Cheat sheets/Spin matrices



\sigma_x = \sigma_1 = \begin{pmatrix} 0&1\\ 1&0 \end{pmatrix} $$



\sigma_y = \sigma_2 = \begin{pmatrix} 0&-i\\ i&0 \end{pmatrix} $$



\sigma_z = \sigma_3 = \begin{pmatrix} 1&0\\ 0&-1 \end{pmatrix} $$