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Suppose I have one electron in the m=0 state, so that its "L vector" is in the x-y plane. The vector can take any orientation in the plane, so its time average is zero and there should be no net angular momentum by this argument, but of course that isn't true because the p-orbital does have angular momentum and the choice of axes is arbitrary. Also since the other p-orbitals are empty, there should be net angular momentum and thus a magnetic moment by the other argument from the class.

I have read that calling L a vector is inaccurate, but even if that is the case I would like to understand it as well as possible in the context of this treatment.