# Hydrogen like atom, perturbation theory

1. Jan 21, 2012

### pstq

Hi all ! I need some help

1. The problem statement, all variables and given/known data

The nucleus of an hydrogen-like atom is usually treated as a point charge Ze. Using the
first order perturbation theory, estimate the error due to this approximation assuming
that the nucleus is a sphere of radius R with a uniform charge distribution.

2. Relevant equations

The potential energy of the electron in the field of homogenous sphere of radius R
and total charge Ze is

$$V(r)= \frac{Ze}{4\pi \epsilon_0} \frac {1}{2R}( \frac{r^2}{R^2}-3),$$ If $R\leq r$
or
$$V(r)= \frac{Ze}{4\pi \epsilon_0} (\frac {-1}{r} )$$ If $r\leq R$

The first order corrections to the wavefunction and the energy are

$E_n ^1= < \psi_n ^0 | V(r) | \psi_n ^0 >$

$\psi_n ^1=\sum _{m ≠ n} \frac {< \psi_m ^0 | V(r) | \psi_n ^0 >}{E_n ^0-E_m ^0}$

3. The attempt at a solution

I have been thinking for a while but I do not understand what I am asked, I have to compute the energy correction or the wavefunction correction? any idea ?

thank you

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