R. Thieberger

Finite element method for solving the two-dimensional schroedinger equation

Abstract A finite element package is presented that is able to treat two-dimensional Schroedinger equation problems over a finite region with an arbitrary potential and homogeneous boundary conditions. In order to present the applicability and accuracy of this approach, two cases for which the exact solution is known are solved: (i) a free particle in a spherical box, (ii) a hydrogen atom enclosed in a finite sphere. The results definitely indicate that the finite element method is both accurate and efficient and could serve as a useful tool in various single particle quantum mechanical problems.

Application of the finite-element method to the hydrogen atom in a box in an electric field

Abstract A better understanding of the problems of a pressurized atom and a crystal, both under the influence of a constant electric field, has been achieved through the numerical solution of the two-dimensional Schroedinger equation. The procedure consists of an adaptation of a finite-element package and gives fast and accurate results.

Models for the hydrogen atom confined within crystalline quartz

Simple cylindrical models of a hydrogen atom confined in crystalline quartz are considered. The isotropic and anisotropic components of the hyperfine splitting are evaluated by solving numerically the appropriate two‐dimensional Schrodinger equation, utilizing an adaptation of a finite element package previously described. The results show good agreement with electron paramagnetic resonance data, and given an estimate of d‐orbital admixture in the H‐atom ground state orbital.

Resonance effects in the Bonhoeffer-van der Pol system

Abstract Resonance effects for the Bonhoeffer-van der Pol system are analysed in the range where a limit cycle and a stable focus coexist. As usual, resonance effects are obtained by external periodic forcing at initial points near the focus, which turns into a small limit cycle. For certain inputs, a transition to the regular limit cycle, assisted by resonance, is obtained.

Monte Carlo calculations for pressure dissociation of H2

A Monte Carlo method is proposed for the calculation of dissociation of diatomic homonuclear molecules as a function of temperature and density. The procedure is illustrated for hydrogen in the temperature range up to 15000 K. The essentially quantum mechanical processes of dissociation and binding are replaced here by a classical model system of free atoms and molecules whose numbers in the system may vary according to certain probabilistic rules. A choice must be made of the interaction potentials between particles, which are then replaced by equivalent hard−spheres repulsive interactions. The attractive part of the singlet H−H interaction, which is responsible for the bound state of two atoms, is replaced by an intrinsic ’’binding’’ energy attributed to each molecule present in the system. Results show that dissociation starts at densities well below the packing density of molecules.

Calculation of the pressure dissociation of H2 by the van der Waals one‐fluid theory

A semiclassical model has been proposed earlier to describe the dissociation of molecular hydrogen as a function of temperature and density. The model system consists of free atoms and molecules allowed to interact through repulsive forces only, while the essentially quantum mechanical effect of binding is replaced by artificially assigning each molecule an intrinsic energy −B, where B is the experimental binding energy. In this work the equations are solved in Percus–Yevick (PY) and van der Waals (vdW) one‐fluid approximations, and compared with previously published Monte Carlo results. The vdW theory reproduces very well the Monte Carlo calculations, while PY is found to be inadequate in this case.

Thomas-Fermi equation with non-spherical boundary conditions

Abstract The confined atom Thomas-Fermi equation with non-spherical boundary conditions is considered. A 2-D finite element code for solving the Thomas-Fermi equation with general boundary conditions is demonstrated. Results for both Dirichlet and Neumann boundary conditions for ellipsoids of revolution are presented.

Revised results for the pressure‐induced rotation‐libration transition in molecular solid hydrogen

The pressure at the transition point from free rotation to libration in solid molecular hydrogen isotopes (para‐H2,and ortho‐D2) at zero temperature is revised in accordance with a new equation of state proposed by Ross, Ree, and Young. The agreement with the experimental values is better than for previous calculations.

Phase Locking of the Bonhoeffer-Van Der Pol Model

In the last decade there has been a great deal of interest in theoretical and experimental studies of periodically forced non linear systems. The neural information propagated along an axon belongs to this category (see e. g. papers in Degn et al., 1987). Stimulated responses of action potentials in squid giant axons can be understood in terms of a dissipative structure that behaves as a nonlinear neural oscillator.