Quantum Arrival and Dwell Times via Idealised Clocks

Abstract: A number of approaches to the problem of defining arrival and dwell time probabilities in quantum theory make use of idealised models of clocks. An interesting question is the extent to which the probabilities obtained in this way are related to standard semiclassical results. In this paper we explore this question using a reasonably general clock model, solved using path integral methods. We find that in the weak coupling regime where the energy of the clock is much less than the energy of the particle it is measuring, the probability for the clock pointer can be expressed in terms of the probability current in the case of arrival times, and the dwell time operator in the case of dwell times, the expected semiclassical results. In the regime of strong system-clock coupling, we find that the arrival time probability is proportional to the kinetic energy density, consistent with an earlier model involving a complex potential. We argue that, properly normalized, this may be the generically expected result in this regime. We show that these conclusions are largely independent of the form of the clock Hamiltonian.