# Derivation of von Neumann Equation for Density Matrices

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?

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.