Absorbing detectors meet scattering theory

TL;DR

This paper directly tests the Absorbing Boundary Condition (ABC) approach to the quantum screen problem by contrasting its predictions with standard scattering theory (ST) in solvable 1D and 2D models. It introduces a contrast between ST and ABC predictions and shows that ABC typically underpredicts detection probabilities, while introducing momentum- and orientation-dependent biases and spurious reflections not observed in ST or experiments. The dependence on the boundary parameter beta and the failure to accommodate generic input states (including coherent superpositions) undermine the universality of ABC, though certain fine-tuned scenarios may mimic ST. The results strongly argue that ABC is not a general solution to the screen problem and prompt empirical exploration for any real detectors that might realize ABC-like dynamics, while reinforcing ST as the benchmark for detector statistics in standard experimental settings.

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.

Figures (7)

• Figure 1: ABC-ST contrast ($\mathscr{C}_{\infty}$) curves for the Gaussian wave packet \ref{['GWP']} with ${k_0=20}$ and a fixed $\Re\beta$ (denoted in the legend). The plot markers indicate $\mathscr{C}_L$ values for ${L=2}$, calculated from Eqs. \ref{['contrastdef']} and \ref{['PABC']} using the time-dependent wave packet $\psi_t^G(x ;k_0,\beta,L)$, Eq. \ref{['GSol']}.
• Figure 2: ABC-ST contrast ($\mathscr{C}_{\infty}$) curves for the superposition of Gaussian wave packets Eq. \ref{['Gsup']}, with ${\Re\beta = 0}$, ${k_0 = 5}$, and a fixed $k_1$ (denoted in the legend). Plot markers indicate $\mathscr{C}_L$ values for ${L = 10}$, calculated using Eqs. \ref{['contrastdef']}, \ref{['PABC']}, and \ref{['Phit']}. Thick curves depict contrasts calculated from Eq. \ref{['Capprox']}.
• Figure 3: Total scattering cross section for ionization of hydrogen atoms by electron impact. Excess energyabove $13.6 \text{ eV}$ versus scattering cross-section (adapted from Bray et al.bray2012electron with permission from the publisher), showing excellent agreement between ST and experiment (see text for additional details).
• Figure 4: A wave packet $\psi_0$ with mean momentum $\vb{k}_0$ approaches either an L-shaped screen $\mathcal{S}$ or a flat screen $\mathcal{S}_\alpha$ tilted at angle $\alpha$ relative to the $x$-axis. The infinitesimal detector segments ${\mathcal{D}(\theta)\subset\mathcal{S}}$ and ${\mathcal{D}_{\alpha}(\theta)\subset\mathcal{S}_{\alpha}}$ both span the same convex cone of apex angle $d\theta$ (shaded area).
• Figure 5: Angular variation of the differential detection probability for a two-dimensional Gaussian wave packet of central momentum ${\vb{k}_0 = \sqrt{3}\,\vu{e}_y - \vu{e}_x}$ approaching the inclined screen $\mathcal{S}_\alpha$. The black curve shows the $\alpha$-independent ST prediction, while colored curves represent predictions of the ABC proposal for ${\beta = 2 i}$ and different tilt angles ${\alpha = \pi/2 + \delta\alpha}$ (see legend). Plot markers indicate numerical values derived from Eq. \ref{['dPABC']} for a finite detector distance of ${L = 15}$.