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Quantum chaos

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Semiclassical Quantization of Rydberg Series in Helium truecm

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. truecm

G. Tanner (and D. Wintgen) truecm

Corrections to Semiclassical Trace Formulas in Terms of Creeping Orbits truecm

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. truecm

P. Rosenqvist, G. Vattay and (A. Wirzba)

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Geometric and Diffractive Orbits in the Scattering from Confocal Hyperbolae truecm

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. truecm

N.D. Whelan

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Effective Group Structure in the Interacting Boson Model truecm

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. truecm

N.D. Whelan

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Weyl Expansion for Symmetric Potentials truecm

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. truecm

B. Lauritzen and N. Whelan truecm

Supersymmetry and Scattering Resonances truecm

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. truecm

N.D. Whelan

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Partial Dynamical Symmetry and Classical Mechanics truecm

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. truecm

N.D. Whelan and (A. Leviatan)

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Semiclassical Expansions in Many-body Systems truecm

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. truecm

B. Lauritzen truecm

Periodic Orbits of Nonscaling Hamiltonian Systems from Quantum Mechanics truecm

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. truecm

(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 truecm

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. truecm

M. Oxborrow and (M. Mihalkovic)



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Next: Chaos experiments Up: RESEARCH PROJECTS Previous: Classical chaos



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