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 Â (position or otherwise), the distribution 𝜋_𝑎⁡(𝑡) of time measurement at a fixed state 𝑎 can be inferred from the distribution 𝜌_𝑡⁡(𝑎) of a state measurement at a fixed time 𝑡 via the transformation 𝜋_𝑎⁡(𝑡)∝ | ^𝜕⁄_𝜕𝑡 ⁢∫^𝑎_−∞𝜌_𝑡⁡(𝑢)𝑑𝑢 | . 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.