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[tex] H = - (\frac{e}{mc}) \vec S \cdot \vec B.[/tex]

But what is c doing in this hamiltonian? Clasically the energy of a magnetic moment in a magnetic field is

[tex] E = - \vec \mu \cdot \vec B = - \frac{e}{2m} \vec S \cdot \vec B[/tex]

and as far as i know what one does to go to QM is to introduce the g-factor. What am I missing here?