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, mμ=207me
According to Bohr's model,
Bohr radius, re1me
And, the energy of a ground state electronic hydrogen atom, Eeme
Also, the energy of a ground state muonic hydrogen atom, Eumu
We have the value of the first Bohr orbit,  re=0.53 A=0.53×10-10m
Let rμ, be the radius of the muonic hydrogen atom.
At equilibrium, we can write the relation as:
mμrμ=mere207 me×rμ=mererμ=0.53×10-10207=2.56×10-13m
Hence, the value of the first Bohr radius of a muonic hydrogen atom is 2.56×10-13m.
Step 2: Find the ground state energy of muon.
We have,
Ee=-13.6 eV
Take the ratio of these energies as:
EeEμ=memμ=me207meEμ=207Ee      =207×(-13.6)=-2.81 keV
Hence, the ground state energy of a muonic hydrogen atom is -2.81 keV.