Modified delta-function potential for hyperfine interactions

Abstract

A modification of the Fermi contact interaction is proposed in which the nuclear moment is represented by a uniformly magnetized spherical shell of radius ${r}_{0}$. In effect, the delta function $\ensuremath{\delta}(r)$ in the Fermi Hamiltonian is replaced by $\ensuremath{\delta}(r\ensuremath{-}{r}_{0})$. The Schr\"odinger equation for a hydrogenlike system thus perturbed is exactly solvable in terms of the Coulomb Green's function. Negative energy eigenvalues have the form ${E}_{\ensuremath{\nu}}=\ensuremath{-}\frac{{Z}^{2}}{2{\ensuremath{\nu}}^{2}}$, with $\ensuremath{\nu}$ a nonintegral quantum number. An asymptotic formula is derived for the quantum defect $\ensuremath{\delta}=\ensuremath{\nu}\ensuremath{-}n$. The $l=0$ eigenfunctions are multiples of the Whittaker functions: ${M}_{\ensuremath{\nu},}^{\frac{1}{2}}(\frac{2Zr}{\ensuremath{\nu}})$ for $r<{r}_{0}$ and ${W}_{\ensuremath{\nu},}^{\frac{1}{2}}(\frac{2Zr}{\ensuremath{\nu}})$ for $r>{r}_{0}$. Explicit forms are given by expansion of the Whittaker functions to first order in quantum defect. In the limit ${r}_{0}\ensuremath{\rightarrow}0$ results pertaining to the original Fermi Hamiltonian are approached. It is shown that a repulsive delta function maintains the unperturbed Coulomb energy while an attractive delta function pulls all bound state energies to $\ensuremath{-}\ensuremath{\infty}$. Perturbation expansions are discussed and comparisons made with earlier calculations. It is shown that second-order and higher perturbation energies diverge as ${r}_{0}\ensuremath{\rightarrow}0$.