6,012 research outputs found
Comment on "Recurrences without closed orbits"
In a recent paper Robicheaux and Shaw [Phys. Rev. A 58, 1043 (1998)]
calculate the recurrence spectra of atoms in electric fields with non-vanishing
angular momentum not equal to 0. Features are observed at scaled actions
``an order of magnitude shorter than for any classical closed orbit of this
system.'' We investigate the transition from zero to nonzero angular momentum
and demonstrate the existence of short closed orbits with L_z not equal to 0.
The real and complex ``ghost'' orbits are created in bifurcations of the
``uphill'' and ``downhill'' orbit along the electric field axis, and can serve
to interpret the observed features in the quantum recurrence spectra.Comment: 2 pages, 1 figure, REVTE
Semiclassical quantization with bifurcating orbits
Bifurcations of classical orbits introduce divergences into semiclassical
spectra which have to be smoothed with the help of uniform approximations. We
develop a technique to extract individual energy levels from semiclassical
spectra involving uniform approximations. As a prototype example, the method is
shown to yield excellent results for photo-absorption spectra for the hydrogen
atom in an electric field in a spectral range where the abundance of
bifurcations would render the standard closed-orbit formula without uniform
approximations useless. Our method immediately applies to semiclassical trace
formulae as well as closed-orbit theory and offers a general technique for the
semiclassical quantization of arbitrary systems
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.
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
- …