Analytic perturbation theory for screened Coulomb potentials: Nonrelativistic case

Abstract

We have developed an analytic perturbation theory for screened Coulomb radial wave functions, based on an expansion of the potential of the form $V(r)=\ensuremath{-}(\frac{a}{r})[1+{V}_{1}\ensuremath{\lambda}r+{V}_{2}{(\ensuremath{\lambda}r)}^{2}+{V}_{3}{(\ensuremath{\lambda}r)}^{3}+\ensuremath{\cdots}]$, where $\ensuremath{\lambda}\ensuremath{\simeq}\ensuremath{\alpha}{Z}^{\frac{1}{3}}$ is a small parameter characterizing the screening. The coefficients ${V}_{k}$ may be chosen such that the above form converges rapidly and gives a good approximation to realistic numerical potentials, such as those of Herman and Skillman, in the interior of the atom. The screened radial wave functions are obtained as a series in $\ensuremath{\lambda}$ with simple analytic coefficients, owing to the special symmetries of the unperturbed Coulomb problem. Both bound and continuum shapes are correctly treated in the region $\ensuremath{\lambda}rl1$. For inner bound states, this includes all of the region where the wave function is large. Similarly, high-energy continuum wave functions will have completed several oscillations in this interval so that by $r\ensuremath{\simeq}{\ensuremath{\lambda}}^{\ensuremath{-}1}$ one has reached the asymptotic region. Consequently, expressions for bound-state normalizations can be given as series in $\ensuremath{\lambda}$, which are accurate, in general, for the $K$ shell and for other low-lying levels of high-$Z$ elements. The continuum normalizations which we obtain are valid for energies on the order of the $K$-shell binding energy above threshold. Bound-state energies and continuum phase shifts are also obtained in these circumstances.