Consider a non-relativistic quantum particle with wave function [Formula: see text] in a bounded [Formula: see text] region [Formula: see text], and suppose that detectors are placed along the boundary [Formula: see text]. Assume the detection process is effectively irreversible, that its mechanism is time independent and also hard, i.e. detections occur only along the boundary [Formula: see text]. Under these conditions, Tumulka [ 32 ] informally argued that the dynamics of [Formula: see text] must be governed by a [Formula: see text] contraction semigroup that weakly solves the Schrödinger equation and proposed modeling the detector by a time-independent local absorbing boundary condition at [Formula: see text]. In this paper, we apply the newly discovered theory of boundary quadruples [ 33 ] to parameterize all [Formula: see text] contraction semigroups whose generators extend the Schrödinger Hamiltonian and prove a variant of Tumulka’s claim: all such evolutions are generated by the placement of a linear absorbing boundary condition on [Formula: see text] along [Formula: see text]. We combine this result with the work of Werner [ 12 ] to show that each [Formula: see text] contraction semigroup naturally admits a Born rule for the time of detection along [Formula: see text], and we prove that a detection will almost surely occur in finite time if detectors have been placed everywhere along [Formula: see text].
Abstract Consider a non-relativistic quantum particle with wave function inside a region
Ω⊂R3 Ω ⊂ R 3 , and suppose that detectors are placed along the boundary
∂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 ψ expressed by a Schrödinger equation in
Ω Ω together with an “absorbing” boundary condition on
∂Ω ∂ Ω first considered by Werner in 1987, viz.,
∂ψ/∂n=iκψ ∂ ψ / ∂ n = i κ ψ with
κ>0 κ > 0 and
∂/∂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 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
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.
This thesis analyzes the dynamics of a non-relativistic spinless particle undergoing "idealized hard detection" along the boundary of some region Ω in physical space. We shall explicitly parameterize the class of pre-detection wave function dynamics that is consistent with a unitary evolution of the particle and the idealized hard detector's joint wave function. We shall also construct explicit unitary models for the particle and the detection apparatus in which the time of the particle's detection along ∂Ω and some physical property σ of the particle at the instant of its detection is recorded by the apparatus with arbitrary precision. For a large class of idealized hard detector models, it is demonstrated that the recorded outcomes of the detection process continuously depend on the choice of idealized hard detector.
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.
The quantum Zeno effect is often regarded as an obstruction to precise arrival time measurements in quantum mechanics. Here, an arbitrarily precise arrival time measurement procedure is constructed using a localized detection process and a suitably chosen boundary condition. The arrival time of an incoming particle is recorded in the position of a clock particle that is emitted by the apparatus upon detection. A non-zero probability of arrival is shown to survive even in the limit as exact arrival time measurement precision is achieved. In this limit, it is also found that the interaction between the incoming particle and the detector is described by an absorbing boundary condition. This justifies the claim of Tumulka that absorbing boundary conditions may be used to model idealized detectors capable of registering particles at the instant of their arrival.
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.