Abstract
In a recent paper Grot, Rovelli, and Tate (GRT) [Phys. Rev. A 54, 4676 (1996)] derived an expression for the probability distribution of intrinsic arrival times at position for a quantum particle with initial wave function freely evolving in one dimension. This was done by quantizing the classical expression for the time of arrival of a free particle at assuming a particular choice of operator ordering, and then regulating the resulting time of arrival operator. For the special case of a minimum-uncertainty-product wave packet at with average wave number and variance they showed that their analytical expression for agreed with the probability current density only to terms of order They dismissed the probability current density as a viable candidate for the exact arrival time distribution on the grounds that it can sometimes be negative. This fact is not a problem within Bohmian mechanics where the arrival time distribution for a particle, either free or in the presence of a potential, is rigorously given by (suitably normalized) [W. R. McKinnon and C. R. Leavens, Phys. Rev. A 51, 2748 (1995); C. R. Leavens, Phys. Lett. A 178, 27 (1993); M. Daumer et al., in On Three Levels: The Mathematical Physics of Micro-, Meso-, and Macro-Approaches to Physics, edited by M. Fannes et al. (Plenum, New York, 1994); M. Daumer, in Bohmian Mechanics and Quantum Theory: An Appraisal, edited by J. T. Cushing et al. (Kluwer Academic, Dordrecht, 1996)]. The two theories are compared in this paper and a case presented for which the results could not differ more: According to GRT’s theory, every particle in the ensemble reaches a point where and are both zero for all while no particle ever reaches according to the theory based on Bohmian mechanics. Some possible implications are discussed.
- Received 15 October 1997
DOI:https://doi.org/10.1103/PhysRevA.58.840
©1998 American Physical Society