It is instructive to verify Eq. (38) by computing the expectation value of \( \hat{H}_0 \) between two single-particle states $$ \begin{equation} \langle \alpha_1|\hat{h}_0|\alpha_2\rangle = \sum_{\alpha\beta} \langle \alpha|\hat{h}_0|\beta\rangle \langle 0|a_{\alpha_1}a_\alpha^{\dagger} a_\beta a_{\alpha_2}^{\dagger}|0\rangle \tag{39} \end{equation} $$