Distribution of the Time at Which an Ideal Detector Clicks

We consider the problem of computing, for a detector surface waiting for a quantum particle to arrive, the probability distribution of the time and place at which the particle gets detected, from the initial wave function of the particle in the non-relativistic regime. Although the standard rules of quantum mechanics offer no operator for the time of arrival, quantum mechanics makes an unambiguous prediction for this distribution, defined by first solving the Schr\"odinger equation for the big quantum system formed by the particle of interest, the detector, a clock, and a device that records the time and place of detection, then making a quantum measurement of the record at a very late time, and finally using the distribution of the recorded time and place. This leads to the question whether there is also a practical, simple rule for computing this distribution, at least approximately (i.e., for an idealized detector). We argue here in favor of a rule based on a 1-particle Schr\"odinger equation with a certain (absorbing) boundary condition at the ideal detecting surface, first considered by Werner in 1987. We present a novel derivation of this rule and describe how it arises as a limit of a "soft" detector represented by an imaginary potential.