J. Shertzer

The finite-element method for energy eigenvalues of quantum mechanical systems

The finite‐element method provides a convenient and flexible procedure for the calculation of energy eigenvalues of quantum mechanical systems. The levels of accuracy that can be attained in the method of finite elements are investigated using various approximations. This is illustrated by first considering two classic examples that form a convenient basis for describing the calculational technique: the radial equation for the hydrogen atom for spherically symmetric states and the simple harmonic oscillator problem in one dimension. These two illustrative examples provide guidelines in the calculation of the energy levels of the hydrogen atom in an arbitrary spatially uniform magnetic field, a problem not solvable by analytical means. The results obtained for the 1s 0 and 2s 0 levels are the most accurate reported so far. This application shows that finite‐element analysis can be employed with advantage for obtaining very accurate results for the energy levels and wavefunctions for quantum mechanical systems.

POSITRONIUM IN CROSSED ELECTRIC AND MAGNETIC FIELDS : THE EXISTENCE OF A LONG-LIVED GROUND STATE

It was earlier reported [Phys. Rev. Lett. 78, 199 (1997)] that long-lived excited states of positronium can be formed in crossed electric and magnetic fields at laboratory field strengths. Unlike the lower-lying states that are localized in the magnetically distorted Coulomb well, these long-lived states which can possess a lifetime up to many years are localized in an outer potential well that is formed for certain values of the pseudomomentum and magnetic field. The present work extends the original analysis and studies the dependence of the spectrum as a function of field strength and pseudomomentum over a wide range of parameters. We predict that in the limit of large pseudomomentum, the ground state of a positronium atom in a magnetic field will become delocalized; for strong fields, the binding energy of this state is quite large, resulting in a ground state that is both stable against direct annihilation and against ionization by low frequency background radiation.

Hyperspherical approach to Coulombic three-body systems with different masses

The authors adopted mass-weighted hyperspherical coordinates to study the properties of Coulombic three-body systems where all three particles are different. Using an adiabatic approximation, they applied the finite-element method to the two-dimensional eigenvalue problems at fixed hyperradius. The authors have calculated the adiabatic hyperspherical potential curves, and examined the wavefunctions (in terms of density plots) and the non-adiabatic coupling terms for a number of three-body systems. By fixing the masses of two of the particles, they examined how these properties vary with the mass of the third particle. The existence of stable bound states versus the masses of the systems is also investigated.

Accurate nonrelativistic expectation values for H+2

Abstract Two separate approaches (a perturbation theory which extends the Born—Oppenheimer approximation of molecular physics and finite element analysis of the three body Coulomb problem) are used to solve the nonrelativistic Schrodinger equation for the hydrogen molecular ion ground state. For both approaches, expectation values are calculated for the bond length, the rotational constant, the permanent quadrupole moment, the Fermi contact parameter, and other operators. These results are compared with “nonadiabatic” variational results, where available.

Stabilization of matter–antimatter atoms in crossed electric and magnetic fields

Abstract It is shown that crossed electric and magnetic fields can be used to stabilize particle–antiparticle systems. Due to the occurence of an outer potential well there exists a class of bound long-lived electron–positron states in which the average positron–electron separation is several thousand angstroms. The near zero probability for the overlap suppresses direct annihilation processes. Transition moments between the ground state in the outer well and the Coulomb states are also extremely small resulting in lifetimes up to the order of one year.

Finite element solution of Laplace's equation for ion-atom chambers

The finite element method is used to solve Laplace’s equation for ion-atom chambers. We first consider a simplified model chamber for which an analytical solution can be obtained; the model chamber serves as a test case to verify the accuracy and convergence of the finite element method. We apply the finite element method to an experimental chamber consisting of five equipotential rings in a grounded cylindrical shell. We determine the strength and homogeneity of the electric field in the region of the chamber where the atoms undergo laser excitation into a Rydberg state.

Electronic energy bands and optical nonlinearity of checker-board superlattices

We investigate the structure of the conduction minibands in a two‐dimensional periodic array of rectangular quantum wires. The finite barrier height in such a GaAs/Ga1−xAlxAs heterostructure gives rise to a ‘‘checker‐board’’ pattern of wells and barriers, i.e., a checker‐board superlattice (CBSL). The energy bands are obtained using the finite element method, in the effective mass approximation, with current continuity at interfaces. The free‐carrier‐induced optical nonlinear susceptibility χ(3) due to band nonparabolicity in the CBSL is obtained in both the rectangular directions. Our calculations predict an increase in χ(3) of about 1–2 orders of magnitude over bulk GaAs, for specific ranges of carrier concentrations.

Long-lived states of positronium in crossed electric and magnetic fields☆

The problem of a two-body Coulomb system in crossed electric and magnetic fields has a long history. Unlike the field-free case, the center of mass motion cannot be separated from the internal motion. When center of mass effects are treated correctly, the crossed fields problem and the problemof transverse motion in a pure magnetic field can be treated in a unified manner. For a neutral system, the total pseudomomentum b K 1⁄4 b P ðe=2ÞB r is a conserved quantity and one can carry out a pseudoseparation of the Hamiltonian. The effective potential for the internal motion depends on j, the eigenvalue of the pseudomomentum.When j exceeds a critical value, a local minimum and maximum form on the potential surface, giving rise to a second potential well. This outer well (OW), which is separated from the Coulomb well (CW) by a barrier, can support bound states. The OW states are characterized by a large interparticle separation. With increasing j, the CW becomes narrower and the CW states are pushed into the continuum; in contrast, the OW moves further from the origin and widens, supporting more and more bound states. We have carried out a systematic study of positronium in crossed fields as a function of field strength and pseudomomentum. By a direct numerical solution of the 3D Schr€ odinger equation, we obtained the complete energy spectrum and corresponding wave functions. We also developed accurate approximation techniques for the low field regime. Our results indicate that the bound states of positronium that reside in the OW are stable against direct annihilation due to the near zero probability for positron–electron overlap. Radiative decay from an OW state to a CW state is also suppressed because the OW and CW wave functions are separated spatially by the barrier. Even at laboratory field strengths, it should be possible to create long-lived positronium. This suggests a mechanism for storing energy using antimatter.

Channel coupling theory of molecular structure: Explanation and elimination of unphysical results

Abstract Earlier (LCAO) structure calculations for H 2 + ungerade and H 2 triplet states based on the channel coupling array (CCA) theory of many-particle collisions have yielded unphysical potential energy curves. It is shown herein for H 2 + that this is a result of not requiring the CCA wavefunction components to obey a certain boundary condition. A simple ansatz obeying this condition is constructed and shown to yield a physically correct, approximate ungerade potential energy.

Long-Lived Positronium and AGN Jets

We suggest that stable states of positronium might exist in the jets of active galactic nuclei (AGN). Electrons and positrons are created near the accretion disks of supermassive black holes at the centers of AGN and are accelerated along magnetic field lines while within the Alfven radius. The conditions in this region are ideal for the creation of bound states of positronium which are stable against annihilation. Traveling at relativistic speeds along the jet, the helical magnetic field enables the atoms to survive for great distances.