Time of arrival in quantum and Bohmian mechanics

In a recent paper Grot, Rovelli, and Tate (GRT) [Phys. Rev. A 54, 4676 (1996)] derived an expression for the probability distribution π(T;X) of intrinsic arrival times $T(X)$ at position $x=X$ for a quantum particle with initial wave function ψ(x,t=0) freely evolving in one dimension. This was done by quantizing the classical expression for the time of arrival of a free particle at $X,$ 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 $t=0$ with average wave number $〈k〉$ and variance Δk they showed that their analytical expression for π(T;X) agreed with the probability current density $J(x=X,t=T)$ only to terms of order Δk/〈k〉. 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 $|J(X,T)|$ (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 $x=X,$ where ψ(x,t) and $J(x,t)$ are both zero for all $t,$ while no particle ever reaches X according to the theory based on Bohmian mechanics. Some possible implications are discussed.