Derivation of von Neumann Equation for Density Matrices

Question

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?

Answer 1

I determined the answer to my question, so I decided to post it.

The weights $w_i$ are constant in time. The reason for this is that all of the time evolution is contained in the kets; all of the $Nw_i$ systems in the $|\alpha_i (t_0)\rangle$ state at time $t_0$ will be in the state $|\alpha_i(t)\rangle$ at a later time $t$, so it is accurate to treat only the time evolution of the kets and keep the $w_i$ constant.