PX262 - G4 - pauli spin matrices

σ_{x} = (\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix}), σ_{y} = (\begin{matrix} 0 & - i \\ i & 0 \end{matrix}), σ_{z} = (\begin{matrix} 1 & 0 \\ 0 & - 1 \end{matrix})

with these spin operators for spin $1 / 2$

{\hat{S}}_{i} = \frac{1}{2} ℏ σ_{i}

\begin{aligned} [{\hat{S}}_{x}, {\hat{S}}_{y}] & = \frac{ℏ^{2}}{4} [(\begin{array}{c} 0 & 1 \\ 1 & 0 \end{array}) (\begin{array}{c} 0 & - i \\ i & 0 \end{array}) - (\begin{array}{c} 0 & - i \\ i & 0 \end{array}) (\begin{array}{c} 0 & 1 \\ 1 & 0 \end{array})] \\ = \frac{ℏ^{2}}{4} [(\begin{array}{c} i & 0 \\ 0 & - i \end{array}) - (\begin{array}{c} - i & 0 \\ 0 & i \end{array})] \\ = i ℏ \frac{ℏ}{2} (\begin{array}{c} 1 & 0 \\ 0 & - 1 \end{array}) \\ ∴ [{\hat{S}}_{x}, {\hat{S}}_{y}] & = i ℏ {\hat{S}}_{z} \end{aligned}

the eigenvalues of ${\hat{S}}_{z}$ are $+ \frac{1}{2} ℏ$ and $- \frac{1}{2} ℏ$ , with the eigenvectors $(\begin{matrix} 1 \\ 0 \end{matrix})$ and $(\begin{matrix} 0 \\ 1 \end{matrix})$
for ${\hat{S}}_{x}$ , the eigenvalues are $\pm \frac{1}{2} ℏ$ , and the eigenvectors are $\frac{1}{\sqrt{2}} (\begin{matrix} 1 \\ i \end{matrix})$ and $\frac{1}{\sqrt{2}} (\begin{matrix} 1 \\ - i \end{matrix})$
the operator:

{\hat{S}}^{2} = {\hat{S}}_{x}^{2} + {\hat{S}}_{y}^{2} + {\hat{S}}_{z}^{2} = \frac{3}{4} ℏ^{2} (\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix})