Table Of ContentStochasticity and Quantum Chaos
Mathematics and Its Applications
Managing Editor:
M. HAZEWINKEL
Centre for Mathematics and Computer Science, Amsterdam, The Netherlands
Volume 317
Stochasticity and
Quantum Chaos
Proceedings of the 3rd Max Boro Symposium,
Sob6tka Castle, September 15-17, 1993
edited by
Zbigniew Raba,
Wojciech Cegla
and
Lech Jak6bczyk
Institute ofTheoretical Physics,
University ofWroclaw,
Wroclaw, Poland
SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.
A C.I.P. CataIogue record for this book is available from the Library of Congress.
ISBN 978-94-010-4076-1 ISBN 978-94-011-0169-1 (eBook)
DOI 10.1007/978-94-011-0169-1
Printed an acid-free paper
AII Rights Reserved
© 1995 Springer Science+Business Media Dordrecht
Originally published by Kluwer Academic Publishers in 1995
Softcover reprint ofthe hardcover Ist edition 1995
No part of the material protected by this copyright notice may be reproduced or
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CONTENTS
Foreword
The quantal fattening of fractals
N.L. Balazs 3
How and when quantum phenomena become real
Ph. Blanchard A. ladczyk 13
Chaotic dynamics in a periodically driven anharmonic oscillator
Yu.L. Bolotin V. Yu. Gonchar M. Ya. Granovsky 31
Coherent and incoherent dynamics in a periodically driven bistable system
T. Dittrich F. Grossmann P. Hanggi B. Oe/schlagel R. Utermann 39
The Ehrenfest theorem for Markov diffusions
P. Garbaczewski 57
The quantum state diffusion model, asymptotic solutions,
thermal equilibrium and Heisenberg picture
N. Gisin 63
Stochastic representation of quantum dynamics
Z. Haba 73
Level repulsion and exceptional points
W.D. Heiss and W.-H. Steeb 91
Type-II intermittency in the presence of additive
and multiplicative noise
H. Herzel F. Argoul A. Arneodo 99
Aspects of Liouville integrability in quantum mechanics
1. Hietarinta 115
vi
KAM techniques for time dependent quantum systems
H.R. Jauslin 123
Dissipation and noise in quantum mechanics
N.G. van Kampen 131
Irregular scattering, number theory, and statistical mechanics
Andreas Knauf 137
Band random matrices, kicked rotator and disordered systems
Luca Molinari 149
Robust scarred states
D. Richards 161
The quasiclassical statistical description of quantum dynamical systems
and quantum chaos
Yu. P. Virchenko 177
Quantum measurement by quantum brain
Mari Jibu and Kunio Yasue 185
Time reversal and Gaussian measures in quantum physics
J. C. Zambrini 195
Stochastic resonance in bistable systems with fluctuating barriers
L. Gammaitoni F. Marchesoni E. Menichella-Saetta S. Santucci 209
Index
217
List of participants
219
FOREWORD
These are the proceedings of the Third Max Born Symposium which took place
at SobOtka Castle in September 1993. The Symposium is organized annually by
the Institute of Theoretical Physics of the University of Wroclaw. Max Born was a
student and later on an assistant at the University of Wroclaw (Wroclaw belonged to
Germany at this time and was called Breslau). The topic of the Max Born Sympo
sium varies each year reflecting the developement of theoretical physics. The subject
of this Symposium "Stochasticity and quantum chaos" may well be considered as
a continuation of the research interest of Max Born. Recall that Born treats his
"Lectures on the mechanics of the atom" (published in 1925) as a nrst volume of
a complete monograph (supposedly to be written by another person). His lectures
concern the quantum mechanics of integrable systems. The quantum mechanics of
non-integrable systems was the subject of the Third Max Born Symposium.
It is known that classical non-integrable Hamiltonian systems show a chaotic
behaviour. On the other hand quantum systems bounded in space are quasiperi
odic. We believe that quantum systems have a reasonable classical limit. It is not
clear how to reconcile the seemingly regular behaviour of quantum systems with
the possible chaotic properties of their classical counterparts. The quantum proper
ties of classically chaotic systems constitute the main subject of these Proceedings.
Other topics discussed are: the quantum mechanics of dissipative systems, quantum
measurement theory, the role of noise in classical and quantum systems.
The Symposium came into being thanks to the enthusiasm of the lecturers and
the substantial help of our colleagues. It has been supported by the Polish Research
Council (KBN) and by the University of Wroclaw. We are thankful to everybody
who contributed to the success of this Symposium. Finally, we would like to thank
Mrs. Anna Jadczyk for her excellent work in preparing the manuscripts.
Wroclaw, February 28th 1994
The Editors
THE QUANTAL FATTENING OF FRACTALS
N.L. BALAZS
Department oj PhY6ic6
State Univenity oj New York at Stony Brook
Stony Brook, NY 11794 - 9800
Abstract. Periodic points of a classical map may belong to a fractal set. It is shown here on a
simple model that upon quantising this classical map the influence of the classical periodic points
upon the quantal results is the same whether the periodic point belongs to a fractal set, or not. A
simple intuitive explanation is given.
1. INTRODUCTION
How does one quantise fractals? Is there a meaning associated with these words?
Fractals were born through the iteration of functions of a complex variable giving
rise to Julia sets (1918)1). They were thus intimately connected with the theory of
(discrete) dynamical systems where the equations of dynamics are replaced by an
iterative scheme, the mapping of the phase space onto itself. It is thus a natural
step to generate fractal sets through the use of a dynamical system, and ask what
happens to these fractal sets if we quantise the classical map.
A large amount of numerical evidence suggests that the classical periodic points (or
orbits) have far reaching effects on the quantum mechanical results. In particular,
we stress three such manifestations.
In the quantal description of a classical map the state of the system is specified by
a vector in a finite dimensional vector space, and the time evolution is generated by
a unitary operator acting on this space. This unitary operator, i.e. its eigenangles
and eigenvalues contains all the information about the dynamical system.
The classical periodic points will leave their imprints on the quantal results in three
different places.
a) There is strong numerical evidence that the presence of hyperbolic periodic points
in the classical description changes the statistics of the nearest neighbour distribu
tion of the eigenangles in a well defined manner2};
b) the eigenvectors may be "scarred" by the periodic points3};
c) in the coherent state representation the correlations between an initial state and
its time evolved version should be large at those pq points which correspond to clas
sical periodic points.
What happens, however, if the classical periodic points are fractals, hence form a
measure zero set? This note will discuss this problem as exemplified on a simple
modeI 4}.
3
Z. Haba et at. (eds.), Stochasticity and Quantum Chaos, 3-11.
© 1995 Kluwer Academic Publishers.
4 N.L. BALAZS
Frg(3a)
Fig. 1. Nearest neighbour spacing distribution (nnsd) for quantum maps with different number
of strips. The number of phase space cells, N, vary slightly, since they must be divisible by the
numb~r of strips. They are succesively 288, 290, 294, 288.
2. THE MODEL
Consider a classical discrete dynamical system that maps the unit square unto itself.
The horizontal axis is labeled by q, the position (in units of the maximum displace
ment); the vertical axis is labeled by p, the momentum (in units of the maximum
momentum). The transformation that represents the model is described geometri
cally as follows. Divide the unit square into three vertical strips. The first and third
strips have the same width a. These vertical strips will be now changed into hori
zontal ones, and stacked on the top of each other. The first strip is made horizontal
through a vertical compression of factor a, followed by a horizontal dilation of the
same factor, becoming thereby a horizontal strip at the bottom. The second vertical
strip is simply rotated into the horizontal position, becoming thereby the second
horizontal strip. The third vertical strip is subjected to the same compression and
dilation as the first, but laid down as the top horizontal strip. Thus, the evolution
of the system is the iteration of the transformation
