What Is the Binding Energy of the Second Electron in a Hydrogen Ion?

[Antonija]

Homework Statement

The negative H− ion is a two-electron system like the He atom. How large is the binding energy of the second electron?

Homework Equations

For hydrogen-like atoms we can use hydrogen wavefunctions so I did it here. First there is potential to be found and then, multipled with charge, it gives energy. Symbols: r₂ is distance from second electron to nucleus, r₁₂ is distance between first and second electron.

The Attempt at a Solution

[/B]
Potential is sum of Coulomb potential between one electron and nucleus, and potential between 2 electrons themselves:


Φ(r₂)= -e/(4πε₀) * 1/r₂ + e/(4πε₀) ∫ (Ψ_1s(r₁))² /r₁₂ d³r₁
I know 1s wavefunction for hydrogen so when I write it, it is proportional to exp(-r₁/a₀)I can put it into my potential but I don't know how to solve integral with wavefunction, as I have r₁₂ in denominator, which is equal to Ir₁-r₂II can use cosine law but still can't solve integral. I don't know how to behave with this in integral. Sorry if my equation writing is bad, I'm posting for the first time so still learning...

 

[mfb]

For hydrogen-like atoms we can use hydrogen wavefunctions so I did it here.

Well, you do not have a hydrogen-like atom.
Your system has some similarity with a helium atom, but with a lower charge of the nucleus. You might be able to transfer some things from the helium system to the hydrogen atom, although I'm not sure how good that approximation will be. It does not look like the problem asks for a full solution of the two-electron wave function.