In this problem we treat the electron–electron repulsion term in the helium Hamiltonian (Equation 5.38) as a perturbation,

(This will not be very accurate, because the perturbation is not small, in comparison to the Coulomb attraction of the nucleus …but it’s a start.)
(a) Find the first-order correction to the ground state,

(You have already done this calculation, if you worked Problem 5.15— only we didn’t call it perturbation theory back then.)
(b) Now treat the first excited state, in which one electron is in the hydrogenic ground state, Ψ100, and the other is in the state Ψ200. Actually, there are two such states, depending on whether the electron spins occupy the singlet configuration (parahelium) or the triplet (orthohelium):

Show that

where

Evaluate these two integrals, put in the actual numbers, and compare your results with Figure 5.2 (the measured energies are -59.2 eV and -58.4eV).

H' = 1 4 e -