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 207m_e orbits around a proton].

Question

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 207m e orbits around a proton].

Philoid · Accepted Answer

Given: Mass of negatively charged muon mμ^– = 207m_e

The Bohr’s radius is given by the relation:

r_e ∝ (1/m_c)

Energy of ground state electronic hydrogen atom E_e∝ m

The value of the first Bohr orbit is given as r_e = 0.53A

Converting into metres, r_e =0.53 × 10^-10m

Let us consider r_μ is the radius of muonic hydrogen atom

At equilibrium

m_μ r_μ = m_e r_e

207m_e × r_μ = m_e r_e

r_μ = (0.53 × 10^-10m)/207

r_μ =2.56 × 10^-13m

The ratio of the energies is given as:

E_e/E_μ = m_e/m_μ = m_e/207m_e

We get

E_μ = 207E_E

Substituting the value of energy, we get

E_μ = 207 × (-13.6)

E_μ =-2.81 keV

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].

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 207m_e orbits around a proton].