A general analysis of the two-body Dirac equation is presented for the case of equal masses interacting via a static Coulomb potential. Radial equations are derived and their analytical structure is discussed. Standard analytical and perturbative methods have failed to provide solutions to the radial equations due to the presence of the singularity on the negative radial axis at roughly the distance of the classical electron radius. The exact radial equations are solved using finite-element analysis, and the low-lying bound states are obtained to an accuracy of one part in 10' . The effect of the singularity is clearly seen in the structure of the finite-element radial components.

The finite-element method has been used to obtain numerical solutions to the Schrodinger equation for the ground state of the helium atom. In contrast to the globally defined trial functions of the standard variational approach, the finite-element algorithm employs locally defined interpolation functions to approximate the unknown wave function. The calculation reported herein used a three-dimensional grid containing nine nodal points along the radial coordinates of the two electrons and four nodal points along the direction corresponding to the cosine of the interelectronic angle.This produced an energy of -2.9032 a.u. , which lies 0.017% above the Frankowski-Pekeris value. The values of (r"), for n = -2, -1, 1, and 2, are closer to those of Frankowski and Pekeris than from all of the variational calculations with the exception of the calculation performed by Weiss, whose energy and ( r" ) values are comparable to those of the finite-element computation.

It was earlier reported ͓Phys. Rev. Lett. 78, 199 ͑1997͔͒ 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.

The Rapid Communications section is intended for the accelerated publication of important new results. Since manuscripts submitted to this section are given priority treatment both in the editorial once and in production, authors should explain in their submittal letter why the work justifies this special handling AR. apid Communication should be no longer than 4 printed pages and must be accompanied by an abstract. Page proofs are sent to authors.A direct numerical solution of the Schrodinger equation for quantum scattering problems is presented. The wave function for each partial wave is expanded in coupled spherical harmonics and the corresponding radial functions are expanded in a local basis set using finite-element analysis, with the appropriate scattering boundary conditions. The method is shown to give very accurate results for elastic phase shifts (S, P, D, and E) and resonance positions for electron-hydrogen scattering.PACS number(s): 34.8G.Bm Various formalisms have been developed over the years