Energy-Time Uncertainty Relation for Absorbing Boundaries

TL;DR

This paper establishes a quantum energy-time uncertainty relation for particles detected at absorbing boundaries, extending the concept to POVMs and considering cases where detection may never occur.

Contribution

It proves a new energy-time uncertainty relation for quantum particles with absorbing boundary conditions, including cases with nonzero probability of non-detection.

Findings

Proves the uncertainty relation  \u2265 /2 for detection time and energy.

Extends the relation to cases with nonzero probability of non-detection.

Shows the relation involves the probability of detection occurring.

Abstract

We prove the uncertainty relation $\sigma_T \, \sigma_E \geq \hbar/2$ between the time $T$ of detection of a quantum particle on the surface $\partial \Omega$ of a region $\Omega\subset \mathbb{R}^3$ containing the particle's initial wave function, using the "absorbing boundary rule" for detection time, and the energy $E$ of the initial wave function. Here, $\sigma$ denotes the standard deviation of the probability distribution associated with a quantum observable and a wave function. Since $T$ is associated with a POVM rather than a self-adjoint operator, the relation is not an instance of the standard version of the uncertainty relation due to Robertson and Schr\"odinger. We also prove that if there is nonzero probability that the particle never reaches $\partial \Omega$ (in which case we write $T=\infty$), and if $\sigma_T$ denotes the standard deviation conditional on the event $T<\infty$, then $\sigma_T \, \sigma_E \geq (\hbar/2) \sqrt{\mathrm{Prob}(T<\infty)}$.