Bohmian trajectories for the half-line barrier

TL;DR

This paper investigates Bohmian trajectories for a particle encountering a half-line barrier with different boundary conditions, revealing how these conditions influence particle paths and diffraction effects.

Contribution

It provides a numerical analysis of Bohmian trajectories in a half-line barrier setup, highlighting differences caused by Dirichlet and Neumann boundary conditions.

Findings

Dirichlet boundary conditions cause stronger particle repulsion.

Trajectories differ qualitatively between boundary conditions.

Comparison with oil drop trajectories illustrates physical intuition.

Abstract

Bohmian trajectories are considered for a particle that is free (i.e. the potential energy is zero), except for a half-line barrier. On the barrier, both Dirichlet and Neumann boundary conditions are considered. The half-line barrier yields one of the simplest cases of diffraction. Using the exact time-dependent propagator found by Schulman, the trajectories are computed numerically for different initial Gaussian wave packets. In particular, it is found that different boundary conditions may lead to qualitatively different sets of trajectories. In the Dirichlet case, the particles tend to be more strongly repelled. The case of an incoming plane wave is also considered. The corresponding Bohmian trajectories are compared with the trajectories of an oil drop hopping on the surface of a vibrating bath.