Conditional probabilities in quantum theory and the tunneling-time controversy

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Abstract
It is argued that there is a sensible way to define conditional probabilities in quantum mechanics, assuming only Bayes’s theorem and standard quantum theory. These probabilities are equivalent to the ‘‘weak measurement’’ predictions due to Aharonov, Albert, and Vaidman [Phys. Rev. Lett. 60, 1351 (1988)] and hence describe the outcomes of real measurements made on subensembles. In particular, this approach is used to address the question of the history of a particle that has tunneled across a barrier. A gedanken experiment is presented to demonstrate the physically testable implications of the results of these calculations, along with graphs of the time evolution of the conditional probability distribution for a tunneling particle and for one undergoing allowed transmission. Numerical results are also presented for the effects of loss in a band-gap medium on transmission and on reflection, as a function of the position of the lossy region; such loss should provide a feasible, though indirect, test of the present conclusions. It is argued that the effects of loss on the pulse delay time are related to the imaginary value of the momentum of a tunneling particle and it is suggested that this might help explain a small discrepancy in an earlier experiment.
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