Generalized Van Der Waals Potential Research Articles

Photoionization Microscopy of Rydberg Hydrogen Atom in a Generalized van der Waals Potential

The ionization dynamics of a Rydberg hydrogen atom in a generalized van der Waals potential is studied using a semiclassical analysis of photoionization microscopy. Interference patterns of the 2-D radial probability density of the electrons escaping from a photoionization process have been calculated to simulate the patterns recorded on a position-sensitive detector. The added contributions from different ionization trajectories from the atom to the detector generate the interference pattern. The interference pattern is sensitive to one of the parameters governing the electron motion in the generalized van der Waals potential. The photoionization microscopy pattern can therefore be controlled by changes in the external potential.

Dynamics of a Rydberg Hydrogen Atom in a Generalized van der Waals Potential and a Magnetic Field

The classical dynamics of a Rydberg hydrogen atom in a generalized van der Waals potential plus a magnetic field is investigated by using the Poincaré surface of section and phase space trajectories method. The dynamical character of this system depends sensitively on the magnetic field strength. The numerical calculations show that for a certain van der Waals potential, its classical dynamics is regular without the external magnetic field. However, with the addition of the external magnetic field, the dynamical property of the Rydberg hydrogen atom begins to change. With the increase of the magnetic field strength, order-chaos-order-chaos types of transition regions are observed for the hydrogen atom. As the magnetic field strength is very large, nearly all the phase space trajectories are chaotic. Under this condition, only chaotic motion appears. This is caused by the diamagnetic Zeeman effect. Our study provides a different perspective on the dynamical behavior of the Rydberg atom in the van der Waals potential and magnetic field.

An analytical treatment of the energy spectrum of hydrogen-like atoms perturbed by a generalized van der Waals potential

Hydrogen-like atoms are two-particle systems governed with spherically symmetric potentials, their non-relativistic Schrödinger equations can be solved analytically yielding exact information about their spectra. Vigorous interest in the investigation of their behavior under the influence of external fields has been increased over the years. The main goal of the present work is to investigate the effect of a generalized van der Waals potential on the energy spectra of the hydrogen-like atoms. Our treatment is based on the Oscillator Representation Method (ORM). The energy spectrum is studied for various parameters over an atom-surface distance 50nm-300nm for different hydrogen-like atoms, like H, He+, Li2+, Be3+, B4+, etc.

BIFURCATIONS OF THE COMMON LEVEL SETS OF ATOMIC HYDROGEN IN VAN DER WAALS POTENTIAL

The classical dynamics of a hydrogen atom in a generalized van der Waals potential is investigated. In order to carry out the analytical and numerical investigations for a range of parametric values, we removed the singularity of the problem using Levi–Civita regularization and converted the problem into that of two coupled sextic anharmonic oscillators. We give a complete description of the real phase space structure of the converted system and give also an explicit periodic solution for singular common-level sets of the first integrals. All generic bifurcations of Liouville tori were determined theoretically. Numerical investigations are carried out for all generic bifurcations and we observe chaos-order-chaos transition when one of the system parameters is varied.

On the nonintegrability of the generalized van der Waals Hamiltonian system

In this paper we aim to prove that, except for the three known cases, the uniparametric family of Hamiltonian systems defined by the generalized van der Waals potential is nonintegrable in the Liouville–Arnold sense. The proof is based on the theorem of Morales and Ramis about nonintegrability by differential Galois theory.

Axially symmetric potentials in the oscillator representation

The Wick-ordering method called the Oscillator Representation in the nonrelativistic Schrodinger equation is proposed to calculate the energy spectrum for axially symmetric potentials allowing the existence of a bound state. In particular, the method is applied to calculate the energy spectrum of (2s) states of a hydrogen atom in a uniform magnetic field of an arbitrary strength. In the perturbation (external field) approximation, the energy spectrum of the so-called quadratic and spherical quadratic Zeeman problem and the problem of a hydrogen atom in a generalized van der Waals potential is calculated analytically. The results of the zeroth approximation of oscillator representation are in good agreement with the exact values

New integrable systems: hydrogen atom in external fields

We report the discovery of two new three-dimensional conservative Hamiltonian systems which are completely integrable. In both cases, two additional integrals of motion in involution are generators of purely dynamical symmetries, i.e., they do not follow from any apparent geometrical symmetry of the system and are fairly complicated functions of canonical coordinates. Physically, the obtained Hamiltonian functions are good approximations for the description of the hydrogen atom in a circularly polarized microwave field (for low frequencies) or in crossed magnetic and electric fields (for low magnetic field). More formally, they correspond to the hydrogen atom in the presence of both a circularly polarized laser field and a generalized Van der Waals potential.

Normalization and the detection of integrability: The generalized Van Der Waals potential

Deprit and Miller have conjectured that normalization of integrable Hamiltonians may produce normal forms exhibiting degenerate equilibria to very high order. Several examples in the class of coupled elliptic oscillators are known. In order to test the utility of normalization as a detector of integrability we normalize, to high order, a perturbed Keplerian system known to have several integrable limits; the generalized van der Waals Hamiltonian for a hydrogen atom. While the separable limits give rise to high order degeneracy we find a non-separable, integrable limit for which the normal form does not exhibit degeneracy. We conclude that normalization may, in certain cases, indicate integrability but is not guaranteed to uncover all integrable limits.

Applicability of Brody distribution in the study of quantum chaos of the hydrogen atom in a generalized van der Waals potential

The study of quantum manifestations of chaos in a generalized van der Waals potential can be carried out using the SO(4, 2) group theoretical approach. We discuss certain salient features of using the method. By analysing some of the recently introduced measures such as the cumulative spacing distribution, T-function and U-function, we demonstrate the efficacy of using the Brody distribution in this problem.

Quantum chaos of the hydrogen atom in a generalized van der Waals potential.

The quantum manifestations of chaos in the hydrogen atom in a generalized van der Waals potential, which includes the celebrated quadratic-Zeeman-effect problem under an appropriate limiting condition, are studied in detail. Using group-theoretical methods, we derive the matrix elements in an algebraic form. As the Hamiltonian is scale invariant, we introduce an appropriate scaling parameter. By considering the unperturbed hydrogen-atom problem, we demonstrate that suitable changes in the scaling parameter stabilize different parts of the spectrum depending upon the parameter's value. For the present generalized van der Waals potential problem, we utilize this property of the scaling parameter effectively to improve the convergence of eigenvalues while diagonalizing the matrices for various parametric values. Then, we vary one of the system parameters in the range [0,3], study the level statistics, and observe a GOE-Poisson-Brody-Poisson-Brody-Poisson-GOE-type (where GOE means Gaussian orthogonal ensemble) of transition regime hitherto unidentified in any of the perturbed hydrogen-atom problems. Our results are not only in agreement with random-matrix-theory predictions but also justify classical and semiclassical investigations.

Atomic analogs of local and normal modes: The hydrogen atom in a generalized van der Waals potential.

The connection between the hydrogen atom in a generalized van der Waals potential and the notion of local and normal modes is investigated analytically and numerically. It is found that the dynamics in the hydrogenic case can be classified as local, normal, or precessional normal in a similar way to the spectroscopic description of degenerate vibrational states of a molecule. Semiclassical quantization of the system is discussed

Semiclassical quantization of the hydrogen atom in a generalized van der Waals potential.

A semiclassical quantization of the hydrogen atom in a generalized van der Waals potential is carried out using the Kustaanheimo-Stiefel transformation and Birkhoff-Gustavson normal-form procedure, employed by Kuwata, Harada, and Hasegawa [J. Phys. A 23, 3227 (1990)] for the diamagnetic Kepler problem. We derive here the generalized approximate Solov'ev constant of motion. By using appropriate action-angle variables in the normal Hamiltonian, we derive four canonically equivalent action integrals that take an especially simple form for the three classically integrable cases and provide exact quantum numbers. For near-integrable cases the semiclassical spectrum can be generated by integrating the appropriate action integrals numerically.

Dynamics of atomic hydrogen in a generalized van der Waals potential.

The classical dynamics of a hydrogen atom in a generalized van der Waals potential is investigated. In order to carry out the analytical and numerical investigations for a range of parametric values, we removed the singularity of the problem using Levi-Civita regularization and converted the problem into that of two coupled sextic anharmonic oscillators. We identify the integrable choices of the oscillator system using the Painlev\'e singularity analysis, and the associated second integrals of motion are derived using the extended Lie transformations. Numerical investigations are carried out for other nonintegrable regions and we observe chaos-order-chaos type of transition regions when one of the system parameters is varied.