Angular momentum is generated by rotations. In quantum mechanics, the operators satisfy the commutation relations $$[\hat{L}_i,\hat{L}_j]=i\hbar\,\varepsilon_{ijk}\hat{L}_k.$$ The allowed total angular momentum values are quantized, and the square of angular momentum is $\hat{L}^2=l(l+1)\hbar^2$.

Spin is intrinsic angular momentum not associated with literal spatial rotation of a classical object. For spin-$\tfrac{1}{2}$ particles, measurements of spin along any axis yield values $\pm\hbar/2$. Spin is essential for understanding magnetism, atomic structure, and particle statistics.

When combining systems, angular momentum values add according to selection rules rather than simple arithmetic alone. For two spins $\tfrac{1}{2}$, the combined system can form a triplet state with total spin $1$ or a singlet state with total spin $0$. This underlies many-body structure and entanglement.