Evaluating the Coulomb Integral for a two-electron atom

[yelneerg]

Homework Statement

Evaluate the so-called Direct (or Coulomb) Integral, C, which occurs in the theory of the 2-electron atom.


C=\int\int d\vec{r}_1d\vec{r}_2\left|\phi_a(1)\right|^2\left|\phi_b(2)\right|^2 \frac{e^2}{\left|\vec{r}_1-\vec{r}_2\right|}

You may assume that both particles are in the ground state.

Homework Equations

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The Attempt at a Solution

I'm really just at a loss as to how to get started. I'm guessing that being in the ground state simplifies something. Maybe expand the \frac{1}{\left|\vec{r}_1-\vec{r}_2\right|} bit?

 

[yelneerg]

I think I may need to use variational principle?