Hello PF, I'm studying for the PGRE and I am having trouble understanding why exactly we use the reduced mass to calculate the energy levels of the positronium atom. Well I think I understand why but I just can't visualize, conceptually, what exactly using the reduced mass is doing for us.
The energy levels of the Hydrogen atom are given by
where the value 13.6eV is the Rydeberg constant and equal to:
Here, the subscript e denotes electron values.
The Attempt at a Solution
In the case of Hydrogen, the proton in the nucleus is so much more massive than the electron that we can approximate it to be stationary while the electron orbits around it. In the case of positronium however, the electron and positron have the same mass and so it is not a very good model to take the positron as being stationary. Then the positron and electron orbit each other. What I don't understand is, how does replacing the mass of the electron in the equation for the Rydeberg constant with the reduced mass of the system account for this fact? Can someone paint me a picture, and explain what exactly this reduced mass system looks like physically? I guess I just never really understood reduced mass problems.
By replacing the mass of the electron with the reduced mass of the positron-electron system, are we in affect changing the problem such that the positron is now a stationary object around which the reduced mass that we calculated is orbiting?
Thanks in advance!