Procedure for absorbing time-dependent wave functions at low kinetic energies and large bandwidths

Abstract

We present a contribution to the discussion of how best to minimize time-evolving wave functions near the edge of a space-time grid. Our suggestion builds on the work of several groups and adopts an optical potential with real and imaginary parts separated in space (though not necessarily in time) to bring about an artificial acceleration and subsequent suppression of the wave-function amplitude as it approaches the boundary of discretized space-time. The emphasis of the proposal is on the absorption of incident wave functions characterized by low kinetic energy (long de Broglie wavelength) and/or a broad spread of momenta. The action of the optical potential is quantified by calculations of transmission and reflection as a function of incident wave-function kinetic energy, relative bandwidth, and de Broglie wavelength. Numerical constraints on the maximum by which a wave packet may be accelerated and the time and space over which the accelerating potential may act are discussed within the terms of the split-operator method for wave-function propagation. The perceived advantages of spatially overlapping real and imaginary potentials with respect to the minimization of broad-bandwidth wave functions are investigated. It is found that the addition of a simple, though not totally arbitrary, energy shift applied in space prior to absorption by a negative imaginary potential greatly enhances the absorption of wave packets constructed from multiple momentum components.