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Register a new userEnergy-Time Uncertainty Relation for Absorbing Boundaries
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Roderich Tumulka
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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 $Omegasubset mathbb{R}^3$ containing the particles 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 Schrodinger. 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 $sigma_T , sigma_E geq (hbar/2) sqrt{mathrm{Prob}(T<infty)}$.
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