Suppose we prepare a system in some properly normalized superposition of the spherical harmonics: A|11> + B|10> + C|1-1>. One of the fundamental results of quantum mechanics is that, if we measure L_z, we will collapse the state of the system onto an eigenstate of the eigenvalue we measure. My question is this: Suppose we measure L_z and we get 0. Which eigenstate of L_z do we collapse to? There are infinitely many spherical harmonics for which m = 0! Thanks.