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Semiclassical Quantization of Rydberg Series in Helium
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Roughly speaking, the spectrum of the helium atom is built up by
an infinite number of more or less interwoven Rydberg series. The
corresponding classical three-body Coulomb problem shows all kinds of
complicated structures, from approximately integrable to fully chaotic.
Using semiclassical techniques we could associate the helium Rydberg
series with the nearly regular motion of one electron far away from
the nucleus. The finer details of the spectrum such as the level repulsion
due to the
overlap of different Rydberg series are caused by the hyperbolic dynamics
near the triple collision. Combining Gutzwiller's periodic orbit theory
with the cycle expansion technique of the semiclassical zeta function using
an infinite alphabet, we are able to connect these two different types of
classical dynamics smoothly. We can thereby reproduce the helium spectrum
(for = 0) from the ground state to energies accessible by the best
available quantum calculations in all details.
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G. Tanner (and D. Wintgen)
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Corrections to Semiclassical Trace Formulas in
Terms of Creeping Orbits
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We incorporate the diffraction phenomena in the semi-classical description of classically chaotic systems.

This leads us to introduction of periodic creeping orbits which are created by
diffraction and thus have no classical particle trajectory analogue.
We are studying their effects in
simple 2-disk and 3-disk 2-dimensional as well as
3-dimensional scattering systems,
and comparing our results to exact quantum mechanical results.
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P. Rosenqvist, G. Vattay and (A. Wirzba)
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Geometric and Diffractive Orbits in the Scattering from Confocal Hyperbolae
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We study the scattering resonances between two confocal hyperbolae and show
that the spectrum is dominated by the effect of a single periodic
orbit. There are two distinct cases depending on whether the orbit is
geometric or diffractive. A generalization of periodic orbit theory allows us
to incorporate the second possibility. In both cases we also perform a WKB
analysis. Although it is found that the semiclassical approximations work
best for resonances
with large energies and narrow widths, there is reasonable agreement
even for resonances with large widths - unlike the two disk scatterer.
We also find agreement with the next order correction to periodic orbit
theory.
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N.D. Whelan
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Effective Group Structure in the Interacting Boson Model
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Spectrum generating algebras are used in various fields of physics as models
to determine quantum structure, including energy levels and transition
strengths.
The advantage of such models is that their group structure allows an extensive
understanding of the system being studied. In addition they
possess a simple classical limit, at least for bosonic systems.
We discuss an algebraic model of nuclear structure, the Interacting Boson
Model (IBM), and show that in one limit its group structure is particularly
simple. For zero angular momentum there is an effective lower dimensional
group structure which describes the system both classically and quantum
mechanically.
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N.D. Whelan
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Weyl Expansion for Symmetric Potentials
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We derive a semiclassical expansion of the smooth part of the density
of states in symmetric potentials. The density of states of each
irreducible representation is separately evaluated
using the Wigner transforms of the projection operators.
For discrete symmetries the expansion yields a formally exact but
asymptotic series in , while for the rotational symmetries
the expansion requires averaging over angular momentum as well as energy.
The validity of the expansion is demonstrated by a numerical calculation
in two dimensions.
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B. Lauritzen and N. Whelan
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Supersymmetry and Scattering Resonances
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The issue of supersymmetric quantum mechanics and how it can be used to
calculate bound states and scattering in one dimension has been of recent
interest. In particular, it has been shown that using supersymmetry results
in improved WKB estimates of transmission coefficients. In addition,
identification of the real poles of the transmission coefficient with the
bound states of an inverted potential yields the exact bound state spectrum in
at least some cases. We study one additional property of
supersymmetry which we have not seen mentioned in the literature. This is the
fact that the supersymmetric mapping between potentials preserves not just the
bound state spectrum but also the unbound scattering resonances and virtual
states.
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N.D. Whelan
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Partial Dynamical Symmetry and Classical Mechanics
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It is well known how dynamical symmetries in algebraic models of
molecules, hadrons and nuclei lead to integrability in the classical
mechanics. It is possible to make a direct comparison of the quantum
operators which are conserved and the corresponding classical invariants
and thereby to enumerate the conserved classical actions. Recently, an
interesting feature in the quantum models was observed. It is possible
to construct Hamiltonians for which a subset of states possess a
symmetry even though the Hamiltonian does not - the opposite of
spontaneous symmetry breaking. This effect is called partial dynamical
symmetry and the symmetric states are called 'special'. We are studying the
structure of the corresponding classical phase space. It is found that there
are tori in the phase space which support the special states and that
the neighbouring region of phase space supports approximately special
states. Also, it is found that there is a global effect in that in
addition to a few states being fully symmetric the rest of the states
are somewhat symmetric. We have not yet obtained a classical explanation
of this effect.
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N.D. Whelan and (A. Leviatan)
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Semiclassical Expansions in Many-body Systems
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The applicability of semiclassical expansions for a interacting many-body
system is investigated.
It is demonstrated that while the usual semiclassical expansion of the
level density is very accurate for a low-dimensional system, it fails at
moderate energies for a system of many degrees of freedom, as in a heavy
nucleus.
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B. Lauritzen
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Periodic Orbits of Nonscaling Hamiltonian Systems from Quantum
Mechanics
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Information about classical periodic orbits is obtained from the quantum
system; the eigenvalues alone give the energy-period plot including the
bifurcations; the eigenvalues and eigenvectors contain the phase space
information.
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(D. C. Meredith), (M. Baranger), (M. Haggerty), B. Lauritzen,
(D. Provost) and (M.A.M. de Aguiar)
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Random Tilings, Zippers, and Decagonal Quasicrystals
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Quasicrystals are metallic alloys, whose diffraction patterns contain sharp
Bragg peaks yet, at the same time, exhibit symmetries (e.g. icosahedral symmetry,
decagonal symmetry) that are incompatible with crystallographic order. The
most astonishing thing about quasicrystals is that they have the gall to exist.
Any rational explanation of this rather disturbing experimental fact must address
the following two questions: What atomic structures do quasicrystals have?
Why are these structures thermodynamically stable and/or kinetically accessible?
My most recent effort, done in conjunction with Marek Mihalkovic
in Grenoble, concerns the formulation of a random-tiling model for the
decagonal (10-fold symmetric)
quasicrystal. Our aim is to see whether random tilings comprising decagons,
pentagonal stars and squashed hexagon can explain the diffuse scattering
that is exhibited by this quasicrystal. To generate the requisite random
tilings a special ``Zipper'' Monte Carlo algorithm must be implemented.
To cut an awfully long story short, let's just say the following: Symantec C++,
MacTraps, QuickDraw, then Double Click On Me.
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M. Oxborrow and (M. Mihalkovic)
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Mon Mar 6 19:42:06 MET 1995