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

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.