So I understand the idea of eigenvalues, eigenvectors, and eigenfunctions corresponding to a given operator on some vector or function space. But I'm just wondering, why are eigenvalues so important in quantum mechanics and physics in general? What I mean is, why are scaled multiples of a given vector/function so important where as other characteristics of an operator seem to be less so? Like for example, the L^2 operator for angular momentum. The eigenvalues of the spherical Laplace operator are l(l+1) and the corresponding eigenvalues for L^2 are: h*sqrt(l(l+1) But why are these the allowable values for angular momentum as opposed to any other arbitrary output of the Laplacian? I hope this makes sense. Thank you.