Second-quantized density and current density operators

In general spinor bases

Given the first-quantized expressions of the (one-electron) density and current density operators and the second quantization procedure, we can write the second-quantized density operator as: \[\begin{equation} \require{physics} \hat{\rho}(\vb{r}) = \sum_{PQ}^M \rho_{PQ}(\vb{r}) \hat{E}_{PQ} \thinspace , \end{equation}\] in which the density operator matrix elements \(\rho_{PQ}(\vb{r})\) (not to be confused with the density matrix) are given by the following Pauli distributions: \[\begin{equation} \rho_{PQ}(\vb{r}) = \phi^\dagger_P(\vb{r}) \phi_Q(\vb{r}) \thinspace . \end{equation}\] In terms of the field operators, we can write the second-quantized density operator as: \[\begin{equation} \hat{\rho}(\vb{r}) = \hat{\phi}^\dagger(\vb{r}) \hat{\phi}(\vb{r}) \end{equation}\] Then, given a normalized reference state \(\ket{\Psi}\), the expectation value of this density operator is given in terms of the 1-DM as: \[\begin{equation} \rho(\vb{r}) = \sum_{PQ}^M D_{PQ} \thinspace \phi^\dagger_P(\vb{r}) \phi_Q(\vb{r}) \thinspace , \end{equation}\] which is the density at the point \(\vb{r}\).

Similarly, we can quantize the current density operator as \[\begin{equation} \hat{\vb{j}}(\vb{r}) = \sum_{PQ}^M \vb{j}_{PQ}(\vb{r}) \hat{E}_{PQ} \thinspace , \end{equation}\] where \(\vb{j}_{PQ}(\vb{r})\) are the matrix elements of the current density operator: \[\begin{equation} \vb{j}_{PQ}(\vb{r}) = - \frac{i}{2} \Bigg( \phi_P^\dagger(\vb{r}) \thinspace [ \grad{\phi_Q(\vb{r})} ] - [ \grad{\phi_P^\dagger(\vb{r})} ] \thinspace \phi_Q(\vb{r}) + 2 i \thinspace \rho_{PQ}(\vb{r}) \thinspace \vb{A}(\vb{r}) \Bigg) \end{equation}\] and the current density is then analogously calculated as a contraction between the current density operator matix elements \(\vb{j}_{PQ}(\vb{r})\) and the \(1\)-DM: \[\begin{equation} \vb{j}(\vb{r}) = \sum_{PQ}^M \vb{j}_{PQ}(\vb{r}) D_{PQ} \thinspace . \end{equation}\]

In a restricted spin-orbital basis

Since the density and current density operators take an identity-like two-component form, we may quantize them in a restricted spin-orbital basis as follows. For the second-quantized density operator, we find: \[\begin{equation} \hat{\rho}(\vb{r}) = \sum_{pq} \rho_{pq}(\vb{r}) \hat{E}_{pq} \thinspace , \end{equation}\] where the density matrix elements \(\rho_{pq}(\vb{r})\) are given by: \[\begin{equation} \rho_{pq}(\vb{r}) = \phi^*_p(\vb{r}) \phi_q(\vb{r}) \thinspace . \end{equation}\] The expectation value of this density operator is given in terms of the 1-DM as: \[\begin{equation} \rho(\vb{r}) = \sum_{pq} D_{pq} \thinspace \phi^*_p(\vb{r}) \phi_q(\vb{r}) \thinspace , \end{equation}\] which is the density at the point \(\vb{r}\).

Similarly, we can quantize the current density operator as \[\begin{equation} \hat{\vb{j}}(\vb{r}) = \sum_{pq} \vb{j}_{pq}(\vb{r}) \hat{E}_{pq} \thinspace , \end{equation}\] where \(\vb{j}_{pq}(\vb{r})\) are the matrix elements of the current density operator: \[\begin{equation} \vb{j}_{pq}(\vb{r}) = - \frac{i}{2} \Bigg( \phi_p^*(\vb{r}) \thinspace [ \grad{\phi_q(\vb{r})} ] - [ \grad{\phi_p^*(\vb{r})} ] \thinspace \phi_q(\vb{r}) + 2 i \thinspace \rho_{pq}(\vb{r}) \thinspace \vb{A}(\vb{r}) \Bigg) \end{equation}\] and the current density is then analogously calculated as a contraction between the current density operator matix elements \(\vb{j}_{pq}(\vb{r})\) and the \(1\)-DM: \[\begin{equation} \vb{j}(\vb{r}) = \sum_{pq} \vb{j}_{pq}(\vb{r}) D_{pq} \thinspace . \end{equation}\]