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.
We investigate the ballistic spreading behavior of the one-dimensional discrete time quantum walks whose time evolution is driven by any balanced quantum coin. We obtain closed-form expressions for the long-time variance of position of quantum walks starting from any initial qubit (spin-
1/2 particle) and position states following a delta-like (local), Gaussian and uniform probability distributions. By averaging over all spin states, we find out that the average variance of a quantum walk starting from a local state is independent of the quantum coin, while from Gaussian and uniform states it depends on the sum of relative phases between spin states given by the quantum coin, being non-dispersive for a Fourier walk and large initial dispersion. We also perform numerical simulations of the average probability distribution and variance along the time to compare them with our analytical results.
We investigate the spreading behavior of the one-dimensional discrete time quantum walk starting from local and Gaussian states. In order to compare both cases, we obtain analytically the average variance of the position by averaging over all spin states. We find out that the average variance for the local case is independent on the quantum coin. However, when the quantum walk starts from a Gaussian state, its average variance depends on the sum of relative phases between spin states given by the quantum coin, and also, on the initial dispersion of the state. We perform numerical simulations of the average probability and variance along the time to confront them with our analytical results.