Question

2. In the derivation of the energy levels in the hydrogen atom one commonly assumes that the nucleus is a point charge. However, in reality the size of the nucleus is of the order of Im = 10-15m. Since this is very much smaller than the typical distance of the electron from the nucleus, which is of the order of a0-0.5A = 0.5 × 10-10m, the finite size of the nucleus can be taken into account perturbatively. (a) Assume that the nucleus of the hydrogen atom is a uniformly charged sphere of radius δ According to Gauss' law fdA-E- Qenclosed/eo, the electric field outside this sphere is the same as for a point charge at the centre, so that for r > δ the potential must be indistinguishable from the Coulomb potential for a point charge Determine the charge Qenclosed (r) enclosed by a spherical surface of radius r < 6, and then use Gauss' law to derive the electric field inside the spherefor r < δ Next determine the potential by using the fact that the force -eEr on the electron is given by the gradient -ôV/dr of the potential. Fix any integration constants such that the potential V(r) is continuous at all r Sketch the potential V (r), find the expression for it, and then find the perturbation △V relative to the Coulomb potential generated by a point-like nucleus (b) Use the ground-state wave function for a hydrogen atom with a point-like nucleus, r100 πα and calculate the shift of the ground-state energy due to the distribution of the nuclear charge over a finite sphere to first order in the perturbation. (c) Expand the result of (b) in powers of δ/00, retain only the first non-vanishing term, and thus show that the ground-state energy shift is approximately The only expansion you need to know is: n!

Answer 1