  We consider a coupled system of Schr\"odinger equations, arising in quantum
mechanics via the so-called time-dependent self-consistent field method. Using
Wigner transformation techniques we study the corresponding classical limit
dynamics in two cases. In the first case, the classical limit is only taken in
one of the two equations, leading to a mixed quantum-classical model which is
closely connected to the well-known Ehrenfest method in molecular dynamics. In
the second case, the classical limit of the full system is rigorously
established, resulting in a system of coupled Vlasov-type equations. In the
second part of our work, we provide a numerical study of the coupled
semi-classically scaled Schr\"odinger equations and of the mixed
quantum-classical model obtained via Ehrenfest's method. A second order (in
time) method is introduced for each case. We show that the proposed methods
allow time steps independent of the semi-classical parameter(s) while still
capturing the correct behavior of physical observables. It also becomes clear
that the order of accuracy of our methods can be improved in a straightforward
way.
