Screened Coulomb Solutions of the Schrödinger Equation

Abstract

Numerical solutions of the Schrödinger equation with the complete screened Coulomb potential (CSCP) are given for $1s$, $2s$, $2p$, $3s$, $3p$, and $3d$ states. The CSCP used is given by $\left\{V\right(r)=,V_i\mathrm{}(r)\mathrm{}\equiv -Z{e}^2[r^{-1\mathrm{}}-{\mathrm{}(D+A)\mathrm{}}^{-1\mathrm{}}], 0\le r\le A\}\{=V_0\mathrm{}(r)\mathrm{}\equiv -Z{e}^2[\frac{D}{\mathrm{}(D+A)]}\left\{\exp \frac{\left[\frac{\mathrm{}(A-r)\mathrm{}}{D}\right]}{r}\right\}, r\ge A\}$ where $D$ is the screening radius and $A$ is the mean minimum radius of the ion atmosphere. The standard transformations $x=\frac{2Zr}{\lambda a_0}$ and $E_{\lambda }=-\frac{Z^2\mu e^4}{2{\hslash }^2{\lambda }^2}$, where $\lambda$ is the CSCP quantum number, yield the well-known form of the Schrödinger equation with $\lambda$ in place of $n$. The numerical solutions are obtained with a nonlinear method that is both accurate and stable. The resulting quantum numbers can be accurately described by simple analytic fits for a wide range of interesting values of $D$. The problem of the number of screened Coulomb states is resolved: the CSCP yields as many states as the Coulomb potential. However, with the CSCP, for states with $\left(\frac{3a_0{n}^2}{2Z}\right)>D$, the separations of the levels are less than the corresponding Coulomb levels, i.e., the density of states near the continuum increases. Removal of $l$-degeneracy, the question of a maximum-bound principal quantum number, and integer quantization of the ground-state quantum numbers are also discussed.