Cheat sheets/Spin matrices

${\sigma }_x={\sigma }_1=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)$

${\sigma }_y={\sigma }_2=\left(\begin{array}{cc}0& -i\\ i& 0\end{array}\right)$

${\sigma }_z={\sigma }_3=\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right)$