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Chapter 39: Problem 51

A so-called muonic atom is a hydrogen atom with the electron replaced by a muon; the muon's mass is 207 times the electron's. Find (a) the size and (b) the ground-state energy of a muonic atom.

Short Answer

Expert verified
The size of the muonic atom is the size of a hydrogen atom divided by 207. The ground-state energy of the muonic atom is 207 times the ground-state energy of a hydrogen atom.

Step by step solution

01

Determine the new radius

Use the formula for radius in the Bohr model \( r_n = n^2*h^2*蔚_0 / (蟺*m*e^2) \) but replacing \( m \) with 207 times the mass of an electron \( m_{mu} = 207*m_e \) as given in the problem, and n=1 for the size of a ground state atom. So, \( r_{mu} = 1^2*h^2*蔚_0 / (蟺*207*m_e*e^2) \). Therefore, the new radius \( r_{mu} \) is the radius of the hydrogen atom divided by 207.
02

Calculate the ground-state energy

Apply the same substitution \( m_{mu} = 207*m_e \) to the formula for the ground-state energy \( E_n = - m * e^4 / (2 * h^2 * n^2 * 蔚_0^2 ) \), where n is equal to 1 for ground state energy. So, \( E_{mu} = - 207*m_e*e^4 / (2*h^2*蔚_0^2) \). Therefore, the ground-state energy is 207 times the energy of a hydrogen atom's ground state energy.
03

Express the final solution

The radius of the muonic atom (or its size) is the radius of a hydrogen atom divided by 207 and similarly its ground-state energy is 207 times the ground-state energy of a hydrogen atom.

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