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Homework Help: Hydrogen like atom, perturbation theory

  1. Jan 21, 2012 #1
    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

    [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]
    [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]

    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
  2. jcsd
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