Question

Why can we only find simultaneous eigenvectors of L^2 and L_z but not L^2 with other components of L? (I mean L_x and L_y)

Answer 1

@ybarrap any ideas?

Answer 2

The idea is that if you could, you'd be violating the Heisenberg Indeterminacy principle.

Answer 3

You can just as easily find L_x or L_y but it is just the convention to choose L_z. Afterall, x,y, and z are all the same depending on your reference frame.