# Average Separation of Proton

1. Nov 3, 2013

### UniverseZ

Average Separation of Muon and Proton

1. The problem statement, all variables and given/known data

A muon is a particle with a mass of 206.7 me where me 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.

2. Relevant equations

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

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

3. 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.

Last edited: Nov 3, 2013
2. Nov 3, 2013

### Staff: Mentor

(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.