Hydrogen like atom, perturbation theory

[pstq]

Hi all ! I need some help

Homework Statement

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.

Homework Equations

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

[tex] V(r)= \frac{Ze}{4\pi \epsilon_0} \frac {1}{2R}( \frac{r^2}{R^2}-3), [/tex] If [itex]R\leq r[/itex]
or
[tex] V(r)= \frac{Ze}{4\pi \epsilon_0} (\frac {-1}{r} ) [/tex] If [itex]r\leq R[/itex]

The first order corrections to the wavefunction and the energy are

[itex] E_n ^1= < \psi_n ^0 | V(r) | \psi_n ^0 >[/itex]

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

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

 

[mark726]

Hello,

Based on the given equations, it seems like you are being asked to estimate the error in the energy of the electron in a hydrogen-like atom due to the approximation of the nucleus as a point charge instead of a sphere with a uniform charge distribution. This can be done using first order perturbation theory, which involves calculating the first order correction to the energy and wavefunction.

To start, you can use the given equations to calculate the first order correction to the energy (E_n^1) and wavefunction (\psi_n^1). Then, you can compare these values to the original energy and wavefunction (E_n^0 and \psi_n^0) to estimate the error caused by the approximation of the nucleus as a point charge. This error will depend on the size of the nucleus (R) and the charge (Ze).

I hope this helps. Let me know if you have any further questions. Good luck with your calculations!