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

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Learn all about the services we offer
124 Pages
English

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Semiclassics beyond the diagonalapproximationDissertationzur Erlangung des Doktorgradesder Naturwissenschaften (Dr. rer. nat.)der naturwissenschaftlichen Fakult˜at II { Physikder Universit˜at Regensburgvorgelegt vonMarko Turekaus Halle (Saale)Februar 2004Promotionsgesuch eingereicht am 05. Februar 2004Promotionskolloquium am 21. April 2004Die Arbeit wurde von Prof. Dr. Klaus Richter angeleitet.Prufungsaussc˜ hu…:Vorsitzender: Prof. Dr. Christian Back1. Gutachter: Prof. Dr. Klaus Richter2. Gutachter: Prof. Dr. Matthias BrackWeiterer Prufer:˜ Prof. Dr. Tilo WettigAbstractThe 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 conjecturehas been conflrmed in many experiments and numerical studies but a formal proofis still lacking. In this thesis we present a semiclassical evaluation of the spectralform factor which goes beyond M.V. Berry’s diagonal approximation. To thisend we extend a method developed by M. Sieber and K. Richter for a speciflcsystem: the motion of a particle on a two-dimensional surface of constant negativecurvature.

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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 suitable in the theory of
classical mechanics was extended towards the more general mathematical concept of
dynamical systems (see e.g. [ASY96], [Rei96] and [GH02]). However, until the mid
1970’s these activities were mostly of purely mathematical nature. It was only then
when digital computers started to become a common scientiflc tool that the interest
in chaotic dynamical systems began to grow signiflcantly. Extensive numerical stud-
ies of systems and computer experiments stimulated the application of
the theory of dynamical systems to a large variety of difierent flelds such as biology
(e.g. predator-prey models), hydrodynamics (e.g. Rayleigh-Bernard convec-
tion), nonlinear electrical circuits and many others (see e.g. [Sch84], [Ott93] and
[LL92]).
As opposed to macroscopic bodies, the dynamics of microscopically small par-1.1 Chaos in classical and quantum mechanics 3
ticles (such as electrons in semiconductor devices) has to be treated within the
framework of quantum theory, see e.g. [Mer98]. It is described in terms of a wave
˜function “(q;t) which is a solution of the Schrodinger equation. The concept
that single points in a phase space represent the state of the system can no longer
be applied because of the Heisenberg uncertainty principle. This principle basi-
fcally states that a single quantum state occupies a flnite phase-space volume (2…~)
˜determined by Planck’s constant ~. Due to the linearity of the Schrodinger
equation with respect to the wave function “(q;t) one would not expect any sim-
ple relation to chaotic behavior, i.e. sensitive dependence on the initial conditions,
as described above. The time evolution of an arbitrary state being a superposi-
tion of energy eigenstates is quasi-periodic. On the other hand one can always
study the classical limit of the quantum dynamics of a given system by ’making’
the particle under consideration macroscopically large again. This limit is given
when the typical wavelengths appearing in the wave functions are negligible com-
pared to all other length scales of the system. The following question then arises
naturally. Consider two difierent closed quantum systems with one of them showing
regular and the other chaotic dynamics in the classical limit. Can one then flnd
˜a criterion based on the Schrodinger equation only, i.e. its energy eigenvalues
E or eigenfunctions “ (q;t), to distinguish these two systems? To put it in othern n
words, is the chaotic nature of the underlying classical system observable within its
quantum mechanical description? The physical phenomena related to this kind of
questions are central to the fleld of quantum chaos [Ber87]. Numerous experiments
andnumericalsimulationsdoindeedshowdifierentstatisticalpropertiesoftheeigen-
functions and eigenenergies if chaotic quantum systems are compared to integrable
systems. This is for example re ected in difierent nearest neighbor distributions or
two-point correlation functions for the energy eigenvalues (for an overview see e.g.
[Les89, St˜o99, Haa01]).
Of particular interest in this fleld is the semiclassical regime. Roughly speaking,
this regime lies in the middle between classical mechanics and quantum mechanics.
Here, one expects that classical objects like trajectories play a role while quantum
efiects like interference are still present. Semiclassics is comparable to the transition
from wave optics to ray optics in the limit of short wavelengths. Formally, the
semiclassical limit can be achieved by letting ~! 0 as all other parameters in the
problem remain unchanged. A very instructive discussion on how the semiclassical
limit emerges from quantum mechanics can be found in [Ber89].
Various semiclassical methods have been developed since the early days of quan-
tum mechanics. For integrable systems a semiclassical quantization can be per-
formed using the action variables that deflne the invariant tori in phase space. One
canmakeacanonicalvariabletransformationsothattheHamiltonianisexpressed
in terms of these actions [Arn01]. The Bohr-Sommerfeld quantization scheme
is then based on the requirement that each of these actions is an integer multiple4 1 Introduction
of Planck’s constant (2…~). However, as Einstein already pointed out in 1917
[Ein17], this quantization procedure is not applicable to chaotic systems.
It was only in the early 1970’s when the flrst links between classically chaotic
Hamiltonian systems and their quantum mechanical counterparts could be made.
M.Gutzwillerderivedaformulaforthesemiclassicallimitofthedensityofstates
in terms of a sum over classical periodic orbits (see [Gut90] and references therein).
Thistraceformulaexpressesthedensityofstates(whichisdirectlyrelatedtotheset
ofquantummechanicaleigenenergies)intermsofclassicalquantitiesliketheactions
and the stabilities of the periodic orbits. The original theory of Gutzwiller gives
onlytheleadingcontributionsin ~withrespecttotheanalyticpartsinthedensityof
states | thus being exact in the semiclassical limit ~! 0. Later on it was extended
to an expansion in this small parameter [GA93, AG93]. However, there are certain
technical problems connected with the trace formula concerning the convergence of
thesumoverperiodicorbits, seee.g. [Ber89]foradiscussionoftheseissues. Despite
these subtleties Gutzwiller’s trace formula is a frequently used tool to study the
quantummechanicalenergyeigenvaluesofchaoticsystemsinthesemiclassicallimit.
Our analysis of the spectral form factor is based on this trace formula.
Not only the energy eigenvalues but also the individual eigenfunctions “ (q)n
˜of the Schrodinger equation are in uenced by the underlying classical dynamics
2[Hel96]. AccordingtoShnirelman’stheoremtheprobabilitydensityj“ (q)j isforn
almostallenergyeigenstatesofaclassicallychaoticsystemgivenbythemicrocanon-
ical distribution [Shn74]. However, there can be exceptions in the form of scarred
wavefunctions[Hel84,Hel89]. Thesescarsareduetoalocalizationofthewavefunc-
tioninthevicinityofaperiodicorbit. Statisticalpropertiesofenergyeigenfunctions
belonging to a certain energy interval were studied by Bogomolny [Bog88] who
showed that energy averaged wave functions can indeed show an enhanced proba-
bility density in the vicinity of classical periodic orbits. However, the flrst model for
wave functions in chaotic systems was developed by Berry [Ber77]. This so-called
random wave model proposes that the wave functions “ (q) are random superpo-n
sitions of plane waves and was successfully applied to a variety of physical systems
(seee.g. [AL97], [BS02]andreferencestherein). Aproofforthismodelcouldnotyet
be found and chaotic wave functions are still subject to ongoing research activities.
Besides the above mentioned interest in fundamental questions concerning the
correspondenceprinciplebetweenquantummechanicsanditsclassicallimitthereare
many practical applications for which a sound understanding of semiclassical meth-
ods and issues concerning quantum chaos is essential. Semiclassical methods have
successfully been applied to atomic and molecular physics, e.g. photo-absorption
spectraofRydbergatomsandatomsinmagneticflelds[FW89]orthesemiclassical
treatment of the Helium atom [WRT92]). Another important fleld where semiclas-
sical methods have been applied with great success is that of mesoscopic electronic
devices [Ric00]. Here the idea is that most of the relevant physically quantities,