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Quantum First Passage Time Problem-A Bohmian Perspective

Siddhant Das

TL;DR

The paper addresses the quantum arrival/first passage time problem by adopting Bohmian mechanics, where particles follow deterministic trajectories guided by the wave function. For a freely evolving Gaussian state, the authors derive the Bohmian radial motion and compute the first passage time distribution $\Pi(\tau)$ via the initial-position measure, equivalently described by the reciprocal-time distribution $\Lambda(\nu)$. They obtain an explicit expression for $\Lambda(\nu)$ consisting of a continuous part and a singular $\delta(\nu)$-term, and they analyze its properties, including asymptotic behaviors as the detector radius $d$ varies, the heavy-tailed nature of the distribution, and the mean reciprocal time $\langle\nu\rangle$ in terms of confluent hypergeometric functions. The results highlight how arrival-time statistics in quantum mechanics can be framed and potentially tested within a trajectory-based formalism, offering concrete predictions distinct from orthodox interpretations and suggesting experimental avenues with Gaussian initial states prepared via controlled potentials.

Abstract

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 $Π(τ)$. Implications of our results are also discussed.

Quantum First Passage Time Problem-A Bohmian Perspective

TL;DR

The paper addresses the quantum arrival/first passage time problem by adopting Bohmian mechanics, where particles follow deterministic trajectories guided by the wave function. For a freely evolving Gaussian state, the authors derive the Bohmian radial motion and compute the first passage time distribution via the initial-position measure, equivalently described by the reciprocal-time distribution . They obtain an explicit expression for consisting of a continuous part and a singular -term, and they analyze its properties, including asymptotic behaviors as the detector radius varies, the heavy-tailed nature of the distribution, and the mean reciprocal time in terms of confluent hypergeometric functions. The results highlight how arrival-time statistics in quantum mechanics can be framed and potentially tested within a trajectory-based formalism, offering concrete predictions distinct from orthodox interpretations and suggesting experimental avenues with Gaussian initial states prepared via controlled potentials.

Abstract

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 . Implications of our results are also discussed.

Paper Structure

This paper contains 4 sections, 35 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Elements of Bohmian mechanics
  3. Formulation
  4. Conclusion

Figures (2)

  • Figure 1: Graphs of the continuous part of $\Lambda(\text{\textnu})$ vs. $\text{\textnu}$ for different values of $d$. Inset: log-linear plot of the same curves.
  • Figure 2: Graph of mean reciprocal passage time $\expval{\text{\textnu}}$ vs. detector radius $d$ (cf Eq. \ref{['mean']}). Inset: Graph of $\alpha$ vs. detector radius $d$ (cf Eq. \ref{['alpha']}).