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pstq
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Hi all ! I need some help
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.
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]
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
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