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Fine structure is down to spin-orbit interaction. So I understand that when the orbital and spin magnetic moments interact, that generates magnetic potential energy. But why does a magnetic potential energy lead to splitting of energy levels?

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Brief answer. Before the spin orbit interaction, every orientation of the orbital magnetic dipole moment with respect to the spin magnetic dipole moment were indistinguishable in energy....they all had the same no matter how they were oriented. Now after the magnetic field of the proton(nucleus) is taken into account (ie spin orbit), different orientations have different energies given by $\Delta E$ = constant($\vec{S}.\vec{L}$).....meaning different angles between $\vec {S}$ and $\vec {L}$ have different orientational potential energies $\Delta E$ which leads to the splitting

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