Simulation of the Hard Core by a Velocity Dependence

Recently, Green has shown explicitly that different potentials which fit the same $S$-wave phase shifts from 0 to 250 MeV (lab system) can give quite different binding energies of nuclear matter. In this paper we show, for two particular potentials, a velocity-dependent one vs one with a hard core, how this different behavior is related to the form of the two-particle wave function at small interparticle distances. The equality of the phase shift vs energy for two potentials requires that the two-particle wave function must have not only the same asymptotic form but also the same value of an effective-range integral, which places some constraint on the form of the interior wave function as well. It is shown that the potentials give also nearly the same results to first order in the "separation method," an approximation to the treatment of nuclear matter in which the wave function is assumed to equal the interior wave function at short distances joining on smoothly with the unperturbed wave function at a distance of about 1 F. However, the change in the particle propagator due to the velocity dependence of the nucleon-nucleus potential gives a positive contribution to the energy, which depends on the wave function at short distances. This contribution is much bigger for the hard-core potential than for the velocity-dependent one, since there is more of a short-range correlation effect in the former case.