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What does one measure, when one measures the arrival times of a quantum particle?

What is the right statistics for the measurements of arrival times of a quantum particle? Although this question is very old, it is still open. Usual experiments are performed in far-field regime and this question becomes unimportant, as a semiclassical analysis suffices. Nevertheless, the development in the detector technology will soon allow for near-field investigations, thus a better understanding of arrival time measurements is needed. Since outcomes of quantum measurements are always described by POVMs, various arrival time POVMs have been proposed. On the other hand many physicists would agree that the arrival time statistics is given by the quantum flux. This urges the question whether a POVM exists, which agrees approximately with the quantum flux values on a reasonable set of wave functions. We answer this question negatively for a very natural set of wave functions, but we remark that the answer is very sensitive to the choice of the set, and provide evidence for the existence of a POVM that agrees with the quantum flux on a more restrictive set.