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Muonic atom energy

  1. Sep 26, 2008 #1
    1. The problem statement, all variables and given/known data
    A muon is captured by a deuteron to form a muonic atom. Find the energy of the ground state and the first excited state.
    find the wavelength when the transition occurs


    2. Relevant equations
    E[tex]_{n}[/tex] = [tex]\frac{-13.6Z^{2}}{n^{2}}[/tex]

    [tex]\frac{1}{\lambda}[/tex] = R{[tex]\frac{1}{n_{f}^{2}}[/tex]-[tex]\frac{1}{n_{i}^{2}}[/tex]}


    3. The attempt at a solution
    I'm not sure what the n's are for each state. I think [tex]n_{f}[/tex] or ground state = 1 but don't know where to go from here
     
    Last edited: Sep 26, 2008
  2. jcsd
  3. Sep 26, 2008 #2
    Those equations would be right if you were just talking about the hydrogen atom. All you'd have to do would be to plug the right numbers in. Z=1, n=1 in the ground state... and what number would you give one energy level above the ground state n=1?

    Unfortunately, it's slightly more complicated than that. Where does the number 13.6 come from? And how is this system different from a hydrogen atom?
     
  4. Sep 26, 2008 #3
    the deuteron is the hydrogen atom, it's [tex]^{2}H[/tex] and the 13.6 is [tex]\frac{ke^{2}}{2a_{o}}[/tex].

    we're not sure how to utilize the equations though. like what is the first excited state? is ground state n=1? how is [tex]n_{i}[/tex] found? once that is found the wavelength should be easy but we're just not sure how determine the excited state, is it n = 2? tough to tell n could = 2,3,4....[tex]\infty[/tex]
     
  5. Sep 27, 2008 #4
    Yes, the ground state is n=1 like I said above, and n=2 is the first excited state.
    The term you used for the bohr radius is only applicable for the hydrogen atom- a bound state of a proton (or deuteron) and an electron. It's a slight approximation because the reduced mass of the system is very nearly the mass of the electron. This approximation is actually slightly more accurate for a deuteron + electron than for a proton + electron.. But what's the mass of the muon???
    Similarly, you can't use the Rydberg formula for anything other than a common or garden hydrogen atom. Find a version of your first equation that includes a correction for the reduced mass of the system, and convert the difference between the two energy levels to a wavelength.
     
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