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arXiv:2508.11368 · The Philosophical Magazine A Journal of Theoretical Experimental and Applied Physics

A solution of the quantum time of arrival problem via mathematical probability theory

Maik Reddiger

Published Feb 22, 2026• 0 citations

Abstract

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.

DOI: 10.1080/14786435.2026.2627725

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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.

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.

The problem of making unambiguous probabilistic predictions about experiments involving waiting "always on" detectors remains a challenge for quantum theory. While most research on this problem studies arrival time, i.e., predicting the distribution of when detection events occur, this paper studies the arrival position problem, which is the complementary challenge of predicting the distribution of where detection events occur. Despite the widespread recognition of the arrival time problem, the inability of standard quantum theory to address the arrival position problem remains a pervasive theoretical blind spot. In this paper, we compare quantitative arrival position predictions derived from prominent proposed solutions to the screen problem. As we show, these models yield distinguishable predictions even in relatively simple experiments achievable with current technology. Notably, many of these discrepancies persist even in the far-field limit, where standard semiclassical approximations are typically assumed to be valid.

We develop a general framework for the construction of probabilities for the time of arrival in quantum systems. The time of arrival is identified with the time instant when a transition in the detector's degrees of freedom takes place. Thus, its definition is embedded within the larger issue of defining probabilities with respect to time for general quantum transitions. The key point in our analysis is that we manage to reduce the problem of defining a quantum time observable to a mathematical model where time is associated to a transition from a subspace of the Hilbert space of the total system to its complementary subspace. This property makes it possible to derive a general expression for the probability for the time of transition, valid for any quantum system, with the only requirement that the time of transition is correlated with a definite macroscopic record. The framework developed here allows for the consideration of any experimental configuration for the measurement of the time of arrival, and it also applies to relativistic systems with interactions described by quantum field theory. We use the method in order to describe time-of-arrival measurements in high-energy particle reactions and for a rigorous derivation of the time-integrated probabilities in particle oscillations.

The quantum Zeno effect is often regarded as an obstruction to precise arrival time measurements in quantum mechanics. Here, an arbitrarily precise arrival time measurement procedure is constructed using a localized detection process and a suitably chosen boundary condition. The arrival time of an incoming particle is recorded in the position of a clock particle that is emitted by the apparatus upon detection. A non-zero probability of arrival is shown to survive even in the limit as exact arrival time measurement precision is achieved. In this limit, it is also found that the interaction between the incoming particle and the detector is described by an absorbing boundary condition. This justifies the claim of Tumulka that absorbing boundary conditions may be used to model idealized detectors capable of registering particles at the instant of their arrival.

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The time-of-arrival problem asks for the probability distribution for when a quantum particle reaches a specified location. It has been the subject of decades of debate, exemplifying the lack of a self-adjoint time observable in quantum theory. In the Page-Wootters framework, time is a relational quantity, emerging from correlations between a system and a clock induced by a global Hamiltonian constraint. We construct a time-of-arrival distribution by inverting the Page-Wootters approach, asking what time a clock reads given that the particle arrives at some fixed position. The result coincides with a common approach to the time-of-arrival problem, suggesting a potential relational interpretation of the latter. In addition to providing a relational description of the time-of-arrival problem, this gives an application of the Page-Wootters formalism to a concrete physical problem, and reveals some complications with its canonical interpretation as a theory of conditional probabilities.