Orthohelium and Parahelium

where is the energy of a hydrogen atom electron whose quantum numbers are , , , the hydrogen ground-state energy, and the expectation value of . It follows from Equation (9.53) (with and ) that

where

and is the Bohr radius. Here, the plus sign in Equation (9.66) corresponds to the spin-singlet state, whereas the minus sign corresponds to the spin-triplet state. The integral --which is known as the

The fact that parahelium energy levels lie slightly above corresponding orthohelium levels is interesting because our original Hamiltonian, (9.52), does not explicitly depend on spin. Nevertheless, there is a spin dependent effect--that is, a helium atom has a lower energy when its electrons possess parallel spins--as a consequence of Fermi-Dirac statistics. To be more exact, the energy is lower in the spin-triplet state because the corresponding spatial wavefunction is antisymmetric, causing the electrons to tend to avoid one another (thereby reducing their electrostatic repulsion).