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I am having a problem regarding fairly basic undergraduate QM physics.

My problem is the following:

Assume 3-dimensional system; we measure angular momenta $L^2$ and $L_z$ to be $6 \hbar^2$ and $\hbar$, respectively. Immediately after, we measure $L_x$. What are the possible values of $L_x$?

This seems like a simple question, and I answered, at first: Since $L^2 |\psi \rangle = \hbar^2l(l+1) |\psi \rangle$, it must be that $l=2$.

Hence, since $l=2$ and $L_z$ can not be simultaneously measured with $L_x$, we can have $L_x | \psi \rangle = (0, \pm 1, \pm 2) | \psi \rangle$. In other words, anything between -2 and 2 (or $-l$ to $l$).

However, I then proceeded to do the calculation another way:

$(L^2 - L_z^2)| \psi \rangle = (L_x^2 + L_y^2) | \psi \rangle = (m_{l_x}^2 + m_{l_y}^2 ) \hbar^2 |\psi \rangle = 5 \hbar^2 | \psi \rangle$

Which implies that $L_x |\psi \rangle = (\pm 1, \pm2)|\psi \rangle$ (and not the previous answer, $L_x |\psi \rangle \neq (0, \pm1, \pm2)| \psi \rangle$).

Where do I go wrong?

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The second way of doing the calculation is wrong. Because, you've assumed that $L_x$ and $L_y$ are measurable simultaneously. They are of course not so. Hence, you're getting a nonsense constraint.

Hope this clears things up.

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