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In a permanent magnet, all the magnetic dipole moments are aligned to generate it. However, apparently the orbital angular momentum of the electron about the nucleus also contributes to the magnetic field, but in quantum mechanics we learn that the atom is a stationary state, so how can orbital angular momentum induce a magnetic field?

Also, we're taught that magnetic fields are generated by moving charges, so when we first learn electrostatics we can ignore magnetism. But the stationary electrons we're concerned with still have magnetic dipole moments so I'd assume even in electrostatics we'd have magnetic fields; are they just assumed to cancel in electrostatics then?

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  • $\begingroup$ Ferromagnetism is caused by spin to spin couplings of essentially stationary electrons, n so-called Amperian currents are involved. The electrons possess both spin and an associated magnetic moments that cannot be reduced to classical motion. $\endgroup$
    – hyportnex
    Feb 25, 2018 at 16:48

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A stationary state of the system as a whole does not mean that the charges themselves are stationary.

This is the case in standard magnetostatics: you have a current distribution that stays constant in time, even if each individual charge is moving, because there is another charge behind it to take its place.

Now, in quantum mechanics you need to be a good deal more careful, because the notion of trajectories is useless (and you cannot think of an atomic orbital as an ensemble of particles). Nevertheless, it is a perfectly well-formed question to ask "is there an electric current in this configuration?" even if you explicitly refrain from asking about the electron's trajectory, and if the orbital has a nonzero orbital angular momentum then the answer will be yes, even if the state itself is stationary.

In particular, it is perfectly consistent with a stationary solution of the Schrödinger equation to have a nonzero probability current: the continuity equation demands that the divergence of that current needs to vanish, but that still allows for the current, and its circulation and its magnetic dipole moment, to be nonzero.

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I don't think they necessarily cancel. It's just in electrostatics we choose to neglect these effects. In most cases the effects of the electrostatic fields are probably stronger than any effects of the intrinsic magnetic moments anyway.

Also, why do you think that stationary states cannot form magnetic fields?

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  • $\begingroup$ Because stationary states don't have time dependence, but magnetic fields are caused by moving charges, which would imply a time dependence on the system. $\endgroup$ Feb 25, 2018 at 8:00
  • $\begingroup$ As stated by Emilio the stationary state does not mean stationary in space. Further, something that we always need to remember in quantum mechanics is that just because we have eigenstates that do not change their energy with time does not mean that the wavefunction itself is constant in time. $\endgroup$ Feb 25, 2018 at 14:11

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