Calculating Energy of Muonic Atom: Electron vs Muon | Homework Solution

[kraigandrews]

Homework Statement

The electron in the Hydrogen atom can be replaced by the heavier muon resulting in a muonic atom. The muonic atom is not stable because the muon lives for 2.2 μs on average and then it decays into an electron and two neutrinos. However some very fast experiments can be performed on the muonic atom.
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What is the energy of the muon in the ground state?
You will need the following masses for this problem:
- electron : 0.5110 Mev/c2,
- muon : 105.7 Mev/c2,
- proton : 938.3 Mev/c2.

Homework Equations

[itex]\mu[/itex]=m_e/(1+(m_e/M));M=proton mass

R_m=([itex]\mu[/itex]/M)[itex]R_{H}\ =\ 109.73731568549(83)\ \times\ 10^{5} m^{-1}[/itex

E=-R_m(hc)/n²

The Attempt at a Solution

So I know to use these equations to find the energy, however, the place where I go wrong is that since the electron has been replaced with a muon I don't know if I need to replace the m_e with the muon mass

 

[jbsteele]

in the equation for \mu or if I can leave it as m_e. If I use the muon mass, then \mu=105.7/(1+(105.7/938.3))=0.11125854R_m=(0.11125854/938.3)[itex]R_{H}\ =\ 109.73731568549(83)\ \times\ 10^{5} m^{-1}[/itexE=-109.73731568549(83) (hc)/n2E=-109.73731568549(83)(6.6260689633x10^-34 x 3.00 x 10^8)/n2E=-109.73731568549(83)(198.4 x 10^-22)/n2E=-2.17288 x 10^-19/n2