Time-of-arrival distributions for continuous quantum systems

• URL: http://arxiv.org/abs/2405.02018v1
• Date: Fri, 3 May 2024 11:33:52 GMT
• Title: Time-of-arrival distributions for continuous quantum systems
• Authors: Mathieu Beau, Maximilien Barbier, Rafael Martellini, Lionel Martellini,
• Abstract summary: We show that the time-of-arrival answer to the long-lasting time-of-arrival problem is readily available in the standard formalism. This finding suggests that the answer to the long-lasting time-of-arrival problem is in fact readily available in the standard formalism.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Using standard results from statistics, we show that for any continuous quantum system (Gaussian or otherwise) and any observable $A$ (position or otherwise), the distribution $ \pi _{a}\left(t\right) $ of a 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}\left( t\right) = \left\vert \frac{\partial }{\partial t} \int_{-\infty }^{a}\rho _{t}\left( u\right) du \right\vert $. This finding suggests that the answer to the long-lasting time-of-arrival problem is in fact readily available in the standard formalism, secretly hidden within the Born rule, and therefore does not require the introduction of an ad-hoc 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.

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