete spectrum which had been always considered as most simple and regular one... addition to a number of recent reviews [9-14], and to these proceedings My presentation below will be from a physicist's point of view even though the whole problem of quantum chaos, as a part of quantum dynamics, is essentially mathematical The main contribution of physicists to the studies of quantum chaos is in extensive numerical (computer) simulations of quantum dynamics, or numerical experiments... dynamical chaos which has destroyed the deterministic image of the classical physics What is the dynamical chaos? Which should be its meaningful definition? This is one of the most controversial questions even in classical mechanics There are two main approaches to the problem; The first one is essentially mathematical [4, 7] The terms dynamical chaos and randomness are abandoned from rigorous statements, and . Connection of Anderson Localization to Quantum Chaos Pseudorandomness of Tm An Aside on Liouville Numbers Comparison of Pseudorandom and Truly Random Cases. Spectrum . 7. 8. The Quantum Steady State Asymptotic Statistical Properties of Quantum Chaos Conclusion: The Quantum Chaos and Traditional Statistical