Spherical Tensors for a Spin-3/2

T_{0,0} = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}

T_{1,0} = I_z = \begin{bmatrix} \frac{3}{2} & 0 & 0 & 0 \\ 0 & \frac{1}{2} & 0 & 0 \\ 0 & 0 & -\frac{1}{2} & 0 \\ 0 & 0 & 0 & -\frac{3}{2} \end{bmatrix}

T_{1,1} = -\frac{1}{\sqrt{2}} I_+ = \begin{bmatrix} 0 & -\sqrt{\frac{3}{2}} & 0 & 0 \\ 0 & 0 & -\sqrt{2} & 0 \\ 0 & 0 & 0 & -\sqrt{\frac{3}{2}} \\ 0 & 0 & 0 & 0 \end{bmatrix}

T_{1,-1} = \frac{1}{\sqrt{2}} I_- = \begin{bmatrix} 0 & 0 & 0 & 0 \\ \sqrt{\frac{3}{2}} & 0 & 0 & 0 \\ 0 & \sqrt{2} & 0 & 0 \\ 0 & 0 & \sqrt{\frac{3}{2}} & 0 \end{bmatrix}

T_{2,0} = \frac{1}{\sqrt{6}} \left[ 3 I_z^2 - I(I+1) \right] = \begin{bmatrix} \sqrt{\frac{3}{2}} & 0 & 0 & 0 \\ 0 & -\sqrt{\frac{3}{2}} & 0 & 0 \\ 0 & 0 & -\sqrt{\frac{3}{2}} & 0 \\ 0 & 0 & 0 & \sqrt{\frac{3}{2}} \end{bmatrix}

T_{2,1} = -\frac{1}{\sqrt{2}} \left[ I_z I_+ + I_+ I_z \right] = \begin{bmatrix} 0 & -\sqrt{3} & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & \sqrt{3} \\ 0 & 0 & 0 & 0 \end{bmatrix}

T_{2,-1} = \frac{1}{\sqrt{2}} \left[ I_z I_- + I_- I_z \right] = \begin{bmatrix} 0 & 0 & 0 & 0 \\ \sqrt{3} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & -\sqrt{3} & 0 \end{bmatrix}

T_{2,2} = \frac{1}{2} I_+^2 = \begin{bmatrix} 0 & 0 & \sqrt{3} & 0 \\ 0 & 0 & 0 & \sqrt{3} \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix}

T_{2,-2} = \frac{1}{2} I_-^2 = \begin{bmatrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ \sqrt{3} & 0 & 0 & 0 \\ 0 & \sqrt{3} & 0 & 0 \end{bmatrix}

T_{3,0} = \frac{1}{\sqrt{10}} \left[ 5 I_z^3 - \left\{ 3I(I+1)-1 \right\} I_z \right] = \begin{bmatrix} \frac{3 \sqrt{10}}{20} & 0 & 0 & 0 \\ 0 & -\frac{9 \sqrt{10}}{20} & 0 & 0 \\ 0 & 0 & \frac{9 \sqrt{10}}{20} & 0 \\ 0 & 0 & 0 & -\frac{3 \sqrt{10}}{20}                  \end{bmatrix}

T_{3,1} = - \frac{1}{2} \left( \frac{3}{10} \right)^{\frac{1}{2}} \frac{1}{2} \left[ 5 I_z^2 - I(I+1) - \frac{1}{2}, I_+ \right]_+ = \begin{bmatrix} 0 & -\frac{3\sqrt{10}}{10} & 0 & 0 \\ 0 & 0 & \frac{3\sqrt{30}}{10} & 0 \\ 0 & 0 & 0 & -\frac{3\sqrt{10}}{10} \\ 0 & 0 & 0 & 0 \end{bmatrix}

T_{3,-1} = \frac{1}{2} \left( \frac{3}{10} \right)^{\frac{1}{2}} \frac{1}{2} \left[ 5 I_z^2 - I(I+1) - \frac{1}{2}, I_- \right]_+ = \begin{bmatrix} 0 & 0 & 0 & 0 \\ \frac{3\sqrt{10}}{10} & 0 & 0 & 0 \\ 0 & -\frac{3\sqrt{30}}{10} & 0 & 0 \\ 0 & 0 & \frac{3\sqrt{10}}{10} & 0 \end{bmatrix}

T_{3,2} = \left( \frac{3}{4} \right)^{\frac{1}{2}} \frac{1}{2} \left[ I_z I_+^2 + I_+^2 I_z \right] = \begin{bmatrix} 0 & 0 & \frac{3}{2} & 0 \\ 0 & 0 & 0 & -\frac{3}{2} \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix}

T_{3,-2} = \left( \frac{3}{4} \right)^{\frac{1}{2}} \frac{1}{2} \left[ I_z I_-^2 + I_-^2 I_z \right] = \begin{bmatrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ \frac{3}{2} & 0 & 0 & 0 \\ 0 & -\frac{3}{2} & 0 & 0 \end{bmatrix}

T_{3,3} = -\frac{1}{2} \left( \frac{1}{2} \right)^{\frac{1}{2}} I_+^3 = \begin{bmatrix} 0 & 0 & 0 & - \frac{3 \sqrt{2}}{2} \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix}

T_{3,-3} = \frac{1}{2} \left( \frac{1}{2} \right)^{\frac{1}{2}} I_-^3 = \begin{bmatrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ \frac{3 \sqrt{2}}{2} & 0 & 0 & 0 \end{bmatrix}

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