98 research outputs found
The nodal structure of doubly-excited resonant states of helium
The authors examine the nodal structure of accurate helium wavefunctions calculated by direct diagonalization of the full six-dimensional problem. It is shown that for fixed interelectronic distance R (or hyperspherical radius R) the symmetric doubly-excited resonant states have well defined lambda , mu nodal structure indicating a near separability in prolate spheroidal coordinates. For fixed lambda , however, a clear mixing of R, mu nodes is demonstrated. This corresponds to a breakdown of the adiabatic approximation and can be understood in terms of the classical two-electron motion
Significance of Ghost Orbit Bifurcations in Semiclassical Spectra
Gutzwiller's trace formula for the semiclassical density of states in a
chaotic system diverges near bifurcations of periodic orbits, where it must be
replaced with uniform approximations. It is well known that, when applying
these approximations, complex predecessors of orbits created in the bifurcation
("ghost orbits") can produce pronounced signatures in the semiclassical spectra
in the vicinity of the bifurcation. It is the purpose of this paper to
demonstrate that these ghost orbits themselves can undergo bifurcations,
resulting in complex, nongeneric bifurcation scenarios. We do so by studying an
example taken from the Diamagnetic Kepler Problem, viz. the period quadrupling
of the balloon orbit. By application of normal form theory we construct an
analytic description of the complete bifurcation scenario, which is then used
to calculate the pertinent uniform approximation. The ghost orbit bifurcation
turns out to produce signatures in the semiclassical spectrum in much the same
way as a bifurcation of real orbits would.Comment: 20 pages, 6 figures, LATEX (IOP style), submitted to J. Phys.
Intermanifold similarities in partial photoionization cross sections of helium
Using the eigenchannel R-matrix method we calculate partial photoionization
cross sections from the ground state of the helium atom for incident photon
energies up to the N=9 manifold. The wide energy range covered by our
calculations permits a thorough investigation of general patterns in the cross
sections which were first discussed by Menzel and co-workers [Phys. Rev. A {\bf
54}, 2080 (1996)]. The existence of these patterns can easily be understood in
terms of propensity rules for autoionization. As the photon energy is increased
the regular patterns are locally interrupted by perturber states until they
fade out indicating the progressive break-down of the propensity rules and the
underlying approximate quantum numbers. We demonstrate that the destructive
influence of isolated perturbers can be compensated with an energy-dependent
quantum defect.Comment: 10 pages, 10 figures, replacement with some typos correcte
Stability ordering of cycle expansions
We propose that cycle expansions be ordered with respect to stability rather
than orbit length for many chaotic systems, particularly those exhibiting
crises. This is illustrated with the strong field Lorentz gas, where we obtain
significant improvements over traditional approaches.Comment: Revtex, 5 incorporated figures, total size 200
Universality of Level Spacing Distributions in Classical Chaos
We suggest that random matrix theory applied to a classical action matrix can
be used in classical physics to distinguish chaotic from non-chaotic behavior.
We consider the 2-D stadium billiard system as well as the 2-D anharmonic and
harmonic oscillator. By unfolding of the spectrum of such matrix we compute the
level spacing distribution, the spectral auto-correlation and spectral
rigidity. We observe Poissonian behavior in the integrable case and Wignerian
behavior in the chaotic case. We present numerical evidence that the action
matrix of the stadium billiard displays GOE behavior and give an explanation
for it. The findings present evidence for universality of level fluctuations -
known from quantum chaos - also to hold in classical physics
Elliptic Quantum Billiard
The exact and semiclassical quantum mechanics of the elliptic billiard is
investigated. The classical system is integrable and exhibits a separatrix,
dividing the phasespace into regions of oscillatory and rotational motion. The
classical separability carries over to quantum mechanics, and the Schr\"odinger
equation is shown to be equivalent to the spheroidal wave equation. The quantum
eigenvalues show a clear pattern when transformed into the classical action
space. The implication of the separatrix on the wave functions is illustrated.
A uniform WKB quantization taking into account complex orbits is shown to be
adequate for the semiclassical quantization in the presence of a separatrix.
The pattern of states in classical action space is nicely explained by this
quantization procedure. We extract an effective Maslov phase varying smoothly
on the energy surface, which is used to modify the Berry-Tabor trace formula,
resulting in a summation over non-periodic orbits. This modified trace formula
produces the correct number of states, even close to the separatrix. The
Fourier transform of the density of states is explained in terms of classical
orbits, and the amplitude and form of the different kinds of peaks is
analytically calculated.Comment: 33 pages, Latex2e, 19 figures,macros: epsfig, amssymb, amstext,
submitted to Annals of Physic
Collective and independent-particle motion in two-electron artificial atoms
Investigations of the exactly solvable excitation spectra of two-electron
quantum dots with a parabolic confinement, for different values of the
parameter R_W expressing the relative magnitudes of the interelectron repulsion
and the zero-point kinetic energy of the confined electrons, reveal for large
R_W a remarkably well-developed ro-vibrational spectrum associated with
formation of a linear trimeric rigid molecule composed of the two electrons and
the infinitely heavy confining dot. This spectrum transforms to one
characteristic of a "floppy" molecule for smaller values of R_W. The
conditional probability distribution calculated for the exact two-electron wave
functions allows for the identification of the ro-vibrational excitations as
rotations and stretching/bending vibrations, and provides direct evidence
pertaining to the formation of such molecules.Comment: Published version. Latex/Revtex, 5 pages with 2 postscript figures
embedded in the text. For related papers, see
http://www.prism.gatech.edu/~ph274c
A factorization of a super-conformal map
A super-conformal map and a minimal surface are factored into a product of
two maps by modeling the Euclidean four-space and the complex Euclidean plane
on the set of all quaternions. One of these two maps is a holomorphic map or a
meromorphic map. These conformal maps adopt properties of a holomorphic
function or a meromorphic function. Analogs of the Liouville theorem, the
Schwarz lemma, the Schwarz-Pick theorem, the Weierstrass factorization theorem,
the Abel-Jacobi theorem, and a relation between zeros of a minimal surface and
branch points of a super-conformal map are obtained.Comment: 21 page
Classical approach in quantum physics
The application of a classical approach to various quantum problems - the
secular perturbation approach to quantization of a hydrogen atom in external
fields and a helium atom, the adiabatic switching method for calculation of a
semiclassical spectrum of hydrogen atom in crossed electric and magnetic
fields, a spontaneous decay of excited states of a hydrogen atom, Gutzwiller's
approach to Stark problem, long-lived excited states of a helium atom recently
discovered with the help of Poincar section, inelastic
transitions in slow and fast electron-atom and ion-atom collisions - is
reviewed. Further, a classical representation in quantum theory is discussed.
In this representation the quantum states are treating as an ensemble of
classical states. This approach opens the way to an accurate description of the
initial and final states in classical trajectory Monte Carlo (CTMC) method and
a purely classical explanation of tunneling phenomenon. The general aspects of
the structure of the semiclassical series such as renormgroup symmetry,
criterion of accuracy and so on are reviewed as well. In conclusion, the
relation between quantum theory, classical physics and measurement is
discussed.Comment: This review paper was rejected from J.Phys.A with referee's comment
"The author has made many worthwhile contributions to semiclassical physics,
but this article does not meet the standard for a topical review"
Quantizing Constrained Systems: New Perspectives
We consider quantum mechanics on constrained surfaces which have
non-Euclidean metrics and variable Gaussian curvature. The old controversy
about the ambiguities involving terms in the Hamiltonian of order hbar^2
multiplying the Gaussian curvature is addressed. We set out to clarify the
matter by considering constraints to be the limits of large restoring forces as
the constraint coordinates deviate from their constrained values. We find
additional ambiguous terms of order hbar^2 involving freedom in the
constraining potentials, demonstrating that the classical constrained
Hamiltonian or Lagrangian cannot uniquely specify the quantization: the
ambiguity of directly quantizing a constrained system is inherently
unresolvable. However, there is never any problem with a physical quantum
system, which cannot have infinite constraint forces and always fluctuates
around the mean constraint values. The issue is addressed from the perspectives
of adiabatic approximations in quantum mechanics, Feynman path integrals, and
semiclassically in terms of adiabatic actions.Comment: 11 pages, 2 figure
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