Quantum threshold reflection is not a consequence of the badlands region of the potential

Abstract

Quantum threshold reflection is a well known quantum phenomenon which prescribes that at threshold, except for special circumstances, a quantum particle scattering from any potential, even if attractive at long range, will be reflected with unit probability. In the past, this property has been associated with the so-called badlands region of the potential, where the semiclassical description of the scattering fails due to a rapid spatial variation of the deBroglie wavelength. This badlands region occurs far from the strong interaction region of the potential and has therefore been used to "explain" the quantum reflection phenomenon. In this paper, we show that the badlands region of the interaction potential is immaterial. The extremely long wavelength of the scattered particle at threshold is much longer than the spatial extension of the badlands region which therefore does not affect the scattering. For this purpose, we review the general proof for the existence of quantum threshold reflection to stress that it is only a consequence of continuity and boundary conditions. The nonlocal character of the scattering implies that the whole interaction potential is involved in the phenomenon. We then provide a detailed numerical study of the threshold scattering of a particle by a Morse potential especially in the time domain. We compare exact quantum computations with incoherent results obtained from a classical Wigner approximation. This study shows that close to threshold the time dependent amplitude of the scattered particle is negligible in the badlands region and that the mean flight time of the particle is not shortened due to a local reflection from the badlands region. This study should serve to definitely rule out the badlands region as a qualitative guide to the properties of quantum threshold reflection.