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

Answer –

We are given that the mass of a negatively charged muon is  m_μ = 207 m_e

Now, According to Bohr’s model bohr’s radius is given by –

   

The energy of a ground state muonic hydrogen atom is such that E_μ α m_μ

We are given the value of the first Bohr orbit,

r_e  = 0.53 A = 0.53 × 10^-10 m

Let r_μ represent the radius of muonic hydrogen atom.

Now, writing the equation at the equilibrium –

   

   

   

Therefore, 2.56 × 10⁻¹³ m is the value of first Bohr radius for a muonic hydrogen atom.

We know that,

E_e= − 13.6 eV

Applying the ratio of the energies we get the relation –

   

   

   

The ground state energy of a muonic hydrogen atom is -2.81keV.