Thanks for your reply csopi. I have corrected my Ψ(t) as below, what I though of first was a fatal mistake.
\psi(t)=e^{(-iEt/\hbar)}\left( \begin{array}{cc} 0 \\
1 \end{array} \right)
Is my assumption that since both energies are the same both can be considered as ground state and...