Angular momentum operator

Summary

The angular momentum operator is the quantum version of the classical angular momentum $L = x \times p$. Its three components are self-adjoint operators that do not commute: $[L_x, L_y] = i\hbar L_z$ and cyclic permutations. They generate the action of rotations on wave functions. Triples of operators with these relations that integrate to an action of rotations come from unitary representations of SU(2). As a consequence, the squared angular momentum takes the values $\hbar^2 j(j+1)$, where $j$ is an integer or half-integer. Each component then takes the values $\hbar m$ with $m = -j, \ldots, j$.