Quadrature-based, coarse-grained treatment of the coordinate representation free particle real-time evolution operator

Abstract

In this paper we report a quadrature evaluation of the coordinate representation, short-time free particle propagator, 〈R‖exp(−iH0τ)‖R′〉. The result is the elimination of most of the highly oscillatory behavior in this quantity yielding in its stead a much smoother function, strongly peaked at R=R′. We view this as a numerical coarse graining of the propagator which leads to the intuitively reasonable result that for short times τ or large mass, the particle should not have a significant amplitude for R points that are far from R′. This leads to an interesting, and potentially useful, banded structure for 〈R‖exp(−iH0τ)‖R′〉. Calculations have been carried out both for zero and nonzero orbital angular momenta, for which we also give the exact analytic results, and the same behavior is found. The quadrature-coarse graining procedure still appears to retain the important quantum effects as demonstrated by subsequent use of the coarse-grained free propagator to calculate the scattering of an electron by a simple central potential. Results are in quantitative agreement with those obtained by alternative, numerically exact methods. The coarse-grained free propagator is, of course, independent of the potential, and we expect that it can provide a very useful tool for computing real-time dynamics for a variety of systems.