Energies and wave functions for a soft-core Coulomb potential

TL;DR

This paper analytically and numerically investigates the properties of a family of soft-core Coulomb potentials, deriving eigenvalues, wave functions, and density concavity, with implications for laser-atom interactions.

Contribution

It provides new analytical solutions for specific cases, applies the potential envelope method for estimates, and proves density concavity for a broad class of potentials.

Findings

Eigenvalues and wave functions are derived analytically for q=1.

Monotonic behavior of potentials and eigenvalues with respect to parameters is shown.

Conjecture on state crossing conditions based on numerical analysis.

Abstract

For the family of model soft Coulomb potentials represented by V(r) = -\frac{Z}{(r^q+\beta^q)^{\frac{1}{q}}}, with the parameters Z>0, \beta>0, q \ge 1, it is shown analytically that the potentials and eigenvalues, E_{\nu\ell}, are monotonic in each parameter. The potential envelope method is applied to obtain approximate analytic estimates in terms of the known exact spectra for pure power potentials. For the case q =1, the Asymptotic Iteration Method is used to find exact analytic results for the eigenvalues E_{\nu\ell} and corresponding wave functions, expressed in terms of Z and \beta. A proof is presented establishing the general concavity of the scaled electron density near the nucleus resulting from the truncated potentials for all q. Based on an analysis of extensive numerical calculations, it is conjectured that the crossing between the pair of states [(\nu,\ell),(\nu',\ell')], is given by the condition \nu'\geq (\nu+1) and \ell' \geq (\ell+3). The significance of these results for the interaction of an intense laser field with an atom is pointed out. Differences in the observed level-crossing effects between the soft potentials and the hydrogen atom confined inside an impenetrable sphere are discussed.