[Math] Schrodinger equation to find general wave function

Question

I am trying to answer this question:

I have tried solving the Schrodinger equation using separation of variables.

However in the later time wave function i can not seem to get the exp(-3i…)

My current workings are:

Answer 1

Expanding

\begin{align*} \Psi(x,0) &= \frac{1}{\sqrt{a}} \sin \frac{\pi x}{a}+ \frac{2}{\sqrt{a}} \sin \frac{\pi x}{a} \cos \frac{\pi x}{a} \\ &= \frac{1}{\sqrt{a}} \sin \frac{\pi x}{a}+ \frac{1}{\sqrt{a}} \sin \frac{2\pi x}{a} \\ \omega_{n} &= \frac{n^2 \pi^2 \hbar}{2ma^2} \\ \psi_{n} (x,t) &= \sqrt{\frac{2}{a}} e^{-\omega_{n} t} \sin \frac{n\pi x}{a} \\ \Psi(x,t) &= \frac{1}{\sqrt{a}} \exp \left( -\frac{n^2 \pi^2 \hbar}{2ma^2} \right) \sin \left( \frac{\pi x}{a} \right)+ \frac{1}{\sqrt{a}} \exp \left( -\frac{4n^2 \pi^2 \hbar}{2ma^2} \right) \sin \left( \frac{2\pi x}{a} \right) \\ &= \frac{\psi_{1}(x,t)}{\sqrt{2}}+\frac{\psi_{2}(x,t)}{\sqrt{2}} \end{align*}