The Problem of Smoothness of Bohmian Trajectories

G. Contopoulos, N. Delis, Christos Efthymiopoulos

Tulsi Dass

Partha Ghose, A. S. Majumdar, S Guha +1 more

Karin Berndl, Detlef D�rr, Sheldon Goldstein +2 more

Nancy Makri

A novel quantum mechanical formulation of time correlation functions is derived, based on Bohmian trajectories integrated along a forward−backward time contour. The derived expression involves a smooth integrand amenable to Monte Carlo integration with a readily available position space density. The Bohmian potential governing the motion of the quantum trajectories is weak and well-behaved, facilitating the computation of Bohmian trajectories.

Paulo Alexandre Machado

Bohmian mechanics is an alternative formulation of quantum mechanics that incorporates the familiar and intuitive picture of particles moving along trajectories and yet predicts the same results as the more widely accepted Copenhagen interpretation. In recent years there has been renewed interest in this Bohmian view, in part for the novel approach that it suggests to certain problems, such as decay processes, both from a theoretical and computational stand point. In this thesis we focus on using the concepts introduced by the Bohmian framework as a practical computational tool. I evaluate a number of implementations of the Bohmian method, get a sense of their strengths and weaknesses and attempt to overcome some stability issues that arise. For problems in one-dimension (lD), accurate solutions of the time-dependent Schrodinger equation produce a wave function from which Bohmian trajectories can be computed by integrating along flux lines. For direct integration of the quantum Hamilton-Jacobi equations, the main problems that arise are related to evauating the quantum potential (QP), especially in regions of low probability density. Sufficient accuracy is required to avoid unphysical trajectory crossings. A number of interpolation schemes were investigated, and smoothed splines with special treatment of edge effects gave the best results. For problems in 2D the alternating direction implicit (ADI) method was employed to produce the wave function. Ways of dealing with unphysical reflections from the boundaries of a finite size domain were studied. The use of cellular automata, especially the lattice-Boltzmann method (LBM) were also considered. Here Bohm trajectories would be propagated by following a small set of rules. The main problem identified is that, unless a scheme can be found in which the quantum potential is self-generating from an equation of continuity, the overhead of computing the QP at each time step, is prohibitive.