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arXiv:0903.1958 · Physics Letters A

Quantum arrival time formula from decoherent histories

J. J. Halliwell, James M. Yearsley

Published Nov 4, 2009• 26 citations
arXivHTMLPDF
DOI
10.1016/j.physleta.2009.10.077

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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 carry out a decoherent-histories analysis of the arrival-time problem, taking advantage of a recently demonstrated connection between time-ordered strings of projection operators and evolution in the presence of a complex potential of step-function form. We concentrate on the limit of a weak potential, in which the resulting arrival-time distribution function is closely related to the quantum-mechanical current. We first consider the analogous classical arrival-time problem involving an absorbing potential, and this sheds some light on certain aspects of the quantum case. We use the path-decomposition expansion to give a derivation of the standard arrival-time distribution defined using a complex potential. This derivation is then used in the decoherent-histories analysis to obtain very simple and plausible expressions for the class operators (describing the amplitudes for crossing the origin during intervals of time). We show that decoherence of histories is obtained for a wide class of initial states (such as simple wave packets and superpositions of wave packets). We find that the decoherent-histories approach gives results with a sensible classical limit that are fully compatible with standard results on the arrival-time problem. We also find some interesting connections between backflow and decoherence.
The decoherent (or consistent) histories approach is applied to the arrival time problem. Since the decoherence condition does not hold for arrival time, a probability distribution of arrival times cannot be defined. Instead, a quasidistribution, which is not positive definite, can be defined in the histories approach. It is shown that well‐known Kijowski's distribution, which is positive definite and often claimed to be the correct distribution of arrival times, is obtained from the quasidistribution when the momentum wave function has certain properties. This suggests that the histories approach still holds some predictive power even when the decoherence condition does not hold.