# Perturbation of hydrogen energy due to nucleus

#### JamesJames

Consider the ground state of the hydrogen atom. Estimate the correction $$\frac{\Delta E}{E_1s}$$ caused by the finite size of the nucleus. Assume that it is a unifromly charged shell with radius b and the potential inside is given by $$\frac{-e^2}{4\pi \epsilon b}$$

Calculate the first order energy eorrection to the ground state and expand in $$\frac{b}{a_0}$$. Keep the leading term and observe $$\frac{\Delta E}{E_1s}$$ for b = 10^-15m.

Ok, I need help in constructing the interaction W (or H'). Once I get that, I would then calculate the expectation value of it by sandwiching it between $$\psi_1s$$. Is this correct and how would I construct the interaction?

Here is what I have so far

H0 = (p^2)/2m - e^2/r and H = H0 for r > r0

H = (p^2)/2m -e^2/(4pi epsilon b) = H0 + H' for r < r0

Then I would solve for H' and use the perturbation equation. Is this correct ?

James

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#### JamesJames

Ok, i tried it and it is not making any sense. H' somehow does not depend on r. What am I doing wrong?

James

#### JamesJames

Anything guys...whatever you can suggest would be great.

James

#### dextercioby

Homework Helper
The perturbation is a constant,indeed...The radius of the nucleus is a constant.And because the $$\psi_{1,0,0} (r,\theta,\phi)$$ is normalized,the integration will be trivial.

Daniel.

#### JamesJames

Are the steps I used correct?

#### JamesJames

Ok, I think I made a mistake. H' does in fact depend on r.

H' = H0 + e^2/r - e^2/(4pi epsilon b)

So I am going to get a constant term plus a term that depends on r so there will be some dependence. Where am I slipping up?

James

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