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Lecture Notes in Physics
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H. Araki
Research Institute for Mathematical Sciences
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Institut ftir Theoretische Physik, ETH
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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
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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
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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
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