Determining the Wave Function From Initial Conditions This is Problem 2.6 (b) in Griffiths, Intro to QM:

A particle in an infinite square well has its initial wave function an even mixture of the first two stationary states:
$\Psi(x,0) = A[\psi_1(x) + \psi_2(x)]$.

Here is the part of the problem that I am having a little trouble with:

(b) Find $\Psi(x,t)$ and $|\Psi(x,t)|^2$. Express the latter as a sinusoidal function of time, as in Example 2.1. To simplify the result, let $\omega \equiv \frac{\pi^2 \hbar}{2ma^2}$

According to the answer key, even after $t=0$, the wave function continues to be a mixture of the first two stationary states. Why is that? I am having a little difficulty understanding this. Why can't it be a new 'mixture?'
Are my questions sufficiently clear?
 A: The time evolution operator of a quantum system is (in units with $\hbar = 1$)
$$ U(t_0,t) = \mathrm{e}^{\mathrm{i}H (t - t_0)}$$
and the "stationary states" are the eigenstates of this operator, i.e. eigenstates of the Hamiltonian. If you are given a collection of stationary states (not a basis of the space, mind you) $\{\lvert \psi_E \rangle\}$ with $H \lvert \psi_E \rangle = E \lvert \psi_E \rangle$, any sum of these will stay a sum of these under time evolution:
$$ U(t_0,t)\sum_E \lvert \psi_E\rangle = \sum_E \mathrm{e}^{\mathrm{i}H (t - t_0)} \lvert \psi_E \rangle = \sum_E\mathrm{e}^{\mathrm{i}E (t - t_0)}\lvert \psi_E\rangle$$
There is simply no room for other stationary states to appear, since every single stationary state's time evolution is given by just a phase.
A: The superposition principle of quantum mechanics is not destroyed by quantum (hamiltonian) unitary evolution operator $U(t_0,t) = \mathrm{e}^{\mathrm{i}H (t - t_0)}$ as per @ACuriousMind's answer.
Event if you dont know of the evolution operator in terms of the hamiltonian (which can be derived easily from the Schrodiger equation), still the fact that the Schrodiger equation is linear (wrt to wavefunction $\psi$) (both time-independent and time-dependent equations), means the linearity of the wavefunction (or in QM terms the superposition) is not destroyed.
Superposition is destroyed only by a quatum measurement (as per the Copenhagen formalism)
