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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?

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    $\begingroup$ You are just thinking classically. In Newtonian physics an angular momentum defines a direction in space that does not change with a change of coordinates. In quantum mechanics, on the other hand, states bound in spherically symmetric potentials are defined by eigenvalues of the operators that commute with the Hamiltonian, For angular momentum those operators are $J^2$ (magnitude squared of total angular momentum) and $J_z$ (component along the arbitrary z-axis). The eigenvalue of $J^2$ is independent of coordinate choice, but the eigenvalue of $J_z$, of course, depends on the z-choice. $\endgroup$ – Lewis Miller Mar 11 '17 at 16:07
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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 enter image description here

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 :)

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  • $\begingroup$ Would the value of m change if I choose l_y or l_x instead of l_z? $\endgroup$ – B. Brekke Mar 11 '17 at 16:17
  • $\begingroup$ @B.Brekke, i added some extra details and an experimental setup to help you understand it better :) $\endgroup$ – gertian Mar 11 '17 at 16:40

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