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Clarification: non-relativistic fine structure of a one-electron atom

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Tuteehub
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( 5 months ago )

The fine structure energy shift (in the non-relativistic limit) for a single-electron atom due to spin-orbit coupling is given by

ΔElj(j+1)l(l+1)s(s+1)2l(l+12)(l+1)ΔEl∝j(j+1)−l(l+1)−s(s+1)2l(l+12)(l+1)

which does not apply when l=0l=0 (as it would be singular). ll is the orbital angular momentum of the electron, ss is its spin angular momentum, and jj is the total angular momentum given by |ls|jl+s|l−s|≤j≤l+s in integer steps (j=l+s)(j=l+s).

 

For the example of hydrogen in the n=2n=2 level:

Am I right in thinking there is not supposed to be any shift for the l=0l=0 s-orbital (j=12j=12)? We can't calculate anything using the formula above in this case, but for the 

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