  An exactly solvable time-dependent quantum mechanical problem is employed to
study the convergence properties of transition amplitudes calculated by using
the Schwinger variational principle. A detailed comparison between the
amplitudes approximated by the perturbative series and by their associated
Schwinger variational principles is performed. The much better performance
obtained by the variational principle is documented through different case
studies. For a given order of the Schwinger principle, it is observed that the
transition amplitudes do not converge to the exact one for large perturbations.
The latter is true even though large combinations of unperturbed states with
constant coefficients are taken as trial wave functions. As a matter of fact,
it is shown that the improvement of the method comes from using better trial
wave functions and increasing the order of the Schwinger principle employed.
