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MathematicalPhysicist
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Homework Statement
Regard the nucleus of charge Ze as a sphere of radius R0 with uniform density.
Assume that R0<<a0 where a0 is Boher radius/
1. Derive an expression for the electrostatic potential V(r) between the nucleus and the electrons in the atom. If V0(r)=-Ze^2/r is the potential from a point charge, find the difference dV=V(r)-V0(r) due to the size of the nucleus.
2. Assume one electron is bound to the nucleus in the lowest bound state. What is its wave function when calculated using the potential V0(r) from a point nucleus?
3. Use first-order perturbation theory to derive an expression for the change in the ground state energy of the electron due to the finite size of the nucleus.
Homework Equations
e- is the electron's charge.
The Attempt at a Solution
1. I believe it's should be [tex]V(r)=\frac{-Ze^2}{r-R0}[/tex] and the difference: [tex]\delta V(r)=-\frac{R_0}{r}\frac{Ze^2}{r-R0}[/tex].
2. I believe the wave function is that of the solution of the hydrogen potential, i.e:
[tex]\psi_{1,0,0}=\sqrt{\frac{1}{\pi a^3_0}} exp(-r/a_0)[/tex], because for n=1,l=0,m=0 it's ground state of the electron.
3. I am not sure, but I think I need to expand dV by r>>R0, but after that I don't know how to procceed.
Any hints?