Question

A proposal has been made to study the properties of an atom composed of a π⁺(m_π⁺ = 237.2m_e) and a µ^-(m_µ^- = 206.77m_e) in order to measure the charge radius of the pion. Assume that all of the pion charge is spread uniformly on a spherical shell at R₀ = 10^-13 cm and that the µ is a point charge. Express the potential as a Coulomb potential for a point charge plus a perturbation and use perturbation theory to calculate a numerical value for the percentage shift in the 1s -2p energy difference A. Neglect spin orbit effects and Lamb shift. Given

Answer 1

The Coulomb potential energy of the muon is

It can be written in the form

is to be treated as a perturbation.

The energy levels and wave functions of the unperturbed system are

As spin orbit and Lamb effects are to be neglected, we need only consider R_nl in perturbation calculations. Thus

Thus a₀ >> R and the factor exp(-2r/a₀) in the integrand above may be neglected. Hence