Angular Momentum & Spin
From $SU(2)$ to the Bloch sphere — quantizing rotations.
Orbital angular momentum $\mathbf L = \mathbf r \times \mathbf p$ obeys $[L_i, L_j] = i\hbar \epsilon_{ijk} L_k$. The same algebra is realized by intrinsic spin, which for spin-$\tfrac{1}{2}$ uses the Pauli matrices
$$\sigma_x = \begin{pmatrix}0&1\\1&0\end{pmatrix}, \; \sigma_y = \begin{pmatrix}0&-i\\i&0\end{pmatrix}, \; \sigma_z = \begin{pmatrix}1&0\\0&-1\end{pmatrix}.$$A general spin state $|\psi\rangle = \cos(\theta/2)|{\uparrow}\rangle + e^{i\varphi}\sin(\theta/2)|{\downarrow}\rangle$ maps to a point on the Bloch sphere with polar angle $\theta$ and azimuth $\varphi$. A rotation by $2\pi$ takes $|\psi\rangle \to -|\psi\rangle$ — the famous spinor sign.