can anyone explain to me how the LS coupling scheme work? thanks
What do you know about the theorem of Clebsch-Gordan...?
Briefly, in atomic physics, LS or spin-orbit coupling comes from the interaction of the electron's magnetic moment with the magnetic field generated by the orbital motion. A rigorous approach to the LS coupling starts with Dirac's Eq., and shows that the coupling is of order v/c. Thus, typically LS coupling effects contribute to the fine structure of atomic spectra. Working out the details of LS coupling energy shifts will stretch your angular momentum chops, considerably.
Why electron of Hydrogen atom in ground state does not experience LS coupling
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