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Do part b just
Transcribed Image Text:Consider the Hamiltonian for a hydrogen-like atom
p2
H =
+ V(r)+:
2me
S.L dV(r)
2mer
dr
where V(r) =-Za/r, Z is the atomic number. We can take the convenience by
setting ħ = c = 1, hence the fine structure constant a =
4T eg fic
a. We know the eigenvalue and eigenstate of hydrogen are
a?me
Ene =
|Wnem) = |Rn,e)|em),
2(n, + l+1)2'
with n = n, + l+1. But with the newly added spin-orbit coupling, it is better
to move in SL-coupling representation. Construct the eigenstates for
{H, J², Jz, L²} with the eigenstates of hydrogen.
b. Find the energy level splitting by spin-orbital coupling (you can utilize
Hellmann-Feynman theorem for the expectation values instead of integration).
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