When we solve the hydrogen's atom problem using the Schrodinger equation, we find the eigenstates of energy in the position representation for example. To do that we use the fact that the observables "H, l^2, l_z" have the same eigenstates. My question is about the l_z operator. The z-Axis is a random one. Lets say that the probability distribution has the form 'A' around that axis. Due to the randomness of the z axis, we can resolve the problem using another axis z'. We will obtain the same form 'A' (Of the probability distribution) around the z' axis. So, for the same problem we have the same probability distributions around 2 different axis. Or to put it differently, we have 2 different probabilities for the same point. I understand that only one can ocour at a time, but i cant understand how can we arbitrarily choose an axis and change the electron's prob distribution around the nucleous as we like! Another way to put the question is: How can we 'force' the H operator to have the same eigenstates with the l_z where z is the random axis we chose? By taking a measurement of l_z on that axis?