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Homework Help: Muonic hydrogen, change in potential of the muon.

  1. Jun 23, 2011 #1


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    1. The problem statement, all variables and given/known data

    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

    2. Relevant equations


    3. 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?
  2. jcsd
  3. Jun 23, 2011 #2


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    Science Advisor

    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.
  4. Jun 23, 2011 #3


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    Thanks. That makes sense.
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