What is Angular momentum operators?

Orbital angular momentum is the operator $\hat{\vect L} = \hat{\vect r}\wedge\hat{\vect p}$, componentwise $\hat L_z = \hat x\hat p_y - \hat y\hat p_x$ and cyclic. From $[\hat x, \hat p_x] = \iu\hbar$:

the components are mutually incompatible — no state has two of them sharp — but the total square is compatible with any one of them: the pair $(\hat L^2, \hat L_z)$ is the standard choice of labels. These commutators are exactly $\iu\hbar$ times the Poisson brackets computed in Example 2.12: Dirac’s dictionary at work.