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arXiv:quant-ph/0202098 · Journal of Physics A Mathematical and General

Bohmian trajectories and Klein's paradox

Gebhard Grübl, Raimund Moser, Klaus Rheinberger

Published Mar 26, 2001• 9 citations
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Abstract

We compute the Bohmian trajectories of the incoming scattering plane waves for Klein's potential step in explicit form. For finite norm incoming scattering solutions we derive their asymptotic space-time localization and we compute some Bohmian trajectories numerically. The paradox, which appears in the traditional treatments of the problem based on the outgoing scattering asymptotics, is absent.

DOI
10.1088/0305-4470/34/13/307

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We compute the Bohmian trajectories of the incoming scattering plane waves for Klein's potential step in explicit form. For finite norm incoming scattering solutions we derive their asymptotic space-time localization and we compute some Bohmian trajectories numerically. The paradox, which appears in the traditional treatments of the problem based on the outgoing scattering asymptotics, is absent.
We study the asymptotic behaviour of Bohmian trajectories in a scattering situation with short-range potential V and for wavefunctions with a scattering and a bound part. It is shown that the set of possible trajectories splits into trajectories whose long-time behaviour is governed by the scattering part of the wavefunction (scattering trajectories) and trajectories whose long time behaviour is governed by the bound part of the wavefunction (bound trajectories). Furthermore, the scattering trajectories behave like trajectories in classical mechanics in the limit t → ∞. As an intermediate step, we show that the asymptotic velocity exists almost surely and is randomly distributed with the density , where is the outgoing asymptote of the scattering part of the wavefunction.
Beginning with the conventional asymptotic description of quantum scattering, we propose a simple wave function that is valid at all distances and compute the corresponding Bohmian trajectories. This 'simulated experiment' allows a concrete visualization of the process throughout the scattering interaction. Not only is this of pedagogic value at the undergraduate level, it also illustrates several important themes in time-dependent scattering theory, such as the importance of wave packets, the role of cross sections in describing measurement and the wave and particle pictures of quantum mechanics.
In this paper the relations between the asymptotic velocity operators of a quantum system and the asymptotic velocities of the associated Bohmian trajectories are studied. In particular it is proved that, under suitable conditions of asymptotic regularity, the probability distribution of the asymptotic velocities of the Bohmian trajectories is equal to the one derived from the asymptotic velocity operators of the associated quantum system. It is also shown that in the relativistic case the distribution of the asymptotic velocities of the Bohmian trajectories is covariant, or equivalently, it does not depend on a preferred foliation (it is well known that this is not the case for the structure of the Bohmian trajectories or for their spatial distribution at a finite time). This result allows us to develop a covariant formulation of relativistic Bohmian mechanics; such a formulation is proposed here merely as a mathematical possibility, while its empirical adequacy will be discussed elsewhere.
In the last few years the hydrodynamic formulation of quantum mechanics, equivalent to the Bohmian equations of motion, has been used to obtain numerical solutions of the Schrodinger equation. Problems, however, have been experienced near wave function nodes (or low probability regions). Here we attempt to compute wave functions and Bohmian trajectories for the interference of one particle or of two identical particles. It turns out that the large number of nodes (i.e. interference minima) makes the hydrodynamic equations impractical, whereas a more straightforward solution of the Schrodinger equation gives very good results.
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.