General Quantization of Motion in Circular Orbits

[wawitz]

For this question, I have to obtain a general quantization of motion in circular orbits by combining the equations (Where U(r) is potential energy):
(mv²)/r= |(dU(r))/dr|

With the angular momentum quantization of: mvr= nℏ

Then use this to calculate the spectrum for circular motion in a potential of U = F₀r.

After combining these equations, along with E = Ke + Pe (for kinetic and potential energies), I obtained this equation:
E_n= 3/2*(n^4/3ℏ^4/3 F₀^2/3+F₀^2/3 n^2/3 ℏ^2/3)/m^1/3

The next step confuses me. To obtain a spectrum, I would have to use the relation ∆E=hc/λ; however this requires a difference of energies for ΔE. Would this mean setting up an equation for E_n+1 – E_n, using the equation I found for E_n?

 

[BvU]

Hello Witz and welcome to PF. There is a pretty strict rule here that you make use of the template. In this case even I can see the usefulness: you have to obtain a general quantization. That's what you have done, so you are finished.

Why contnue? What is triggering the spectrum adventure ?
(You have an idea, why not follow up on it?)