[quantum_state]

They are just eigen states of the angular momentum operator.

[setopt]

That’s a very quantum physics-centric explanation though (username checks out…).

A more general definition is that they are eigenfunctions of the Laplacian operator on a sphere, which arise in many contexts.

[jasomill]

More generally, see also the Laplace-Beltrami and Laplace-de Rham operators, both defined on (pseudo-)Riemannian manifolds not necessarily embedded in ℝ³.

Even more generally, see the huge body of beautiful mathematics that has arisen from the study of elliptic differential operators in general (de Rham cohomology, Hodge theory, the Atiyah-Singer index theorem, …).

[Koshkin]

Did you mean that they are basis functions for irreducible representations of SO(3)?