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b. Find the energy level splitting by spin-orbital coupling (you can utilize Hellmann-Feynman theorem for the expectation values instead of integration).

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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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).
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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