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Physical Review A

Time of arrival in quantum and Bohmian mechanics

C. Richard Leavens

Published Aug 1, 1998• 93 citations
Publisher

Abstract

In a recent paper Grot, Rovelli, and Tate (GRT) [Phys. Rev. A 54, 4676 (1996)] derived an expression for the probability distribution $\ensuremath{\pi}(T;X)$ of intrinsic arrival times $T(X)$ at position $x=X$ for a quantum particle with initial wave function $\ensuremath{\psi}(x,t=0)$ 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 $\ensuremath{\Delta}k$ they showed that their analytical expression for $\ensuremath{\pi}(T;X)$ agreed with the probability current density $J(x=X,t=T)$ only to terms of order $\ensuremath{\Delta}k/〈k〉.$ 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 $\ensuremath{\psi}(x,t)$ 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.

DOI
10.1103/physreva.58.840

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In a recent paper Grot, Rovelli, and Tate (GRT) [Phys. Rev. A 54, 4676 (1996)] derived an expression for the probability distribution \ensuremathπ(T;X)\ensuremath{\pi}(T;X)\ensuremathπ(T;X) of intrinsic arrival times T(X)T(X)T(X) at position x=Xx=Xx=X for a quantum particle with initial wave function \ensuremathψ(x,t=0)\ensuremath{\psi}(x,t=0)\ensuremathψ(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,X,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=0t=0t=0 with average wave number 〈k〉〈k〉〈k〉 and variance \ensuremathΔk\ensuremath{\Delta}k\ensuremathΔk they showed that their analytical expression for \ensuremathπ(T;X)\ensuremath{\pi}(T;X)\ensuremathπ(T;X) agreed with the probability current density J(x=X,t=T)J(x=X,t=T)J(x=X,t=T) only to terms of order \ensuremathΔk/〈k〉.\ensuremath{\Delta}k/〈k〉.\ensuremathΔk/〈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)∣|J(X,T)|∣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,x=X,x=X, where \ensuremathψ(x,t)\ensuremath{\psi}(x,t)\ensuremathψ(x,t) and J(x,t)J(x,t)J(x,t) are both zero for all t,t,t, while no particle ever reaches XXX according to the theory based on Bohmian mechanics. Some possible implications are discussed.
Abstract How to compute the probability distribution of a detection time, i.e., of the time which a detector registers as the arrival time of a quantum particle, is a long-debated problem. In this regard, Bohmian mechanics provides in a straightforward way the distribution of the time at which the particle actually does arrive at a given surface in 3-space in the absence of detectors. However, as we discuss here, since the presence of detectors can change the evolution of the wave function and thus the particle trajectories, it cannot be taken for granted that the arrival time of the Bohmian trajectories in the absence of detectors agrees with the one in the presence of detectors, and even less with the detection time. In particular, we explain why certain distributions that Das and Dürr (Sci. Rep. 9: 2242, 2019) presented as the distribution of the detection time in a case with spin, based on assuming that all three times mentioned coincide, are actually not what Bohmian mechanics predicts.
We review recent progress in understanding the arrival time problem in quantum mechanics, from the point of view of the decoherent histories approach to quantum theory. We begin by discussing the arrival time problem, focussing in particular on the role of the probability current in the expected classical solution. After a brief introduction to decoherent histories we review the use of complex potentials in the construction of appropriate class operators. We then discuss the arrival time problem for a particle coupled to an environment, and review how the arrival time probability can be expressed in terms of a POVM in this case. We turn finally to the question of decoherence of the corresponding histories, and we show that this can be achieved for simple states in the case of a free particle, and for general states for a particle coupled to an environment.
We study the problem of computing the probability for the time of arrival of a quantum particle at a given spatial position. We consider a solution to this problem based on the spectral decomposition of the particle's (Heisenberg) state into the eigenstates of a suitable operator, which we denote as the ``time-of-arrival'' operator. We discuss the general properties of this operator. We construct the operator explicitly in the simple case of a free nonrelativistic particle and compare the probabilities it yields with the ones estimated indirectly in terms of the flux of the Schr\"odinger current. We derive a well-defined uncertainty relation between time of arrival and energy; this result shows that the well-known arguments against the existence of such a relation can be circumvented. Finally, we define a ``time representation'' of the quantum mechanics of a free particle, in which the time of arrival is diagonal. Our results suggest that, contrary to what is commonly assumed, quantum mechanics exhibits a hidden equivalence between independent (time) and dependent (position) variables, analogous to the one revealed by the parametrized formalism in classical mechanics. \textcopyright{} 1996 The American Physical Society.
In a previous paper [V. Delgado and J. G. Muga, Phys. Rev. A 56, 3425 (1997)] we introduced a self-adjoint operator T(X)\mathcal{T}^(X)T(X) whose eigenstates can be used to define consistently a probability distribution of the time of arrival at a given spatial point. In the present work we show that the probability distribution previously proposed can be well understood on classical grounds in the sense that it is given by the expectation value of a certain positive-definite operator J(+)(X){J}^{(+)}(X)J(+)(X), which is nothing but a straightforward quantum version of the modulus of the classical current. For quantum states highly localized in momentum space about a certain momentum ${p}_{0}\ensuremath{\ne}0$, the expectation value of J(+)(X){J}^{(+)}(X)J(+)(X) becomes indistinguishable from the quantum probability current. This fact may provide a justification for the common practice of using the latter quantity as a probability distribution of arrival times.