Compatibility

I'm trying to wrap my head around the idea of compatible versus incompatible observables (in quantum mechanics, natch). If two observables are compatible, that means they can share a common set of eigenfunctions. (Does this mean they CAN share a common set, or can one be a subset, or do they have to be the same set? Hm.) If they have a common set of eigenfunctions, that means their stationary states are described by the same wavefunctions.

So one example of this is the commutation/noncommutation of the various operators related to angular momentum. For example, the square of the angular momentum commutes with all of its components: [L^2, L_x or L_y or L_z] = 0. This means that the square of the angular momentum and any of its components have stationary states described by the same wavefunctions. However, the components do not commute with each other: if i != j != k then [L_i, L_j] = i*hbar*L_k (for even permutations). So the components of the angular momentum do not have stationary states described by the same wavefunctions as the other two components.

This is a very strange idea. If you know the eigenvalue returned by the eigenequation for one component, you can't know either of the other two. Okay, mathematically, I see why that is, in terms of commutation relations...but intuitively, I find that kind of baffling.

More lack of insight later.