Angular momentum operators satisfy the commutation relations $$[\hat L_x,\hat L_y]=i\hbar \hat L_z$$ and cyclic permutations. Their eigenvalues are $$L^2 = \ell(\ell+1)\hbar^2, \quad L_z = m\hbar.$$

Quantization follows from the algebra of rotations and the requirement that wavefunctions be single valued under physical rotations.

Spin is intrinsic angular momentum, not literal spinning in the classical sense. Electrons have spin $s=\frac{1}{2}$ and two possible outcomes for $S_z$: $\pm \frac{\hbar}{2}$.

Spin-$\frac{1}{2}$ states are described by two-component spinors. Their transformations under rotation differ from ordinary vectors and require $4\pi$ rotations to return to the original state vector exactly.

Spin is central to magnetic resonance, quantum computing qubits, spectroscopy, and solid-state magnetism.