Perturbation Theory: Calculating for the correction on the ground state energy

[jhosamelly]

Homework Statement

Homework Equations

$E_{1}=<ψ_{1}|V(r)|ψ_{1}>$

The Attempt at a Solution

That is equal to the integral ∫ψVψd^3r

So I'll just perform the integral, correct ? But r is not constant here right? So, I' ll keep it inside the integral? How should I continue? Please help. Thanks.

 

[TSny]

$E_{1}=<ψ_{1}|V(r)|ψ_{1}>$That is equal to the integral ∫ψVψd^3r

So I'll just perform the integral, correct ? But r is not constant here right? So, I' ll keep it inside the integral?

Right, $r$ is a variable of integration.

Something to think about: In the integral should you use the potential energy $V(r)$ as stated in the problem or the perturbation $\delta V(r)$ of the potential energy (due to switching from the potential energy of a point nucleus to the potential energy of a finite-sized nucleus)?

 

[jhosamelly]

Right, $r$ is a variable of integration.

Something to think about: In the integral should you use the potential energy $V(r)$ as stated in the problem or the perturbation $\delta V(r)$ of the potential energy (due to switching from the potential energy of a point nucleus to the potential energy of a finite-sized nucleus)?

I really don't get your point sorry. I'm guessing I should use the perturbation of the potential. But how can I get that?

 

[TSny]

The hydrogen atom is usually solved treating the nucleus as concentrated in a point. The ground state wavefunction that you specified was derived under this assumption.

Now you want to treat the nucleus more realistically as having a finite size and calculate a correction to the ground state energy in going from a point nucleus to the finite nucleus. The Hamiltonian for a finite nucleus can be thought of as the Hamiltonian for the point nucleus plus a "perturbation". So, the perturbation is just the difference between the Hamiltonian for a finite nucleus and the Hamiltonian for a point nucleus. You should convince yourself that the perturbation is just the change $\delta V(r)$ in the potential energy function when going from the point nucleus to the finite nucleus.

Can you find a mathematical expression for $\delta V(r)$? The potential energy for a finite nucleus is given in the problem. So, you need to remember what the potential energy function is for a point nucleus.

 

[andrien]

you should just break up the integral at r=R.see if it works.