Detecting screens modeled by Schrödinger operators that generate $C_0$ contraction semigroups

Abstract

Consider a non-relativistic quantum particle with wave function $\ddot{I}ˆ$ in a bounded $C^2$ region $\hat{I}\text{c◯}\subset {\mathbb{R}}^n$, and suppose detectors are placed along the boundary $\partial \hat{I}\text{c◯}$. Assume the detection process is irreversible, its mechanism is time independent and also hard, i.e., detections occur only along the boundary $\partial \hat{I}\text{c◯}$. Under these conditions Tumulka informally argued that the dynamics of $\ddot{I}ˆ$ must be governed by a $C_0$ contraction semigroup that weakly solves the Schrödinger equation and proposed modeling the detector by a time-independent local absorbing boundary condition at $\partial \hat{I}\text{c◯}$. In this paper, we apply the newly discovered theory of boundary quadruples to parameterize all $C_0$ contraction semigroups whose generators extend the Schrödinger Hamiltonian, and prove a variant of Tumulka's claim: all such evolutions are generated by the placement of a linear absorbing boundary condition on $\ddot{I}ˆ$ along $\partial \hat{I}\text{c◯}$. We combine this result with the work of Werner to show that each $C_0$ contraction semigroup naturally admits a Born rule for the time of detection along $\partial \hat{I}\text{c◯}$, and we prove that a detection will almost surely occur in finite time if detectors have been placed everywhere along $\partial \hat{I}\text{c◯}$.