Today in class we learned that the commutator $[L_x, L_y] = i\hbar L_z$ Where $L_x, L_y, L_z$ are operators. A consequence of this seems to be according to Heisenberg's uncertainty relation $\Delta L_x \Delta L_y \geq \frac{1}{2}\cdot |<[L_x, L_y]>| $ that the uncertainty in $L_x$ and $L_y$ is dependent on $L_z$.
Now I'm considering the case where $L_z$, which is the value, not the operator, is exactly zero. This means that the mean value of $L_z$ is also zero. Hence the right hand side is zero. Suddenly it seems like I can measure both $L_x$ and $L_y$ exactly. But this contradicts what I have learnt, which is that you can only measure one of the components exactly.
Edit: I have an idea that $L_z=0 => L_x=L_y=0$. But I don't know if this is correct.