Canonical pairs, Spatially Confined Motion and the Quantum Time of Arrival Problem

Abstract

It has always been believed that no self-adjoint and canonical time of arrival operator can be constructed within the confines of standard quantum mechanics. In this Letter we demonstrate the otherwise. We do so by pointing out that there is no a priori reason in demanding that canonical pairs form a system of imprimitivities. We then proceed to show that a class of self-adjoint and canonical time of arrival (TOA) operators can be constructed for a spatially confined free particle. And then discuss the relatiobship between the non-self-adjointess of the TOA operator for the unconfined particle and the self-adjointness of the confined one.