Time-Evolution of Ensembles

The evolution of systems in both pure and mixed states can be described according to the following equation:

$$\frac{\partial \hat{\rho}}{\partial t} = -\frac{[\hat{\rho},\hat{H}]}{i\hbar}.$$ (17)

Exercise 3: Using the equation of motion for a state vector $\vert\psi>$ (i.e., Eq. (1)), show that Eq. (17) describes the time evolution of $\hat{\rho}$ for a pure state.

Exercise 4: Using the linearity of Eq. (1), show that Eq. (17) also describes the time evolution of $\hat{\rho}$ for a mixed state.