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arXiv:2509.07518 · Physical Review A

Absorbing detectors versus scattering theory

Will Cavendish, Siddhant Das

Published Oct 28, 2025• 2 citations
arXivHTMLPDF

Abstract

Any proposed solution to the screen problem in quantum mechanics—the challenge of predicting the joint distribution of particle arrival times and impact positions—must align with the extensive data obtained from scattering experiments. In this paper, we conduct a direct consistency check of the absorbing boundary condition (ABC) proposal, a prominent approach to address the screen problem, against the predictions derived from scattering theory (ST). Through a series of exactly solvable one- and two-dimensional examples, we demonstrate that the ABC proposal's predictions are in tension with the well-established results of ST. Specifically, it predicts sharp momentum- and screen-orientation-dependent detection probabilities, along with secondary reflections that contradict existing experimental data. We conclude that while it remains possible that physical detectors described by the ABC proposal could be found in the future, the proposal is empirically inadequate as a general solution to the screen problem, as it is inconsistent with the behavior of detectors in standard experimental settings.

DOI
10.1103/332p-4nl1

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Any proposed solution to the screen problem in quantum mechanics—the challenge of predicting the joint distribution of particle arrival times and impact positions—must align with the extensive data obtained from scattering experiments. In this paper, we conduct a direct consistency check of the absorbing boundary condition (ABC) proposal, a prominent approach to address the screen problem, against the predictions derived from scattering theory (ST). Through a series of exactly solvable one- and two-dimensional examples, we demonstrate that the ABC proposal's predictions are in tension with the well-established results of ST. Specifically, it predicts sharp momentum- and screen-orientation-dependent detection probabilities, along with secondary reflections that contradict existing experimental data. We conclude that while it remains possible that physical detectors described by the ABC proposal could be found in the future, the proposal is empirically inadequate as a general solution to the screen problem, as it is inconsistent with the behavior of detectors in standard experimental settings.
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.
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.
There are several inequivalent proposals in the literature for how to compute the probability distribution of the time that a detector registers for the arrival of a quantum particle. For three of these proposals, based on two kinds of absorbing boundary conditions and imaginary potentials, we compute the predicted distribution for an experimental setup involving a single non-relativistic quantum particle with spin 0 or 1/2 in a wave guide along the zzz axis with the detector waiting downstream. We find that the distribution shows signs of partial reflection of the wave function off of the detector; for a spin-1/2 wave function, it is independent of the initial spin orientation for the parameters tested but does depend, for boundary conditions coupling to the spin, on the width of the wave guide. We also compare our predictions with the competing ones of Das and Dürr [arXiv:1802.07141].
In quantum mechanics, among the proposals for how to compute the probability distribution of detection time and place along a given spatial surface or in a given volume, two leading candidates involve absorbing boundary conditions (ABCs) and complex absorbing potentials (CAPs) for the Schrödinger equation. Here, we extend earlier results on how the two are related. It was previously shown that ABCs arise as a limiting case of CAPs, but for that a (reflecting) Neumann boundary condition was assumed at the end of the region with the CAP. Here, we obtain ABCs in a different limit using a Dirichlet boundary condition at the end of the CAP region, which seems more practically relevant and realizable.
The problem of detection time distribution concerns a quantum particle surrounded by detectors and consists of computing the probability distribution of where and when the particle will be detected. While the correct answer can be obtained in principle by solving the Schrodinger equation of particle and detectors together, a more practical answer should involve a simple rule representing the behavior of idealized detectors. We have argued elsewhere [http://arxiv.org/abs/1601.03715] that the most natural rule for this purpose is the "absorbing boundary rule," based on the 1-particle Schrodinger equation with a certain "absorbing" boundary condition, first considered by Werner in 1987, at the ideal detecting surface. Here we develop a relativistic variant of this rule using the Dirac equation and also a boundary condition. We treat one or several detectable particles, in flat or curved space-time, with stationary or moving detectors.
Abstract Despite the astonishing successes of quantum mechanics, due to some fundamental problems such as the measurement problem and quantum arrival time problem, the predictions of the theory are in some cases not quite clear and unique. Especially, there are various predictions for the joint spatiotemporal distribution of particle detection events on a screen, which are derived from different formulations and interpretations of the quantum theory. Although the differences are typically small, we show that these predictions can be experimentally distinguished by a proposed unconventional double-slit configuration, which is realizable using present-day single-atom interferometry. This experiment would enrich our understanding of the foundations of quantum mechanics.