Question

Obtain the first Bohrs radius and the ground state energy of a muonic hydrogen atom [i.e., an atom in which a negatively charged muon $\left({\mu }^{-}\right)$ of mass about $207m_e$ orbits around a proton].

Answer 1

The charge of the muon $m=207m_e$,

Bohr radius is given as:

$r_e=1/m_e$

We know that for the first Bohr orbit

$r_e=0.53\times {10}^{-10}m$

So, at equilibrium, $mr=m_e{r}_e$

The radius pf the muonic hydrogen atom

$r=2.56{\times }^{-13}m$

We also know that,

$E_e\propto m_e;E_e=-13.6eV$

$\Rightarrow \frac{E_e}{E}=\frac{m_e}{m}\Rightarrow E=-2.81keV$