Energies and wave functions for a soft-core Coulomb potential

For the family of model soft-core Coulomb potentials represented by $V\left(r\right)=-\left[Z/{\left(r^q+{\beta }^q\right)}^{1/q}\right]$, 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 $\left[\left(\nu ,\ell \right),\left({\nu }^{\prime },{\ell }^{\prime }\right)\right]$ is given by the conditions ${\nu }^{\prime }\ge \left(\nu +1\right)$ and ${\ell }^{\prime }\ge \left(\ell +3\right)$. The significance of these results for the interaction of intense laser field with an atom is pointed out. Differences in the observed level-crossing effects between the soft-core potentials and the hydrogen atom confined inside an impenetrable sphere are discussed.

Figure 1

(Color online) The family of potentials $V\left(r\right)=-1/{\left(r^q+1\right)}^q$ for $1\le q<\infty$ (in a.u.). The graphs are nonintersecting for $r>0$, and are ordered, decreasing as $q$ increases.

Figure 2

Un-normalized wave functions (32) for the Schrödinger equation, Eq. (2), with soft-core Coulomb potential, $q=1$, where $a=\frac{1}{l+3}$ and $\beta$ given by Eq. (31) for different values of $l$ (in a.u.).

Figure 3

Un-normalized wave functions (32) for the Schrödinger equation, Eq. (2), with soft-core Coulomb potential, $q=1$, where $a=\frac{1}{l+3}$ and $\beta$ given by Eq. (33) for different values of $l$ (in a.u.).

Figure 4

(Color online) Crossings of energy levels in $V_1\left(r\right)=-1/\left(r+\beta \right)$ for the $6s$ level with $7f$, $7g$, and $7i$ levels as a function of $\beta$ (in a.u.). The energy-level ordering over the range $\beta =20–40$ is changed relative to that corresponding to lower $\beta$ values.

Figure 5

(Color online) Crossings of energy levels in $V_2\left(r\right)=-1/{\left(r^2+{\beta }^2\right)}^{1/2}$ for pair of levels $\left(4p-5g\right)$, $\left(5p-6g\right)$, $\left(6p-7g\right)$ as a function of $\beta$ (in a.u.). The energy-level ordering over the range $\beta =20–40$ undergoes changes as a consequence of crossovers.