Is the magnetic quantum number dependent on my choice of coordinate system? I'm reading on wikipedia: https://en.wikipedia.org/wiki/Magnetic_quantum_number
that
"The axis used for the polar coordinates in this analysis is chosen arbitrarily. The quantum number $m$ refers to the projection of the angular momentum in this arbitrarily chosen direction"
I don't think I understand this. If m is the projection, to me it seems like I can change the axis, and make the projection, hence m, any number between $0$ and $l$. But I don't think it is possible to change a physical quantity like that by just changing the coordinate system. What is wrong with my understanding?
 A: If we denote $\hat{L}^2$ as the total angular momentum (squared) and $l_i$ its projection onto the i-axis than:
$$[\hat{L}^2, l_i] = 0 \phantom{text}\text{but}\phantom{text}[l_i,l_j] \neq 0$$
This means that, given some state $\psi$ we cannot measure its total angular momentum and all projections. The "best" thing that we can do is to make a measurement of its total angular momentum $L^2$ and pick out one projection e.g. $l_x$ and call this last quantity m.
What they mean with "The axis used for the polar coordinates in this analysis is chosen arbitrarily" is that we could also have measured $l_y$, $l_z$ or any linear combination of $l_x,l_y,l_z$ and called it m.
Measuring some projection say $l_x$ changes the state of the system such that there is no memory of its $l_y$ and $l_z$ values. This is demonstrated in the Stern Gerlach experiment

The source emits a beam of electrons. We measure its projected spin onto the z axis(which corresponds to the choice $l_z=m$) and eliminate those with $l_z = -1/2$. 
Than we try to measure its projected spin onto the x axis(which corresponds to the choice $l_x=m$. We observe both $\pm1/2$ as was to be expected. 
In a final stage we remeasure its projection onto the z axis. We observe both $\pm1/2$. 
What happens here is the following
$$Z_+ = \frac{1}{\sqrt{2}}(X_+ +X_-)$$
$$X_+ = \frac{1}{\sqrt{2}}(Z_+ -Z_-)$$
so after the first measurement of $Z_+$ we know that $l_x$ can be either + or - 1/2. Once we measure $l_x$ to be +1/2 we lose all previous information about $l_z$ so that it may be $\pm 1/2$ again.
So indeed, making a measurement of $l_x$ changes the value of $l_y$ and this has been proven experimentally as explained above.
And because you define m to be the projection that you just measured you are indeed changing m by performing these measurements.
I hope this helps :)
