Quantum probability distribution of arrival times and probability current density

Agreement with a conclusive experimental test is the ultimate requirement for establishing the validity of any theoretical proposal. However, discriminating experimentally among different alternatives is not always a straightforward matter. Indeed, under arbitrary experimental conditions it might be no possible to distinguish sufficiently among different conceivable alternatives as to guarantee the correctness of a particular proposal. In the present paper we focus on the proposal made in previous papers [V. Delgado and J. G. Muga, Phys. Rev. A 56, 3425 (1997); V. Delgado, Phys. Rev. A 57, 762 (1998)] for a quantum probability distribution of the time of arrival at a certain spatial point, with the purpose of determining under what circumstances appreciable differences with the probability current density can be expected. To this end we have performed a systematic quantitative analysis of the corresponding relative differences as a function of the initial state describing the quantum particle, both in the case of free evolution and in the presence of an intermediate potential barrier. Our investigation reflects that important discrepancies can only be expected in the purely quantum regime, for states of genuinely quantum character, having no classical analogue. These results indicate that in order to discriminate conclusively among the different alternatives the corresponding experimental test should be performed in the quantum regime and with sufficiently high resolution as to resolve small quantum effects. PACS number(s): 03.65.Bz, 03.65.Ca