Are muon energies quantized? I'm working on a problem that asks:
For a central charge $Ze$, obtain an expression for the radius $r_n$ of the $n$th muonic orbit. Express this as a multiple of the radius $a_o$ of the first Bohr (electron) orbit in hydrogen.
This is based on the assumption that we have slowed down a high speed muon and trapped it in orbit around the nucleus of an atom. 
I believe I know how to do this assuming that the angular momentum of the muon is quantized because its energy is quantized. If it's not quantized then what the heck do you do?
 A: You are working with a bound state . This means your energies are negative and the muon cannot escape the potential. You will have quantized energies in this situation. This is entirely analogous to the hydrogenic energies that are solved analytically in many textbooks. You can approach this problem by starting with the time-independent Schrodinger equation 
$$\hat{H}\,\psi  = E \,\psi$$ 
and then use separation of variables on your wavefunction. If your energies were not quantized then the muon would not be bound and thus your would have a free muon. I don't believe bound states exist that have a continuum spectrum of energies but I could be mistaken.
I believe your problem wants you to use the Bohr-Sommerfeld quantization condition in order to express the radius in terms of Bohr radii. That is
$$L = m_{\mu} v \,\, r = n \hbar$$
which you can explicitly put into an $F = m a$ equation where $F$ is the Coulomb force and you can express $a$ in terms of $\frac{v^2}{r}$ for uniform circular motion.
