Correcting the quantum clock: Conditional sojourn times

Abstract

Can the quantum-mechanical sojourn time be clocked without theclock affecting the sojourn time? Here we re-examine thepreviously proposed non-unitary clock, involvingabsorption/amplification by an added infinitesimal imaginarypotential (iVi), and find it not to preserve, ingeneral, the positivity of the sojourn time, conditional oneventual reflection or transmission. The sojourn time is found tobe affected by the scattering concomitant with the mismatch,however small, due to the very clock potential (iVi)introduced for the purpose, as also by any prompt scatteringinvolving partial waves that have not traversed the region ofinterest. We propose a formal procedure whereby the sojourn timeso clocked can be corrected for these spurious scattering effects.The resulting conditional sojourn times are thenpositive-definite for an arbitrary potential, and have the properhigh- and low-energy limits. We note that such a correction alsoapplies to the Larmor clock and identical results are obtained byapplying our procedure in that case as well.