The operator $$ \begin{equation} \hat{N} = \sum_\alpha a_\alpha^{\dagger} a_\alpha \tag{30} \end{equation} $$ is called the number operator since it counts the number of particles in a given state vector when it acts on the different single-particle states. It acts on one single-particle state at the time and falls therefore under category one-body operators. Next we look at another important one-body operator, namely \( \hat{H}_0 \) and study its operator form in the occupation number representation.