High-Order Perturbation Theory for the Bound States of an Electron in a Screened Coulomb Potential

Abstract

The solution to the nonrelativistic Schr\odinger equation for a bound electron in an attractive screened Coulomb potential is investigated using the large-$Z$ ($Z$ is nuclear charge) asymptotic expansion theory. Both the basic asymptotic and perturbation solutions are found. The problem of finding the $k\mathrm{th}$ order perturbation wave function and energy for any state is reduced to solving, recursively, a set of $k$ linear algebraic equations in $k$ unknowns. The asymptotic expansions for the energy and wave functions are presented to the tenth order in perturbation theory for the $1S$ state and to fifth order for the general $n$, $l=n\ensuremath{-}1$ quantum state. Results for the $2S$ states are also given. Comparison of the perturbation-theory results with those of numerical integrations for the energy show excellent agreement. It is shown that a finite screening radius gives rise to a finite number of bound states, a result which contradicts some recently published work. Application of the screened Coulomb potential model to intensity cutoffs in the spectra of solar and laboratory hydrogen plasmas is discussed.