What are the average and transition energies for a muonic hydrogen atom?

[UniverseZ]

Average Separation of Muon and Proton

Homework Statement

A muon is a particle with a mass of 206.7 m_e where m_e is the electron mass, and with the same charge as an electron. It can be captured in "orbit" by a proton to form a muonic hydrogen atom.
a. Find the average separation <r> of proton and muon in the ground state of the system (in meters).

b. Determine the minimum and maximum bound state transition energies to the ground state of the system, in eV.

Homework Equations

normal hydrogen atom
R₁₀=2a^-3/2e^-r/a

Y₀⁰ = √(1/4pi)
Y₀¹ = √(3/(4pi)cos(θ)
Y₁^±1 = √(3/(8pi))sinθe^(±iθ)

The Attempt at a Solution

a. For the electron, the average separation would be ∫R₁₀rR₁₀r²dr
Now for the muon, is it simply a plug-in of the Bohr radius a = 4∏ε₀[STRIKE]h[/STRIKE]/(206.7*m*e²)?

b. I'm actually not sure how to approach this. Would it be something like doing <Y₀¹|H|Y₀⁰>? I could be totally off.

Thanks for any input.

 

[mfb]

(a) is similar to the electron problem, and the distance just scales as the Bohr radius, indeed.
(b) what are the energy levels of the muon?

You don't need any wavefunctions here.