Dear All,
regarding the fine splitting of the energy levels of the electron in the Hydrogen atom,
if n = 2 then l = 0 or 1.
Moreover if l= 0 then j = 1/2 while if l =1 then j = 1/2 or j = 3/2.
However the slitting depends only on n and j.
Consequently, the two states |n=2, l=0, j=1/2> and |n=2, l=1, j=1/2> have the same energy while the state |n=2, l=1, j=3/2> has a slightly different energy.
Best wishes,
LS