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arXiv:1901.08672 · Physical Review A

Exotic Bohmian arrival times of spin-1/2 particles: An analytical treatment

Siddhant Das, Markus Nöth, Detlef Dürr

Published May 28, 2019• 32 citations
arXivHTMLPDF

Abstract

It is well known that orthodox quantum mechanics does not make unambiguous predictions for the statistics in arrival time (or time-of-flight) experiments. Bohmian mechanics (or de Broglie--Bohm theory) offers a distinct conceptual advantage in this regard, owing to the well-defined concepts of point particles and trajectories embedded in this theory. We revisit a recently proposed experiment [S. Das and D. D\"urr, Sci. Rep. 9, 2242 (2019)], the numerical analysis of which revealed a striking spin dependence in the (Bohmian) time-of-arrival distributions of a spin-1/2 particle. We present here a mathematically tractable variant of the same experiment, where the predicted effects can be established rigorously. We also obtain some results that can be compared with experiment.

DOI
10.1103/physreva.99.052124

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It is well known that orthodox quantum mechanics does not make unambiguous predictions for the statistics in arrival time (or time-of-flight) experiments. Bohmian mechanics (or de Broglie--Bohm theory) offers a distinct conceptual advantage in this regard, owing to the well-defined concepts of point particles and trajectories embedded in this theory. We revisit a recently proposed experiment [S. Das and D. D"urr, Sci. Rep. 9, 2242 (2019)], the numerical analysis of which revealed a striking spin dependence in the (Bohmian) time-of-arrival distributions of a spin-1/2 particle. We present here a mathematically tractable variant of the same experiment, where the predicted effects can be established rigorously. We also obtain some results that can be compared with experiment.
Abstract How to compute the probability distribution of a detection time, i.e., of the time which a detector registers as the arrival time of a quantum particle, is a long-debated problem. In this regard, Bohmian mechanics provides in a straightforward way the distribution of the time at which the particle actually does arrive at a given surface in 3-space in the absence of detectors. However, as we discuss here, since the presence of detectors can change the evolution of the wave function and thus the particle trajectories, it cannot be taken for granted that the arrival time of the Bohmian trajectories in the absence of detectors agrees with the one in the presence of detectors, and even less with the detection time. In particular, we explain why certain distributions that Das and Dürr (Sci. Rep. 9: 2242, 2019) presented as the distribution of the detection time in a case with spin, based on assuming that all three times mentioned coincide, are actually not what Bohmian mechanics predicts.
The prediction of arrival time or first passage time statistics of a quantum particle is an open problem, which challenges the foundations of quantum theory. One of the most promising and insightful approaches to this problem stems from the de Broglie-Bohm pilot-wave theory (a.k.a Bohmian mechanics). Applying the fundamental postulates of this theory, we analyze a simplified first passage time experiment and derive the empirical passage time distribution Π(τ)Π(τ)Π(τ). Implications of our results are also discussed.
Motivated by a recent prediction [Commun. Phys. 6, 195 (2023).] that time-of-flight experiments with ultracold atoms could test different interpretations of quantum mechanics, this work investigates the arrival times predicted by the stochastic interpretation, whereby quantum particles follow definite but nondeterministic and nondifferentiable trajectories. The distribution of arrival times is obtained from a Fokker-Planck equation and confirmed by direct simulation of trajectories. It is found to be, in general, different from the distribution predicted by the Bohmian interpretation, in which quantum particles follow definite deterministic and differentiable trajectories. This result suggests that trajectory-based interpretations of quantum mechanics could be experimentally discriminated.
Relativistic extensions of de Broglie-Bohm theory postulate a preferred foliation of space-time, an additional structure essential for defining simultaneous configurations on Minkowski space-time, but conventionally believed to be empirically undetectable at quantum equilibrium. In this paper, we outline an experimental protocol for empirically detecting the preferred foliation, which is assumed to be flat for simplicity. Building on the arrival-time distributions for spin-1/2 particles predicted by Das and Dürr, we show that in an EPRB-type experiment with spacelike-separated spin and arrival-time measurements, the observed arrival-time statistics will depend crucially on the temporal order of these measurements relative to the preferred foliation of space-time. This dependence offers a potential experimental signature of the preferred foliation postulated by relativistic Bohmian models. Moreover, it implies the possibility of superluminal signaling.
The de Broglie–Bohm interpretation of quantum mechanics with and without spin-dependence is used to determine electron trajectories, arrival-time distribution of electrons and the mean arrival time of spin-1/2 particles (including electrons) in the presence of a uniform field and a barrier separately. The difference for the mean arrival times, which are calculated with different guidance equations, is examined versus mass of the arriving particle and versus group velocity of the arriving electron wave packet and also versus the width of the barrier. Numerical calculations show that these differences are of the order of 10 −18 –10 −17 s. Another feature of the modified guidance law is that Bohmian trajectories cross each other, but this does not contradict the single-valuedness of the wavefunction.
The arrival time statistics of spin-1/2 particles governed by Pauli's equation, and defined by their Bohmian trajectories, show unexpected and very well articulated features. Comparison with other proposed statistics of arrival times that arise from either the usual (convective) quantum flux or from semiclassical considerations suggest testing the notable deviations in an arrival time experiment, thereby probing the predictive power of Bohmian trajectories. The suggested experiment, including the preparation of the wave functions, could be done with present-day experimental technology.