Table Of ContentQuantum versus Chaos
Fundamental Theories of Physics
An International Book Series on The Fundamental Theories of Physics:
Their Clarification, Development and Application
Editor: ALWYN VAN DER MERWE
University of Denver, U.S.A.
Editorial Advisory Board:
LAWRENCE P. HORWITZ, Tel-Aviv University, Israel
BRIAN D. JOSEPHSON, University of Cambridge, U.K.
CLIVE KILMISTER, University of London, U.K.
PEKKA J. LAHTI, University of Turku, Finland
GÜNTER LUDWIG, Philipps-Universität, Marburg, Germany
ASHER PERES, Israel Institute of Technology, Israel
NATHAN ROSEN, Israel Institute of Technology, Israel
EDUARD PRUGOVECKI, University of Toronto, Canada
MENDEL SACHS, State University of New York at Buffalo, U.S.A.
ABDUS SALAM, International Centre for Theoretical Physics, Trieste, Italy
HANS-JÜRGEN TREDER, Zentralinstitut für Astrophysik der Akademie der
Wissenschafen, Germany
Volume 87
Quantum versus Chaos
Questions Emerging from Mesoscopic Cosmos
by
Katsuhiro Nakamura
Faculty of Engineering,
Osaka City University,
Osaka, Japan
KLUWER ACADEMIC PUBLISHERS
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eBookISBN: 0-306-47121-3
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Table of Contents
Preface ix
Chapter 1. Genesis of chaos and breakdown of
quantization of adiabatic invariants 1
1.1. Introduction 1
1.2. Collapse of KAM tori and onset of chaos 2
1.3. Diagnostic characters of chaos 7
1.4. Suppression of chaos in quantum dynamics 9
1.5. Breakdown of quantization of adiabatic invariants 11
References 14
Chapter 2. Semiclassical quantization of chaos:
trace formula 15
2.1. Green's function and Feynman's path integral method 15
2.2. Quantization of integrable systems 17
2.3. Quantization of chaos: trace formula 19
2.4. Application of trace formula to autocorrelation functions 22
2.5. Significance and limitation of trace formula 29
References 30
Chapter 3. Pseudo-chaos without classical counterpart
in 1-dimensional quantum transport 31
3.1. Introduction 32
3.2. Quantum transport in superlattice and pseudo-chaos 32
3.3. Resonant tunneling in double-barrier structure and
pseudo-chaos 39
3.4. General remarks 43
References 44
Chapter 4. Chaos and quantum transport in open magnetic
billiards: from stadium to Sinai billiards 45
4.1. Introduction 46
V
vi Table of Contents
4.2. Magneto-conductance in stadium billiard: experimental
results 47
4.3. Transition from chaos to tori 50
4.4. Quantum-mechanical and semiclassical theories 54
4.5. Comparison in stadium billiards between theory and
experiment 59
4.6. Open Sinai billiard in magnetic field: distribution of
Lyapunov exponents and ghost orbits 66
4.7. Comparison in Sinai billiard between quantal and classical
theories 70
4.8. Summary 75
References 76
Chapter 5. Chaotic scattering on hyperbolic billiards:
success of semiclassical theory 79
5.1. Introduction 79
5.2. Exact quantum theory 83
5.3. Semiclassical theory 85
5.4. Distribution of complex resonances 92
5.5. Experiment on antidot arrays in magnetic field 97
References 100
Chapter 6. Nonadiabaticity-induced quantum chaos 102
6.1. Avoided level crossings and gauge structure 102
6.2. Nonadiabatic transitions and gauge structure 107
6.3. Forces induced by Born-Oppenheimer approximation 117
6.4. Nonadiabaticity-induced chaos 119
References 125
Chapter 7. Level dynamics and statistical mechanics 127
7.1. Level dynamics: from Brownian motion to generalized
Calogero-Moser-Sutherland (gCM/gCS) system 127
7.2. Soliton turbulence: a new interpretation of irregular
spectra 136
7.3. Statistical mechanics of gCM system 142
7.4. Statistical mechanics of gCS system in intermediate
regime 151
7.5. Extension to case of several parameters 154
References 160
Table of Contents vii
Chapter 8. Towards time-discrete quantum mechanics 162
8.1. Stable and unstable manifolds in time-discrete classical
dynamics 162
8.2. Breakdown of perturbation theory 166
8.3. Internal equation and Stokes phenomenon 168
8.4. Asymptotic expansion beyond all orders and homoclinic
structures 175
8.5. Time discretization and quantum dynamics 181
8.6. Time-discrete unitary quantum dynamics 183
8.7. Time-discrete non-unitary quantum dynamics 186
8.8. Problems to be examined 196
References 197
Chapter 9. Conclusions and prospects 198
References 207
Index 209
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Preface
The framework of quantum mechanics in the adiabatic limit where no
quantum transition occurs is traced back to the quantization condition
of adiabatic invariants, i.e., of action variables. In fact, the
interpretation of this condition as the commutation rule for a pair of
canonical variables led to the construction of Heisenberg's matrix
equation; another interpretation of this condition, as that for confining
a standing wave, led to the birth of Schröd inger's wave mechanics. In
classically-chaotic systems, however, the stable tori are broken up and
we can conceive no action to be quantized. Therefore, we cannot prevent
ourselves from being suspicious of using the present formalism of
quantum mechanics beyond the logically acceptable (i.e., classically-
integrable) regime. Actually, quantum dynamics of classically-chaotic
systems yields only quasi-periodic and recurrent behaviors, thereby
losing the classical-quantum correspondence. There prevails a general
belief in the incompatibility between quantum andchaos. Presumably,
a generalized variant of quantum mechanics should be established so
as to accommodate the temporal chaos.
The range of validity of the present formalism of quantum
mechanics will be elucidated by an accumulation of experiments on the
mesoscopic or nanoscale cosmos. Owing to recent progress in advanced
technology, nanoscale quantum dots such as chaotic stadium and
integrable circle billiards have been fabricated at interfaces of
semiconductor heterojunctions, and quantum transport in these
systems is under active experimental investigation. Anomalous
fluctuation properties as well as interesting fine spectral structures
that have already been reported are indicating symptoms of chaos.
Quantum transport in mesoscopic systems will serve as a nice
candidate for elucidating the effectivenes and noneffectiveness of
quantum mechanics when applied to classically-chaotic systems. The
experimental results could even provide a clue towards the creation of
a generalized quantum mechanics, just as blackbody cavity radiation
at the turn of the last century did for the creation of present-day
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