A system having Hamiltonian \( \hat{H} \) follows the eigenvalue equation, \( \hat{H} \psi_n = E_n \psi_n \), with
\[
E_n = \left(n + \frac{1}{2}\right)
\]
If the state of the system is prepared as,
\[
\Psi = N(\psi_1 + \psi_2 + \psi_3 - \psi_4 - \psi_5),
\]
where N is the normalization constant, then the expectation value of energy is