2,570 research outputs found
Uniform semiclassical approximations on a topologically non-trivial configuration space: The hydrogen atom in an electric field
Semiclassical periodic-orbit theory and closed-orbit theory represent a
quantum spectrum as a superposition of contributions from individual classical
orbits. Close to a bifurcation, these contributions diverge and have to be
replaced with a uniform approximation. Its construction requires a normal form
that provides a local description of the bifurcation scenario. Usually, the
normal form is constructed in flat space. We present an example taken from the
hydrogen atom in an electric field where the normal form must be chosen to be
defined on a sphere instead of a Euclidean plane. In the example, the necessity
to base the normal form on a topologically non-trivial configuration space
reveals a subtle interplay between local and global aspects of the phase space
structure. We show that a uniform approximation for a bifurcation scenario with
non-trivial topology can be constructed using the established uniformization
techniques. Semiclassical photo-absorption spectra of the hydrogen atom in an
electric field are significantly improved when based on the extended uniform
approximations
Photoabsorption spectra of the diamagnetic hydrogen atom in the transition regime to chaos: Closed orbit theory with bifurcating orbits
With increasing energy the diamagnetic hydrogen atom undergoes a transition
from regular to chaotic classical dynamics, and the closed orbits pass through
various cascades of bifurcations. Closed orbit theory allows for the
semiclassical calculation of photoabsorption spectra of the diamagnetic
hydrogen atom. However, at the bifurcations the closed orbit contributions
diverge. The singularities can be removed with the help of uniform
semiclassical approximations which are constructed over a wide energy range for
different types of codimension one and two catastrophes. Using the uniform
approximations and applying the high-resolution harmonic inversion method we
calculate fully resolved semiclassical photoabsorption spectra, i.e.,
individual eigenenergies and transition matrix elements at laboratory magnetic
field strengths, and compare them with the results of exact quantum
calculations.Comment: 26 pages, 9 figures, submitted to J. Phys.
Semiclassical quantization of the hydrogen atom in crossed electric and magnetic fields
The S-matrix theory formulation of closed-orbit theory recently proposed by
Granger and Greene is extended to atoms in crossed electric and magnetic
fields. We then present a semiclassical quantization of the hydrogen atom in
crossed fields, which succeeds in resolving individual lines in the spectrum,
but is restricted to the strongest lines of each n-manifold. By means of a
detailed semiclassical analysis of the quantum spectrum, we demonstrate that it
is the abundance of bifurcations of closed orbits that precludes the resolution
of finer details. They necessitate the inclusion of uniform semiclassical
approximations into the quantization process. Uniform approximations for the
generic types of closed-orbit bifurcation are derived, and a general method for
including them in a high-resolution semiclassical quantization is devised
The hydrogen atom in an electric field: Closed-orbit theory with bifurcating orbits
Closed-orbit theory provides a general approach to the semiclassical
description of photo-absorption spectra of arbitrary atoms in external fields,
the simplest of which is the hydrogen atom in an electric field. Yet, despite
its apparent simplicity, a semiclassical quantization of this system by means
of closed-orbit theory has not been achieved so far. It is the aim of this
paper to close that gap. We first present a detailed analytic study of the
closed classical orbits and their bifurcations. We then derive a simple form of
the uniform semiclassical approximation for the bifurcations that is suitable
for an inclusion into a closed-orbit summation. By means of a generalized
version of the semiclassical quantization by harmonic inversion, we succeed in
calculating high-quality semiclassical spectra for the hydrogen atom in an
electric field
Uniform approximations for non-generic bifurcation scenatios including bifurcations of ghost orbits
Gutzwiller's trace formula allows interpreting the density of states of a
classically chaotic quantum system in terms of classical periodic orbits. It
diverges when periodic orbits undergo bifurcations, and must be replaced with a
uniform approximation in the vicinity of the bifurcations. As a characteristic
feature, these approximations require the inclusion of complex ``ghost
orbits''. By studying an example taken from the Diamagnetic Kepler Problem,
viz. the period-quadrupling of the balloon-orbit, we demonstrate that these
ghost orbits themselves can undergo bifurcations, giving rise to non-generic
complicated bifurcation scenarios. We extend classical normal form theory so as
to yield analytic descriptions of both bifurcations of real orbits and ghost
orbit bifurcations. We then show how the normal form serves to obtain a uniform
approximation taking the ghost orbit bifurcation into account. We find that the
ghost bifurcation produces signatures in the semiclassical spectrum in much the
same way as a bifurcation of real orbits does.Comment: 56 pages, 21 figure, LaTeX2e using amsmath, amssymb, epsfig, and
rotating packages. To be published in Annals of Physic
Symmetry-breaking thermally induced collapse of dipolar Bose-Einstein condensates
We investigate a Bose-Einstein condensate with additional long-range dipolar
interaction in a cylindrically symmetric trap within a variational framework.
Compared to the ground state of this system, little attention has as yet been
payed to its unstable excited states. For thermal excitations, however, the
latter is of great interest, because it forms the "activated complex" that
mediates the collapse of the condensate. For a certain value of the s-wave
scatting length our investigations reveal a bifurcation in the transition
state, leading to the emergence of two additional and symmetry-breaking excited
states. Because these are of lower energy than their symmetric counterpart, we
predict the occurrence of a symmetry-breaking thermally induced collapse of
dipolar condensates. We show that its occurrence crucially depends on the trap
geometry and calculate the thermal decay rates of the system within leading
order transition state theory with the help of a uniform rate formula near the
rank-2 saddle which allows to smoothly pass the bifurcation.Comment: 6 pages, 3 figure
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