  Using a new stability estimate for the difference of the square roots of two
solutions of the Vlasov$\unicode{x2013}$Poisson equation, we obtain the
convergence in the $L^2$ norm of the Wigner transform of a solution of the
Hartree equation with Coulomb potential to a solution of the
Vlasov$\unicode{x2013}$Poisson equation, with a rate of convergence
proportional to $\hbar$. This improves the $\hbar^{3/4-\varepsilon}$ rate of
convergence in $L^2$ obtained in [L.~Lafleche, C.~Saffirio: Analysis & PDE, to
appear]. Another reason of interest of this paper is the new method,
reminiscent of the ones used to prove the mean-field limit from the many-body
Schr\"odinger equation towards the Hartree$\unicode{x2013}$Fock equation for
mixed states.
