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Absorbing boundary condition as limiting case of imaginary potentials

2022; Institute of Physics; Volume: 75; Issue: 1 Linguagem: Inglês

10.1088/1572-9494/ac9bea

1572-9494

Roderich Tumulka,

Statistical Mechanics and Entropy

Abstract Imaginary potentials such as V ( x ) = −i σ 1 Ω ( x ) (with σ > 0 a constant, Ω a subset of 3-space, and 1 Ω 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.