<s>[INST] Schrodinger equation to find general wave function 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:

 [/INST] 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*}
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