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arXiv:quant-ph/0101132

On Bohmian trajectories in two-particle interference devices

‎Louis Marchildon

Published Jan 29, 2001• 7 citations
arXivHTMLPDF

Abstract

Claims have been made that, in two-particle interference experiments involving bosons, Bohmian trajectories may entail observable consequences incompatible with standard quantum mechanics. By general arguments and by an examination of specific instances, we show that this is not the case.
DOI
10.48550/arxiv.quant-ph/0101132

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A number of assertions have recently been made that in two-particle interference devices, Bohmian trajectories may not reproduce exactly all statistical predictions of quantum mechanics. Specifically, let two identical bosons go through identical slits arranged symmetrically with respect to a plane, with wave functions transforming into each other under the symmetry operation. Then it is claimed that pairs of particles following Bohmian trajectories and registered on a screen opposite the slits may have observable spatial distribution properties not predicted by quantum mechanics. We point out that alleged discrepancies in fact originate from a specific sampling assumption on Bohmian particles that violates their general probability distribution.
Abstract From “surreal” trajectories to which-way measurements, Basil Hiley had a lesson: claims of falsifying the Bohmian model do not withstand scrutiny provided the model is applied correctly. In this work we compute de Broglie-Bohm trajectories for particles tunneling in coupled waveguides relevant to a recent experiment having claimed to challenge the Bohmian model. We show that the Bohmian model – correctly applied – gives results identical to the standard quantum approach, first by working out a simple one-dimensional model, and then by computing Bohmian trajectories for the full two-dimensional problem representing a quantum particle propagating inside coupled waveguides. We further recall the contextual nature of the Bohmian trajectories whereby the trajectories of a closed system differ from the ones observed when an interaction with a measurement apparatus takes places.
Standard quantum mechanics famously disallows to simultaneously measure particle position and momentum with arbitrary precision. In contrast, Bohmian mechanics, a hidden-variable interpretation, at the same time ascribes reality to the positions and momenta of quantum particles at the cost of a nonlocal ontology [1]. It thus allows to conceive of definite particle trajectories, while being fully compatible with the standard theory in all empirical predictions. To record such trajectories [2] we consider two optical settings using weak measurements in order to investigate different features of particle propagation in the Bohmian picture.
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.
We propose an experimental time-of-flight test of the photon "trajectory" predictions of the Bohmian version of quantum mechanics. The photon trajectories in free space, as predicted by Bohmian mechanics, deviate from expected straight-line paths, bending and turning corners in the presence of quantum interference. We propose an experiment using two such corner-turns that should, in principle, allow these hypothetical Bohmian photons to arrive at a timing detector earlier than conventional photons traveling on the straight-line paths predicted by standard quantum mechanics.