Energy-Time Uncertainty Relation for Absorbing Boundaries

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 $\sigma_T \, \sigma_E \geq (\hbar/2) \sqrt{\mathrm{Prob}(T<\infty)}$.