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arXiv:2005.14514

Energy-Time Uncertainty Relation for Absorbing Boundaries

Roderich Tumulka

Published May 29, 2020• 0 citations
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Abstract

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

DOI
10.48550/arxiv.2005.14514

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We prove the uncertainty relation σT σE≥ℏ/2σ_T \, σ_E \geq \hbar/2σT​σE​≥ℏ/2 between the time TTT of detection of a quantum particle on the surface ∂Ω\partial Ω∂Ω of a region Ω⊂R3Ω\subset \mathbb{R}^3Ω⊂R3 containing the particle's initial wave function, using the "absorbing boundary rule" for detection time, and the energy EEE of the initial wave function. Here, σσσ denotes the standard deviation of the probability distribution associated with a quantum observable and a wave function. Since TTT 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 ∂Ω\partial Ω∂Ω (in which case we write T=∞T=\inftyT=∞), and if σTσ_TσT​ denotes the standard deviation conditional on the event T<∞T<\inftyT<∞, then σT σE≥(ℏ/2)Prob(T<∞)σ_T \, σ_E \geq (\hbar/2) \sqrt{\mathrm{Prob}(T<\infty)}σT​σE​≥(ℏ/2)Prob(T<∞)​.
Abstract Consider a non-relativistic quantum particle with wave function inside a region Ω⊂R3\Omega \subset \mathbb {R}^3Ω⊂R3 Ω ⊂ R 3 , and suppose that detectors are placed along the boundary ∂Ω\partial \Omega ∂Ω ∂ Ω . The question how to compute the probability distribution of the time at which the detector surface registers the particle boils down to finding a reasonable mathematical definition of an ideal detecting surface; a particularly convincing definition, called the absorbing boundary rule, involves a time evolution for the particle’s wave function ψ\psi ψ ψ expressed by a Schrödinger equation in Ω\Omega Ω Ω together with an “absorbing” boundary condition on ∂Ω\partial \Omega ∂Ω ∂ Ω first considered by Werner in 1987, viz., ∂ψ/∂n=iκψ\partial \psi /\partial n=i\kappa \psi ∂ψ/∂n=iκψ ∂ ψ / ∂ n = i κ ψ with κ>0\kappa >0κ>0 κ > 0 and ∂/∂n\partial /\partial n∂/∂n ∂ / ∂ n the normal derivative. We provide here a discussion of the rigorous mathematical foundation of this rule. First, for the viability of the rule it plays a crucial role that these two equations together uniquely define the time evolution of ψ\psi ψ ψ ; we point out here how, under some technical assumptions on the regularity (i.e., smoothness) of the detecting surface, the Lumer-Phillips theorem implies that the time evolution is well defined and given by a contraction semigroup. Second, we show that the collapse required for the N-particle version of the problem is well defined. We also prove that the joint distribution of the detection times and places, according to the absorbing boundary rule, is governed by a positive-operator-valued measure.
We address the question of how to compute the probability distribution of the time at which a detector clicks in the situation of nnn nonrelativistic quantum particles in a volume $\mathrm{\ensuremath{\Omega}}\ensuremath{\subset}{\mathbb{R}}^{3}$ in physical space and detectors placed along the boundary $\ensuremath{\partial}\mathrm{\ensuremath{\Omega}}$ of $\mathrm{\ensuremath{\Omega}}$. We previously [Tumulka, Ann. Phys. (NY) 442, 168910 (2022)] argued in favor of a rule for the one-particle case that involves a Schr\"odinger equation with an absorbing boundary condition on $\ensuremath{\partial}\mathrm{\ensuremath{\Omega}}$ introduced by Werner; we call this rule the ``absorbing boundary rule.'' Here, we describe the natural extension of the absorbing boundary rule to the nnn-particle case. A key element of this extension is that, upon a detection event, the wave function gets collapsed by inserting the detected position, at the time of detection, into the wave function, thus yielding a wave function of $n\ensuremath{-}1$ particles. We also describe an extension of the absorbing boundary rule to the case of moving detectors.
We prove an uncertainty relation for energy and arrival time, where the arrival of a particle at a detector is modeled by an absorbing term added to the Hamiltonian. In this well-known scheme the probability for the particle's arrival at the counter is identified with the loss of normalization for an initial wave packet. Under the sole assumption that the absorbing term vanishes on the initial wavefunction, we show that and , where 〈 T 〉 denotes the mean arrival time and p is the probability for the particle to be eventually absorbed. Nearly minimal uncertainty can be achieved in a two-level system, and we propose a trapped ion experiment to realize this situation.
The problem of detection time distribution concerns a quantum particle surrounded by detectors and consists of computing the probability distribution of where and when the particle will be detected. While the correct answer can be obtained in principle by solving the Schrodinger equation of particle and detectors together, a more practical answer should involve a simple rule representing the behavior of idealized detectors. We have argued elsewhere [http://arxiv.org/abs/1601.03715] that the most natural rule for this purpose is the "absorbing boundary rule," based on the 1-particle Schrodinger equation with a certain "absorbing" boundary condition, first considered by Werner in 1987, at the ideal detecting surface. Here we develop a relativistic variant of this rule using the Dirac equation and also a boundary condition. We treat one or several detectable particles, in flat or curved space-time, with stationary or moving detectors.
Although time measurements are routinely performed in laboratories, their theoretical description is still an open problem. Correspondingly, the status of the energy-time uncertainty relation is unsettled. In the first part of this work the necessity of positive operator valued measures (POVM) as descriptions of every quantum experiment is reviewed, as well as the suggestive role played by the probability current in time measurements. Furthermore, it is shown that no POVM exists, which approximately agrees with the probability current on a very natural set of wave functions; nevertheless, the choice of the set is crucial, and on more restrictive sets the probability current does provide a good arrival time prediction. Some ideas to experimentally detect quantum effects in time measurements are discussed. In the second part of the work the energy-time uncertainty relation is considered, in particular for a model of alpha decay for which the variance of the energy can be calculated explicitly, and the variance of time can be estimated. This estimate is tight for systems with long lifetimes, in which case the uncertainty relation is shown to be satisfied. Also the linewidth-lifetime relation is shown to hold, but contrary to the common expectation, it is found that the two relations behave independently, and therefore it is not possible to interpret one as a consequence of the other. To perform the mentioned analysis quantitative scattering estimates are necessary. To this end, bounds of the form ∥1Re−iHtψ∥22≤Ct−3\|1_Re^{-iHt}ψ\|_2^2 \leq C t^{-3}∥1R​e−iHtψ∥22​≤Ct−3 have been derived, where ψψψ denotes the initial state, HHH the Hamiltonian, RRR a positive constant, and CCC is explicitly known. As intermediate step, bounds on the derivatives of the SSS-matrix in the form $\|1_K S^{(n)}\|_\infty \leq C_{n,K} $ have been established, with n=1,2,3n=1,2,3n=1,2,3, and the constants Cn,KC_{n,K}Cn,K​ explicitly known.
There are several inequivalent proposals in the literature for how to compute the probability distribution of the time that a detector registers for the arrival of a quantum particle. For three of these proposals, based on two kinds of absorbing boundary conditions and imaginary potentials, we compute the predicted distribution for an experimental setup involving a single non-relativistic quantum particle with spin 0 or 1/2 in a wave guide along the zzz axis with the detector waiting downstream. We find that the distribution shows signs of partial reflection of the wave function off of the detector; for a spin-1/2 wave function, it is independent of the initial spin orientation for the parameters tested but does depend, for boundary conditions coupling to the spin, on the width of the wave guide. We also compare our predictions with the competing ones of Das and Dürr [arXiv:1802.07141].