structural:reformatting
Perturbation theory for Helium

I = ∫ ∫ | ψ 100 (r 1) | 2 | ψ n l m (r 2) | 2 k e 2 | r 1 − r 2 | d 3 r 1 d 3 r 2 {\displaystyle I=\int \int |\psi _{100}(\mathbf {r} _{1})|^{2}|\psi _{nlm}(\mathbf {r} _{2})|^{2}{\frac {ke^{2}}{|\mathbf {r} _{1}-\mathbf {r} _{2}|}}d^{3}\mathbf {r} _{1}d^{3}\mathbf {r} _{2}} J = ∫ ∫ ψ 100 (r 1) ψ n l m (r 2) k e 2 | r 1 − r 2 | ψ n l m * (r 1) ψ 100 * (r 2) d 3 r 1 d 3 r 2 {\displaystyle J=\int \int \psi _{100}(\mathbf {r} _{1})\psi _{nlm}(\mathbf {r} _{2}){\frac {ke^{2}}{|\mathbf {r} _{1}-\mathbf {r} _{2}|}}\psi _{nlm}^{\text{*}}(\mathbf {r} _{1})\psi _{100}^{\text{*}}(\mathbf {r} _{2})d^{3}\mathbf {r} _{1}d^{3}\mathbf {r} _{2}} The first integral is said to be analogous to classical potential due to Coulomb interaction, where the squares of wavefunctions are interpreted as electron density. However, no such classical analog exists for the J term. Using Green’s theorem, one can show that the J terms are always positive. From these, the diagram for energy level splitting can be roughly sketched. It also follows that for these states of helium, the energy of parallel spins cannot be more than that of antiparallel spins.