# Orbital magnetic moment

I am taking a class which discusses orbital angular momentum in a pseudo-quantum way, and it was explained that the orbital angular momentum is zero if the time average of the individual "L vectors" sum to zero. I am considering p-orbitals. The argument is that, if there are 3 electrons in the subshell, the orbital angular momentum sums to zero and the atom has no orbital magnetic moment. The magnetic moment arises when there are one or two empty p-orbitals, in which case the angular momentum does not sum to zero.

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.