Using a new stability estimate for the difference of the square roots of two solutions of the Vlasov--Poisson equation, we obtain the convergence in L^2 of the Wigner transform of a solution of the Hartree equation with Coulomb potential to a solution of the Vlasov--Poisson equation, with a rate of convergence proportional to \hbar, improving the rate \hbar^{3/4-\varepsilon} obtained in [L. Lafleche, C. Saffirio: Analysis & PDE, to appear] for this particular case. Another reason of interest of this new method is that it is thought of as being closer to methods used to prove the mean-field limit from the many-body Schrödinger equation towards the Hartree--Fock equation for mixed states.
