Abstract
If an ensemble of particles moving parallel to the x axis in the positive direction impinges on a piecewise continuous potential confined to the interval [−a,a] (a>0) it will divide into a transmitted ensemble and a reflected ensemble. It is shown that the classical results for the means, variances, and covariance of position and velocity of the transmitted and reflected ensembles hold in quantum mechanics if the position and momentum probability densities of the incident ensemble are assumed to be sufficiently localized, both in momentum and position. It is also assumed that the support of the probability density of momentum is bounded and positive, and on it the moduli and arguments of the transmission and reflection coefficients are sufficiently well behaved.
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