See the attached file.
The Schrödinger equation is:
i hbar d|psi>/dt = H|psi> (1)
The initial state is:
|psi(t=0)> = N(|1> + |2>)
where |1> and |2> are two eigenstates of the Hamiltonian:
H|1> = E1 |1>
H|2> = E2 |2>
And E1 is not equal to E2 (by definition as the two states are nondegenerate)
The normalization constant N = 1/sqrt(2) is |1> and |2> are normalized.
We can solve Eq. (1) as follows. Since the equation is a linear differential equation, a superposition of two solutions is another solution. Such a superposition will satisfy the superposition of the initial conditions of the individual solutions. Now, we can write any state as a superposition of eigenstates of the Hamiltonian. So, all we need to do is to solve Eq. (1) for the initial condition were ...
A detailed solution is given.