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We consider a coupled system of Schr\"odinger equations, arising in quantummechanics via the so-called time-dependent self-consistent field method. UsingWigner transformation techniques we study the corresponding classical limitdynamics in two cases. In the first case, the classical limit is only taken inone of the two equations, leading to a mixed quantum-classical model which isclosely connected to the well-known Ehrenfest method in molecular dynamics. Inthe second case, the classical limit of the full system is rigorouslyestablished, resulting in a system of coupled Vlasov-type equations. In thesecond part of our work, we provide a numerical study of the coupledsemi-classically scaled Schr\"odinger equations and of the mixedquantum-classical model obtained via Ehrenfest's method. A second order (intime) method is introduced for each case. We show that the proposed methodsallow time steps independent of the semi-classical parameter(s) while stillcapturing the correct behavior of physical observables. It also becomes clearthat the order of accuracy of our methods can be improved in a straightforwardway.

On the classical limit of a time-dependent self-consistent field system: analysis and computation