Time-of-arrival distributions for continuous quantum systems and application to quantum backflow

Abstract

Using standard results from statistics, we show that for any continuous quantum system (Gaussian or otherwise) and any observable $\widehat{A}$ (position or otherwise), the distribution $\pi_{a}\left(t\right)$ of time measurement at a fixed state $a$ can be inferred from the distribution $\rho_{t}\left( a\right)$ of a state measurement at a fixed time $t$ via the transformation $\pi_{a}(t) \propto \left\vert \frac{\partial }{\partial t} \int_{-\infty }^a \rho_t(u) du \right\vert$. This finding suggests that the answer to the long-lasting time-of-arrival problem is in fact secretly hidden within the Born rule, and therefore does not require the introduction of a time operator or a commitment to a specific (e.g., Bohmian) ontology. The generality and versatility of the result are illustrated by applications to the time-of-arrival at a given location for a free particle in a superposed state and to the time required to reach a given velocity for a free-falling quantum particle. Our approach also offers a potentially promising new avenue toward the design of an experimental protocol for the yet-to-be-performed observation of the phenomenon of quantum backflow.