Absorbing Boundary Condition as Limiting Case of Imaginary Potentials

Abstract

Imaginary potentials such as $V(x)=-iv{1}_{\Omega }(x)$ (with $v>0$ a constant, $\Omega$ a subset of 3-space, and $1_{\Omega }$ 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 $\Omega$. In order to model a "hard" detector (i.e., one that registers a particle as soon as it enters $\Omega$), one may think of taking the limit $v\to \infty$ of increasing detector strength $v$. However, as pointed out by Allcock, in this limit the particle never enters $\Omega$; its wave function gets reflected at the boundary $\partial \Omega$ of $\Omega$ in the same way as by a Dirichlet boundary condition on $\partial \Omega$. 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 $\partial \Omega$. 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.

12 pages LaTeX, no figures

References in corpus (4)

• Detection of a quantum particle on a lattice under repeated projective measurements
• Quantum Dynamics under continuous projective measurements: non-Hermitian description and the continuous space limit
• Existence of Schrodinger Evolution with Absorbing Boundary Condition
• Hilbert space partitioning for non-Hermitian Hamiltonians: From off-resonance to Zeno subspaces

Cited by in corpus (8)

• Quantum Dynamics under continuous projective measurements: non-Hermitian description and the continuous space limit
• Existence of Schrodinger Evolution with Absorbing Boundary Condition
• On a Derivation of the Absorbing Boundary Rule
• Inequivalence of stochastic and Bohmian arrival times in time-of-flight experiments
• Absorbing detectors meet scattering theory
• Detection Time Distribution Predicted Using Absorbing Boundary Conditions and Imaginary Potentials
• A solution of the quantum time of arrival problem via mathematical probability theory
• Energy-Time Uncertainty Relation for Absorbing Boundaries