513.001 Molecular and Solid State Physics

Variational wavefunction

This is an important problem but a difficult one. Calculate the expectation value of the energy of the two-electron wave function

for the helium Hamiltonian

You will need to integrate over r₁ and r₂ so this is a six dimensional integral. It is possible to integrate term by term. The first term only depends on r₁ so the integral over r₂ is trivial and the integral over r₁ is similar to that done in problem 1.4. The second, third, and fourth terms can all be solved like the first term. Only the last term gives some trouble.

Since α is not given, this will result in an expression for the energy as a function of α. Minimize this expression to find the lowest energy.

Here a₀ is the Bohr radius.