Characteristic times in the motion of a particle

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Abstract
Quantum mechanics does not provide direct tools for calculating time quantities related to the motion of a particle. In this paper we introduce a meaningful ‘‘time,’’ the ‘‘stay time’’ in a space region, and we propose a method for calculating its statistical distribution. The stay time is obtained by a method based on Feynman’s path integrals, which is similar to the one devised by Sokolovski and Baskin. We add a perturbative potential to the region being considered in order to induce variations in the wave function from which we can draw information about the time spent in the region. Unlike Sokolovski and Baskin, however, we obtain a real stay time and real greater order moments of its distribution. We also analyze other two ‘‘event times,’’ the ‘‘time of presence’’ at a given position and the ‘‘time of passage’’ through a surface. These times, which were introduced by Olkhovski and Recami, are obtained directly from the time evolution of the probability density and the probability current density. We find some relations between such times and the stay time, which show the consistency of the proposed method. Our approach is internally self-consistent, allows a general analysis of the characteristic times in the motion of a quantum particle, and is effective in explaining the results of other studies, in particular in the field of the tunneling times of potential barriers.