Time of arrival in quantum and Bohmian mechanics

Details Time of arrival in quantum and Bohmian mechanics Time of arrival in quantum and Bohmian mechanics

Aug 1998

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1998phrva..58..840l&link_type=abstract

Physical Review A, vol. 58, Issue 2, pp. 840-847

Mathematics

Probability

37

Formalism, Quantum Gravity

Scientific paper

In a recent paper Grot, Rovelli, and Tate (GRT) [Phys. Rev. A 54, 4676 (1996)] derived an expression for the probability distribution π(TX) 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 and variance Δk they showed that their analytical expression for π(TX) agreed with the probability current density J(x=X,t=T) only to terms of order Δk/. 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.

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