#### Atomic spinors (spin-orbit eigenfunctions) expressed in a basis of atomicspin orbitals, and matrix elements of the angular momentum and Zeemanoperators

Below are sets of ${Ĥ}^{\text{SO}}=\zeta \stackrel{^}{L}\cdot \stackrel{^}{S}$ eigenfunctions $|j,{m}_{j}⟩$ expressed in a basis $|\ell ,{m}_{\ell },{m}_{s}⟩$ of atomic spin orbitals for a given angular momentum quantum number $\ell =1$, 2, or 3. These functions were generated with a Mathematica notebook similar to the Mathematica notebook discussed on this page. We also give these eigenfunctions subject to time reversal, and matrix elements of the spin and orbital angular momenum operators, and of the Zeeman (magnetic ﬁeld perturbation) operator ${Ĥ}_{\alpha }^{\text{Z}}={\stackrel{^}{L}}_{\alpha }+2{Ŝ}_{\alpha }$ with $\alpha =x,y,z$. The deﬁnition of the orbital angular momentum eigenfunctions in the basis uses the Condon-Shortley phase. As a consequence, time reversal changes ${m}_{\ell }$ to $-{m}_{\ell }$ but also introduces a negative sign in the spin orbital basis if ${m}_{\ell }$ is odd. The coeﬃcients representing the eigenfunctions of ${Ĥ}^{\text{SO}}$ and their time-reversed counterparts reﬂect this. The $|j,{m}_{j}⟩$ functions provided here were veriﬁed to be simultaneous eigenfunctions of ${Ĥ}^{\text{SO}}$, ${ĵ}^{2}$, and ${ĵ}_{z}$, with phases such that ${ĵ}_{±}|j,{m}_{j}⟩={\alpha }_{±}|j,{m}_{j}±1⟩$, with ${\alpha }_{±}$ being a positive real number as long as ${m}_{j}±1$ is in the allowed range of the projection quantum number, zero otherwise.

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### $\bullet$porbitals $\ell =1$

SO Hamiltonian ${Ĥ}^{\text{SO}}=\zeta \stackrel{^}{L}\cdot \stackrel{^}{S}$ in the basis of $\ell =1$ spin-orbitals $|l,{m}_{\ell },{m}_{s}⟩$ (excluding SO coupling constant $\zeta$)

 $\begin{array}{ccccccc}\hfill & \hfill |1,1,\frac{1}{2}⟩\hfill & \hfill |1,1,-\frac{1}{2}⟩\hfill & \hfill |1,0,\frac{1}{2}⟩\hfill & \hfill |1,0,-\frac{1}{2}⟩\hfill & \hfill |1,-1,\frac{1}{2}⟩\hfill & \hfill |1,-1,-\frac{1}{2}⟩\hfill \\ ̲& ̲& ̲& ̲& ̲& ̲& ̲\\ |1,1,\frac{1}{2}⟩\hfill & \hfill \frac{1}{2}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |1,1,-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill -\frac{1}{2}\hfill & \hfill \frac{1}{\sqrt{2}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |1,0,\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill \frac{1}{\sqrt{2}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |1,0,-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{1}{\sqrt{2}}\hfill & \hfill 0\hfill \\ |1,-1,\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{1}{\sqrt{2}}\hfill & \hfill -\frac{1}{2}\hfill & \hfill 0\hfill \\ |1,-1,-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{1}{2}\hfill \\ \hfill \end{array}$

${Ĥ}^{\text{SO}}$ Eigenfunctions $|j,{m}_{j}⟩$ in the basis of $\ell =1$ spin-orbitals $|l,{m}_{\ell },{m}_{s}⟩$
(eigenvalues: $-1$ for $j=1∕2$, $+1∕2$ for $j=3∕2$, times $\zeta$)

 $\begin{array}{ccccccc}\hfill & \hfill |\frac{1}{2},-\frac{1}{2}⟩\hfill & \hfill |\frac{1}{2},\frac{1}{2}⟩\hfill & \hfill |\frac{3}{2},-\frac{3}{2}⟩\hfill & \hfill |\frac{3}{2},-\frac{1}{2}⟩\hfill & \hfill |\frac{3}{2},\frac{1}{2}⟩\hfill & \hfill |\frac{3}{2},\frac{3}{2}⟩\hfill \\ ̲& ̲& ̲& ̲& ̲& ̲& ̲\\ |1,1,\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \\ |1,1,-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill -\sqrt{\frac{2}{3}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{1}{\sqrt{3}}\hfill & \hfill 0\hfill \\ |1,0,\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill \frac{1}{\sqrt{3}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \sqrt{\frac{2}{3}}\hfill & \hfill 0\hfill \\ |1,0,-\frac{1}{2}⟩\hfill & \hfill -\frac{1}{\sqrt{3}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \sqrt{\frac{2}{3}}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |1,-1,\frac{1}{2}⟩\hfill & \hfill \sqrt{\frac{2}{3}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{1}{\sqrt{3}}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |1,-1,-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill \end{array}$

Kramers-conjugates $|j,{m}_{j}{⟩}^{\prime }$ of the ${Ĥ}^{\text{SO}}$ Eigenfunctions in the basis of $\ell =1$ spin-orbitals $|l,{m}_{\ell },{m}_{s}⟩$

 $\begin{array}{ccccccc}\hfill & \hfill {|\frac{1}{2},-\frac{1}{2}⟩}^{\prime }\hfill & \hfill {|\frac{1}{2},\frac{1}{2}⟩}^{\prime }\hfill & \hfill {|\frac{3}{2},-\frac{3}{2}⟩}^{\prime }\hfill & \hfill {|\frac{3}{2},-\frac{1}{2}⟩}^{\prime }\hfill & \hfill {|\frac{3}{2},\frac{1}{2}⟩}^{\prime }\hfill & \hfill {|\frac{3}{2},\frac{3}{2}⟩}^{\prime }\hfill \\ ̲& ̲& ̲& ̲& ̲& ̲& ̲\\ |1,1,\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |1,1,-\frac{1}{2}⟩\hfill & \hfill -\sqrt{\frac{2}{3}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{1}{\sqrt{3}}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |1,0,\frac{1}{2}⟩\hfill & \hfill \frac{1}{\sqrt{3}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\sqrt{\frac{2}{3}}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |1,0,-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill \frac{1}{\sqrt{3}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \sqrt{\frac{2}{3}}\hfill & \hfill 0\hfill \\ |1,-1,\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill -\sqrt{\frac{2}{3}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{1}{\sqrt{3}}\hfill & \hfill 0\hfill \\ |1,-1,-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -1\hfill \\ \hfill \end{array}$

Operator ${Ŝ}_{x}$ in the basis of $\ell =1$ spin-orbitals $|l,{m}_{\ell },{m}_{s}⟩$

 $\begin{array}{ccccccc}\hfill & \hfill |1,1,\frac{1}{2}⟩\hfill & \hfill |1,1,-\frac{1}{2}⟩\hfill & \hfill |1,0,\frac{1}{2}⟩\hfill & \hfill |1,0,-\frac{1}{2}⟩\hfill & \hfill |1,-1,\frac{1}{2}⟩\hfill & \hfill |1,-1,-\frac{1}{2}⟩\hfill \\ ̲& ̲& ̲& ̲& ̲& ̲& ̲\\ |1,1,\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill \frac{1}{2}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |1,1,-\frac{1}{2}⟩\hfill & \hfill \frac{1}{2}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |1,0,\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{1}{2}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |1,0,-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{1}{2}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |1,-1,\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{1}{2}\hfill \\ |1,-1,-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{1}{2}\hfill & \hfill 0\hfill \\ \hfill \end{array}$

Operator ${Ŝ}_{y}$ in the basis of $\ell =1$ spin-orbitals $|l,{m}_{\ell },{m}_{s}⟩$

 $\begin{array}{ccccccc}\hfill & \hfill |1,1,\frac{1}{2}⟩\hfill & \hfill |1,1,-\frac{1}{2}⟩\hfill & \hfill |1,0,\frac{1}{2}⟩\hfill & \hfill |1,0,-\frac{1}{2}⟩\hfill & \hfill |1,-1,\frac{1}{2}⟩\hfill & \hfill |1,-1,-\frac{1}{2}⟩\hfill \\ ̲& ̲& ̲& ̲& ̲& ̲& ̲\\ |1,1,\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill -\frac{i}{2}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |1,1,-\frac{1}{2}⟩\hfill & \hfill \frac{i}{2}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |1,0,\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{i}{2}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |1,0,-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{i}{2}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |1,-1,\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{i}{2}\hfill \\ |1,-1,-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{i}{2}\hfill & \hfill 0\hfill \\ \hfill \end{array}$

Operator ${Ŝ}_{z}$ in the basis of $\ell =1$ spin-orbitals $|l,{m}_{\ell },{m}_{s}⟩$

 $\begin{array}{ccccccc}\hfill & \hfill |1,1,\frac{1}{2}⟩\hfill & \hfill |1,1,-\frac{1}{2}⟩\hfill & \hfill |1,0,\frac{1}{2}⟩\hfill & \hfill |1,0,-\frac{1}{2}⟩\hfill & \hfill |1,-1,\frac{1}{2}⟩\hfill & \hfill |1,-1,-\frac{1}{2}⟩\hfill \\ ̲& ̲& ̲& ̲& ̲& ̲& ̲\\ |1,1,\frac{1}{2}⟩\hfill & \hfill \frac{1}{2}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |1,1,-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill -\frac{1}{2}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |1,0,\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{1}{2}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |1,0,-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{1}{2}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |1,-1,\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{1}{2}\hfill & \hfill 0\hfill \\ |1,-1,-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{1}{2}\hfill \\ \hfill \end{array}$

Operator ${Ŝ}_{x}$ in the basis of $\ell =1$ SO eigenfunctions $|j,{m}_{j}⟩$

 $\begin{array}{ccccccc}\hfill & \hfill |\frac{1}{2},-\frac{1}{2}⟩\hfill & \hfill |\frac{1}{2},\frac{1}{2}⟩\hfill & \hfill |\frac{3}{2},-\frac{3}{2}⟩\hfill & \hfill |\frac{3}{2},-\frac{1}{2}⟩\hfill & \hfill |\frac{3}{2},\frac{1}{2}⟩\hfill & \hfill |\frac{3}{2},\frac{3}{2}⟩\hfill \\ ̲& ̲& ̲& ̲& ̲& ̲& ̲\\ |\frac{1}{2},-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill -\frac{1}{6}\hfill & \hfill \frac{1}{\sqrt{6}}\hfill & \hfill 0\hfill & \hfill -\frac{1}{3\sqrt{2}}\hfill & \hfill 0\hfill \\ |\frac{1}{2},\frac{1}{2}⟩\hfill & \hfill -\frac{1}{6}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{1}{3\sqrt{2}}\hfill & \hfill 0\hfill & \hfill -\frac{1}{\sqrt{6}}\hfill \\ |\frac{3}{2},-\frac{3}{2}⟩\hfill & \hfill \frac{1}{\sqrt{6}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{1}{2\sqrt{3}}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |\frac{3}{2},-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill \frac{1}{3\sqrt{2}}\hfill & \hfill \frac{1}{2\sqrt{3}}\hfill & \hfill 0\hfill & \hfill \frac{1}{3}\hfill & \hfill 0\hfill \\ |\frac{3}{2},\frac{1}{2}⟩\hfill & \hfill -\frac{1}{3\sqrt{2}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{1}{3}\hfill & \hfill 0\hfill & \hfill \frac{1}{2\sqrt{3}}\hfill \\ |\frac{3}{2},\frac{3}{2}⟩\hfill & \hfill 0\hfill & \hfill -\frac{1}{\sqrt{6}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{1}{2\sqrt{3}}\hfill & \hfill 0\hfill \\ \hfill \end{array}$

Operator ${Ŝ}_{y}$ in the basis of $\ell =1$ SO eigenfunctions $|j,{m}_{j}⟩$

 $\begin{array}{ccccccc}\hfill & \hfill |\frac{1}{2},-\frac{1}{2}⟩\hfill & \hfill |\frac{1}{2},\frac{1}{2}⟩\hfill & \hfill |\frac{3}{2},-\frac{3}{2}⟩\hfill & \hfill |\frac{3}{2},-\frac{1}{2}⟩\hfill & \hfill |\frac{3}{2},\frac{1}{2}⟩\hfill & \hfill |\frac{3}{2},\frac{3}{2}⟩\hfill \\ ̲& ̲& ̲& ̲& ̲& ̲& ̲\\ |\frac{1}{2},-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill -\frac{i}{6}\hfill & \hfill -\frac{i}{\sqrt{6}}\hfill & \hfill 0\hfill & \hfill -\frac{i}{3\sqrt{2}}\hfill & \hfill 0\hfill \\ |\frac{1}{2},\frac{1}{2}⟩\hfill & \hfill \frac{i}{6}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{i}{3\sqrt{2}}\hfill & \hfill 0\hfill & \hfill -\frac{i}{\sqrt{6}}\hfill \\ |\frac{3}{2},-\frac{3}{2}⟩\hfill & \hfill \frac{i}{\sqrt{6}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{i}{2\sqrt{3}}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |\frac{3}{2},-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill \frac{i}{3\sqrt{2}}\hfill & \hfill -\frac{i}{2\sqrt{3}}\hfill & \hfill 0\hfill & \hfill \frac{i}{3}\hfill & \hfill 0\hfill \\ |\frac{3}{2},\frac{1}{2}⟩\hfill & \hfill \frac{i}{3\sqrt{2}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{i}{3}\hfill & \hfill 0\hfill & \hfill \frac{i}{2\sqrt{3}}\hfill \\ |\frac{3}{2},\frac{3}{2}⟩\hfill & \hfill 0\hfill & \hfill \frac{i}{\sqrt{6}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{i}{2\sqrt{3}}\hfill & \hfill 0\hfill \\ \hfill \end{array}$

Operator ${Ŝ}_{z}$ in the basis of $\ell =1$ SO eigenfunctions $|j,{m}_{j}⟩$

 $\begin{array}{ccccccc}\hfill & \hfill |\frac{1}{2},-\frac{1}{2}⟩\hfill & \hfill |\frac{1}{2},\frac{1}{2}⟩\hfill & \hfill |\frac{3}{2},-\frac{3}{2}⟩\hfill & \hfill |\frac{3}{2},-\frac{1}{2}⟩\hfill & \hfill |\frac{3}{2},\frac{1}{2}⟩\hfill & \hfill |\frac{3}{2},\frac{3}{2}⟩\hfill \\ ̲& ̲& ̲& ̲& ̲& ̲& ̲\\ |\frac{1}{2},-\frac{1}{2}⟩\hfill & \hfill \frac{1}{6}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{\sqrt{2}}{3}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |\frac{1}{2},\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill -\frac{1}{6}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{\sqrt{2}}{3}\hfill & \hfill 0\hfill \\ |\frac{3}{2},-\frac{3}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{1}{2}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |\frac{3}{2},-\frac{1}{2}⟩\hfill & \hfill \frac{\sqrt{2}}{3}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{1}{6}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |\frac{3}{2},\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill \frac{\sqrt{2}}{3}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{1}{6}\hfill & \hfill 0\hfill \\ |\frac{3}{2},\frac{3}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{1}{2}\hfill \\ \hfill \end{array}$

Operator ${\stackrel{^}{L}}_{x}$ in the basis of $\ell =1$ spin-orbitals $|l,{m}_{\ell },{m}_{s}⟩$

 $\begin{array}{ccccccc}\hfill & \hfill |1,1,\frac{1}{2}⟩\hfill & \hfill |1,1,-\frac{1}{2}⟩\hfill & \hfill |1,0,\frac{1}{2}⟩\hfill & \hfill |1,0,-\frac{1}{2}⟩\hfill & \hfill |1,-1,\frac{1}{2}⟩\hfill & \hfill |1,-1,-\frac{1}{2}⟩\hfill \\ ̲& ̲& ̲& ̲& ̲& ̲& ̲\\ |1,1,\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{1}{\sqrt{2}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |1,1,-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{1}{\sqrt{2}}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |1,0,\frac{1}{2}⟩\hfill & \hfill \frac{1}{\sqrt{2}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{1}{\sqrt{2}}\hfill & \hfill 0\hfill \\ |1,0,-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill \frac{1}{\sqrt{2}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{1}{\sqrt{2}}\hfill \\ |1,-1,\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{1}{\sqrt{2}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |1,-1,-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{1}{\sqrt{2}}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill \end{array}$

Operator ${\stackrel{^}{L}}_{y}$ in the basis of $\ell =1$ spin-orbitals $|l,{m}_{\ell },{m}_{s}⟩$

 $\begin{array}{ccccccc}\hfill & \hfill |1,1,\frac{1}{2}⟩\hfill & \hfill |1,1,-\frac{1}{2}⟩\hfill & \hfill |1,0,\frac{1}{2}⟩\hfill & \hfill |1,0,-\frac{1}{2}⟩\hfill & \hfill |1,-1,\frac{1}{2}⟩\hfill & \hfill |1,-1,-\frac{1}{2}⟩\hfill \\ ̲& ̲& ̲& ̲& ̲& ̲& ̲\\ |1,1,\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{i}{\sqrt{2}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |1,1,-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{i}{\sqrt{2}}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |1,0,\frac{1}{2}⟩\hfill & \hfill \frac{i}{\sqrt{2}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{i}{\sqrt{2}}\hfill & \hfill 0\hfill \\ |1,0,-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill \frac{i}{\sqrt{2}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{i}{\sqrt{2}}\hfill \\ |1,-1,\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{i}{\sqrt{2}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |1,-1,-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{i}{\sqrt{2}}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill \end{array}$

Operator ${\stackrel{^}{L}}_{z}$ in the basis of $\ell =1$ spin-orbitals $|l,{m}_{\ell },{m}_{s}⟩$

 $\begin{array}{ccccccc}\hfill & \hfill |1,1,\frac{1}{2}⟩\hfill & \hfill |1,1,-\frac{1}{2}⟩\hfill & \hfill |1,0,\frac{1}{2}⟩\hfill & \hfill |1,0,-\frac{1}{2}⟩\hfill & \hfill |1,-1,\frac{1}{2}⟩\hfill & \hfill |1,-1,-\frac{1}{2}⟩\hfill \\ ̲& ̲& ̲& ̲& ̲& ̲& ̲\\ |1,1,\frac{1}{2}⟩\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |1,1,-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |1,0,\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |1,0,-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |1,-1,\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -1\hfill & \hfill 0\hfill \\ |1,-1,-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -1\hfill \\ \hfill \end{array}$

Operator ${\stackrel{^}{L}}_{x}$ in the basis of $\ell =1$ SO eigenfunctions $|j,{m}_{j}⟩$

 $\begin{array}{ccccccc}\hfill & \hfill |\frac{1}{2},-\frac{1}{2}⟩\hfill & \hfill |\frac{1}{2},\frac{1}{2}⟩\hfill & \hfill |\frac{3}{2},-\frac{3}{2}⟩\hfill & \hfill |\frac{3}{2},-\frac{1}{2}⟩\hfill & \hfill |\frac{3}{2},\frac{1}{2}⟩\hfill & \hfill |\frac{3}{2},\frac{3}{2}⟩\hfill \\ ̲& ̲& ̲& ̲& ̲& ̲& ̲\\ |\frac{1}{2},-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill \frac{2}{3}\hfill & \hfill -\frac{1}{\sqrt{6}}\hfill & \hfill 0\hfill & \hfill \frac{1}{3\sqrt{2}}\hfill & \hfill 0\hfill \\ |\frac{1}{2},\frac{1}{2}⟩\hfill & \hfill \frac{2}{3}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{1}{3\sqrt{2}}\hfill & \hfill 0\hfill & \hfill \frac{1}{\sqrt{6}}\hfill \\ |\frac{3}{2},-\frac{3}{2}⟩\hfill & \hfill -\frac{1}{\sqrt{6}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{1}{\sqrt{3}}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |\frac{3}{2},-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill -\frac{1}{3\sqrt{2}}\hfill & \hfill \frac{1}{\sqrt{3}}\hfill & \hfill 0\hfill & \hfill \frac{2}{3}\hfill & \hfill 0\hfill \\ |\frac{3}{2},\frac{1}{2}⟩\hfill & \hfill \frac{1}{3\sqrt{2}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{2}{3}\hfill & \hfill 0\hfill & \hfill \frac{1}{\sqrt{3}}\hfill \\ |\frac{3}{2},\frac{3}{2}⟩\hfill & \hfill 0\hfill & \hfill \frac{1}{\sqrt{6}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{1}{\sqrt{3}}\hfill & \hfill 0\hfill \\ \hfill \end{array}$

Operator ${\stackrel{^}{L}}_{y}$ in the basis of $\ell =1$ SO eigenfunctions $|j,{m}_{j}⟩$

 $\begin{array}{ccccccc}\hfill & \hfill |\frac{1}{2},-\frac{1}{2}⟩\hfill & \hfill |\frac{1}{2},\frac{1}{2}⟩\hfill & \hfill |\frac{3}{2},-\frac{3}{2}⟩\hfill & \hfill |\frac{3}{2},-\frac{1}{2}⟩\hfill & \hfill |\frac{3}{2},\frac{1}{2}⟩\hfill & \hfill |\frac{3}{2},\frac{3}{2}⟩\hfill \\ ̲& ̲& ̲& ̲& ̲& ̲& ̲\\ |\frac{1}{2},-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill \frac{2i}{3}\hfill & \hfill \frac{i}{\sqrt{6}}\hfill & \hfill 0\hfill & \hfill \frac{i}{3\sqrt{2}}\hfill & \hfill 0\hfill \\ |\frac{1}{2},\frac{1}{2}⟩\hfill & \hfill -\frac{2i}{3}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{i}{3\sqrt{2}}\hfill & \hfill 0\hfill & \hfill \frac{i}{\sqrt{6}}\hfill \\ |\frac{3}{2},-\frac{3}{2}⟩\hfill & \hfill -\frac{i}{\sqrt{6}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{i}{\sqrt{3}}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |\frac{3}{2},-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill -\frac{i}{3\sqrt{2}}\hfill & \hfill -\frac{i}{\sqrt{3}}\hfill & \hfill 0\hfill & \hfill \frac{2i}{3}\hfill & \hfill 0\hfill \\ |\frac{3}{2},\frac{1}{2}⟩\hfill & \hfill -\frac{i}{3\sqrt{2}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{2i}{3}\hfill & \hfill 0\hfill & \hfill \frac{i}{\sqrt{3}}\hfill \\ |\frac{3}{2},\frac{3}{2}⟩\hfill & \hfill 0\hfill & \hfill -\frac{i}{\sqrt{6}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{i}{\sqrt{3}}\hfill & \hfill 0\hfill \\ \hfill \end{array}$

Operator ${\stackrel{^}{L}}_{z}$ in the basis of $\ell =1$ SO eigenfunctions $|j,{m}_{j}⟩$

 $\begin{array}{ccccccc}\hfill & \hfill |\frac{1}{2},-\frac{1}{2}⟩\hfill & \hfill |\frac{1}{2},\frac{1}{2}⟩\hfill & \hfill |\frac{3}{2},-\frac{3}{2}⟩\hfill & \hfill |\frac{3}{2},-\frac{1}{2}⟩\hfill & \hfill |\frac{3}{2},\frac{1}{2}⟩\hfill & \hfill |\frac{3}{2},\frac{3}{2}⟩\hfill \\ ̲& ̲& ̲& ̲& ̲& ̲& ̲\\ |\frac{1}{2},-\frac{1}{2}⟩\hfill & \hfill -\frac{2}{3}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{\sqrt{2}}{3}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |\frac{1}{2},\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill \frac{2}{3}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{\sqrt{2}}{3}\hfill & \hfill 0\hfill \\ |\frac{3}{2},-\frac{3}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |\frac{3}{2},-\frac{1}{2}⟩\hfill & \hfill -\frac{\sqrt{2}}{3}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{1}{3}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |\frac{3}{2},\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill -\frac{\sqrt{2}}{3}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{1}{3}\hfill & \hfill 0\hfill \\ |\frac{3}{2},\frac{3}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \\ \hfill \end{array}$

Operator ${\stackrel{^}{L}}_{x}+2{Ŝ}_{x}$ in the basis of $\ell =1$ SO eigenfunctions $|j,{m}_{j}⟩$

 $\begin{array}{ccccccc}\hfill & \hfill |\frac{1}{2},-\frac{1}{2}⟩\hfill & \hfill |\frac{1}{2},\frac{1}{2}⟩\hfill & \hfill |\frac{3}{2},-\frac{3}{2}⟩\hfill & \hfill |\frac{3}{2},-\frac{1}{2}⟩\hfill & \hfill |\frac{3}{2},\frac{1}{2}⟩\hfill & \hfill |\frac{3}{2},\frac{3}{2}⟩\hfill \\ ̲& ̲& ̲& ̲& ̲& ̲& ̲\\ |\frac{1}{2},-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill \frac{1}{3}\hfill & \hfill \frac{1}{\sqrt{6}}\hfill & \hfill 0\hfill & \hfill -\frac{1}{3\sqrt{2}}\hfill & \hfill 0\hfill \\ |\frac{1}{2},\frac{1}{2}⟩\hfill & \hfill \frac{1}{3}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{1}{3\sqrt{2}}\hfill & \hfill 0\hfill & \hfill -\frac{1}{\sqrt{6}}\hfill \\ |\frac{3}{2},-\frac{3}{2}⟩\hfill & \hfill \frac{1}{\sqrt{6}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{2}{\sqrt{3}}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |\frac{3}{2},-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill \frac{1}{3\sqrt{2}}\hfill & \hfill \frac{2}{\sqrt{3}}\hfill & \hfill 0\hfill & \hfill \frac{4}{3}\hfill & \hfill 0\hfill \\ |\frac{3}{2},\frac{1}{2}⟩\hfill & \hfill -\frac{1}{3\sqrt{2}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{4}{3}\hfill & \hfill 0\hfill & \hfill \frac{2}{\sqrt{3}}\hfill \\ |\frac{3}{2},\frac{3}{2}⟩\hfill & \hfill 0\hfill & \hfill -\frac{1}{\sqrt{6}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{2}{\sqrt{3}}\hfill & \hfill 0\hfill \\ \hfill \end{array}$

Operator ${\stackrel{^}{L}}_{y}+2{Ŝ}_{y}$ in the basis of $\ell =1$ SO eigenfunctions $|j,{m}_{j}⟩$

 $\begin{array}{ccccccc}\hfill & \hfill |\frac{1}{2},-\frac{1}{2}⟩\hfill & \hfill |\frac{1}{2},\frac{1}{2}⟩\hfill & \hfill |\frac{3}{2},-\frac{3}{2}⟩\hfill & \hfill |\frac{3}{2},-\frac{1}{2}⟩\hfill & \hfill |\frac{3}{2},\frac{1}{2}⟩\hfill & \hfill |\frac{3}{2},\frac{3}{2}⟩\hfill \\ ̲& ̲& ̲& ̲& ̲& ̲& ̲\\ |\frac{1}{2},-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill \frac{i}{3}\hfill & \hfill -\frac{i}{\sqrt{6}}\hfill & \hfill 0\hfill & \hfill -\frac{i}{3\sqrt{2}}\hfill & \hfill 0\hfill \\ |\frac{1}{2},\frac{1}{2}⟩\hfill & \hfill -\frac{i}{3}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{i}{3\sqrt{2}}\hfill & \hfill 0\hfill & \hfill -\frac{i}{\sqrt{6}}\hfill \\ |\frac{3}{2},-\frac{3}{2}⟩\hfill & \hfill \frac{i}{\sqrt{6}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{2i}{\sqrt{3}}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |\frac{3}{2},-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill \frac{i}{3\sqrt{2}}\hfill & \hfill -\frac{2i}{\sqrt{3}}\hfill & \hfill 0\hfill & \hfill \frac{4i}{3}\hfill & \hfill 0\hfill \\ |\frac{3}{2},\frac{1}{2}⟩\hfill & \hfill \frac{i}{3\sqrt{2}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{4i}{3}\hfill & \hfill 0\hfill & \hfill \frac{2i}{\sqrt{3}}\hfill \\ |\frac{3}{2},\frac{3}{2}⟩\hfill & \hfill 0\hfill & \hfill \frac{i}{\sqrt{6}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{2i}{\sqrt{3}}\hfill & \hfill 0\hfill \\ \hfill \end{array}$

Operator ${\stackrel{^}{L}}_{z}+2{Ŝ}_{z}$ in the basis of $\ell =1$ SO eigenfunctions $|j,{m}_{j}⟩$

 $\begin{array}{ccccccc}\hfill & \hfill |\frac{1}{2},-\frac{1}{2}⟩\hfill & \hfill |\frac{1}{2},\frac{1}{2}⟩\hfill & \hfill |\frac{3}{2},-\frac{3}{2}⟩\hfill & \hfill |\frac{3}{2},-\frac{1}{2}⟩\hfill & \hfill |\frac{3}{2},\frac{1}{2}⟩\hfill & \hfill |\frac{3}{2},\frac{3}{2}⟩\hfill \\ ̲& ̲& ̲& ̲& ̲& ̲& ̲\\ |\frac{1}{2},-\frac{1}{2}⟩\hfill & \hfill -\frac{1}{3}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{\sqrt{2}}{3}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |\frac{1}{2},\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill \frac{1}{3}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{\sqrt{2}}{3}\hfill & \hfill 0\hfill \\ |\frac{3}{2},-\frac{3}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -2\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |\frac{3}{2},-\frac{1}{2}⟩\hfill & \hfill \frac{\sqrt{2}}{3}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{2}{3}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |\frac{3}{2},\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill \frac{\sqrt{2}}{3}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{2}{3}\hfill & \hfill 0\hfill \\ |\frac{3}{2},\frac{3}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 2\hfill \\ \hfill \end{array}$

### $\bullet$dorbitals $\ell =2$

SO Hamiltonian ${Ĥ}^{\text{SO}}=\zeta \stackrel{^}{L}\cdot \stackrel{^}{S}$ in the basis of $\ell =2$ spin-orbitals $|l,{m}_{\ell },{m}_{s}⟩$ (excluding SO coupling constant $\zeta$)

 $\begin{array}{ccccccccccc}\hfill & \hfill |2,2,\frac{1}{2}⟩\hfill & \hfill |2,2,-\frac{1}{2}⟩\hfill & \hfill |2,1,\frac{1}{2}⟩\hfill & \hfill |2,1,-\frac{1}{2}⟩\hfill & \hfill |2,0,\frac{1}{2}⟩\hfill & \hfill |2,0,-\frac{1}{2}⟩\hfill & \hfill |2,-1,\frac{1}{2}⟩\hfill & \hfill |2,-1,-\frac{1}{2}⟩\hfill & \hfill |2,-2,\frac{1}{2}⟩\hfill & \hfill |2,-2,-\frac{1}{2}⟩\hfill \\ ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲\\ |2,2,\frac{1}{2}⟩\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |2,2,-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill -1\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |2,1,\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill \frac{1}{2}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |2,1,-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{1}{2}\hfill & \hfill \sqrt{\frac{3}{2}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |2,0,\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \sqrt{\frac{3}{2}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |2,0,-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \sqrt{\frac{3}{2}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |2,-1,\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \sqrt{\frac{3}{2}}\hfill & \hfill -\frac{1}{2}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |2,-1,-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{1}{2}\hfill & \hfill 1\hfill & \hfill 0\hfill \\ |2,-2,\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill -1\hfill & \hfill 0\hfill \\ |2,-2,-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \\ \hfill \end{array}$

${Ĥ}^{\text{SO}}$ Eigenfunctions in the basis of $\ell =2$ spin-orbitals $|l,{m}_{\ell },{m}_{s}⟩$
(eigenvalues: $-3∕2$ for $j=3∕2$, $+1$ for $j=5∕2$, times $\zeta$)

 $\begin{array}{ccccccccccc}\hfill & \hfill |\frac{3}{2},-\frac{3}{2}⟩\hfill & \hfill |\frac{3}{2},-\frac{1}{2}⟩\hfill & \hfill |\frac{3}{2},\frac{1}{2}⟩\hfill & \hfill |\frac{3}{2},\frac{3}{2}⟩\hfill & \hfill |\frac{5}{2},-\frac{5}{2}⟩\hfill & \hfill |\frac{5}{2},-\frac{3}{2}⟩\hfill & \hfill |\frac{5}{2},-\frac{1}{2}⟩\hfill & \hfill |\frac{5}{2},\frac{1}{2}⟩\hfill & \hfill |\frac{5}{2},\frac{3}{2}⟩\hfill & \hfill |\frac{5}{2},\frac{5}{2}⟩\hfill \\ ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲\\ |2,2,\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \\ |2,2,-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{2}{\sqrt{5}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{1}{\sqrt{5}}\hfill & \hfill 0\hfill \\ |2,1,\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{1}{\sqrt{5}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{2}{\sqrt{5}}\hfill & \hfill 0\hfill \\ |2,1,-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\sqrt{\frac{3}{5}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \sqrt{\frac{2}{5}}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |2,0,\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \sqrt{\frac{2}{5}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \sqrt{\frac{3}{5}}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |2,0,-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill -\sqrt{\frac{2}{5}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \sqrt{\frac{3}{5}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |2,-1,\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill \sqrt{\frac{3}{5}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \sqrt{\frac{2}{5}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |2,-1,-\frac{1}{2}⟩\hfill & \hfill -\frac{1}{\sqrt{5}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{2}{\sqrt{5}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |2,-2,\frac{1}{2}⟩\hfill & \hfill \frac{2}{\sqrt{5}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{1}{\sqrt{5}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |2,-2,-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill \end{array}$

Kramers-conjugates $|j,{m}_{j}{⟩}^{\prime }$ of the ${Ĥ}^{\text{SO}}$ Eigenfunctions in the basis of $\ell =2$ spin-orbitals $|l,{m}_{\ell },{m}_{s}⟩$

 $\begin{array}{ccccccccccc}\hfill & \hfill {|\frac{3}{2},-\frac{3}{2}⟩}^{\prime }\hfill & \hfill {|\frac{3}{2},-\frac{1}{2}⟩}^{\prime }\hfill & \hfill {|\frac{3}{2},\frac{1}{2}⟩}^{\prime }\hfill & \hfill {|\frac{3}{2},\frac{3}{2}⟩}^{\prime }\hfill & \hfill {|\frac{5}{2},-\frac{5}{2}⟩}^{\prime }\hfill & \hfill {|\frac{5}{2},-\frac{3}{2}⟩}^{\prime }\hfill & \hfill {|\frac{5}{2},-\frac{1}{2}⟩}^{\prime }\hfill & \hfill {|\frac{5}{2},\frac{1}{2}⟩}^{\prime }\hfill & \hfill {|\frac{5}{2},\frac{3}{2}⟩}^{\prime }\hfill & \hfill {|\frac{5}{2},\frac{5}{2}⟩}^{\prime }\hfill \\ ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲\\ |2,2,\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |2,2,-\frac{1}{2}⟩\hfill & \hfill \frac{2}{\sqrt{5}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{1}{\sqrt{5}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |2,1,\frac{1}{2}⟩\hfill & \hfill -\frac{1}{\sqrt{5}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{2}{\sqrt{5}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |2,1,-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill -\sqrt{\frac{3}{5}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\sqrt{\frac{2}{5}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |2,0,\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill \sqrt{\frac{2}{5}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\sqrt{\frac{3}{5}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |2,0,-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \sqrt{\frac{2}{5}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \sqrt{\frac{3}{5}}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |2,-1,\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\sqrt{\frac{3}{5}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \sqrt{\frac{2}{5}}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |2,-1,-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{1}{\sqrt{5}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{2}{\sqrt{5}}\hfill & \hfill 0\hfill \\ |2,-2,\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{2}{\sqrt{5}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{1}{\sqrt{5}}\hfill & \hfill 0\hfill \\ |2,-2,-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \\ \hfill \end{array}$

Operator ${Ŝ}_{x}$ in the basis of $\ell =2$ spin-orbitals $|l,{m}_{\ell },{m}_{s}⟩$

 $\begin{array}{ccccccccccc}\hfill & \hfill |2,2,\frac{1}{2}⟩\hfill & \hfill |2,2,-\frac{1}{2}⟩\hfill & \hfill |2,1,\frac{1}{2}⟩\hfill & \hfill |2,1,-\frac{1}{2}⟩\hfill & \hfill |2,0,\frac{1}{2}⟩\hfill & \hfill |2,0,-\frac{1}{2}⟩\hfill & \hfill |2,-1,\frac{1}{2}⟩\hfill & \hfill |2,-1,-\frac{1}{2}⟩\hfill & \hfill |2,-2,\frac{1}{2}⟩\hfill & \hfill |2,-2,-\frac{1}{2}⟩\hfill \\ ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲\\ |2,2,\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill \frac{1}{2}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |2,2,-\frac{1}{2}⟩\hfill & \hfill \frac{1}{2}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |2,1,\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{1}{2}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |2,1,-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{1}{2}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |2,0,\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{1}{2}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |2,0,-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{1}{2}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |2,-1,\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{1}{2}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |2,-1,-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{1}{2}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |2,-2,\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{1}{2}\hfill \\ |2,-2,-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{1}{2}\hfill & \hfill 0\hfill \\ \hfill \end{array}$

Operator ${Ŝ}_{y}$ in the basis of $\ell =2$ spin-orbitals $|l,{m}_{\ell },{m}_{s}⟩$

 $\begin{array}{ccccccccccc}\hfill & \hfill |2,2,\frac{1}{2}⟩\hfill & \hfill |2,2,-\frac{1}{2}⟩\hfill & \hfill |2,1,\frac{1}{2}⟩\hfill & \hfill |2,1,-\frac{1}{2}⟩\hfill & \hfill |2,0,\frac{1}{2}⟩\hfill & \hfill |2,0,-\frac{1}{2}⟩\hfill & \hfill |2,-1,\frac{1}{2}⟩\hfill & \hfill |2,-1,-\frac{1}{2}⟩\hfill & \hfill |2,-2,\frac{1}{2}⟩\hfill & \hfill |2,-2,-\frac{1}{2}⟩\hfill \\ ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲\\ |2,2,\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill -\frac{i}{2}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |2,2,-\frac{1}{2}⟩\hfill & \hfill \frac{i}{2}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |2,1,\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{i}{2}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |2,1,-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{i}{2}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |2,0,\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{i}{2}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |2,0,-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{i}{2}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |2,-1,\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{i}{2}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |2,-1,-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{i}{2}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |2,-2,\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{i}{2}\hfill \\ |2,-2,-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{i}{2}\hfill & \hfill 0\hfill \\ \hfill \end{array}$

Operator ${Ŝ}_{z}$ in the basis of $\ell =2$ spin-orbitals $|l,{m}_{\ell },{m}_{s}⟩$

 $\begin{array}{ccccccccccc}\hfill \hfill & \hfill |2,2,\frac{1}{2}⟩\hfill & \hfill |2,2,-\frac{1}{2}⟩\hfill & \hfill |2,1,\frac{1}{2}⟩\hfill & \hfill |2,1,-\frac{1}{2}⟩\hfill & \hfill |2,0,\frac{1}{2}⟩\hfill & \hfill |2,0,-\frac{1}{2}⟩\hfill & \hfill |2,-1,\frac{1}{2}⟩\hfill & \hfill |2,-1,-\frac{1}{2}⟩\hfill & \hfill |2,-2,\frac{1}{2}⟩\hfill & \hfill |2,-2,-\frac{1}{2}⟩\hfill \\ ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲\\ \hfill |2,2,\frac{1}{2}⟩\hfill & \hfill \frac{1}{2}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill |2,2,-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill -\frac{1}{2}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill |2,1,\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{1}{2}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill |2,1,-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{1}{2}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill |2,0,\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{1}{2}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill |2,0,-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{1}{2}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill |2,-1,\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{1}{2}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill |2,-1,-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{1}{2}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill |2,-2,\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{1}{2}\hfill & \hfill 0\hfill \\ \hfill |2,-2,-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{1}{2}\hfill \\ \hfill \hfill \end{array}$

Operator ${Ŝ}_{x}$ in the basis of $\ell =2$ SO eigenfunctions $|j,{m}_{j}⟩$

 $\begin{array}{ccccccccccc}\hfill & \hfill |\frac{3}{2},-\frac{3}{2}⟩\hfill & \hfill |\frac{3}{2},-\frac{1}{2}⟩\hfill & \hfill |\frac{3}{2},\frac{1}{2}⟩\hfill & \hfill |\frac{3}{2},\frac{3}{2}⟩\hfill & \hfill |\frac{5}{2},-\frac{5}{2}⟩\hfill & \hfill |\frac{5}{2},-\frac{3}{2}⟩\hfill & \hfill |\frac{5}{2},-\frac{1}{2}⟩\hfill & \hfill |\frac{5}{2},\frac{1}{2}⟩\hfill & \hfill |\frac{5}{2},\frac{3}{2}⟩\hfill & \hfill |\frac{5}{2},\frac{5}{2}⟩\hfill \\ ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲\\ |\frac{3}{2},-\frac{3}{2}⟩\hfill & \hfill 0\hfill & \hfill -\frac{\sqrt{3}}{10}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{1}{\sqrt{5}}\hfill & \hfill 0\hfill & \hfill -\frac{1}{5\sqrt{2}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |\frac{3}{2},-\frac{1}{2}⟩\hfill & \hfill -\frac{\sqrt{3}}{10}\hfill & \hfill 0\hfill & \hfill -\frac{1}{5}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{\sqrt{3}}{5}\hfill & \hfill 0\hfill & \hfill -\frac{\sqrt{\frac{3}{2}}}{5}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |\frac{3}{2},\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill -\frac{1}{5}\hfill & \hfill 0\hfill & \hfill -\frac{\sqrt{3}}{10}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{\sqrt{\frac{3}{2}}}{5}\hfill & \hfill 0\hfill & \hfill -\frac{\sqrt{3}}{5}\hfill & \hfill 0\hfill \\ |\frac{3}{2},\frac{3}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{\sqrt{3}}{10}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{1}{5\sqrt{2}}\hfill & \hfill 0\hfill & \hfill -\frac{1}{\sqrt{5}}\hfill \\ |\frac{5}{2},-\frac{5}{2}⟩\hfill & \hfill \frac{1}{\sqrt{5}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{1}{2\sqrt{5}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |\frac{5}{2},-\frac{3}{2}⟩\hfill & \hfill 0\hfill & \hfill \frac{\sqrt{3}}{5}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{1}{2\sqrt{5}}\hfill & \hfill 0\hfill & \hfill \frac{\sqrt{2}}{5}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |\frac{5}{2},-\frac{1}{2}⟩\hfill & \hfill -\frac{1}{5\sqrt{2}}\hfill & \hfill 0\hfill & \hfill \frac{\sqrt{\frac{3}{2}}}{5}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{\sqrt{2}}{5}\hfill & \hfill 0\hfill & \hfill \frac{3}{10}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |\frac{5}{2},\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill -\frac{\sqrt{\frac{3}{2}}}{5}\hfill & \hfill 0\hfill & \hfill \frac{1}{5\sqrt{2}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{3}{10}\hfill & \hfill 0\hfill & \hfill \frac{\sqrt{2}}{5}\hfill & \hfill 0\hfill \\ |\frac{5}{2},\frac{3}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{\sqrt{3}}{5}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{\sqrt{2}}{5}\hfill & \hfill 0\hfill & \hfill \frac{1}{2\sqrt{5}}\hfill \\ |\frac{5}{2},\frac{5}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{1}{\sqrt{5}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{1}{2\sqrt{5}}\hfill & \hfill 0\hfill \\ \hfill \end{array}$

Operator ${Ŝ}_{y}$ in the basis of $\ell =2$ SO eigenfunctions $|j,{m}_{j}⟩$

 $\begin{array}{ccccccccccc}\hfill & \hfill |\frac{3}{2},-\frac{3}{2}⟩\hfill & \hfill |\frac{3}{2},-\frac{1}{2}⟩\hfill & \hfill |\frac{3}{2},\frac{1}{2}⟩\hfill & \hfill |\frac{3}{2},\frac{3}{2}⟩\hfill & \hfill |\frac{5}{2},-\frac{5}{2}⟩\hfill & \hfill |\frac{5}{2},-\frac{3}{2}⟩\hfill & \hfill |\frac{5}{2},-\frac{1}{2}⟩\hfill & \hfill |\frac{5}{2},\frac{1}{2}⟩\hfill & \hfill |\frac{5}{2},\frac{3}{2}⟩\hfill & \hfill |\frac{5}{2},\frac{5}{2}⟩\hfill \\ ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲\\ |\frac{3}{2},-\frac{3}{2}⟩\hfill & \hfill 0\hfill & \hfill -\frac{i\sqrt{3}}{10}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{i}{\sqrt{5}}\hfill & \hfill 0\hfill & \hfill -\frac{i}{5\sqrt{2}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |\frac{3}{2},-\frac{1}{2}⟩\hfill & \hfill \frac{i\sqrt{3}}{10}\hfill & \hfill 0\hfill & \hfill -\frac{i}{5}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{i\sqrt{3}}{5}\hfill & \hfill 0\hfill & \hfill -\frac{1}{5}i\sqrt{\frac{3}{2}}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |\frac{3}{2},\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill \frac{i}{5}\hfill & \hfill 0\hfill & \hfill -\frac{i\sqrt{3}}{10}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{1}{5}i\sqrt{\frac{3}{2}}\hfill & \hfill 0\hfill & \hfill -\frac{i\sqrt{3}}{5}\hfill & \hfill 0\hfill \\ |\frac{3}{2},\frac{3}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{i\sqrt{3}}{10}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{i}{5\sqrt{2}}\hfill & \hfill 0\hfill & \hfill -\frac{i}{\sqrt{5}}\hfill \\ |\frac{5}{2},-\frac{5}{2}⟩\hfill & \hfill \frac{i}{\sqrt{5}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{i}{2\sqrt{5}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |\frac{5}{2},-\frac{3}{2}⟩\hfill & \hfill 0\hfill & \hfill \frac{i\sqrt{3}}{5}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{i}{2\sqrt{5}}\hfill & \hfill 0\hfill & \hfill \frac{i\sqrt{2}}{5}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |\frac{5}{2},-\frac{1}{2}⟩\hfill & \hfill \frac{i}{5\sqrt{2}}\hfill & \hfill 0\hfill & \hfill \frac{1}{5}i\sqrt{\frac{3}{2}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{i\sqrt{2}}{5}\hfill & \hfill 0\hfill & \hfill \frac{3i}{10}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |\frac{5}{2},\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill \frac{1}{5}i\sqrt{\frac{3}{2}}\hfill & \hfill 0\hfill & \hfill \frac{i}{5\sqrt{2}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{3i}{10}\hfill & \hfill 0\hfill & \hfill \frac{i\sqrt{2}}{5}\hfill & \hfill 0\hfill \\ |\frac{5}{2},\frac{3}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{i\sqrt{3}}{5}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{i\sqrt{2}}{5}\hfill & \hfill 0\hfill & \hfill \frac{i}{2\sqrt{5}}\hfill \\ |\frac{5}{2},\frac{5}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{i}{\sqrt{5}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{i}{2\sqrt{5}}\hfill & \hfill 0\hfill \\ \hfill \end{array}$

Operator ${Ŝ}_{z}$ in the basis of $\ell =2$ SO eigenfunctions $|j,{m}_{j}⟩$

 $\begin{array}{ccccccccccc}\hfill & \hfill |\frac{3}{2},-\frac{3}{2}⟩\hfill & \hfill |\frac{3}{2},-\frac{1}{2}⟩\hfill & \hfill |\frac{3}{2},\frac{1}{2}⟩\hfill & \hfill |\frac{3}{2},\frac{3}{2}⟩\hfill & \hfill |\frac{5}{2},-\frac{5}{2}⟩\hfill & \hfill |\frac{5}{2},-\frac{3}{2}⟩\hfill & \hfill |\frac{5}{2},-\frac{1}{2}⟩\hfill & \hfill |\frac{5}{2},\frac{1}{2}⟩\hfill & \hfill |\frac{5}{2},\frac{3}{2}⟩\hfill & \hfill |\frac{5}{2},\frac{5}{2}⟩\hfill \\ ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲\\ |\frac{3}{2},-\frac{3}{2}⟩\hfill & \hfill \frac{3}{10}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{2}{5}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |\frac{3}{2},-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill \frac{1}{10}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{\sqrt{6}}{5}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |\frac{3}{2},\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{1}{10}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{\sqrt{6}}{5}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |\frac{3}{2},\frac{3}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{3}{10}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{2}{5}\hfill & \hfill 0\hfill \\ |\frac{5}{2},-\frac{5}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{1}{2}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |\frac{5}{2},-\frac{3}{2}⟩\hfill & \hfill \frac{2}{5}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{3}{10}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |\frac{5}{2},-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill \frac{\sqrt{6}}{5}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{1}{10}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |\frac{5}{2},\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{\sqrt{6}}{5}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{1}{10}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |\frac{5}{2},\frac{3}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{2}{5}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{3}{10}\hfill & \hfill 0\hfill \\ |\frac{5}{2},\frac{5}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{1}{2}\hfill \\ \hfill \end{array}$

Operator ${\stackrel{^}{L}}_{x}$ in the basis of $\ell =2$ spin-orbitals $|l,{m}_{\ell },{m}_{s}⟩$

 $\begin{array}{ccccccccccc}\hfill & \hfill |2,2,\frac{1}{2}⟩\hfill & \hfill |2,2,-\frac{1}{2}⟩\hfill & \hfill |2,1,\frac{1}{2}⟩\hfill & \hfill |2,1,-\frac{1}{2}⟩\hfill & \hfill |2,0,\frac{1}{2}⟩\hfill & \hfill |2,0,-\frac{1}{2}⟩\hfill & \hfill |2,-1,\frac{1}{2}⟩\hfill & \hfill |2,-1,-\frac{1}{2}⟩\hfill & \hfill |2,-2,\frac{1}{2}⟩\hfill & \hfill |2,-2,-\frac{1}{2}⟩\hfill \\ ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲\\ |2,2,\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |2,2,-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |2,1,\frac{1}{2}⟩\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \sqrt{\frac{3}{2}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |2,1,-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \sqrt{\frac{3}{2}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |2,0,\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \sqrt{\frac{3}{2}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \sqrt{\frac{3}{2}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |2,0,-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \sqrt{\frac{3}{2}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \sqrt{\frac{3}{2}}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |2,-1,\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \sqrt{\frac{3}{2}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ |2,-1,-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \sqrt{\frac{3}{2}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \\ |2,-2,\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |2,-2,-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill \end{array}$

Operator ${\stackrel{^}{L}}_{y}$ in the basis of $\ell =2$ spin-orbitals $|l,{m}_{\ell },{m}_{s}⟩$

 $\begin{array}{ccccccccccc}\hfill & \hfill |2,2,\frac{1}{2}⟩\hfill & \hfill |2,2,-\frac{1}{2}⟩\hfill & \hfill |2,1,\frac{1}{2}⟩\hfill & \hfill |2,1,-\frac{1}{2}⟩\hfill & \hfill |2,0,\frac{1}{2}⟩\hfill & \hfill |2,0,-\frac{1}{2}⟩\hfill & \hfill |2,-1,\frac{1}{2}⟩\hfill & \hfill |2,-1,-\frac{1}{2}⟩\hfill & \hfill |2,-2,\frac{1}{2}⟩\hfill & \hfill |2,-2,-\frac{1}{2}⟩\hfill \\ ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲\\ |2,2,\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -i\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |2,2,-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -i\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |2,1,\frac{1}{2}⟩\hfill & \hfill i\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -i\sqrt{\frac{3}{2}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |2,1,-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill i\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -i\sqrt{\frac{3}{2}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |2,0,\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill i\sqrt{\frac{3}{2}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -i\sqrt{\frac{3}{2}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |2,0,-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill i\sqrt{\frac{3}{2}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -i\sqrt{\frac{3}{2}}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |2,-1,\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill i\sqrt{\frac{3}{2}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -i\hfill & \hfill 0\hfill \\ |2,-1,-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill i\sqrt{\frac{3}{2}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -i\hfill \\ |2,-2,\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill i\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |2,-2,-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill i\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill \end{array}$

Operator ${\stackrel{^}{L}}_{z}$ in the basis of $\ell =2$ spin-orbitals $|l,{m}_{\ell },{m}_{s}⟩$

 $\begin{array}{ccccccccccc}\hfill & \hfill |2,2,\frac{1}{2}⟩\hfill & \hfill |2,2,-\frac{1}{2}⟩\hfill & \hfill |2,1,\frac{1}{2}⟩\hfill & \hfill |2,1,-\frac{1}{2}⟩\hfill & \hfill |2,0,\frac{1}{2}⟩\hfill & \hfill |2,0,-\frac{1}{2}⟩\hfill & \hfill |2,-1,\frac{1}{2}⟩\hfill & \hfill |2,-1,-\frac{1}{2}⟩\hfill & \hfill |2,-2,\frac{1}{2}⟩\hfill & \hfill |2,-2,-\frac{1}{2}⟩\hfill \\ ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲\\ |2,2,\frac{1}{2}⟩\hfill & \hfill 2\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |2,2,-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 2\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |2,1,\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |2,1,-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |2,0,\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |2,0,-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |2,-1,\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |2,-1,-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |2,-2,\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -2\hfill & \hfill 0\hfill \\ |2,-2,-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -2\hfill \\ \hfill \end{array}$

Operator ${\stackrel{^}{L}}_{x}$ in the basis of $\ell =2$ SO eigenfunctions $|j,{m}_{j}⟩$

 $\begin{array}{ccccccccccc}\hfill & \hfill |\frac{3}{2},-\frac{3}{2}⟩\hfill & \hfill |\frac{3}{2},-\frac{1}{2}⟩\hfill & \hfill |\frac{3}{2},\frac{1}{2}⟩\hfill & \hfill |\frac{3}{2},\frac{3}{2}⟩\hfill & \hfill |\frac{5}{2},-\frac{5}{2}⟩\hfill & \hfill |\frac{5}{2},-\frac{3}{2}⟩\hfill & \hfill |\frac{5}{2},-\frac{1}{2}⟩\hfill & \hfill |\frac{5}{2},\frac{1}{2}⟩\hfill & \hfill |\frac{5}{2},\frac{3}{2}⟩\hfill & \hfill |\frac{5}{2},\frac{5}{2}⟩\hfill \\ ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲\\ |\frac{3}{2},-\frac{3}{2}⟩\hfill & \hfill 0\hfill & \hfill \frac{3\sqrt{3}}{5}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{1}{\sqrt{5}}\hfill & \hfill 0\hfill & \hfill \frac{1}{5\sqrt{2}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |\frac{3}{2},-\frac{1}{2}⟩\hfill & \hfill \frac{3\sqrt{3}}{5}\hfill & \hfill 0\hfill & \hfill \frac{6}{5}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{\sqrt{3}}{5}\hfill & \hfill 0\hfill & \hfill \frac{\sqrt{\frac{3}{2}}}{5}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |\frac{3}{2},\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill \frac{6}{5}\hfill & \hfill 0\hfill & \hfill \frac{3\sqrt{3}}{5}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{\sqrt{\frac{3}{2}}}{5}\hfill & \hfill 0\hfill & \hfill \frac{\sqrt{3}}{5}\hfill & \hfill 0\hfill \\ |\frac{3}{2},\frac{3}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{3\sqrt{3}}{5}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{1}{5\sqrt{2}}\hfill & \hfill 0\hfill & \hfill \frac{1}{\sqrt{5}}\hfill \\ |\frac{5}{2},-\frac{5}{2}⟩\hfill & \hfill -\frac{1}{\sqrt{5}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{2}{\sqrt{5}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |\frac{5}{2},-\frac{3}{2}⟩\hfill & \hfill 0\hfill & \hfill -\frac{\sqrt{3}}{5}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{2}{\sqrt{5}}\hfill & \hfill 0\hfill & \hfill \frac{4\sqrt{2}}{5}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |\frac{5}{2},-\frac{1}{2}⟩\hfill & \hfill \frac{1}{5\sqrt{2}}\hfill & \hfill 0\hfill & \hfill -\frac{\sqrt{\frac{3}{2}}}{5}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{4\sqrt{2}}{5}\hfill & \hfill 0\hfill & \hfill \frac{6}{5}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |\frac{5}{2},\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill \frac{\sqrt{\frac{3}{2}}}{5}\hfill & \hfill 0\hfill & \hfill -\frac{1}{5\sqrt{2}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{6}{5}\hfill & \hfill 0\hfill & \hfill \frac{4\sqrt{2}}{5}\hfill & \hfill 0\hfill \\ |\frac{5}{2},\frac{3}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{\sqrt{3}}{5}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{4\sqrt{2}}{5}\hfill & \hfill 0\hfill & \hfill \frac{2}{\sqrt{5}}\hfill \\ |\frac{5}{2},\frac{5}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{1}{\sqrt{5}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{2}{\sqrt{5}}\hfill & \hfill 0\hfill \\ \hfill \end{array}$

Operator ${\stackrel{^}{L}}_{y}$ in the basis of $\ell =2$ SO eigenfunctions $|j,{m}_{j}⟩$

 $\begin{array}{ccccccccccc}\hfill & \hfill |\frac{3}{2},-\frac{3}{2}⟩\hfill & \hfill |\frac{3}{2},-\frac{1}{2}⟩\hfill & \hfill |\frac{3}{2},\frac{1}{2}⟩\hfill & \hfill |\frac{3}{2},\frac{3}{2}⟩\hfill & \hfill |\frac{5}{2},-\frac{5}{2}⟩\hfill & \hfill |\frac{5}{2},-\frac{3}{2}⟩\hfill & \hfill |\frac{5}{2},-\frac{1}{2}⟩\hfill & \hfill |\frac{5}{2},\frac{1}{2}⟩\hfill & \hfill |\frac{5}{2},\frac{3}{2}⟩\hfill & \hfill |\frac{5}{2},\frac{5}{2}⟩\hfill \\ ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲\\ |\frac{3}{2},-\frac{3}{2}⟩\hfill & \hfill 0\hfill & \hfill \frac{3i\sqrt{3}}{5}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{i}{\sqrt{5}}\hfill & \hfill 0\hfill & \hfill \frac{i}{5\sqrt{2}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |\frac{3}{2},-\frac{1}{2}⟩\hfill & \hfill -\frac{3i\sqrt{3}}{5}\hfill & \hfill 0\hfill & \hfill \frac{6i}{5}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{i\sqrt{3}}{5}\hfill & \hfill 0\hfill & \hfill \frac{1}{5}i\sqrt{\frac{3}{2}}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |\frac{3}{2},\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill -\frac{6i}{5}\hfill & \hfill 0\hfill & \hfill \frac{3i\sqrt{3}}{5}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{1}{5}i\sqrt{\frac{3}{2}}\hfill & \hfill 0\hfill & \hfill \frac{i\sqrt{3}}{5}\hfill & \hfill 0\hfill \\ |\frac{3}{2},\frac{3}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{3i\sqrt{3}}{5}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{i}{5\sqrt{2}}\hfill & \hfill 0\hfill & \hfill \frac{i}{\sqrt{5}}\hfill \\ |\frac{5}{2},-\frac{5}{2}⟩\hfill & \hfill -\frac{i}{\sqrt{5}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{2i}{\sqrt{5}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |\frac{5}{2},-\frac{3}{2}⟩\hfill & \hfill 0\hfill & \hfill -\frac{i\sqrt{3}}{5}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{2i}{\sqrt{5}}\hfill & \hfill 0\hfill & \hfill \frac{4i\sqrt{2}}{5}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |\frac{5}{2},-\frac{1}{2}⟩\hfill & \hfill -\frac{i}{5\sqrt{2}}\hfill & \hfill 0\hfill & \hfill -\frac{1}{5}i\sqrt{\frac{3}{2}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{4i\sqrt{2}}{5}\hfill & \hfill 0\hfill & \hfill \frac{6i}{5}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |\frac{5}{2},\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill -\frac{1}{5}i\sqrt{\frac{3}{2}}\hfill & \hfill 0\hfill & \hfill -\frac{i}{5\sqrt{2}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{6i}{5}\hfill & \hfill 0\hfill & \hfill \frac{4i\sqrt{2}}{5}\hfill & \hfill 0\hfill \\ |\frac{5}{2},\frac{3}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{i\sqrt{3}}{5}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{4i\sqrt{2}}{5}\hfill & \hfill 0\hfill & \hfill \frac{2i}{\sqrt{5}}\hfill \\ |\frac{5}{2},\frac{5}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{i}{\sqrt{5}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{2i}{\sqrt{5}}\hfill & \hfill 0\hfill \\ \hfill \end{array}$

Operator ${\stackrel{^}{L}}_{z}$ in the basis of $\ell =2$ SO eigenfunctions $|j,{m}_{j}⟩$

 $\begin{array}{ccccccccccc}\hfill & \hfill |\frac{3}{2},-\frac{3}{2}⟩\hfill & \hfill |\frac{3}{2},-\frac{1}{2}⟩\hfill & \hfill |\frac{3}{2},\frac{1}{2}⟩\hfill & \hfill |\frac{3}{2},\frac{3}{2}⟩\hfill & \hfill |\frac{5}{2},-\frac{5}{2}⟩\hfill & \hfill |\frac{5}{2},-\frac{3}{2}⟩\hfill & \hfill |\frac{5}{2},-\frac{1}{2}⟩\hfill & \hfill |\frac{5}{2},\frac{1}{2}⟩\hfill & \hfill |\frac{5}{2},\frac{3}{2}⟩\hfill & \hfill |\frac{5}{2},\frac{5}{2}⟩\hfill \\ ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲\\ |\frac{3}{2},-\frac{3}{2}⟩\hfill & \hfill -\frac{9}{5}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{2}{5}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |\frac{3}{2},-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill -\frac{3}{5}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{\sqrt{6}}{5}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |\frac{3}{2},\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{3}{5}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{\sqrt{6}}{5}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |\frac{3}{2},\frac{3}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{9}{5}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{2}{5}\hfill & \hfill 0\hfill \\ |\frac{5}{2},-\frac{5}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -2\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |\frac{5}{2},-\frac{3}{2}⟩\hfill & \hfill -\frac{2}{5}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{6}{5}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |\frac{5}{2},-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill -\frac{\sqrt{6}}{5}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{2}{5}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |\frac{5}{2},\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{\sqrt{6}}{5}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{2}{5}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |\frac{5}{2},\frac{3}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{2}{5}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{6}{5}\hfill & \hfill 0\hfill \\ |\frac{5}{2},\frac{5}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 2\hfill \\ \hfill \end{array}$

Operator ${\stackrel{^}{L}}_{x}+2{Ŝ}_{x}$ in the basis of $\ell =2$ SO eigenfunctions $|j,{m}_{j}⟩$

 $\begin{array}{ccccccccccc}\hfill & \hfill |\frac{3}{2},-\frac{3}{2}⟩\hfill & \hfill |\frac{3}{2},-\frac{1}{2}⟩\hfill & \hfill |\frac{3}{2},\frac{1}{2}⟩\hfill & \hfill |\frac{3}{2},\frac{3}{2}⟩\hfill & \hfill |\frac{5}{2},-\frac{5}{2}⟩\hfill & \hfill |\frac{5}{2},-\frac{3}{2}⟩\hfill & \hfill |\frac{5}{2},-\frac{1}{2}⟩\hfill & \hfill |\frac{5}{2},\frac{1}{2}⟩\hfill & \hfill |\frac{5}{2},\frac{3}{2}⟩\hfill & \hfill |\frac{5}{2},\frac{5}{2}⟩\hfill \\ ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲\\ |\frac{3}{2},-\frac{3}{2}⟩\hfill & \hfill 0\hfill & \hfill \frac{2\sqrt{3}}{5}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{1}{\sqrt{5}}\hfill & \hfill 0\hfill & \hfill -\frac{1}{5\sqrt{2}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |\frac{3}{2},-\frac{1}{2}⟩\hfill & \hfill \frac{2\sqrt{3}}{5}\hfill & \hfill 0\hfill & \hfill \frac{4}{5}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{\sqrt{3}}{5}\hfill & \hfill 0\hfill & \hfill -\frac{\sqrt{\frac{3}{2}}}{5}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |\frac{3}{2},\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill \frac{4}{5}\hfill & \hfill 0\hfill & \hfill \frac{2\sqrt{3}}{5}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{\sqrt{\frac{3}{2}}}{5}\hfill & \hfill 0\hfill & \hfill -\frac{\sqrt{3}}{5}\hfill & \hfill 0\hfill \\ |\frac{3}{2},\frac{3}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{2\sqrt{3}}{5}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{1}{5\sqrt{2}}\hfill & \hfill 0\hfill & \hfill -\frac{1}{\sqrt{5}}\hfill \\ |\frac{5}{2},-\frac{5}{2}⟩\hfill & \hfill \frac{1}{\sqrt{5}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{3}{\sqrt{5}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |\frac{5}{2},-\frac{3}{2}⟩\hfill & \hfill 0\hfill & \hfill \frac{\sqrt{3}}{5}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{3}{\sqrt{5}}\hfill & \hfill 0\hfill & \hfill \frac{6\sqrt{2}}{5}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |\frac{5}{2},-\frac{1}{2}⟩\hfill & \hfill -\frac{1}{5\sqrt{2}}\hfill & \hfill 0\hfill & \hfill \frac{\sqrt{\frac{3}{2}}}{5}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{6\sqrt{2}}{5}\hfill & \hfill 0\hfill & \hfill \frac{9}{5}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |\frac{5}{2},\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill -\frac{\sqrt{\frac{3}{2}}}{5}\hfill & \hfill 0\hfill & \hfill \frac{1}{5\sqrt{2}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{9}{5}\hfill & \hfill 0\hfill & \hfill \frac{6\sqrt{2}}{5}\hfill & \hfill 0\hfill \\ |\frac{5}{2},\frac{3}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{\sqrt{3}}{5}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{6\sqrt{2}}{5}\hfill & \hfill 0\hfill & \hfill \frac{3}{\sqrt{5}}\hfill \\ |\frac{5}{2},\frac{5}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{1}{\sqrt{5}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{3}{\sqrt{5}}\hfill & \hfill 0\hfill \\ \hfill \end{array}$

Operator ${\stackrel{^}{L}}_{y}+2{Ŝ}_{y}$ in the basis of $\ell =2$ SO eigenfunctions $|j,{m}_{j}⟩$

 $\begin{array}{ccccccccccc}\hfill & \hfill |\frac{3}{2},-\frac{3}{2}⟩\hfill & \hfill |\frac{3}{2},-\frac{1}{2}⟩\hfill & \hfill |\frac{3}{2},\frac{1}{2}⟩\hfill & \hfill |\frac{3}{2},\frac{3}{2}⟩\hfill & \hfill |\frac{5}{2},-\frac{5}{2}⟩\hfill & \hfill |\frac{5}{2},-\frac{3}{2}⟩\hfill & \hfill |\frac{5}{2},-\frac{1}{2}⟩\hfill & \hfill |\frac{5}{2},\frac{1}{2}⟩\hfill & \hfill |\frac{5}{2},\frac{3}{2}⟩\hfill & \hfill |\frac{5}{2},\frac{5}{2}⟩\hfill \\ ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲\\ |\frac{3}{2},-\frac{3}{2}⟩\hfill & \hfill 0\hfill & \hfill \frac{2i\sqrt{3}}{5}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{i}{\sqrt{5}}\hfill & \hfill 0\hfill & \hfill -\frac{i}{5\sqrt{2}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |\frac{3}{2},-\frac{1}{2}⟩\hfill & \hfill -\frac{2i\sqrt{3}}{5}\hfill & \hfill 0\hfill & \hfill \frac{4i}{5}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{i\sqrt{3}}{5}\hfill & \hfill 0\hfill & \hfill -\frac{1}{5}i\sqrt{\frac{3}{2}}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |\frac{3}{2},\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill -\frac{4i}{5}\hfill & \hfill 0\hfill & \hfill \frac{2i\sqrt{3}}{5}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{1}{5}i\sqrt{\frac{3}{2}}\hfill & \hfill 0\hfill & \hfill -\frac{i\sqrt{3}}{5}\hfill & \hfill 0\hfill \\ |\frac{3}{2},\frac{3}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{2i\sqrt{3}}{5}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{i}{5\sqrt{2}}\hfill & \hfill 0\hfill & \hfill -\frac{i}{\sqrt{5}}\hfill \\ |\frac{5}{2},-\frac{5}{2}⟩\hfill & \hfill \frac{i}{\sqrt{5}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{3i}{\sqrt{5}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |\frac{5}{2},-\frac{3}{2}⟩\hfill & \hfill 0\hfill & \hfill \frac{i\sqrt{3}}{5}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{3i}{\sqrt{5}}\hfill & \hfill 0\hfill & \hfill \frac{6i\sqrt{2}}{5}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |\frac{5}{2},-\frac{1}{2}⟩\hfill & \hfill \frac{i}{5\sqrt{2}}\hfill & \hfill 0\hfill & \hfill \frac{1}{5}i\sqrt{\frac{3}{2}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{6i\sqrt{2}}{5}\hfill & \hfill 0\hfill & \hfill \frac{9i}{5}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |\frac{5}{2},\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill \frac{1}{5}i\sqrt{\frac{3}{2}}\hfill & \hfill 0\hfill & \hfill \frac{i}{5\sqrt{2}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{9i}{5}\hfill & \hfill 0\hfill & \hfill \frac{6i\sqrt{2}}{5}\hfill & \hfill 0\hfill \\ |\frac{5}{2},\frac{3}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{i\sqrt{3}}{5}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{6i\sqrt{2}}{5}\hfill & \hfill 0\hfill & \hfill \frac{3i}{\sqrt{5}}\hfill \\ |\frac{5}{2},\frac{5}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{i}{\sqrt{5}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{3i}{\sqrt{5}}\hfill & \hfill 0\hfill \\ \hfill \end{array}$

Operator ${\stackrel{^}{L}}_{z}+2{Ŝ}_{z}$ in the basis of $\ell =2$ SO eigenfunctions $|j,{m}_{j}⟩$

 $\begin{array}{ccccccccccc}\hfill & \hfill |\frac{3}{2},-\frac{3}{2}⟩\hfill & \hfill |\frac{3}{2},-\frac{1}{2}⟩\hfill & \hfill |\frac{3}{2},\frac{1}{2}⟩\hfill & \hfill |\frac{3}{2},\frac{3}{2}⟩\hfill & \hfill |\frac{5}{2},-\frac{5}{2}⟩\hfill & \hfill |\frac{5}{2},-\frac{3}{2}⟩\hfill & \hfill |\frac{5}{2},-\frac{1}{2}⟩\hfill & \hfill |\frac{5}{2},\frac{1}{2}⟩\hfill & \hfill |\frac{5}{2},\frac{3}{2}⟩\hfill & \hfill |\frac{5}{2},\frac{5}{2}⟩\hfill \\ ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲\\ |\frac{3}{2},-\frac{3}{2}⟩\hfill & \hfill -\frac{6}{5}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{2}{5}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |\frac{3}{2},-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill -\frac{2}{5}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{\sqrt{6}}{5}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |\frac{3}{2},\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{2}{5}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{\sqrt{6}}{5}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |\frac{3}{2},\frac{3}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{6}{5}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{2}{5}\hfill & \hfill 0\hfill \\ |\frac{5}{2},-\frac{5}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -3\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |\frac{5}{2},-\frac{3}{2}⟩\hfill & \hfill \frac{2}{5}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{9}{5}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |\frac{5}{2},-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill \frac{\sqrt{6}}{5}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{3}{5}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |\frac{5}{2},\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{\sqrt{6}}{5}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{3}{5}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |\frac{5}{2},\frac{3}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{2}{5}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{9}{5}\hfill & \hfill 0\hfill \\ |\frac{5}{2},\frac{5}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 3\hfill \\ \hfill \end{array}$

### $\bullet$forbitals $\ell =3$

SO Hamiltonian ${Ĥ}^{\text{SO}}=\zeta \stackrel{^}{L}\cdot \stackrel{^}{S}$ in the basis of $\ell =3$ spin-orbitals $|l,{m}_{\ell },{m}_{s}⟩$ (excluding SO coupling constant $\zeta$)

 $\begin{array}{ccccccccccccccc}\hfill & \hfill |3,3,\frac{1}{2}⟩\hfill & \hfill |3,3,-\frac{1}{2}⟩\hfill & \hfill |3,2,\frac{1}{2}⟩\hfill & \hfill |3,2,-\frac{1}{2}⟩\hfill & \hfill |3,1,\frac{1}{2}⟩\hfill & \hfill |3,1,-\frac{1}{2}⟩\hfill & \hfill |3,0,\frac{1}{2}⟩\hfill & \hfill |3,0,-\frac{1}{2}⟩\hfill & \hfill |3,-1,\frac{1}{2}⟩\hfill & \hfill |3,-1,-\frac{1}{2}⟩\hfill & \hfill |3,-2,\frac{1}{2}⟩\hfill & \hfill |3,-2,-\frac{1}{2}⟩\hfill & \hfill |3,-3,\frac{1}{2}⟩\hfill & \hfill |3,-3,-\frac{1}{2}⟩\hfill \\ ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲\\ |3,3,\frac{1}{2}⟩\hfill & \hfill \frac{3}{2}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |3,3,-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill -\frac{3}{2}\hfill & \hfill \sqrt{\frac{3}{2}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |3,2,\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill \sqrt{\frac{3}{2}}\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |3,2,-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -1\hfill & \hfill \sqrt{\frac{5}{2}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |3,1,\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \sqrt{\frac{5}{2}}\hfill & \hfill \frac{1}{2}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |3,1,-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{1}{2}\hfill & \hfill \sqrt{3}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |3,0,\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \sqrt{3}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |3,0,-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \sqrt{3}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |3,-1,\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \sqrt{3}\hfill & \hfill -\frac{1}{2}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |3,-1,-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{1}{2}\hfill & \hfill \sqrt{\frac{5}{2}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |3,-2,\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \sqrt{\frac{5}{2}}\hfill & \hfill -1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |3,-2,-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill \sqrt{\frac{3}{2}}\hfill & \hfill 0\hfill \\ |3,-3,\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \sqrt{\frac{3}{2}}\hfill & \hfill -\frac{3}{2}\hfill & \hfill 0\hfill \\ |3,-3,-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{3}{2}\hfill \end{array}$

${Ĥ}^{\text{SO}}$ Eigenfunctions in the basis of $\ell =3$ spin-orbitals $|l,{m}_{\ell },{m}_{s}⟩$
(eigenvalues: $-2$ for $j=5∕2$, $+3∕2$ for $j=7∕2$, times $\zeta$)

 $\begin{array}{ccccccccccccccc}\hfill & \hfill |\frac{5}{2},-\frac{5}{2}⟩\hfill & \hfill |\frac{5}{2},-\frac{3}{2}⟩\hfill & \hfill |\frac{5}{2},-\frac{1}{2}⟩\hfill & \hfill |\frac{5}{2},\frac{1}{2}⟩\hfill & \hfill |\frac{5}{2},\frac{3}{2}⟩\hfill & \hfill |\frac{5}{2},\frac{5}{2}⟩\hfill & \hfill |\frac{7}{2},-\frac{7}{2}⟩\hfill & \hfill |\frac{7}{2},-\frac{5}{2}⟩\hfill & \hfill |\frac{7}{2},-\frac{3}{2}⟩\hfill & \hfill |\frac{7}{2},-\frac{1}{2}⟩\hfill & \hfill |\frac{7}{2},\frac{1}{2}⟩\hfill & \hfill |\frac{7}{2},\frac{3}{2}⟩\hfill & \hfill |\frac{7}{2},\frac{5}{2}⟩\hfill & \hfill |\frac{7}{2},\frac{7}{2}⟩\hfill \\ ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲\\ |3,3,\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \\ |3,3,-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\sqrt{\frac{6}{7}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{1}{\sqrt{7}}\hfill & \hfill 0\hfill \\ |3,2,\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{1}{\sqrt{7}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \sqrt{\frac{6}{7}}\hfill & \hfill 0\hfill \\ |3,2,-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\sqrt{\frac{5}{7}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \sqrt{\frac{2}{7}}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |3,1,\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \sqrt{\frac{2}{7}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \sqrt{\frac{5}{7}}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |3,1,-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{2}{\sqrt{7}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \sqrt{\frac{3}{7}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |3,0,\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \sqrt{\frac{3}{7}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{2}{\sqrt{7}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |3,0,-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\sqrt{\frac{3}{7}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{2}{\sqrt{7}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |3,-1,\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{2}{\sqrt{7}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \sqrt{\frac{3}{7}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |3,-1,-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill -\sqrt{\frac{2}{7}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \sqrt{\frac{5}{7}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |3,-2,\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill \sqrt{\frac{5}{7}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \sqrt{\frac{2}{7}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |3,-2,-\frac{1}{2}⟩\hfill & \hfill -\frac{1}{\sqrt{7}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \sqrt{\frac{6}{7}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |3,-3,\frac{1}{2}⟩\hfill & \hfill \sqrt{\frac{6}{7}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{1}{\sqrt{7}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |3,-3,-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill \end{array}$

Kramers-conjugates $|j,{m}_{j}{⟩}^{\prime }$ of the ${Ĥ}^{\text{SO}}$ Eigenfunctions in the basis of $\ell =3$ spin-orbitals $|l,{m}_{\ell },{m}_{s}⟩$

 $\begin{array}{ccccccccccccccc}\hfill & \hfill |\frac{5}{2},-\frac{5}{2}\text{}⟩\text{’}\hfill & \hfill |\frac{5}{2},-\frac{3}{2}\text{}⟩\text{’}\hfill & \hfill |\frac{5}{2},-\frac{1}{2}\text{}⟩\text{’}\hfill & \hfill |\frac{5}{2},\frac{1}{2}\text{}⟩\text{’}\hfill & \hfill |\frac{5}{2},\frac{3}{2}\text{}⟩\text{’}\hfill & \hfill |\frac{5}{2},\frac{5}{2}\text{}⟩\text{’}\hfill & \hfill |\frac{7}{2},-\frac{7}{2}\text{}⟩\text{’}\hfill & \hfill |\frac{7}{2},-\frac{5}{2}\text{}⟩\text{’}\hfill & \hfill |\frac{7}{2},-\frac{3}{2}\text{}⟩\text{’}\hfill & \hfill |\frac{7}{2},-\frac{1}{2}\text{}⟩\text{’}\hfill & \hfill |\frac{7}{2},\frac{1}{2}\text{}⟩\text{’}\hfill & \hfill |\frac{7}{2},\frac{3}{2}\text{}⟩\text{’}\hfill & \hfill |\frac{7}{2},\frac{5}{2}\text{}⟩\text{’}\hfill & \hfill |\frac{7}{2},\frac{7}{2}\text{}⟩\text{’}\hfill \\ ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲\\ |3,3,\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |3,3,-\frac{1}{2}⟩\hfill & \hfill -\sqrt{\frac{6}{7}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{1}{\sqrt{7}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |3,2,\frac{1}{2}⟩\hfill & \hfill \frac{1}{\sqrt{7}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\sqrt{\frac{6}{7}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |3,2,-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill \sqrt{\frac{5}{7}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \sqrt{\frac{2}{7}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |3,1,\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill -\sqrt{\frac{2}{7}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \sqrt{\frac{5}{7}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |3,1,-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{2}{\sqrt{7}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\sqrt{\frac{3}{7}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |3,0,\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \sqrt{\frac{3}{7}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{2}{\sqrt{7}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |3,0,-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \sqrt{\frac{3}{7}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{2}{\sqrt{7}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |3,-1,\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{2}{\sqrt{7}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \sqrt{\frac{3}{7}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |3,-1,-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\sqrt{\frac{2}{7}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\sqrt{\frac{5}{7}}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |3,-2,\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \sqrt{\frac{5}{7}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\sqrt{\frac{2}{7}}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |3,-2,-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{1}{\sqrt{7}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \sqrt{\frac{6}{7}}\hfill & \hfill 0\hfill \\ |3,-3,\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\sqrt{\frac{6}{7}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{1}{\sqrt{7}}\hfill & \hfill 0\hfill \\ |3,-3,-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -1\hfill \\ \hfill \end{array}$

Operator ${Ŝ}_{x}$ in the basis of $\ell =3$ spin-orbitals $|l,{m}_{\ell },{m}_{s}⟩$

 $\begin{array}{ccccccccccccccc}\hfill & \hfill |3,3,\frac{1}{2}⟩\hfill & \hfill |3,3,-\frac{1}{2}⟩\hfill & \hfill |3,2,\frac{1}{2}⟩\hfill & \hfill |3,2,-\frac{1}{2}⟩\hfill & \hfill |3,1,\frac{1}{2}⟩\hfill & \hfill |3,1,-\frac{1}{2}⟩\hfill & \hfill |3,0,\frac{1}{2}⟩\hfill & \hfill |3,0,-\frac{1}{2}⟩\hfill & \hfill |3,-1,\frac{1}{2}⟩\hfill & \hfill |3,-1,-\frac{1}{2}⟩\hfill & \hfill |3,-2,\frac{1}{2}⟩\hfill & \hfill |3,-2,-\frac{1}{2}⟩\hfill & \hfill |3,-3,\frac{1}{2}⟩\hfill & \hfill |3,-3,-\frac{1}{2}⟩\hfill \\ ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲\\ |3,3,\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill \frac{1}{2}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |3,3,-\frac{1}{2}⟩\hfill & \hfill \frac{1}{2}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |3,2,\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{1}{2}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |3,2,-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{1}{2}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |3,1,\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{1}{2}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |3,1,-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{1}{2}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |3,0,\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{1}{2}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |3,0,-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{1}{2}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |3,-1,\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{1}{2}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |3,-1,-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{1}{2}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |3,-2,\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{1}{2}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |3,-2,-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{1}{2}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |3,-3,\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{1}{2}\hfill \\ |3,-3,-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{1}{2}\hfill & \hfill 0\hfill \\ \hfill \end{array}$

Operator ${Ŝ}_{y}$ in the basis of $\ell =3$ spin-orbitals $|l,{m}_{\ell },{m}_{s}⟩$

 $\begin{array}{ccccccccccccccc}\hfill & \hfill |3,3,\frac{1}{2}⟩\hfill & \hfill |3,3,-\frac{1}{2}⟩\hfill & \hfill |3,2,\frac{1}{2}⟩\hfill & \hfill |3,2,-\frac{1}{2}⟩\hfill & \hfill |3,1,\frac{1}{2}⟩\hfill & \hfill |3,1,-\frac{1}{2}⟩\hfill & \hfill |3,0,\frac{1}{2}⟩\hfill & \hfill |3,0,-\frac{1}{2}⟩\hfill & \hfill |3,-1,\frac{1}{2}⟩\hfill & \hfill |3,-1,-\frac{1}{2}⟩\hfill & \hfill |3,-2,\frac{1}{2}⟩\hfill & \hfill |3,-2,-\frac{1}{2}⟩\hfill & \hfill |3,-3,\frac{1}{2}⟩\hfill & \hfill |3,-3,-\frac{1}{2}⟩\hfill \\ ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲\\ |3,3,\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill -\frac{i}{2}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |3,3,-\frac{1}{2}⟩\hfill & \hfill \frac{i}{2}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |3,2,\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{i}{2}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |3,2,-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{i}{2}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |3,1,\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{i}{2}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |3,1,-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{i}{2}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |3,0,\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{i}{2}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |3,0,-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{i}{2}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |3,-1,\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{i}{2}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |3,-1,-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{i}{2}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |3,-2,\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{i}{2}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |3,-2,-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{i}{2}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |3,-3,\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{i}{2}\hfill \\ |3,-3,-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{i}{2}\hfill & \hfill 0\hfill \\ \hfill \end{array}$

Operator ${Ŝ}_{z}$ in the basis of $\ell =3$ spin-orbitals $|l,{m}_{\ell },{m}_{s}⟩$

 $\begin{array}{ccccccccccccccc}\hfill & \hfill |3,3,\frac{1}{2}⟩\hfill & \hfill |3,3,-\frac{1}{2}⟩\hfill & \hfill |3,2,\frac{1}{2}⟩\hfill & \hfill |3,2,-\frac{1}{2}⟩\hfill & \hfill |3,1,\frac{1}{2}⟩\hfill & \hfill |3,1,-\frac{1}{2}⟩\hfill & \hfill |3,0,\frac{1}{2}⟩\hfill & \hfill |3,0,-\frac{1}{2}⟩\hfill & \hfill |3,-1,\frac{1}{2}⟩\hfill & \hfill |3,-1,-\frac{1}{2}⟩\hfill & \hfill |3,-2,\frac{1}{2}⟩\hfill & \hfill |3,-2,-\frac{1}{2}⟩\hfill & \hfill |3,-3,\frac{1}{2}⟩\hfill & \hfill |3,-3,-\frac{1}{2}⟩\hfill \\ ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲\\ |3,3,\frac{1}{2}⟩\hfill & \hfill \frac{1}{2}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |3,3,-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill -\frac{1}{2}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |3,2,\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{1}{2}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |3,2,-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{1}{2}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |3,1,\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{1}{2}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |3,1,-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{1}{2}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |3,0,\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{1}{2}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |3,0,-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{1}{2}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |3,-1,\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{1}{2}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |3,-1,-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{1}{2}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |3,-2,\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{1}{2}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |3,-2,-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{1}{2}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |3,-3,\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{1}{2}\hfill & \hfill 0\hfill \\ |3,-3,-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{1}{2}\hfill \\ \hfill \end{array}$

Operator ${Ŝ}_{x}$ in the basis of $\ell =3$ SO eigenfunctions $|j,{m}_{j}⟩$

 $\begin{array}{ccccccccccccccc}\hfill & \hfill |\frac{5}{2},-\frac{5}{2}⟩\hfill & \hfill |\frac{5}{2},-\frac{3}{2}⟩\hfill & \hfill |\frac{5}{2},-\frac{1}{2}⟩\hfill & \hfill |\frac{5}{2},\frac{1}{2}⟩\hfill & \hfill |\frac{5}{2},\frac{3}{2}⟩\hfill & \hfill |\frac{5}{2},\frac{5}{2}⟩\hfill & \hfill |\frac{7}{2},-\frac{7}{2}⟩\hfill & \hfill |\frac{7}{2},-\frac{5}{2}⟩\hfill & \hfill |\frac{7}{2},-\frac{3}{2}⟩\hfill & \hfill |\frac{7}{2},-\frac{1}{2}⟩\hfill & \hfill |\frac{7}{2},\frac{1}{2}⟩\hfill & \hfill |\frac{7}{2},\frac{3}{2}⟩\hfill & \hfill |\frac{7}{2},\frac{5}{2}⟩\hfill & \hfill |\frac{7}{2},\frac{7}{2}⟩\hfill \\ ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲\\ |\frac{5}{2},-\frac{5}{2}⟩\hfill & \hfill 0\hfill & \hfill -\frac{\sqrt{5}}{14}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \sqrt{\frac{3}{14}}\hfill & \hfill 0\hfill & \hfill -\frac{1}{7\sqrt{2}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |\frac{5}{2},-\frac{3}{2}⟩\hfill & \hfill -\frac{\sqrt{5}}{14}\hfill & \hfill 0\hfill & \hfill -\frac{\sqrt{2}}{7}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{\sqrt{\frac{15}{2}}}{7}\hfill & \hfill 0\hfill & \hfill -\frac{\sqrt{\frac{3}{2}}}{7}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |\frac{5}{2},-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill -\frac{\sqrt{2}}{7}\hfill & \hfill 0\hfill & \hfill -\frac{3}{14}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{\sqrt{5}}{7}\hfill & \hfill 0\hfill & \hfill -\frac{\sqrt{3}}{7}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |\frac{5}{2},\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{3}{14}\hfill & \hfill 0\hfill & \hfill -\frac{\sqrt{2}}{7}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{\sqrt{3}}{7}\hfill & \hfill 0\hfill & \hfill -\frac{\sqrt{5}}{7}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |\frac{5}{2},\frac{3}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{\sqrt{2}}{7}\hfill & \hfill 0\hfill & \hfill -\frac{\sqrt{5}}{14}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{\sqrt{\frac{3}{2}}}{7}\hfill & \hfill 0\hfill & \hfill -\frac{\sqrt{\frac{15}{2}}}{7}\hfill & \hfill 0\hfill \\ |\frac{5}{2},\frac{5}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{\sqrt{5}}{14}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{1}{7\sqrt{2}}\hfill & \hfill 0\hfill & \hfill -\sqrt{\frac{3}{14}}\hfill \\ |\frac{7}{2},-\frac{7}{2}⟩\hfill & \hfill \sqrt{\frac{3}{14}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{1}{2\sqrt{7}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |\frac{7}{2},-\frac{5}{2}⟩\hfill & \hfill 0\hfill & \hfill \frac{\sqrt{\frac{15}{2}}}{7}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{1}{2\sqrt{7}}\hfill & \hfill 0\hfill & \hfill \frac{\sqrt{3}}{7}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |\frac{7}{2},-\frac{3}{2}⟩\hfill & \hfill -\frac{1}{7\sqrt{2}}\hfill & \hfill 0\hfill & \hfill \frac{\sqrt{5}}{7}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{\sqrt{3}}{7}\hfill & \hfill 0\hfill & \hfill \frac{\sqrt{15}}{14}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |\frac{7}{2},-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill -\frac{\sqrt{\frac{3}{2}}}{7}\hfill & \hfill 0\hfill & \hfill \frac{\sqrt{3}}{7}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{\sqrt{15}}{14}\hfill & \hfill 0\hfill & \hfill \frac{2}{7}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |\frac{7}{2},\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{\sqrt{3}}{7}\hfill & \hfill 0\hfill & \hfill \frac{\sqrt{\frac{3}{2}}}{7}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{2}{7}\hfill & \hfill 0\hfill & \hfill \frac{\sqrt{15}}{14}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |\frac{7}{2},\frac{3}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{\sqrt{5}}{7}\hfill & \hfill 0\hfill & \hfill \frac{1}{7\sqrt{2}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{\sqrt{15}}{14}\hfill & \hfill 0\hfill & \hfill \frac{\sqrt{3}}{7}\hfill & \hfill 0\hfill \\ |\frac{7}{2},\frac{5}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{\sqrt{\frac{15}{2}}}{7}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{\sqrt{3}}{7}\hfill & \hfill 0\hfill & \hfill \frac{1}{2\sqrt{7}}\hfill \\ |\frac{7}{2},\frac{7}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\sqrt{\frac{3}{14}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{1}{2\sqrt{7}}\hfill & \hfill 0\hfill \\ \hfill \end{array}$

Operator ${Ŝ}_{y}$ in the basis of $\ell =3$ SO eigenfunctions $|j,{m}_{j}⟩$

 $\begin{array}{ccccccccccccccc}\hfill & \hfill |\frac{5}{2},-\frac{5}{2}⟩\hfill & \hfill |\frac{5}{2},-\frac{3}{2}⟩\hfill & \hfill |\frac{5}{2},-\frac{1}{2}⟩\hfill & \hfill |\frac{5}{2},\frac{1}{2}⟩\hfill & \hfill |\frac{5}{2},\frac{3}{2}⟩\hfill & \hfill |\frac{5}{2},\frac{5}{2}⟩\hfill & \hfill |\frac{7}{2},-\frac{7}{2}⟩\hfill & \hfill |\frac{7}{2},-\frac{5}{2}⟩\hfill & \hfill |\frac{7}{2},-\frac{3}{2}⟩\hfill & \hfill |\frac{7}{2},-\frac{1}{2}⟩\hfill & \hfill |\frac{7}{2},\frac{1}{2}⟩\hfill & \hfill |\frac{7}{2},\frac{3}{2}⟩\hfill & \hfill |\frac{7}{2},\frac{5}{2}⟩\hfill & \hfill |\frac{7}{2},\frac{7}{2}⟩\hfill \\ ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲\\ |\frac{5}{2},-\frac{5}{2}⟩\hfill & \hfill 0\hfill & \hfill -\frac{i\sqrt{5}}{14}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -i\sqrt{\frac{3}{14}}\hfill & \hfill 0\hfill & \hfill -\frac{i}{7\sqrt{2}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |\frac{5}{2},-\frac{3}{2}⟩\hfill & \hfill \frac{i\sqrt{5}}{14}\hfill & \hfill 0\hfill & \hfill -\frac{i\sqrt{2}}{7}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{1}{7}i\sqrt{\frac{15}{2}}\hfill & \hfill 0\hfill & \hfill -\frac{1}{7}i\sqrt{\frac{3}{2}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |\frac{5}{2},-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill \frac{i\sqrt{2}}{7}\hfill & \hfill 0\hfill & \hfill -\frac{3i}{14}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{i\sqrt{5}}{7}\hfill & \hfill 0\hfill & \hfill -\frac{i\sqrt{3}}{7}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |\frac{5}{2},\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{3i}{14}\hfill & \hfill 0\hfill & \hfill -\frac{i\sqrt{2}}{7}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{i\sqrt{3}}{7}\hfill & \hfill 0\hfill & \hfill -\frac{i\sqrt{5}}{7}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |\frac{5}{2},\frac{3}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{i\sqrt{2}}{7}\hfill & \hfill 0\hfill & \hfill -\frac{i\sqrt{5}}{14}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{1}{7}i\sqrt{\frac{3}{2}}\hfill & \hfill 0\hfill & \hfill -\frac{1}{7}i\sqrt{\frac{15}{2}}\hfill & \hfill 0\hfill \\ |\frac{5}{2},\frac{5}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{i\sqrt{5}}{14}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{i}{7\sqrt{2}}\hfill & \hfill 0\hfill & \hfill -i\sqrt{\frac{3}{14}}\hfill \\ |\frac{7}{2},-\frac{7}{2}⟩\hfill & \hfill i\sqrt{\frac{3}{14}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{i}{2\sqrt{7}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |\frac{7}{2},-\frac{5}{2}⟩\hfill & \hfill 0\hfill & \hfill \frac{1}{7}i\sqrt{\frac{15}{2}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{i}{2\sqrt{7}}\hfill & \hfill 0\hfill & \hfill \frac{i\sqrt{3}}{7}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |\frac{7}{2},-\frac{3}{2}⟩\hfill & \hfill \frac{i}{7\sqrt{2}}\hfill & \hfill 0\hfill & \hfill \frac{i\sqrt{5}}{7}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{i\sqrt{3}}{7}\hfill & \hfill 0\hfill & \hfill \frac{i\sqrt{15}}{14}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |\frac{7}{2},-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill \frac{1}{7}i\sqrt{\frac{3}{2}}\hfill & \hfill 0\hfill & \hfill \frac{i\sqrt{3}}{7}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{i\sqrt{15}}{14}\hfill & \hfill 0\hfill & \hfill \frac{2i}{7}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |\frac{7}{2},\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{i\sqrt{3}}{7}\hfill & \hfill 0\hfill & \hfill \frac{1}{7}i\sqrt{\frac{3}{2}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{2i}{7}\hfill & \hfill 0\hfill & \hfill \frac{i\sqrt{15}}{14}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |\frac{7}{2},\frac{3}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{i\sqrt{5}}{7}\hfill & \hfill 0\hfill & \hfill \frac{i}{7\sqrt{2}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{i\sqrt{15}}{14}\hfill & \hfill 0\hfill & \hfill \frac{i\sqrt{3}}{7}\hfill & \hfill 0\hfill \\ |\frac{7}{2},\frac{5}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{1}{7}i\sqrt{\frac{15}{2}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{i\sqrt{3}}{7}\hfill & \hfill 0\hfill & \hfill \frac{i}{2\sqrt{7}}\hfill \\ |\frac{7}{2},\frac{7}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill i\sqrt{\frac{3}{14}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{i}{2\sqrt{7}}\hfill & \hfill 0\hfill \\ \hfill \end{array}$

Operator ${Ŝ}_{z}$ in the basis of $\ell =3$ SO eigenfunctions $|j,{m}_{j}⟩$

 $\begin{array}{ccccccccccccccc}\hfill & \hfill |\frac{5}{2},-\frac{5}{2}⟩\hfill & \hfill |\frac{5}{2},-\frac{3}{2}⟩\hfill & \hfill |\frac{5}{2},-\frac{1}{2}⟩\hfill & \hfill |\frac{5}{2},\frac{1}{2}⟩\hfill & \hfill |\frac{5}{2},\frac{3}{2}⟩\hfill & \hfill |\frac{5}{2},\frac{5}{2}⟩\hfill & \hfill |\frac{7}{2},-\frac{7}{2}⟩\hfill & \hfill |\frac{7}{2},-\frac{5}{2}⟩\hfill & \hfill |\frac{7}{2},-\frac{3}{2}⟩\hfill & \hfill |\frac{7}{2},-\frac{1}{2}⟩\hfill & \hfill |\frac{7}{2},\frac{1}{2}⟩\hfill & \hfill |\frac{7}{2},\frac{3}{2}⟩\hfill & \hfill |\frac{7}{2},\frac{5}{2}⟩\hfill & \hfill |\frac{7}{2},\frac{7}{2}⟩\hfill \\ ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲\\ |\frac{5}{2},-\frac{5}{2}⟩\hfill & \hfill \frac{5}{14}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{\sqrt{6}}{7}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |\frac{5}{2},-\frac{3}{2}⟩\hfill & \hfill 0\hfill & \hfill \frac{3}{14}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{\sqrt{10}}{7}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |\frac{5}{2},-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{1}{14}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{2\sqrt{3}}{7}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |\frac{5}{2},\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{1}{14}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{2\sqrt{3}}{7}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |\frac{5}{2},\frac{3}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{3}{14}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{\sqrt{10}}{7}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |\frac{5}{2},\frac{5}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{5}{14}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{\sqrt{6}}{7}\hfill & \hfill 0\hfill \\ |\frac{7}{2},-\frac{7}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{1}{2}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |\frac{7}{2},-\frac{5}{2}⟩\hfill & \hfill \frac{\sqrt{6}}{7}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{5}{14}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |\frac{7}{2},-\frac{3}{2}⟩\hfill & \hfill 0\hfill & \hfill \frac{\sqrt{10}}{7}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{3}{14}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |\frac{7}{2},-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{2\sqrt{3}}{7}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{1}{14}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |\frac{7}{2},\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{2\sqrt{3}}{7}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{1}{14}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |\frac{7}{2},\frac{3}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{\sqrt{10}}{7}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{3}{14}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |\frac{7}{2},\frac{5}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{\sqrt{6}}{7}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{5}{14}\hfill & \hfill 0\hfill \\ |\frac{7}{2},\frac{7}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{1}{2}\hfill \\ \hfill \end{array}$

Operator ${\stackrel{^}{L}}_{x}$ in the basis of $\ell =3$ spin-orbitals $|l,{m}_{\ell },{m}_{s}⟩$

 $\begin{array}{ccccccccccccccc}\hfill & \hfill |3,3,\frac{1}{2}⟩\hfill & \hfill |3,3,-\frac{1}{2}⟩\hfill & \hfill |3,2,\frac{1}{2}⟩\hfill & \hfill |3,2,-\frac{1}{2}⟩\hfill & \hfill |3,1,\frac{1}{2}⟩\hfill & \hfill |3,1,-\frac{1}{2}⟩\hfill & \hfill |3,0,\frac{1}{2}⟩\hfill & \hfill |3,0,-\frac{1}{2}⟩\hfill & \hfill |3,-1,\frac{1}{2}⟩\hfill & \hfill |3,-1,-\frac{1}{2}⟩\hfill & \hfill |3,-2,\frac{1}{2}⟩\hfill & \hfill |3,-2,-\frac{1}{2}⟩\hfill & \hfill |3,-3,\frac{1}{2}⟩\hfill & \hfill |3,-3,-\frac{1}{2}⟩\hfill \\ ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲\\ |3,3,\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \sqrt{\frac{3}{2}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |3,3,-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \sqrt{\frac{3}{2}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |3,2,\frac{1}{2}⟩\hfill & \hfill \sqrt{\frac{3}{2}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \sqrt{\frac{5}{2}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |3,2,-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill \sqrt{\frac{3}{2}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \sqrt{\frac{5}{2}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |3,1,\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \sqrt{\frac{5}{2}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \sqrt{3}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |3,1,-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \sqrt{\frac{5}{2}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \sqrt{3}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |3,0,\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \sqrt{3}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \sqrt{3}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |3,0,-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \sqrt{3}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \sqrt{3}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |3,-1,\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \sqrt{3}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \sqrt{\frac{5}{2}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |3,-1,-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \sqrt{3}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \sqrt{\frac{5}{2}}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |3,-2,\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \sqrt{\frac{5}{2}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \sqrt{\frac{3}{2}}\hfill & \hfill 0\hfill \\ |3,-2,-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \sqrt{\frac{5}{2}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \sqrt{\frac{3}{2}}\hfill \\ |3,-3,\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \sqrt{\frac{3}{2}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |3,-3,-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \sqrt{\frac{3}{2}}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill \end{array}$

Operator ${\stackrel{^}{L}}_{y}$ in the basis of $\ell =3$ spin-orbitals $|l,{m}_{\ell },{m}_{s}⟩$

 $\begin{array}{ccccccccccccccc}\hfill & \hfill |3,3,\frac{1}{2}⟩\hfill & \hfill |3,3,-\frac{1}{2}⟩\hfill & \hfill |3,2,\frac{1}{2}⟩\hfill & \hfill |3,2,-\frac{1}{2}⟩\hfill & \hfill |3,1,\frac{1}{2}⟩\hfill & \hfill |3,1,-\frac{1}{2}⟩\hfill & \hfill |3,0,\frac{1}{2}⟩\hfill & \hfill |3,0,-\frac{1}{2}⟩\hfill & \hfill |3,-1,\frac{1}{2}⟩\hfill & \hfill |3,-1,-\frac{1}{2}⟩\hfill & \hfill |3,-2,\frac{1}{2}⟩\hfill & \hfill |3,-2,-\frac{1}{2}⟩\hfill & \hfill |3,-3,\frac{1}{2}⟩\hfill & \hfill |3,-3,-\frac{1}{2}⟩\hfill \\ ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲\\ |3,3,\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -i\sqrt{\frac{3}{2}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |3,3,-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -i\sqrt{\frac{3}{2}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |3,2,\frac{1}{2}⟩\hfill & \hfill i\sqrt{\frac{3}{2}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -i\sqrt{\frac{5}{2}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |3,2,-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill i\sqrt{\frac{3}{2}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -i\sqrt{\frac{5}{2}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |3,1,\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill i\sqrt{\frac{5}{2}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -i\sqrt{3}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |3,1,-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill i\sqrt{\frac{5}{2}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -i\sqrt{3}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |3,0,\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill i\sqrt{3}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -i\sqrt{3}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |3,0,-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill i\sqrt{3}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -i\sqrt{3}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |3,-1,\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill i\sqrt{3}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -i\sqrt{\frac{5}{2}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |3,-1,-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill i\sqrt{3}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -i\sqrt{\frac{5}{2}}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |3,-2,\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill i\sqrt{\frac{5}{2}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -i\sqrt{\frac{3}{2}}\hfill & \hfill 0\hfill \\ |3,-2,-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill i\sqrt{\frac{5}{2}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -i\sqrt{\frac{3}{2}}\hfill \\ |3,-3,\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill i\sqrt{\frac{3}{2}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |3,-3,-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill i\sqrt{\frac{3}{2}}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill \end{array}$

Operator ${\stackrel{^}{L}}_{z}$ in the basis of $\ell =3$ spin-orbitals $|l,{m}_{\ell },{m}_{s}⟩$

 $\begin{array}{ccccccccccccccc}\hfill & \hfill |3,3,\frac{1}{2}⟩\hfill & \hfill |3,3,-\frac{1}{2}⟩\hfill & \hfill |3,2,\frac{1}{2}⟩\hfill & \hfill |3,2,-\frac{1}{2}⟩\hfill & \hfill |3,1,\frac{1}{2}⟩\hfill & \hfill |3,1,-\frac{1}{2}⟩\hfill & \hfill |3,0,\frac{1}{2}⟩\hfill & \hfill |3,0,-\frac{1}{2}⟩\hfill & \hfill |3,-1,\frac{1}{2}⟩\hfill & \hfill |3,-1,-\frac{1}{2}⟩\hfill & \hfill |3,-2,\frac{1}{2}⟩\hfill & \hfill |3,-2,-\frac{1}{2}⟩\hfill & \hfill |3,-3,\frac{1}{2}⟩\hfill & \hfill |3,-3,-\frac{1}{2}⟩\hfill \\ ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲\\ |3,3,\frac{1}{2}⟩\hfill & \hfill 3\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |3,3,-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 3\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |3,2,\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 2\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |3,2,-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 2\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |3,1,\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |3,1,-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |3,0,\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |3,0,-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |3,-1,\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |3,-1,-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |3,-2,\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -2\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |3,-2,-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -2\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |3,-3,\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -3\hfill & \hfill 0\hfill \\ |3,-3,-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -3\hfill \\ \hfill \end{array}$

Operator ${\stackrel{^}{L}}_{x}$ in the basis of $\ell =3$ SO eigenfunctions $|j,{m}_{j}⟩$

 $\begin{array}{ccccccccccccccc}\hfill & \hfill |\frac{5}{2},-\frac{5}{2}⟩\hfill & \hfill |\frac{5}{2},-\frac{3}{2}⟩\hfill & \hfill |\frac{5}{2},-\frac{1}{2}⟩\hfill & \hfill |\frac{5}{2},\frac{1}{2}⟩\hfill & \hfill |\frac{5}{2},\frac{3}{2}⟩\hfill & \hfill |\frac{5}{2},\frac{5}{2}⟩\hfill & \hfill |\frac{7}{2},-\frac{7}{2}⟩\hfill & \hfill |\frac{7}{2},-\frac{5}{2}⟩\hfill & \hfill |\frac{7}{2},-\frac{3}{2}⟩\hfill & \hfill |\frac{7}{2},-\frac{1}{2}⟩\hfill & \hfill |\frac{7}{2},\frac{1}{2}⟩\hfill & \hfill |\frac{7}{2},\frac{3}{2}⟩\hfill & \hfill |\frac{7}{2},\frac{5}{2}⟩\hfill & \hfill |\frac{7}{2},\frac{7}{2}⟩\hfill \\ ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲\\ |\frac{5}{2},-\frac{5}{2}⟩\hfill & \hfill 0\hfill & \hfill \frac{4\sqrt{5}}{7}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\sqrt{\frac{3}{14}}\hfill & \hfill 0\hfill & \hfill \frac{1}{7\sqrt{2}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |\frac{5}{2},-\frac{3}{2}⟩\hfill & \hfill \frac{4\sqrt{5}}{7}\hfill & \hfill 0\hfill & \hfill \frac{8\sqrt{2}}{7}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{\sqrt{\frac{15}{2}}}{7}\hfill & \hfill 0\hfill & \hfill \frac{\sqrt{\frac{3}{2}}}{7}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |\frac{5}{2},-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill \frac{8\sqrt{2}}{7}\hfill & \hfill 0\hfill & \hfill \frac{12}{7}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{\sqrt{5}}{7}\hfill & \hfill 0\hfill & \hfill \frac{\sqrt{3}}{7}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |\frac{5}{2},\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{12}{7}\hfill & \hfill 0\hfill & \hfill \frac{8\sqrt{2}}{7}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{\sqrt{3}}{7}\hfill & \hfill 0\hfill & \hfill \frac{\sqrt{5}}{7}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |\frac{5}{2},\frac{3}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{8\sqrt{2}}{7}\hfill & \hfill 0\hfill & \hfill \frac{4\sqrt{5}}{7}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{\sqrt{\frac{3}{2}}}{7}\hfill & \hfill 0\hfill & \hfill \frac{\sqrt{\frac{15}{2}}}{7}\hfill & \hfill 0\hfill \\ |\frac{5}{2},\frac{5}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{4\sqrt{5}}{7}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{1}{7\sqrt{2}}\hfill & \hfill 0\hfill & \hfill \sqrt{\frac{3}{14}}\hfill \\ |\frac{7}{2},-\frac{7}{2}⟩\hfill & \hfill -\sqrt{\frac{3}{14}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{3}{\sqrt{7}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |\frac{7}{2},-\frac{5}{2}⟩\hfill & \hfill 0\hfill & \hfill -\frac{\sqrt{\frac{15}{2}}}{7}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{3}{\sqrt{7}}\hfill & \hfill 0\hfill & \hfill \frac{6\sqrt{3}}{7}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |\frac{7}{2},-\frac{3}{2}⟩\hfill & \hfill \frac{1}{7\sqrt{2}}\hfill & \hfill 0\hfill & \hfill -\frac{\sqrt{5}}{7}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{6\sqrt{3}}{7}\hfill & \hfill 0\hfill & \hfill \frac{3\sqrt{15}}{7}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |\frac{7}{2},-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill \frac{\sqrt{\frac{3}{2}}}{7}\hfill & \hfill 0\hfill & \hfill -\frac{\sqrt{3}}{7}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{3\sqrt{15}}{7}\hfill & \hfill 0\hfill & \hfill \frac{12}{7}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |\frac{7}{2},\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{\sqrt{3}}{7}\hfill & \hfill 0\hfill & \hfill -\frac{\sqrt{\frac{3}{2}}}{7}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{12}{7}\hfill & \hfill 0\hfill & \hfill \frac{3\sqrt{15}}{7}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |\frac{7}{2},\frac{3}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{\sqrt{5}}{7}\hfill & \hfill 0\hfill & \hfill -\frac{1}{7\sqrt{2}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{3\sqrt{15}}{7}\hfill & \hfill 0\hfill & \hfill \frac{6\sqrt{3}}{7}\hfill & \hfill 0\hfill \\ |\frac{7}{2},\frac{5}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{\sqrt{\frac{15}{2}}}{7}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{6\sqrt{3}}{7}\hfill & \hfill 0\hfill & \hfill \frac{3}{\sqrt{7}}\hfill \\ |\frac{7}{2},\frac{7}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \sqrt{\frac{3}{14}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{3}{\sqrt{7}}\hfill & \hfill 0\hfill \\ \hfill \end{array}$

Operator ${\stackrel{^}{L}}_{y}$ in the basis of $\ell =3$ SO eigenfunctions $|j,{m}_{j}⟩$

 $\begin{array}{ccccccccccccccc}\hfill & \hfill |\frac{5}{2},-\frac{5}{2}⟩\hfill & \hfill |\frac{5}{2},-\frac{3}{2}⟩\hfill & \hfill |\frac{5}{2},-\frac{1}{2}⟩\hfill & \hfill |\frac{5}{2},\frac{1}{2}⟩\hfill & \hfill |\frac{5}{2},\frac{3}{2}⟩\hfill & \hfill |\frac{5}{2},\frac{5}{2}⟩\hfill & \hfill |\frac{7}{2},-\frac{7}{2}⟩\hfill & \hfill |\frac{7}{2},-\frac{5}{2}⟩\hfill & \hfill |\frac{7}{2},-\frac{3}{2}⟩\hfill & \hfill |\frac{7}{2},-\frac{1}{2}⟩\hfill & \hfill |\frac{7}{2},\frac{1}{2}⟩\hfill & \hfill |\frac{7}{2},\frac{3}{2}⟩\hfill & \hfill |\frac{7}{2},\frac{5}{2}⟩\hfill & \hfill |\frac{7}{2},\frac{7}{2}⟩\hfill \\ ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲\\ |\frac{5}{2},-\frac{5}{2}⟩\hfill & \hfill 0\hfill & \hfill \frac{4i\sqrt{5}}{7}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill i\sqrt{\frac{3}{14}}\hfill & \hfill 0\hfill & \hfill \frac{i}{7\sqrt{2}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |\frac{5}{2},-\frac{3}{2}⟩\hfill & \hfill -\frac{4i\sqrt{5}}{7}\hfill & \hfill 0\hfill & \hfill \frac{8i\sqrt{2}}{7}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{1}{7}i\sqrt{\frac{15}{2}}\hfill & \hfill 0\hfill & \hfill \frac{1}{7}i\sqrt{\frac{3}{2}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |\frac{5}{2},-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill -\frac{8i\sqrt{2}}{7}\hfill & \hfill 0\hfill & \hfill \frac{12i}{7}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{i\sqrt{5}}{7}\hfill & \hfill 0\hfill & \hfill \frac{i\sqrt{3}}{7}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |\frac{5}{2},\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{12i}{7}\hfill & \hfill 0\hfill & \hfill \frac{8i\sqrt{2}}{7}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{i\sqrt{3}}{7}\hfill & \hfill 0\hfill & \hfill \frac{i\sqrt{5}}{7}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |\frac{5}{2},\frac{3}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{8i\sqrt{2}}{7}\hfill & \hfill 0\hfill & \hfill \frac{4i\sqrt{5}}{7}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{1}{7}i\sqrt{\frac{3}{2}}\hfill & \hfill 0\hfill & \hfill \frac{1}{7}i\sqrt{\frac{15}{2}}\hfill & \hfill 0\hfill \\ |\frac{5}{2},\frac{5}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{4i\sqrt{5}}{7}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{i}{7\sqrt{2}}\hfill & \hfill 0\hfill & \hfill i\sqrt{\frac{3}{14}}\hfill \\ |\frac{7}{2},-\frac{7}{2}⟩\hfill & \hfill -i\sqrt{\frac{3}{14}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{3i}{\sqrt{7}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |\frac{7}{2},-\frac{5}{2}⟩\hfill & \hfill 0\hfill & \hfill -\frac{1}{7}i\sqrt{\frac{15}{2}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{3i}{\sqrt{7}}\hfill & \hfill 0\hfill & \hfill \frac{6i\sqrt{3}}{7}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |\frac{7}{2},-\frac{3}{2}⟩\hfill & \hfill -\frac{i}{7\sqrt{2}}\hfill & \hfill 0\hfill & \hfill -\frac{i\sqrt{5}}{7}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{6i\sqrt{3}}{7}\hfill & \hfill 0\hfill & \hfill \frac{3i\sqrt{15}}{7}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |\frac{7}{2},-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill -\frac{1}{7}i\sqrt{\frac{3}{2}}\hfill & \hfill 0\hfill & \hfill -\frac{i\sqrt{3}}{7}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{3i\sqrt{15}}{7}\hfill & \hfill 0\hfill & \hfill \frac{12i}{7}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |\frac{7}{2},\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{i\sqrt{3}}{7}\hfill & \hfill 0\hfill & \hfill -\frac{1}{7}i\sqrt{\frac{3}{2}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{12i}{7}\hfill & \hfill 0\hfill & \hfill \frac{3i\sqrt{15}}{7}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |\frac{7}{2},\frac{3}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{i\sqrt{5}}{7}\hfill & \hfill 0\hfill & \hfill -\frac{i}{7\sqrt{2}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{3i\sqrt{15}}{7}\hfill & \hfill 0\hfill & \hfill \frac{6i\sqrt{3}}{7}\hfill & \hfill 0\hfill \\ |\frac{7}{2},\frac{5}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{1}{7}i\sqrt{\frac{15}{2}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{6i\sqrt{3}}{7}\hfill & \hfill 0\hfill & \hfill \frac{3i}{\sqrt{7}}\hfill \\ |\frac{7}{2},\frac{7}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -i\sqrt{\frac{3}{14}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{3i}{\sqrt{7}}\hfill & \hfill 0\hfill \\ \hfill \end{array}$

Operator ${\stackrel{^}{L}}_{z}$ in the basis of $\ell =3$ SO eigenfunctions $|j,{m}_{j}⟩$

 $\begin{array}{ccccccccccccccc}\hfill & \hfill |\frac{5}{2},-\frac{5}{2}⟩\hfill & \hfill |\frac{5}{2},-\frac{3}{2}⟩\hfill & \hfill |\frac{5}{2},-\frac{1}{2}⟩\hfill & \hfill |\frac{5}{2},\frac{1}{2}⟩\hfill & \hfill |\frac{5}{2},\frac{3}{2}⟩\hfill & \hfill |\frac{5}{2},\frac{5}{2}⟩\hfill & \hfill |\frac{7}{2},-\frac{7}{2}⟩\hfill & \hfill |\frac{7}{2},-\frac{5}{2}⟩\hfill & \hfill |\frac{7}{2},-\frac{3}{2}⟩\hfill & \hfill |\frac{7}{2},-\frac{1}{2}⟩\hfill & \hfill |\frac{7}{2},\frac{1}{2}⟩\hfill & \hfill |\frac{7}{2},\frac{3}{2}⟩\hfill & \hfill |\frac{7}{2},\frac{5}{2}⟩\hfill & \hfill |\frac{7}{2},\frac{7}{2}⟩\hfill \\ ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲& ̲\\ |\frac{5}{2},-\frac{5}{2}⟩\hfill & \hfill -\frac{20}{7}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{\sqrt{6}}{7}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |\frac{5}{2},-\frac{3}{2}⟩\hfill & \hfill 0\hfill & \hfill -\frac{12}{7}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{\sqrt{10}}{7}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ |\frac{5}{2},-\frac{1}{2}⟩\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \hfill \end{array}$