Detection-time distribution for several quantum particles

Abstract

We address the question of how to compute the probability distribution of the time at which a detector clicks in the situation of $n$ nonrelativistic quantum particles in a volume $\mathrm{\ensuremath{\Omega}}\ensuremath{\subset}{\mathbb{R}}^{3}$ in physical space and detectors placed along the boundary $\ensuremath{\partial}\mathrm{\ensuremath{\Omega}}$ of $\mathrm{\ensuremath{\Omega}}$. We previously [Tumulka, Ann. Phys. (NY) 442, 168910 (2022)] argued in favor of a rule for the one-particle case that involves a Schr\"odinger equation with an absorbing boundary condition on $\ensuremath{\partial}\mathrm{\ensuremath{\Omega}}$ introduced by Werner; we call this rule the ``absorbing boundary rule.'' Here, we describe the natural extension of the absorbing boundary rule to the $n$-particle case. A key element of this extension is that, upon a detection event, the wave function gets collapsed by inserting the detected position, at the time of detection, into the wave function, thus yielding a wave function of $n\ensuremath{-}1$ particles. We also describe an extension of the absorbing boundary rule to the case of moving detectors.