12.17 Obtain the first Bohr’s radius and the ground state energy of a muonic hydrogen atom [i.e., an atom in which is a negatively charged muon (µ) of mass about 207me orbits around a proton].

Hint: Apply Bohr's model of an atom.
Step 1: Find the first Bohr's orbit of a muon.
Mass of a negatively charged muon, ${m}_{\mu }=207{m}_{e}$
According to Bohr's model,
And, the energy of a ground state electronic hydrogen atom, ${E}_{e}\propto {m}_{e}$
Also, the energy of a ground state muonic hydrogen atom, ${E}_{u}\propto {m}_{u}$
We have the value of the first Bohr orbit,
Let ${r}_{\mu }$, be the radius of the muonic hydrogen atom.
At equilibrium, we can write the relation as:
Hence, the value of the first Bohr radius of a muonic hydrogen atom is $2.56×{10}^{-13}m$.
Step 2: Find the ground state energy of muon.
We have,
Take the ratio of these energies as:
Hence, the ground state energy of a muonic hydrogen atom is -2.81 keV.