Semiclassical approximation for a momentum dependent one-body potential

Abstract

Recently a semiclassical approximation was applied by Jennings et al. for a system of noninteracting fermions in a local one body potential. This is a way to calculate shell corrections alternative to the Strutinski method. We have generalized this method to a spherical but a momentum dependent potential of the form $V(r)+\frac{1}{2}[{p}^{ 2}W(r)+W(r){p}^{ 2}]$. Explicit expressions are developed for the number of particles and the smoothed sum of single particle energies in terms of the Fermi energy and the one body potential and its first two derivatives. They are calculated for selected values of the parameters and compared with the sum of single particle energies obtained by numerical solution of the Schr\"odinger equation. The difference between the two is essentially the shell correction. The shell correction can be roughly estimated using a simple modification of the isotropic harmonic oscillator model.[NUCLEAR STRUCTURE Shell corrections, momentum dependent potentials.]