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 $\delta \mathrm{}\left(r\right)\mathrm{}$ in the Fermi Hamiltonian is replaced by $\delta (r-r_0)$. The Schrödinger equation for a hydrogenlike system thus perturbed is exactly solvable in terms of the Coulomb Green's function. Negative energy eigenvalues have the form $E_{\nu }=-\frac{Z^2}{2{\nu }^2}$, with $\nu$ a nonintegral quantum number. An asymptotic formula is derived for the quantum defect $\delta =\nu -n$. The $l=0\mathrm{}$ eigenfunctions are multiples of the Whittaker functions: $M_{\nu ,}^{\frac{1}{2}}\left(\frac{2Zr}{\nu }\right)$ for $r<\mathrm{}{r}_0$ and $W_{\nu ,}^{\frac{1}{2}}\left(\frac{2Zr}{\nu }\right)$ for $r>\mathrm{}{r}_0$. Explicit forms are given by expansion of the Whittaker functions to first order in quantum defect. In the limit $r_0\to 0$ 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 $-\infty$. Perturbation expansions are discussed and comparisons made with earlier calculations. It is shown that second-order and higher perturbation energies diverge as $r_0\to 0$.