Semiclassics beyond the diagonal approximation [Elektronische Ressource] / vorgelegt von Marko Turek

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• Physik

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• Publié par: universitat_regensburg
• Publié le: 01 janvier 2004
• Nombre de lectures: 32
• Langue: English
• Poids de l'ouvrage: 1 Mo

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Semiclassics beyond the diagonal
approximation
Dissertation
zur Erlangung des Doktorgrades
der Naturwissenschaften (Dr. rer. nat.)
der naturwissenschaftlichen Fakult˜at II { Physik
der Universit˜at Regensburg
vorgelegt von
Marko Turek
aus Halle (Saale)
Februar 2004Promotionsgesuch eingereicht am 05. Februar 2004
Promotionskolloquium am 21. April 2004
Die Arbeit wurde von Prof. Dr. Klaus Richter angeleitet.
Prufungsaussc˜ hu…:
Vorsitzender: Prof. Dr. Christian Back
1. Gutachter: Prof. Dr. Klaus Richter
2. Gutachter: Prof. Dr. Matthias Brack
Weiterer Prufer:˜ Prof. Dr. Tilo WettigAbstract
The statistical properties of the energy spectrum of classically chaotic closed quan-
tumsystemsarethecentralsubjectofthisthesis. IthasbeenconjecturedbyO.Bo-
higas, M.-J. Giannoni and C. Schmit that the spectral statistics of chaotic sys-
tems is universal and can be described by random-matrix theory. This conjecture
has been conflrmed in many experiments and numerical studies but a formal proof
is still lacking. In this thesis we present a semiclassical evaluation of the spectral
form factor which goes beyond M.V. Berry’s diagonal approximation. To this
end we extend a method developed by M. Sieber and K. Richter for a speciflc
system: the motion of a particle on a two-dimensional surface of constant negative
curvature. In particular we prove that these semiclassical methods reproduce the
random-matrix theory predictions for the next to leading order correction also for a
much wider class of systems, namely non-uniformly hyperbolic systems with f‚ 2
degrees of freedom. We achieve this result by extending the conflguration-space
approach of M. Sieber and K. Richter to a canonically invariant phase-spaceh.
Zusammenfassung
Das zentrale Thema dieser Arbeit sind die statistischen Eigenschaften des En-
ergiespektrums geschlossener Quantensysteme deren klassische Analoga durch chao-
tische Dynamik gekennzeichnet sind. Fur˜ diese Systeme stellten O. Bohigas,
M.-J. Giannoni und C. Schmit die Vermutung auf, da… die spektrale Statistik
universell ist und den Vorhersagen der Zufallsmatrixtheorie folgt. Diese Vermu-
tung wurde bereits durch eine Vielzahl von Experimenten und numerischen Un-
tersuchungen best˜atigt, ein formaler Beweis konnte bisher jedoch nicht gefunden
werden. In dieser Arbeit wird der spektrale Formfaktor auf der Grundlage semi-
klassischer Methoden berechnet, die ub˜ er M.V. Berrys Diagonaln˜aherung hinaus
gehen. Die Grundlage dafur˜ stellt die Erweiterung einer Methode von M. Sieber
undK. Richter dar, welche fur˜ die Bewegung eines Teilchens auf einer zweidimen-
sionalen Fl˜ache konstanter negativer Krumm˜ ung entwickelt wurde. Insbesondere
wird in der vorliegenden Arbeit gezeigt, da… die Anwendung dieser semiklassischen
Methoden auf die viel gr˜o…ere Klasse nicht-uniformer hyperbolischer Systeme mit
beliebiger Anzahl von Freiheitsgraden ebenfalls die Vorhersagen der Zufallsmatrix-
theorie reproduziert. Zu diesem Zweck wird eine kanonisch invariante Phasenraum-
methode entwickelt, welche den Ortsraumzugang von M. Sieber undK. Richter
erweitert.Contents
1 Introduction 1
1.1 Chaos in classical and quantum mechanics . . . . . . . . . . . . . . . 1
1.2 Random-matrix theory and BGS conjecture . . . . . . . . . . . . . . 5
1.3 Model systems in quantum chaos . . . . . . . . . . . . . . . . . . . . 8
1.4 Purpose and outline of the work . . . . . . . . . . . . . . . . . . . . . 10
2 Chaotic systems and spectral statistics 13
2.1 Dynamical systems and chaos . . . . . . . . . . . . . . . . . . . . . . 13
2.2 Spectral statistics in complex systems . . . . . . . . . . . . . . . . . . 20
2.3 Semiclassical approach to spectral statistics . . . . . . . . . . . . . . 23
2.4 Matrix element statistics . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.5 Beyond the diagonal approximation: conflguration-space approach . . 29
3 Crossing angle distribution in billiard systems 35
3.1 Crossing angle in the uniformly hyperbolic billiard . . . . 35
3.2 Model system: Limac»on billiards . . . . . . . . . . . . . . . . . . . . 38
3.3 Crossing angle distribution in the cardioid . . . . . . . . . . . . . . . 44
4 Phase-space approach for two-dimensional systems 57
4.1 Correlated orbits and the ’encounter region’ . . . . . . . . . . . . . . 57
4.2 Action difierence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.3 Maslov index and weight of the partner orbit . . . . . . . . . . . . . 70
4.4 Counting the partner orbits and calculation of the form factor . . . . 73vi CONTENTS
5 Extensions and applications of the phase-space approach 83
5.1 Higher-dimensional systems . . . . . . . . . . . . . . . . . . . . . . . 83
5.2 GOE { GUE transition . . . . . . . . . . . . . . . . . . . . . . . . . . 92
5.3 Matrix element uctuations . . . . . . . . . . . . . . . . . . . . . . . 95
6 Conclusions and outlook 101
6.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
6.2 Open questions and outlook . . . . . . . . . . . . . . . . . . . . . . . 104
A Conversion between volume and surface integral 105
Literature 107
Acknowledgments 117CHAPTER 1
Introduction
1.1 Chaos in classical and quantum mechanics
The chaotic motion of macroscopic bodies as well as the quantum mechanical prop-
erties of microscopic particles have been intensively studied for more or less one
hundred years now. Nevertheless it took more than flfty years until the flrst signifl-
cant attempts were made to bring the two flelds together. The traditional theory for
classical mechanics goes back to Newton, Lagrange and Hamilton. According
to this theory the dynamical state of any macroscopic body is described by its po-
sition q and its velocity q_ or momentum p at a given time t. The motion of thist tt
macroscopic object can then be described quantitatively by solving the equations
of motion. The solution uniquely determines the position and the momentum at
any later time t for given initial conditions (q ;p ) at time t = 0. Therefore, the0 0
state of a classical body (or a system of many bodies) can be uniquely character-
ized in terms of a point x = (q;p) in the associated phase space and the dynamics
of the body is then given by the trajectory x in that phase space. This impliest
that the motion as described in the framework of classical mechanics is completely
deterministic. However, this does not mean that the motion represented by the so-
lution x necessarily shows a simple and regular behavior as a function of time. Ast
one can imagine, the motion of many particles interacting with each other, e.g. via
their gravitational or electromagnetic forces, can easily become extremely complex.
In this case it would be hopeless to look for a speciflc solution of the equations of
motion and one typically employs statistical theories for the characterization of this
type of systems. But also systems with only a few degrees of freedom can show2 1 Introduction
a very complex dynamical behavior. This can be caused by non-linearities in the
equations of motion. For example, already the problem of describing the dynam-
ics of three interacting bodies can lead to very complex solutions as flrst shown by
¶Poincare in 1892 [Poi92]. This complex behavior is related to the fact that the
dynamics shows a very sensitive dependence on the initial conditions. By this one
(1) (2)
means that two trajectories starting at close points x and x in phase space0 0
(1) (2)
diverge from each other very rapidly, i.e. exponentially. The distancejx ¡ x jt t
between two initially close trajectories grows approximately as» exp‚t with timet
until it reaches more or less the system size. Here,‚> 0 is the so-called Lyapunov
exponentwhichcharacterizes thetimescale of theexponentialgrowth. If abounded
and energy conserving system is considered this sensitive dependence on the initial
conditions leads to a chaotic motion. This especially implies that it is impossible
to predict the dynamics of a chaotic system for long times ‚t? 1 as the initial
conditions can always be measured with a certain accuracy only.
A deflnition of a classical system with regular motion can be given in terms of
the invariants of motion [Arn01]. Assume that there are f degrees of freedom for
the dynamics, e.g. f = 3 for the motion of a single particle in the three dimensional
space. For closed systems without dissipation the total energy E is conserved. If
there are further f¡1 independent functions h(q ;p ) that are invariant under thet t
classical dynamics then the system is called integrable and shows regular dynamics.
These constants of motion can be chosen to be actions. They restrict the motion in
phase space to tori which form anf dimensional hypersurface in the 2f dimensional
phase space. Hence the time evolution of a state is either periodic or quasi-periodic.
Ifontheotherhandtherearenofurtherconservedquantitiesbesidestheenergythen
the motion in phase space is only restricted to a 2f¡1 dimensional hypersurface. In
this case the dynamics can be either completely chaotic or partially chaotic, which
is then called mixed.
¶After the early work byPoincare on the three body problem several signiflcant
contributions were made to the fleld of chaotic dynamics, e.g. by Birkhoff, Kol-
mogorov, Smale and others, and the original description suita