Imaginary potentials such as V(x)=−iσ1Ω(x)(withσ>0 a constant,Ωa subset of 3-space,and 1Ωits characteristic function)have been used in quantum mechanics as models of a detector.They represent the effect of a‘...Imaginary potentials such as V(x)=−iσ1Ω(x)(withσ>0 a constant,Ωa subset of 3-space,and 1Ωits 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 volumeΩ.In order to model a‘hard’detector(i.e.one that registers a particle as soon as it entersΩ),one may think of taking the limitσ→∞of increasing detector strengthσ.However,as pointed out by Allcock,in this limit the particle never entersΩ;its wave function gets reflected at the boundary∂ΩofΩin 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.展开更多
文摘Imaginary potentials such as V(x)=−iσ1Ω(x)(withσ>0 a constant,Ωa subset of 3-space,and 1Ωits 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 volumeΩ.In order to model a‘hard’detector(i.e.one that registers a particle as soon as it entersΩ),one may think of taking the limitσ→∞of increasing detector strengthσ.However,as pointed out by Allcock,in this limit the particle never entersΩ;its wave function gets reflected at the boundary∂ΩofΩin 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.
