Asymptotic Behavior of Bohmian Trajectories in Scattering Situations

TL;DR

This paper analyzes the long-term behavior of Bohmian trajectories in quantum scattering, showing a split into bound and scattering trajectories with scattering ones resembling classical mechanics, and establishes the distribution of asymptotic velocities.

Contribution

It demonstrates the asymptotic split of Bohmian trajectories into bound and scattering types and characterizes the distribution of their asymptotic velocities.

Findings

Trajectories split into bound and scattering types.

Scattering trajectories behave classically at long times.

Asymptotic velocity distribution matches the outgoing wave function's spectral measure.

Abstract

We study the asymptotic behavior of Bohmian trajectories in a scattering situation with short range potential and for wave functions with a scattering and a bound part. It is shown that the set of possible trajectories splits into trajectories whose long time behavior is governed by the scattering part of the wave function (scattering trajectories) and trajectories whose long time behavior is governed by the bound part of the wave function (bound trajectories). Furthermore the scattering trajectories behave like trajectories in classical mechanics in the long time limit. As an intermediate step we show that the asymptotic velocity v_{infty}:=lim_{t to infty}Q/t exists almost surely and is randomly distributed with the density |hat{Psi}^{out}|^2, where Psi^{out} is the outgoing asymptote of the scattering part of the wave function.