Restricted spin operators

For a single spatial orbital \(p\), we can define a spin raising operator \[\begin{equation} \require{physics} \hat{S}^+_p % = \hat{a}^\dagger_{p \alpha} \hat{a}_{p \beta} % \thinspace , \end{equation}\] which flips an \(\alpha\)-electron in the spatial orbital \(p\) into a \(\beta\)-electron. The Hermitian adjoint of the spin raising operator is the spin lowering operator: \[\begin{equation} \hat{S}^-_p % = \qty(\hat{S}^+_p)^\dagger % = \hat{a}^\dagger_{p \beta} \hat{a}_{p \alpha} % \thinspace , \end{equation}\] which is seen to flip a \(\beta\)-electron in the spatial orbital \(p\) into an \(\alpha\)-electron. We can also define the operator for the \(z\)-component of the spin angular momentum for the spatial orbital \(p\) as \[\begin{align} \hat{S}^z_p % & = \frac{1}{2} \qty( \hat{a}^\dagger_{p \alpha} \hat{a}_{p \alpha} % - \hat{a}^\dagger_{p \beta} \hat{a}_{p \beta} ) \\ % & = \frac{1}{2} \qty( % \hat{N}_{p \alpha} % - \hat{N}_{p \beta} % ) \thinspace . \end{align}\]

For some commutators with the elementary second quantization operators, we can find that \[\begin{align} & \comm{\hat{S}^+_p}{ \hat{a}^\dagger_{q \alpha} } = 0 \\ & \comm{\hat{S}^-_p}{ \hat{a}^\dagger_{q \alpha} } % = \delta_{pq} \hat{a}^\dagger_{p \beta} \\ % & \comm{\hat{S}^+_p}{ \hat{a}_{q \alpha} } % = - \delta_{pq} \hat{a}_{p \beta} \\ % & \comm{\hat{S}^-_p}{ \hat{a}_{q \alpha} } = 0 \end{align}\] and \[\begin{align} & \comm{\hat{S}^+_p}{ \hat{a}^\dagger_{q \beta} } % = \delta_{pq} \hat{a}^\dagger_{p \alpha} \\ % & \comm{\hat{S}^-_p}{ \hat{a}^\dagger_{q \beta} } = 0 \\ % & \comm{\hat{S}^+_p}{ \hat{a}_{q \beta} } = 0 \\ % & \comm{\hat{S}^-_p}{ \hat{a}_{q \beta} } = % - \delta_{pq} \hat{a}_{p \alpha} % \thinspace . \end{align}\]

Using the Pauli spin matrices (cfr. equation \(\eqref{eq:Pauli_spin_matrices}\)), we can derive the so-called Pauli raising and Pauli lowering matrices: \[\begin{align} & \sigma_+ = \sigma_x + i \sigma_y % = \begin{pmatrix} 0 & 2 \\ 0 & 0 \end{pmatrix} \\ % & \sigma_- = \sigma_x - i \sigma_y % = \begin{pmatrix} 0 & 0 \\ 2 & 0 \end{pmatrix} % \thinspace , \end{align}\] with the commutation relations \[\begin{equation} \comm{\sigma_+}{\sigma_-} % = 4 \sigma_z % % \qquad \qquad % % \comm{\sigma_z}{\sigma_{\pm}} % = \pm 2 \sigma_{\pm} % \thinspace . \end{equation}\] Using these Pauli matrices, we can define the spin operators most generally as: \[\begin{equation} \hat{S}^\omega_p % = \frac{1}{2} \sum_{\sigma \tau} % \hat{a}^\dagger_{p \sigma} % \sigma^\omega_{\sigma \tau} % \hat{a}_{p \tau} % \thinspace , \end{equation}\] in which \(\omega\) labels \(x,y,z,+,-\). For the commutators of the second-quantized spin operators, we find \[\begin{align} & \comm{\hat{S}^+_p}{\hat{S}^-_q} = 2 \delta_{pq} \hat{S}^z_p \\ & \comm{\hat{S}^z_p}{\hat{S}^\pm_q} = \pm \delta_{pq} \hat{S}^\pm_p \thinspace , \end{align}\] such that we can say that \(\set{\hat{S}^+_p, \hat{S}^-_p, \hat{S}^z_p}\) spans the \(\mathfrak{su}(2)\) Lie algebra for every spin orbital \(p\) (cfr. section \(\ref{sec:lie_algebras}\)).

We can also define a total spin raising operator as \[\begin{equation} \hat{S}^+ = \sum_p^K \hat{S}^+_p % \thinspace , \end{equation}\] a the total spin lowering operator still being the Hermitian adjoint of the total spin rising operator: \[\begin{equation} \hat{S}^- = \qty(\hat{S}^+)^\dagger = \sum_p^K \hat{S}^-_p \end{equation}\] and the total \(z\)-projection of the spin angular momentum: \[\begin{equation} \hat{S}^z = \sum_p^K \hat{S}^z_p % \thinspace . \end{equation}\] For some elementary commutators, we can find that \[\begin{align} & \comm{\hat{S}^+}{ \hat{a}^\dagger_{p \alpha} } = 0 \\ % & \comm{\hat{S}^-}{ \hat{a}^\dagger_{p \alpha} } = % \hat{a}^\dagger_{p \beta} \\ % & \comm{\hat{S}^+}{ \hat{a}_{p \alpha} } = % - \hat{a}_{p \beta} \\ % & \comm{\hat{S}^-}{ \hat{a}_{p \alpha} } = 0 \end{align}\] and \[\begin{align} & \comm{\hat{S}^+}{ \hat{a}^\dagger_{p \beta} } = % \hat{a}^\dagger_{p \alpha} \\ % & \comm{\hat{S}^-}{ \hat{a}^\dagger_{p \beta} } = 0 \\ % & \comm{\hat{S}^+}{ \hat{a}_{p \beta} } = 0 \\ % & \comm{\hat{S}^-}{ \hat{a}_{p \beta} } =% - \hat{a}_{p \alpha} % \thinspace . \end{align}\] For the commutators of the set of `total’ spin operators, we find \[\begin{align} & \comm{\hat{S}^+}{\hat{S}^-} = 2 \hat{S}^z \\ % & \comm{\hat{S}^z}{\hat{S}^\pm} = \pm \hat{S}^\pm % \thinspace , \end{align}\] such that the total spin operators themselves also span an \(\mathfrak{su}(2)\) Lie algebra and the operator for the total spin can be written as the Casimir operator: \[\begin{align} \hat{S}^2 &= % \hat{S}^+ \hat{S}^- - \hat{S}^z + \qty(\hat{S}^z)^2 \\ % &= \hat{S}^- \hat{S}^+ + \hat{S}^z + \qty(\hat{S}^z)^2 % \thinspace . \end{align}\] We should emphasize that the total spin operator is not the sum of the separate Casimir operators for every orbital \(p\): \[\begin{equation} \hat{S}^2 \neq \sum_p^K \hat{S}^2_p % \thinspace . \end{equation}\]