Calculating Muonic Atom Energy: Ground State and Excited State Wavelength

[nick227]

Homework Statement

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

Homework Equations

E$$_{n}$$ = $$\frac{-13.6Z^{2}}{n^{2}}$$

$$\frac{1}{\lambda}$$ = R{$$\frac{1}{n_{f}^{2}}$$-$$\frac{1}{n_{i}^{2}}$$}

The Attempt at a Solution

I'm not sure what the n's are for each state. I think $$n_{f}$$ or ground state = 1 but don't know where to go from here

 

[muppet]

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?

 

[phyguy321]

the deuteron is the hydrogen atom, it's $$^{2}H$$ and the 13.6 is $$\frac{ke^{2}}{2a_{o}}$$.

we're not sure how to utilize the equations though. like what is the first excited state? is ground state n=1? how is $$n_{i}$$ 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...$$\infty$$

 

[muppet]

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.