  We describe the spin and charge dynamics of the system of two electrons
confined within a double quantum dot defined in a quantum wire. The spin
dynamics is driven by the electron motion in presence of the spin-orbit
interaction and the randomly varying local Overhauser field due to the nuclear
spins. The Schroedinger equation is solved with the time-dependent
configuration interaction method that allows for an exact description of the
system dynamics. The procedures of the spin separation, exchange and read-out
by the spin to charge conversion all induced by the detuning variation are
simulated. The rates of the potential variation that are necessary for the spin
separation and spin to charge conversion in the context of the Landau-Zener
transitions are determined. The average over random configurations of the
hyperfine field produce spin exchange results which qualitatively agree with
the experimental data.
