Semiclassical initial value calculations of collinear helium atom

Semiclassical calculations using the Herman-Kluk initial value treatment are performed to determine energy eigenvalues of bound and resonance states of the collinear helium atom. Both the $eZe$ configuration (where the classical motion is fully chaotic) and the $Zee$ configuration (where the classical dynamics is nearly integrable) are treated. The classical motion is regularized to remove singularities that occur when the electrons collide with the nucleus. Very good agreement is obtained with quantum energies for bound and resonance states calculated by the complex rotation method.Comment: 24 pages, 3 figures. Submitted to J. Phys.

Numerical study of scars in a chaotic billiard

We study numerically the scaling properties of scars in stadium billiard. Using the semiclassical criterion, we have searched systematically the scars of the same type through a very wide range, from ground state to as high as the 1 millionth state. We have analyzed the integrated probability density along the periodic orbit. The numerical results confirm that the average intensity of certain types of scars is independent of $\hbar$ rather than scales with $\sqrt{\hbar}$. Our findings confirm the theoretical predictions of Robnik (1989).Comment: 7 pages in Revtex 3.1, 5 PS figures available upon request. To appear in Phys. Rev. E, Vol. 55, No. 5, 199

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.

The helium atom in a strong magnetic field

We investigate the electronic structure of the helium atom in a magnetic field b etween B=0 and B=100a.u. The atom is treated as a nonrelativistic system with two interactin g electrons and a fixed nucleus. Scaling laws are provided connecting the fixed-nucleus Hamiltonia n to the one for the case of finite nuclear mass. Respecting the symmetries of the electronic Ham iltonian in the presence of a magnetic field, we represent this Hamiltonian as a matrix with res pect to a two-particle basis composed of one-particle states of a Gaussian basis set. The corresponding generalized eigenvalue problem is solved numerically, providing in the present paper results for vanish ing magnetic quantum number M=0 and even or odd z-parity, each for both singlet and triplet spin symmetry. Total electronic energies of the ground state and the first few excitations in each su bspace as well as their one-electron ionization energies are presented as a function of the magnetic fie ld, and their behaviour is discussed. Energy values for electromagnetic transitions within the M=0 sub space are shown, and a complete table of wavelengths at all the detected stationary points with respect to their field dependence is given, thereby providing a basis for a comparison with observed ab sorption spectra of magnetic white dwarfs.Comment: 21 pages, 4 Figures, acc.f.publ.in J.Phys.

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

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

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

Scars of Invariant Manifolds in Interacting Chaotic Few-Body Systems

We present a novel extension of the concept of scars for the wave functions of classically chaotic few-body systems of identical particles with rotation and permutation symmetry. Generically there exist manifolds in classical phase space which are invariant under the action of a common subgroup of these two symmetries. Such manifolds are associated with highly symmetric configurations. If sufficiently stable, the quantum motion on such manifolds displays a notable enhancement of the revival in the autocorrelation function which is not directly associated with individual periodic orbits. Rather, it indicates some degree of localization around an invariant manifold which has collective characteristics that should be experimentally observable.Comment: 4 pages, RevTeX, 4 PS/EPS-figures, uses psfig.sty, quantum computation changed, to be published in Physical Review Letter

Hydrogen atom moving across a strong magnetic field: analytical approximations

Analytical approximations are constructed for binding energies, quantum-mechanical sizes and oscillator strengths of main radiative transitions of hydrogen atoms arbitrarily moving in magnetic fields 10^{12}-10^{13} G. Examples of using the obtained approximations for determination of maximum transverse velocity of an atom and for evaluation of absorption spectra in magnetic neutron star atmospheres are presented.Comment: 17 pages, 3 figures, 5 tables, LaTeX with IOP style files (included). In v.2, Fig.1 and Table 5 have been corrected. In v.3, a misprint in the fit for oscillator strengths, Eq.(21), has been correcte

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