In quantum mechanics, among the proposals for how to compute the probability distribution of detection time and place along a given spatial surface or in a given volume, two leading candidates involve absorbing boundary conditions (ABCs) and complex absorbing potentials (CAPs) for the Schrödinger equation. Here, we extend earlier results on how the two are related. It was previously shown that ABCs arise as a limiting case of CAPs, but for that a (reflecting) Neumann boundary condition was assumed at the end of the region with the CAP. Here, we obtain ABCs in a different limit using a Dirichlet boundary condition at the end of the CAP region, which seems more practically relevant and realizable.
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.
Abstract. Quantum dynamics, typically expressed in the form of a time-dependent Schrödinger equation with a Hermitian Hamiltonian, is a natural application for quantum computing. However, when simulating quantum dynamics that involves the emission of electrons, it is necessary to use artificial boundary conditions (ABCs) to confine the computation within a fixed domain. The introduction of ABCs alters the Hamiltonian structure of the dynamics, and existing quantum algorithms cannot be directly applied since the evolution is no longer unitary. The current paper utilizes a recently introduced Schrödingerization method that converts non-Hermitian dynamics into a Schrödinger form for the artificial boundary problems [S. Jin, N. Liu, and Y. Yu, Quantum Simulation of Partial Differential Equations via Schrödingerisation, preprint, arXiv:2212.13969 , 2022], [S. Jin, N. Liu, and Y. Yu, Phys. Rev. A, 108 (2023), 032603]. We implement this method for three types of ABCs, including the complex absorbing potential technique, perfectly matched layer methods, and Dirichlet-to-Neumann approach. We analyze the query complexity of these algorithms and perform numerical experiments to demonstrate the validity of this approach. This helps to bridge the gap between available quantum algorithms and computational models for quantum dynamics in unbounded domains.
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.
Any proposed solution to the screen problem in quantum mechanics—the challenge of predicting the joint distribution of particle arrival times and impact positions—must align with the extensive data obtained from scattering experiments. In this paper, we conduct a direct consistency check of the absorbing boundary condition (ABC) proposal, a prominent approach to address the screen problem, against the predictions derived from scattering theory (ST). Through a series of exactly solvable one- and two-dimensional examples, we demonstrate that the ABC proposal's predictions are in tension with the well-established results of ST. Specifically, it predicts sharp momentum- and screen-orientation-dependent detection probabilities, along with secondary reflections that contradict existing experimental data. We conclude that while it remains possible that physical detectors described by the ABC proposal could be found in the future, the proposal is empirically inadequate as a general solution to the screen problem, as it is inconsistent with the behavior of detectors in standard experimental settings.
This chapter deals with the time-independent Schrödinger equation. The introduction of complex ‘Hamiltonians’ is essential for treating resonances that are inherently different from genuine (physical) bound states. True square-integrable bound-state eigenvectors are associated with discrete negative eigenenergies and Hermitean Hamiltonians. Resonant states are also square-integrable, but they are linked to complex energies with positive real parts that belong to a continuum. The concept of the absorbing boundary condition comes to rescue the situation via the introduction of complex ‘Hamiltonians’ with normalizable scattering wavefunctions as proper physical states. A method which can provide an adequate spectral representation of the total Green operator or resolvent G^(ω) would be one of the key inputs into an invaluable practical quantum-mechanical theory for scattering (E = 0) and spectroscopy (E < 0).