For a particular energy level in hydrogen, with quantum numbers n and l, one will find when considering the spin-orbit interaction, the level is split into two fine structure levels with energy separation:(adsbygoogle = window.adsbygoogle || []).push({});

[tex]\Delta E_{s.o.}=\beta_{nl}(l+1/2)[/tex]

I was trying to prove this result. The spin of an electron is 1/2. Therefore there are two possible values for the total angular momentum, as the spin can be either +-1/2. Therefore using (and the relevant energy spin-orbit equation):

[tex]\Delta E_{s.o.}=E_{j=l+1/2}-E_{j=l-1/2}[/tex]

gives the first result. However, when you follow the proof through I am confused because in the energy spin-orbit equation (not stated here) you use s=1/2 for both possible energy states. However you use l=+-1/2 for the angular momentum. Why is this? Surely you'd use s=+-1/2 for the electron spin also?

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# Hydrogen spin-orbit interaction

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