Say I have a state $|\Psi_0\rangle$.
I measure observable $\hat{A}$, the wavefunction collapses to one its eigenstates. I can write $|\Psi_0\rangle = \sum_j \alpha_j|\psi^A_j\rangle$, where $|\psi^A_j\rangle$ are the eigenstates of $\hat{A}$.
Suppose I do not have access to the results of the experiment though.
I now apply a second operator $\hat{B}$.
I can see two ways of proceeding to know how the system will evolve in time:
1) I can write $|\Psi^B_t\rangle = \hat{B}|\Psi_t\rangle = \sum_j \alpha_j\hat{B}|\psi^A_j(t)\rangle$, where I have to work out how each of the $|\psi^A_j\rangle$ evolves with time;
2) Defining the density matrix $\rho = \sum_i p_i|\psi^A\rangle\langle\psi^A|$, to then use $\frac{\partial \rho}{\partial t} = \frac{1}{i\hbar}[\rho, H]$.
I agree option #2 is easier, but option #1 is still feasible.
Is there any situation where I cannot use method 1 (pure state time evolution) but have to use use the density matrix approach?