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arXiv:1601.03871 · Physical Review A

Detection-time distribution for several quantum particles

Roderich Tumulka

Published Oct 24, 2022• 17 citations
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Abstract

We address the question of how to compute the probability distribution of the time at which a detector clicks in the situation of $n$ 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 $n$-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.

DOI
10.1103/physreva.106.042220

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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 \ensuremathΩ\ensuremath⊂R3\mathrm{\ensuremath{\Omega}}\ensuremath{\subset}{\mathbb{R}}^{3}\ensuremathΩ\ensuremath⊂R3 in physical space and detectors placed along the boundary \ensuremath∂\ensuremathΩ\ensuremath{\partial}\mathrm{\ensuremath{\Omega}}\ensuremath∂\ensuremathΩ of \ensuremathΩ\mathrm{\ensuremath{\Omega}}\ensuremathΩ. 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∂\ensuremathΩ\ensuremath{\partial}\mathrm{\ensuremath{\Omega}}\ensuremath∂\ensuremathΩ 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−1n\ensuremath{-}1n\ensuremath−1 particles. We also describe an extension of the absorbing boundary rule to the case of moving detectors.
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.
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.
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].
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<∞)​.
We are concerned with the justification of the statement, commonly (explicitly or implicitly) used in quantum scattering theory, that for a free non-relativistic quantum particle with initial wave function Ψ0(x)Ψ_0(\boldsymbol{x})Ψ0​(x), surrounded by detectors along a sphere of large radius RRR, the probability distribution of the detection time and place has asymptotic density (i.e., scattering cross section) σ(x,t)=m3ℏ−3Rt−4∣Ψ^0(mx/ℏt)∣2σ(\boldsymbol{x},t)= m^3 \hbar^{-3} R t^{-4} |\widehatΨ_0(m\boldsymbol{x}/\hbar t)|^2σ(x,t)=m3ℏ−3Rt−4∣Ψ0​(mx/ℏt)∣2 with Ψ^0\widehatΨ_0Ψ0​ the Fourier transform of Ψ0Ψ_0Ψ0​. We give two derivations of this formula, based on different macroscopic models of the detection process. The first one consists of a negative imaginary potential of strength λ>0λ>0λ>0 in the detector volume (i.e., outside the sphere of radius RRR) in the limit R→∞,λ→0,Rλ→∞R\to\infty,λ\to 0, Rλ\to \inftyR→∞,λ→0,Rλ→∞. The second one consists of repeated nearly-projective measurements of (approximately) the observable 1∣x∣>R1_{|\boldsymbol{x}|>R}1∣x∣>R​ at times T,2T,3T,…\mathscr{T},2\mathscr{T},3\mathscr{T},\ldotsT,2T,3T,… in the limit R→∞,T→∞,T/R→0R\to\infty,\mathscr{T}\to\infty,\mathscr{T}/R\to 0R→∞,T→∞,T/R→0; this setup is similar to that of the quantum Zeno effect, except that there one considers T→0\mathscr{T}\to 0T→0 instead of T→∞\mathscr{T}\to\inftyT→∞. We also provide a comparison to Bohmian mechanics: while in the absence of detectors, the arrival times and places of the Bohmian trajectories on the sphere of radius RRR have asymptotic distribution density given by the same formula as σσσ, their deviation from the detection times and places is not necessarily small, although it is small compared to RRR, so the effect of the presence of detectors on the particle can be neglected in the far-field regime. We also cover the generalization to surfaces with non-spherical shape, to the case of NNN non-interacting particles, to time-dependent surfaces, and to the Dirac equation.
Abstract Imaginary potentials such as V(x) = −iσ1Ω(x) (with σ > 0 a constant, Ω a subset of 3-space, and 1Ω its characteristic function) have been used in quantum mechanics as models of a detector. They represent the effect of a ‘soft’ detector that takes a while to notice a particle in the detector volume Ω. In order to model a ‘hard’ detector (i.e. one that registers a particle as soon as it enters Ω), one may think of taking the limit σ → ∞ of increasing detector strength σ. However, as pointed out by Allcock, in this limit the particle never enters Ω; its wave function gets reflected at the boundary ∂Ω of Ω in the same way as by a Dirichlet boundary condition on ∂Ω. This phenomenon, a cousin of the ‘quantum Zeno effect,’ might suggest that a hard detector is mathematically impossible. Nevertheless, a mathematical description of a hard detector has recently been put forward in the form of the ‘absorbing boundary rule’ involving an absorbing boundary condition on the detecting surface ∂Ω. We show here that in a suitable (non-obvious) limit, the imaginary potential V yields a non-trivial distribution of detection time and place in agreement with the absorbing boundary rule. That is, a hard detector can be obtained as a limit, but it is a different limit than Allcock considered.