论文标题
吸收边界的能源时间不确定性关系
Energy-Time Uncertainty Relation for Absorbing Boundaries
论文作者
论文摘要
We prove the uncertainty relation $σ_T \, σ_E \geq \hbar/2$ between the time $T$ of detection of a quantum particle on the surface $\partial Ω$ of a region $Ω\subset \mathbb{R}^3$ containing the particle's initial wave function, using the "absorbing boundary rule" for detection time, and the energy $E$ of the initial wave function.在这里，$σ$表示与可观察到的波函数相关的概率分布的标准偏差。由于$ t $与POVM相关，而不是自偶会运算符，因此关系不是Robertson和Schrödinger引起的不确定性关系的标准版本的实例。我们还证明，如果粒子永远不会达到$ \ partial的ω$（在这种情况下，我们写$ t = \ infty $），并且如果$σ_T$表示$ t <\ infty $，则表示标准偏差，则表示$ t <\ infty $，则表示$ t <\ nact $ c： \ sqrt {\ mathrm {prog}（t <\ infty）} $。
We prove the uncertainty relation $σ_T \, σ_E \geq \hbar/2$ between the time $T$ of detection of a quantum particle on the surface $\partial Ω$ of a region $Ω\subset \mathbb{R}^3$ 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, $σ$ 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 Schrödinger. We also prove that if there is nonzero probability that the particle never reaches $\partial Ω$ (in which case we write $T=\infty$), and if $σ_T$ denotes the standard deviation conditional on the event $T<\infty$, then $σ_T \, σ_E \geq (\hbar/2) \sqrt{\mathrm{Prob}(T<\infty)}$.
