angular-momentum operators

In quantum mechanics every measurable quantity is represented by an operator — a mathematical instruction that acts on the wavefunction — and angular momentum is no exception. The operators Lx, Ly, and Lz stand for the three components of orbital angular momentum along the x, y, and z directions, and a fourth operator L² stands for the squared total magnitude. Built from position and momentum as L = r × p, they encode rotation the way the momentum operator encodes straight-line motion.

What you can know about a state is dictated by these operators. The allowed measured values are their eigenvalues: measure Lz and you get one of the values m·ℏ; measure L² and you get ℓ(ℓ+1)·ℏ². The spherical harmonics are precisely the wavefunctions on which Lz and L² act cleanly, returning a definite number — which is why those two operators, and not the others, label angular-momentum states.

The deep feature of these operators is that they do not all play nicely together. Lx, Ly, and Lz fail to commute, meaning the order in which you apply them matters. Physically this says you cannot pin down all three components of angular momentum at once: choosing to know Lz sharply forces Lx and Ly to remain undetermined. This is the uncertainty principle wearing its rotational costume, and it follows rigorously from the operators' algebra.