# Accuracy of First Order Theory

Problem 20: The finite size of the proton perturbs the energy of the n = 1 orbit in a hydrogen atom.

(i) Given the proton to be a uniformly charge sphere of radius R = 1 fm, determine the first order energy shift, making reasonable approximations. How accurate should the first order theory be in this case?

(ii) Do the calculation again for the hypothesis that the nuclear charge is uniformly distributed over the surface of a spherical shell of radius R.

(iii) Once again do the calculations for a muon (mass = 106 MeV/c2) in orbit around Pb nucleus (Z = 82, A = 207) with uniform and spherical-shell charge density distributions. Assume the nuclear radius scales like R ~ A1/3. How accurate should the first order theory be in this case?

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#### Solution Preview

The potential energy outside the nucleus is:

U(r) = - kq^2/r r > r_0

Were k is the Coloumb constant.

The charge inside the sphere as a ...

#### Solution Summary

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