Bohmian Trajectories of the Time-oscillating Schr\"odinger Equations

TL;DR

This paper investigates Bohmian trajectories for Schr"odinger equations with rapidly oscillating potentials, proving convergence to effective trajectories, which aids efficient simulation in oscillating fields.

Contribution

It establishes the convergence of Bohmian trajectories in oscillating potentials and links them to effective Schr"odinger equations, enhancing simulation methods.

Findings

Bohmian trajectories converge locally in measure.

Limit trajectories match those of the effective Schr"odinger equation.

Results facilitate efficient simulation in oscillating potential fields.

Abstract

Bohmian mechanics is a non-relativistic quantum theory based on a particle approach. In this paper we study the Schr\"odinger equation with rapidly oscillating potential and the associated Bohmian trajectory. We prove that the corresponding Bohmian trajectory converges locally in measure, and the limit coincides with the Bohmian trajectory for the effective Schr\"{o}dinger equation on a finite time interval. This is beneficial for the efficient simulation of the Bohmian trajectories in oscillating potential fields.