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arXiv:2107.08777 · The European Physical Journal Plus

Arrival time from the general theory of quantum time distributions

Tajron Jurić, Hrvoje Nikolić

Published May 1, 2022• 8 citations
arXivHTMLPDF
DOI
10.1140/epjp/s13360-022-02854-w

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Physical Review A•Nov 29, 2010•5 citations
The quantum time-of-arrival problem is discussed within the standard formulation of nonrelativistic quantum mechanics with parametric time. It is shown that a general class of arrival-time probability distributions results from the assumption that the arrival process of a quantum particle is similar in nature to other time-dependent arrival-type processes occurring, e.g., in population biology or queue theory. A simple but illustrative example related to the well-known Wigner discussion of the time-energy uncertainty relation is given and the numerical results obtained are compared with Kijowski's distribution [Rep. Math. Phys. 6, 362 (1974)] of arrival times for a free quantum particle.
Time of arrival refers to the time a particle takes after emission to impinge upon a suitably idealised detector surface. Within quantum theory, no generally accepted solution exists so far for the corresponding probability distribution of arrival times. In this work we derive a general solution for a single body without spin impacting on a so called ideal detector in the absence of any other forces or obstacles. A solution of the so called screen problem for this case is also given. After discussing the shortcomings of the so called ‘absorbing boundary condition’, which is arguably the natural approach within quantum mechanics, we construct the ideal detector model via mathematical probability theory. This detector model assures that the probability flux through the detector surface is always positive, so that the corresponding distributions can be derived via an approach originally suggested by Daumer, Dürr, Goldstein, and Zanghì. The resulting dynamical model is based on an adaption of the Madelung equations and is, strictly speaking, not compatible with quantum mechanics. Still, it is well-described within geometric quantum theory. Geometric quantum theory is a novel adaption of quantum mechanics, which makes the latter consistent with mathematical probability theory. Implications to the general theory of measurement and avenues for future research are also provided. Future mathematical work should focus on finding an appropriate distributional formulation of the evolution equations and studying the well-posedness of the corresponding Cauchy problem.
We reconsider Baute et al.'s free crossing states [Phys. Rev. A 61, 022118 (2000)] and show that if we require a generalization in the interacting case that goes in complete parallel with the free-particle case, then this generalized crossing state cannot be arbitrary but is determined by the null space of the particle's quantum time-of-arrival operator. Nonetheless, the free crossing states appear as the leading term in the asymptotic expansion of our generalized crossing state in the limit of large momentum. We then examine the quantum time-of-arrival problem of a spinless particle in a uniform gravitational field. Mass-dependent time-of-arrival probability distributions emerge, signifying quantum departures from the weak equivalence principle. However, in the classical limit of large mass and vanishing uncertainty in position, the mass dependence of the quantum time-of-arrival distribution becomes exponentially small and the mean quantum time of arrival reduces to the classical time of arrival.