Muonic hydrogen, change in potential of the muon.

[L-x]

Homework Statement

Taking the proton to be a uniformly charged sphere of radius 1 fm, find an
expression for the change in potential experienced by the muon as compared with that
in the case of a point-like proton. Use first order perturbation theory to calculate the
ground-state energy shift in muonic hydrogen

Homework Equations

E=-grad(V)

The Attempt at a Solution

The electric field due to a uniformly charged sphere is exactly the same as the field due to a point charge of the region outside the sphere (by a simple application of gauss's law). For this reason I see no need for the potential to change. What am I missing?

 

[phyzguy]

What you say is true - the potential outside of the proton is unchanged. However, the potential inside the proton is different in the two cases. Since the wave function at the origin is non-zero, there is some finite probability for the muon to be found inside the proton. So you can take the standard 1S solution as the unperturbed state, and apply a perturbation which is a difference between the potential of a point charge and the potential of a sphere. The perturbation is only non-zero inside the proton.

 

[L-x]

Thanks. That makes sense.