A solution of the quantum time of arrival problem via mathematical probability theory

TL;DR

This paper develops a mathematical probability theory-based model to solve the quantum time of arrival problem, providing a general solution for a particle impacting an ideal detector and discussing implications for quantum measurement theory.

Contribution

It introduces a novel detector model using probability theory that ensures positive probability flux, and adapts Madelung equations within geometric quantum theory for the quantum time of arrival problem.

Findings

Derived a general solution for the quantum time of arrival for a single particle

Constructed an ideal detector model ensuring positive probability flux

Linked the model to geometric quantum theory for consistency

Abstract

Time of arrival refers to the time a particle takes after emission to impinge upon a suitably idealized detector surface. Within quantum theory, no generally accepted solution exists so far for the corresponding probability distribution of arrival times. In this work we derive a general solution for a single body without spin impacting on a so called ideal detector in the absence of any other forces or obstacles. A solution of the so called screen problem for this case is also given. After discussing the shortcomings of the so called "absorbing boundary condition", which is arguably the natural approach within quantum mechanics, we construct the ideal detector model via mathematical probability theory. This detector model assures that the probability flux through the detector surface is always positive, so that the corresponding distributions can be derived via an approach originally suggested by Daumer, D\"urr, Goldstein, and Zangh\`i. The resulting dynamical model is based on an adaption of the Madelung equations and is, strictly speaking, not compatible with quantum mechanics. Still, it is well-described within geometric quantum theory. Geometric quantum theory is a novel adaption of quantum mechanics, which makes the latter consistent with mathematical probability theory. Implications to the general theory of measurement and avenues for future research are also provided. Future mathematical work should focus on finding an appropriate distributional formulation of the evolution equations and studying the well-posedness of the corresponding Cauchy problem.