Wave-packet propagation in momentum space: Calculation of sharp-energy S-matrix elements

Abstract

This paper examines momentum-space methods as a means of implementing a scattering-theoretic, long-time lemma on the extraction of sharp-energy S-matrix elements from a wave-packet description of collisions. In order to concentrate on the momentum space and computational aspects, the collision system studied is that of two particles; each has the mass of a nucleon. The formulation of the problem in momentum space avoids any spreading of the packets and allows for a straightforward analysis, which proceeds as follows. First, a time discretization is introduced, so that a conditionally stable, recursive, time-evolution scheme can be employed. The momentum dependence of the full wave packet is next expressed via an expansion in locally defined interpolating polynomials (here, piecewise quadratics), as in the finite-element method. Once the time evolution has progressed sufficiently, the S-matrix element S0(q) can be extracted from the ratio of the qth momentum components of the full and free wave packets. It is essential here that the numerically propagated free wave packet be used in this ratio, since otherwise numerical errors induced in the full wave packet are not canceled, and S0(q) can become as large as 2 or more. Wave packets with central momenta q0 equal to 1, 2, and 4 fm-1 (energies ranging from about 30 to 500 MeV) have been studied, and the behavior of the wave packets and S0(q) for several time intervals, extraction times, numbers of mesh points, etc., have been explored. In general, results with errors of less than at most a few percent are easily obtainable. © 1992 The American Physical Society.