ABSTRACTS
| Matthias Brack (Regensburg) |
SU(2) Symmetry Breaking in Some Hamiltonian Systems
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I will discuss perturbative and uniform trace formulae for the
semiclassical description of Henon-Heiles and related potentials
in which the SU(2) symmetry of the underlying harmonic oscillator
is broken and requires extensions of the standard Gutzwiller
approach.
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| Stephen Creagh (Nottingham) |
Dynamics in Wavefunction Statistics and Tunnelling
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Statistical models based on random matrix theory provide
a natural means of characterising the lifetimes of complex
systems. The dissociation by tunnelling of molecules with
chaotic dynamics and the transmission properties of quantum
dots are particular examples. We show that the internal
dynamics of such systems can, through the mechanism of scarring,
lead to strong modifications of the normal random-matrix-theory
statistics. We provide an explicit quantitative prediction
of such deviations following the approach recently developed by
Kaplan, Heller and coworkers. These predictions are tested
successfully on two-dimensional potentials.
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| Paul Dando (UCL) |
Closed Orbit Theory for Molecules in Fields
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Semiclassical closed-orbit theory was developed initially as a
qualitative and quantitative tool to interpret the dynamics of excited
hydrogen atoms in static external fields: large-scale structures in
the photo-absorption spectrum were explained in terms of classical
orbits that close at the nucleus. Recently, we have extended the
closed-orbit formalism to allow for a semiclassical treatment of
Rydberg molecules in a strong magnetic field. An outline of our
theoretical approach will be given and comparisons of our
semiclassical calculations using this formalism with fully quantum
mechanical calculations of the photo-absorption spectrum of a model
Rydberg molecule in a static magnetic field will be presented. The
connection between classical orbits and the various features observed
in the quantum spectrum will be explained and their physical
interpretation discussed.
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| John Delos (William and Mary) |
(1) Semiclassical Physics from Aristotle to Schroedinger
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| John Delos (William and Mary) |
(2) Classical Orbits and Quantum Spectra of Atoms in Fields
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| Bertrand Georgeot (Toulouse) |
Quantum Computing: Many-Body Effects and Simulation of Chaos
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| Quantum computers hold great promises, since the massive
parallelism due to the superposition principle of quantum
mechanics may enormously increase the speed of a computational
process. Still, they are prone to errors and in particular
the presence of residual imperfections and disorder will be discussed.
They lead to many-body effects that may hamper the computation, but
still leave a reasonable parameter window for operability
of the computer. Methods for the
efficient simulation of classical and quantum chaotic
systems on such a quantum processor will be also presented.
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| Toshihiro Iwai (Kyoto) |
Boundary Conditions at Singular Configurations of Many Bodies
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The center-of-mass system of many bodies admits a natural action of the rotation
group SO(3). Singular configurations in the title of this talk means those
configurations at which the rotation group SO(3) acting on the center-of-mass
system has non-trivial isotropy subgroups. Practically, the singular
configurations under consideration are collinear configurations and
a multiple collision. If singular configurations are gotton rid of in the
center-of-mass system, the restricted center-of-mass system is made into an
SO(3) bundle, and the bundle picture works well in the study of quantum
mechanics of many bodies. For example, reducing the rotational degrees of
freedom is well performed in this picture. However, if one wants to take
singular configurations into account, the bundle picture fails to work.
We need another reduction method.
The Peter-Weyl theorem on the unitary irreducible representations of compact
Lie groups provides a method for reducing quantum systems with symmetry
described by a compact Lie group. Since the Peter-Weyl method works
irrespectively of whether the Lie group acts freely on a manifold in
question (i.e. isotropy subgroups are trivial everywhere) or not,
one can use the method for reducing the quantum system. One can also apply
the method to impose suitable boundary conditions for wave functions at
singular configurations. In particular, three-body systems will be considered
to obtain boundary conditions for wave functions at collinear configurations
and at a triple collision.
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| Charles Jaffe (West Virginia) |
Transition States: Molecular, Atomic and Celestial
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The current definitions of the transition states will be reviewed. Examples
from molecular physics, atomic physics and celestial mechanics illustrating the
limitations of these definitions will be presented. Solutions for these
difficulties will be proposed.
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| Patricio Leboeuf (Orsay) |
Thermodynamics of Small Fermi Systems: Quantum Fluctuations
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We investigate the probability distribution of the quantum fluctuations of
thermodynamic functions of finite, ballistic, phase--coherent Fermi gases.
Depending on the chaotic or integrable nature of the underlying classical
dynamics, on the thermodynamic function considered, and on temperature, we
find that the probability distributions are dominated either (i) by the
local fluctuations of the single--particle spectrum on the scale of the
mean level spacing, or (ii) by the long--range modulations of that
spectrum produced by the short periodic orbits. In case (i) the
probability distributions are computed using the appropriate local
universality class, uncorrelated levels for integrable systems and random
matrix theory for chaotic ones. In case (ii) all the moments of the
distributions can be explicitly computed in terms of periodic orbit
theory, and are system--dependent, non--universal functions. The
dependence with temperature and with the number of particles of the
fluctuations is explicitly computed in all cases, and the different
relevant energy scales are displayed.
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| Igor Kozin (Aberdeen) |
Relative Equilibria and Periodic Orbits in Molecules
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Using several examples we show how relative equilibria
and periodic orbits can be used to interpret and predict
features of quantum spectra of molecules.
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| Robert Littlejohn (Berkeley) |
Gauge Theory of Vibrations of Polyatomic Molecules
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TBA
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| Robert Mackay (Warwick) |
Persistence of Spectral Projections for Large Networks of Quantum
Units, and Quantum Discrete Breathers
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Many experiments on molecular crystals suggest they can support spatially
localised excitations. Although approximate theory for these was developed
over 30 years ago, I am not aware of a rigorous explanation. I propose to
to explain them by proving persistence of spectral projections for appropriate
classes of system uniformly in system size. Although I have not yet filled
in a few details, I think the strategy of proof is good.
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| Jorg Main (Stuttgart) |
Use of Harmonic Inversion Techniques in Semiclassical Quantization
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Harmonic inversion is introduced as a powerful tool for semiclassical
quantization, which solves the fundamental convergence problems in
periodic orbit and closed orbit theory. The advantage of semiclassical
quantization by harmonic inversion is the universality and wide
applicability of the method to open and bound systems with underlying
regular, chaotic, and even mixed classical dynamics. The method also
allows the semiclassical calculation of diagonal matrix elements and,
e.g. for atoms in external fields, individual non-diagonal transition
strengths. We report the latest state of the art of the technique
including recent extensions to consider uniform approximations and the
optimization of the efficiency by, e.g. using a functional equation.
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| Kevin Mitchell (William and Mary) |
Chaotic Ionization of Hydrogen in Parallel Fields
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Classical ionization of an excited hydrogen atom in parallel electric
and magnetic fields is a useful and experimentally accessible model of
chaotic decay and scattering. The dynamics can be reduced to an
area-preserving chaotic map of the phase plane. Decay is studied by
examining segments of a line of initial conditions that ionize at
various iterates of the map. We observe sequences of segments which
escape at successive iterates of the map. These sequences, called
``epistrophes'', decay geometrically at large iterate number but have
unpredictable initial behavior. The epistrophes impart a fractal
structure to the plot of ionization time, but their unpredictable
beginnings break any true asymptotic self-similarity, leaving a weaker
``epistrophic self-similarity.'' The existence and characterization
of epistrophes follow from a general geometric analysis of homoclinic
tangles. Consequently, our results apply to a large class of chaotic
maps.
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| San Vu Ngoc (Grenoble) |
Redistribution of Eigenvalues in Polyads via Quantum Monodromy
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We consider a 2 degrees of freedom semiclassical integrable system
depending on an exernal parameter. If no monodromy is present
there is a well-defined notion of "polyads", ie. clusters of eigenvalues.
The appearance of monodromy due to a generic hamiltonian Hopf
bifurcation has the effect of redistributing the eigenvalues between
these polyads, as already remarked by Zhilinskii and Sadowskii.
We show how to describe this redistribution in terms of the monodromy
index, using Duistermaat-Heckman theory and the geometry of
moment polytopes.
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| Igor Pavlichenkov (Moscow) |
Quantum Phase Transitions in Rotational Bands
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Phase transitions that occur at zero temperature when non-thermal parameter
is changed are called quantum phase transitions (QPT). The talk is devoted
to QPT in finite (mesoscopic) systems and their manifestation in rotational
spectra. The quantum fluctuations which drive these transitions and the
concept of a broken symmetry are a guideline for the discussion.
I begin with a general introduction to QPT and their connection with
bifurcations. This is done in the context of the bifurcation in rotational
spectra of symmetric three-atom molecules XY$_2$. Subsequently, I
consider a more complicated problem of the superfluid-to-normal phase
transition in the rotational bands of superdeformed atomic nuclei.
Numerous examples of critical phenomena in rotational bands show an
intimate connection between internal and rotational motion and lead to
a general theorem: {\it A change in the internal structure of a quantum
system manifests itself in the modification of its rotational spectrum.}
Finally, I briefly discuss the ensemble of magnetically trapped Fermi
atoms. This ultracold degenerate gas of weakly interacting fermions is
an ideal system for quantitative study of QPT because full control over
the trapped potential and the interaction is available.
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| Jonathan Robbins (Bristol) |
Spin-Statistics and Quantum Mechanics
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An account is given of the spin-statistics relation in quantum
mechanics, related to but distinct from earlier work (Berry and
Robbins, Proc. Roy. Soc. Lond. A 453, 1771 - 1790 (1997)).
Following previous authors, quantum mechanics for identical
particles with spin is formulated in terms of a Hilbert space of
sections of vector bundles over a configuration space of (unordered)
points in R^3. The statistics are determined by the holonomies of the
bundles.
We introduce a class of bundles whose characteristic properties are
motivated by physical considerations as well as those of mathematical
simplicity. With a recent result of Atiyah it is shown that these
bundles exists, and that they necessarily engender the correct
relation between spin and statistics.
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| Jan-Michael Rost (Dresden) |
Dimensional Effects in Few-Electron Quantum Dots
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While the dynamics for three-dimensional axially symmetric two-electron
quantum dots with parabolic confinement potentials is in general
non-separable we have found
an exact separability with three quantum numbers for specific values
of the magnetic field. Furthermore, it is shown that the magnetic
properties such as the magnetic
moment and the susceptibility are sensitive to the presence and
strength of a vertical confinement. Using a semiclassical approach the
calculation of the eigenvalues
reduces to simple quadratures providing a transparent and almost
analytical quantization of the quantum dot energy levels which differ
from the exact energies only by a few percent.
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| Howard Taylor (USC, Los Angeles) |
A Simple Scheme for Extracting Internal Motions from Spectroscopic Hamiltonians
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The problem addressed is to determine the internal vibrational motions that
when quantized yield the vibrational bound and resonant states of a molecule.
In particular we consider systems with two or more vibrational resonances due
to frequencies in rational ratio, where motions at variance with simple normal
or local modes exist. We restrict ourselves to problems where a spectroscopic
normal form Hamiltonians (Heff) can be obtained either from fits to experimental
lines, or with the use of a PES, to calculated eigenvalues. The application of
canonical perturbation theory to a system with a known PES can also supply such
an Heff. Presently we also require that the number of degrees of freedom minus
the number of constants of motion, one being the polyad quantum number (P) that
will exist when resonances are present, is no greater than three. We treat the
non-trivial cases where more than one resonance exists and hence chaos can occur
and where the reduced phase space, obtained by canonical transform using
constants of motion, corresponds to two or three degrees of freedom. We further
limit ourselves to the challenging cases where the dynamics is not susceptible
to any simple adiabatic or other separation scheme and where wave functions and
trajectories in the full dimension are too complex to be interpreted by any
graphical representations.
In such cases it will be demonstrated that most if not all of the dynamics
can be uncovered and that dynamic quantum numbers representing quasiconserved
quantities can be assigned given only the already existing eigenfunction-basis
transformation matrix used in fitting the Heff to the experimentally or
theoretically generated spectrum. The method in conception depends on the
ability previously gained(1), in studying problems where nonlinear dynamics
was used to find the motions underlying the simple patterns seen in plots of
the density and the phases of semiclassical eigenfunctions created from the
information in the transformation matrix calculated in the above "fit". These
eigenfunctions are parametric in the constants of the motion (the polyad number
P in particular) and lie for DCO on a 2D toroidal configuration space
described by two angle variables. Since the features of these 2D wave
functions are generally simple to recognize once one is comfortable working
in this unusual space; just viewing the patterns allows the sorting of the
interleaved states of different dynamics into suits (like a deck of cards)
each based on different dynamics. Then nodal counts and/or phase advances
(since we are in a space for angles) allows sequential quantum number
assignment. The contours of the 2D angle space wave function density and
phases can be used for each suite to infer the type of classical internal
motion that the atoms are undergoing in normal mode, local mode or
displacement coordinate space. These motions are those that when quantized
gives rise to the levels in the suit. A discussion, as to why wave functions
represented in compact angle (of action-angle) spaces as opposed to the usual
open coordinate spaces are so much simpler to interpret is given.
The assignments and dynamics of DCO are presented(4). Because of a strong
1:1:2 resonance assignment and interpretation alluded previous workers who
computed eigenfunctions from both a high quality potential surface(2) and
from a spectroscopic Hamiltonian(3). Again we stress no serious computation
was needed to extract dynamics and to assign once the Heff was available.
Graphical representations of the phase and densities of eigenfunctions in
reduced configuration (angle) space, the principles of nonlinear dynamic and
semiclassical ideas on how wave functions accumulate about phase space
organizing structure are the keys to the analysis.
1. (a) M.P. Jacobson, C. Jung, H.S. Taylor and R.W. Field, J. Chem. Phys., 111, 600 (1999).
(b) C. Jung, H.S. Taylor and M. Jacobson, J. Phys. Chem. A., 105, 681
(2001).
2. H.-M. Keller, H. Floethmann, A. J. Dobbyn, R. Schinke, H.-J. Werner,
C. Bauer and P. Rosmus, J.Chem. Phys. 105, 4983 (1996).
3. A. Troellsch and F. Temps, Zeitschreft for Physikalische Chemie, 215, 207
(2000).
4. E.Atligan, C. Jung and H.S. Taylor, J. Phys. Chem. (2002) in press.
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| Turgay Uzer (Georgia Tech) |
Phase Space Transition States
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Dynamical systems theory is used to construct a phase-space
version of Transition State Theory. Special multidimensional separatrices
are found which act as impenetrable barriers in phase space between
trajectories that which lead to reactions and those which do not. The
elusive momentum-dependent transition state between reactants and products
is thereby characterized using a practical algorithm.
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| Holger Waalkens (Bremen) |
Quantum Monodromy in the Two Centers Problem
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The motion of a particle subject to the attraction of two space fixed
Newtonian centers belongs to the class of well studied integrable
systems. The corresponding quantum system is of particular interest in
molecular physics because it represents the model of a diatom with a
single electron in Born-Oppenheimer approximation. In this talk the
bifurcation diagram of bound motion of the three degrees of freedom
system will be discussed. It will be shown that the bifurcation diagram
divides the three dimensional space of the constants of motion into two
regions of regular values of which one region lies inside of the other
wherefore the other region is not simply connected. In addition to that,
the latter region is pierced by an isolated line of the bifurcation
diagram. It arises a complicated case of monodromy with two sources. In
particular, the former source of monodromy bases on a subset of the
bifurcation diagram which is of codimension 1 in the space of the
constants of motion. This case has not been studied before. The
monodromy will be quantitatively described in terms of 3x3 monodromy
matrices calculated from smoothly fitting together classical actions
whose local existence is guaranteed by the Liouville-Arnold theorem. The
implication on the quantum system will be discussed in terms of the
numerically computed quantum spectrum.
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| Boris Zhilinskii (Dunkerque) |
Hamiltonian Monodromy as a Lattice Defect
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Monodromy in classical and quantum Hamiltonian systems is analysed
from the point of view of defects of crystal lattices. Simple
geometrical construction of different local defects
will be given and applied to the characterization of monodromies.
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