Detecting screens modeled by Schr\"odinger operators that generate $C_0$ contraction semigroups

TL;DR

This paper characterizes all quantum dynamics governed by Schrödinger operators with absorbing boundary conditions that model irreversible detection processes on the boundary of a region, extending Tumulka's informal argument.

Contribution

It applies boundary quadruples theory to fully parameterize contraction semigroups generated by Schrödinger operators with absorbing boundary conditions, confirming Tumulka's claim.

Findings

All such dynamics are generated by linear absorbing boundary conditions.

Each contraction semigroup admits a Born rule for detection times.

Detection occurs almost surely in finite time with boundary detectors.

Abstract

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