Consider an ensemble of systems where each system is in one of a set of states $|\alpha_i\rangle$, with proportions $w_i$, such that the density operator is
$$ \hat{\rho} = \sum_i w_i |\alpha_i\rangle \langle \alpha_i|. $$
I'd like to derive the von Neumann equation to describe the time evolution of the density matrix. Now, I imagine that the entire point of such an exercise is that the weights as well as the states change in time; the states evolve via the Schrodinger equation, so I would think to stick in the above definition. Doing so, I get
$$ i\hbar\frac{\partial \hat{\rho}}{\partial t} = -[\hat{\rho},\hat{H}] + i\hbar \sum_i \frac{\partial w_i}{\partial t} |\alpha_i \rangle \langle \alpha_i|, $$
which is obviously not the von Neumann equation due to the last term on the right. What happens to that term? Am I missing an important reason that the $w_i$ do not evolve in time?