Orbital occupation number operators

In analogy to the occupation number operators (cfr. equation \(\eqref{eq:ON_operator}\)), we can define orbital occupation number operators as \[\begin{equation} \require{physics} \hat{N}_{p \sigma} % = \hat{a}^\dagger_{p \sigma} \hat{a}_{p \sigma} % \thinspace . \end{equation}\] The orbital occupation number operator is then \[\begin{equation} \hat{N}_p = \hat{N}_{p \alpha} + \hat{N}_{p \beta} % \thinspace , \end{equation}\] so that the total number operator becomes \[\begin{equation} \hat{N} = \sum_p^K \hat{N}_p = \sum_p^K \hat{E}_{pp} % \thinspace , \end{equation}\] which is of course a sum of spin-number operators: \[\begin{equation} \hat{N} = \sum_\sigma \hat{N}_\sigma % \thinspace , \end{equation}\] in which \(\hat{N}_\sigma\) is the spin-\(\sigma\) number operator: \[\begin{equation} \hat{N}_\sigma = \sum_p^K \hat{N}_{p \sigma} % \thinspace . \end{equation}\]

Occupation number vectors are the eigenvectors of the orbital occupation number operators: \[\begin{equation} \hat{N}_{p \sigma} \ket{\vb{k}} = k_{p \sigma} \ket{\vb{k}} % \thinspace , \end{equation}\] they are Hermitian: \[\begin{equation} \hat{N}_{p \sigma}^\dagger = \hat{N}_{p \sigma} % \thinspace , \end{equation}\] commute among themselves \[\begin{equation} \comm{\hat{N}_{p \sigma}}{\hat{N}_{q \tau}} = 0 % \thinspace , \end{equation}\] and are idempotent \[\begin{equation} \hat{N}^2_{p \sigma} = \hat{N}_{p \sigma} % \thinspace , \end{equation}\] fully analogous to section \(\ref{sec:ON_operators}\), as we have just given the operators a different subscript by relabeling them. Note that the orbital occupation number operator is not idempotent: \[\begin{equation} \hat{N}_p^2 \neq \hat{N}_p % \thinspace . \end{equation}\]

Some useful elementary commutators are \[\begin{align} & \comm{ % \hat{N}_{p \sigma} % }{ % \hat{a}^\dagger_{q \tau} % } % = \delta_{pq} \delta_{\sigma \tau} \hat{a}^\dagger_{q \tau} % = \delta_{pq} \delta_{\sigma \tau} \hat{a}^\dagger_{p \sigma} \\ % & \comm{ % \hat{N}_{p \sigma} % }{ % \hat{a}_{q \tau} % } % = - \delta_{pq} \delta_{\sigma \tau} \hat{a}_{q \tau} % = - \delta_{pq} \delta_{\sigma \tau} \hat{a}_{p \sigma} % \thinspace , \end{align}\] from which follows that \[\begin{equation} \comm{ % \hat{N}_{p \sigma} % }{ % \hat{E}^\tau_{qr} % } = \delta_{\sigma \tau} (\delta_{pq} - \delta_{pr}) \hat{E}_{qr}^\tau % \thinspace . \end{equation}\] We then have (for \(p \neq q\)) \[\begin{align} & \comm{ % \hat{N}_{p \alpha} % }{ % \hat{E}^\alpha_{pq} % } = \hat{E}^\alpha_{pq} \\ % & \comm{ % \hat{N}_{q \alpha} % }{ % \hat{E}^\alpha_{pq} % } = - \hat{E}^\alpha_{pq} % \thinspace , \end{align}\] but we should proceed with caution, since \[\begin{equation} \hat{E}^\alpha_{pq} \hat{N}_{p \alpha} % = 0 % = \hat{N}_{q \alpha} \hat{E}^\alpha_{pq} \label{eq:E_N_zero} \end{equation}\] and \[\begin{equation} \hat{E}^\alpha_{pq} = % \hat{N}_{p \alpha} \hat{E}^\alpha_{pq} % = \hat{E}^\alpha_{pq} \hat{N}_{q \alpha} % \label{eq:E_N_E} \thinspace . \end{equation}\] Furthermore, we have \[\begin{align} & \comm{\hat{N}}{ % \hat{a}^\dagger_{p \sigma} % } = \hat{a}^\dagger_{p \sigma} \\ % & \comm{\hat{N}}{ % \hat{a}_{p \sigma} % } = - \hat{a}_{p \sigma} % \thinspace . \end{align}\]