Quantum angular momentum is represented by three operators $L_x,L_y,L_z$ whose components do not commute: $$[L_x,L_y]=i\hbar L_z,$$ with cyclic permutations. Because the components are incompatible observables, a state cannot generally have definite values of all three simultaneously.

The squared magnitude $$L^2=L_x^2+L_y^2+L_z^2$$ commutes with each component, so $L^2$ and one chosen component, conventionally $L_z$, can have simultaneous eigenstates $|\ell,m\rangle$: $$L^2|\ell,m\rangle=\hbar^2\ell(\ell+1)|\ell,m\rangle,$$ $$L_z|\ell,m\rangle=\hbar m|\ell,m\rangle,$$ where $$\ell=0,1,2,\ldots,\qquad m=-\ell,-\ell+1,\ldots,\ell.$$

For $\ell=1$, the magnitude is $\sqrt2\hbar$ and the allowed $z$ components are $-\hbar,0,+\hbar$. The vector picture of a classical angular momentum with arbitrary projection is replaced by discrete measurement outcomes.

Ladder operators $$L_\pm=L_x\pm iL_y$$ change $m$ by one unit while leaving $\ell$ unchanged. They organize each angular-momentum multiplet algebraically.

Orbital angular momentum in three-dimensional wave mechanics obeys this structure, but the algebra is more general. Intrinsic spin and total angular momentum use the same commutation relations even when no literal particle orbit exists.