Interference in Bohmian Mechanics with Complex Action

TL;DR

This paper introduces a complex action formulation of Bohmian mechanics that improves numerical stability near wavefunction nodes by allowing trajectory crossing, thus addressing a key computational challenge.

Contribution

The authors develop a complex action version of Bohmian mechanics that enables trajectory crossing and better handles wavefunction nodes, improving numerical methods.

Findings

Complex Bohmian mechanics reduces quantum force magnitude.

Trajectory crossing allows modeling of nodes as multiple trajectories.

Method successfully applied to Eckart barrier reflection amplitude.

Abstract

In recent years, intensive effort has gone into developing numerical tools for exact quantum mechanical calculations that are based on Bohmian mechanics. As part of this effort we have recently developed as alternative formulation of Bohmian mechanics in which the quantum action, S, is taken to be complex [JCP {125}, 231103 (2006)]. In the alternative formulation there is a significant reduction in the magnitude of the quantum force as compared with the conventional Bohmian formulation, at the price of propagating complex trajectories. In this paper we show that Bohmian mechanics with complex action is able to overcome the main computational limitation of conventional Bohmian methods -- the propagation of wavefunctions once nodes set in. In the vicinity of nodes, the quantum force in conventional Bohmian formulations exhibits rapid oscillations that pose severe difficulties for existing numerical schemes. We show that within complex Bohmian mechanics, multiple complex initial conditions can lead to the same real final position, allowing for the description of nodes as a sum of the contribution from two or more crossing trajectories. The idea is illustrated on the reflection amplitude from a one-dimensional Eckart barrier. We believe that trajectory crossing, although in contradiction to the conventional Bohmian trajectory interpretation, provides an important new tool for dealing with the nodal problem in Bohmian methods.