Quantum Arrival and Dwell Times via Idealised Clocks

TL;DR

This paper investigates how idealised quantum clocks can be used to define arrival and dwell time probabilities, showing their relation to semiclassical results in different coupling regimes using path integral methods.

Contribution

It introduces a general clock model and analyzes the connection between quantum clock probabilities and classical semiclassical results across coupling regimes.

Findings

Weak coupling regime yields probabilities related to probability current and dwell time operator.

Strong coupling regime results in arrival time probability proportional to kinetic energy density.

Conclusions are largely independent of the clock Hamiltonian form.

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.