Coulomb States Research Articles

Screened Coulomb Solutions of the Schrödinger Equation

Numerical solutions of the Schr\"odinger 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 ${V(r)=,{V}_{i}(r)\ensuremath{\equiv}\ensuremath{-}Z{e}^{2}[{r}^{\ensuremath{-}1}\ensuremath{-}{(D+A)}^{\ensuremath{-}1}], 0\ensuremath{\le}r\ensuremath{\le}A}{={V}_{0}(r)\ensuremath{\equiv}\ensuremath{-}Z{e}^{2}[\frac{D}{(D+A)]}{\mathrm{exp}\frac{[\frac{(A\ensuremath{-}r)}{D}]}{r}}, r\ensuremath{\ge}A}$ where $D$ is the screening radius and $A$ is the mean minimum radius of the ion atmosphere. The standard transformations $x=\frac{2Zr}{\ensuremath{\lambda}{a}_{0}}$ and ${E}_{\ensuremath{\lambda}}=\ensuremath{-}\frac{{Z}^{2}\ensuremath{\mu}{e}^{4}}{2{\ensuremath{\hbar}}^{2}{\ensuremath{\lambda}}^{2}}$, where $\ensuremath{\lambda}$ is the CSCP quantum number, yield the well-known form of the Schr\"odinger equation with $\ensuremath{\lambda}$ 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 $(\frac{3{a}_{0}{n}^{2}}{2Z})gD$, 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.

Finite Electronic Partition Function from Screened Coulomb Interactions

A definition of a finite partition function for bound electronic states is presented for a hydrogenic ion, with the associated problems of the fall in intensity of spectral lines and the lowering of the effective ionization potential. The partition function is partly based on numerical solutions of the Schr\"odinger equation (SE) with the complete screened Coulomb potential (CSCP), where the $1s$, $2s$, $2p$, $3s$, $3p$, and $3d$ states are considered. The CSCP is given by ${V(r),={V}_{i}(r)=\ensuremath{-}Z{e}^{2}\left(\frac{1}{r}\ensuremath{-}\frac{1}{D+A}\right), 0<~r<~A}{={V}_{0}(r)=\ensuremath{-}Z{e}^{2}\frac{D}{D+A}\frac{\mathrm{exp}[\frac{(A\ensuremath{-}r)}{D}]}{r}, r>~A}$ where $D$ is the screening radius and $A$ is the mean minimum radius of the ion atmosphere. The standard transformations $x=\frac{2Zr}{\ensuremath{\lambda}{a}_{0}}$ and ${E}_{\ensuremath{\lambda}}=\ensuremath{-}\frac{{Z}^{2}\ensuremath{\mu}{e}^{4}}{2{\ensuremath{\hbar}}^{2}{\ensuremath{\lambda}}^{2}}$, where $\ensuremath{\lambda}$ is the CSCP quantum number, yield the standard form of the SE equation with $\ensuremath{\lambda}$ in place of $n$. The numerical solutions are obtained with a nonlinear method that is both accurate and stable. Since the accurate numerical solutions do not yield an explicit maximum-bound principal quantum number, another property of the screened solutions is used to define a decreasing probability of an electron occupying a particular Coulomb eigenstate. The quantity derived is the relative probability of the screened to the unscreened Coulomb state, given by ${\ensuremath{\Phi}}_{n,l}=\frac{{\ensuremath{\lambda}}^{2l+1}N(\ensuremath{\lambda}, l, d, a)}{{n}^{2l+1}N(n, l)},$ where the $N'\mathrm{s}$ are the normalizations in $x$ space, calculated with variations of $D$ and $A$ only. This relative occupation probability of the bound state is used to modify the Boltzmann factor in the standard expression for the electronic partition function. Correlations with observations are discussed: there is excellent agreement with the fall in intensity of hydrogen lines in the solar atmosphere and good agreement with lines from laboratory hydrogen at 21\ifmmode^\circ\else\textdegree\fi{}K. The effective ionization potential of hydrogen is calculated using the maximum detected level of hydrogen observed in the solar chromosphere. A simple analytical fit to the $\ensuremath{\Phi}$ function and a useful approximate analytical expression for the screened Coulomb partition function are given.

Screened Coulomb Solutions of the Schrödinger Equation

Numerical solutions of the Schr\"odinger 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 ${V(r)=,{V}_{i}(r)\ensuremath{\equiv}\ensuremath{-}Z{e}^{2}[{r}^{\ensuremath{-}1}\ensuremath{-}{(D+A)}^{\ensuremath{-}1}], 0\ensuremath{\le}r\ensuremath{\le}A}{={V}_{0}(r)\ensuremath{\equiv}\ensuremath{-}Z{e}^{2}[\frac{D}{(D+A)]}{\mathrm{exp}\frac{[\frac{(A\ensuremath{-}r)}{D}]}{r}}, r\ensuremath{\ge}A}$ where $D$ is the screening radius and $A$ is the mean minimum radius of the ion atmosphere. The standard transformations $x=\frac{2Zr}{\ensuremath{\lambda}{a}_{0}}$ and ${E}_{\ensuremath{\lambda}}=\ensuremath{-}\frac{{Z}^{2}\ensuremath{\mu}{e}^{4}}{2{\ensuremath{\hbar}}^{2}{\ensuremath{\lambda}}^{2}}$, where $\ensuremath{\lambda}$ is the CSCP quantum number, yield the well-known form of the Schr\"odinger equation with $\ensuremath{\lambda}$ 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 $(\frac{3{a}_{0}{n}^{2}}{2Z})>D$, 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.