Relation of fine structure constant to splitting of spectral li

[Emspak]

Homework Statement

Estimate the magnitude of the fine structure splitting in H-α in THz

Homework Equations

Rydberg -- R_y \left(\frac{1}{n_f^2}-\frac{1}{n_i^2}\right) = \Delta E

The Attempt at a Solution

This isn't really a request for solution help, and more a justification. I know that if I apply the above equation I get (-13.6eV) (5/36)~ 1.9 eV and from there I get a wavelength (~656 nm). So far so good. I also know that the fine structure constant α² multiplied by that will give me a change in frequency. That is the splution sheet says "~α² (1.9eV) " and proceeds to say that if you multiply the reciprocal of that by 242 THz you get 6.8 GHz.

Well an good, but I am trying to justify this better. I know that 656nm ~ 1.9 GHz. I just want to figure out how to make the step. That is, it seems right, but if someone said "Hey this is the equation they are using" that would help.

I also ask this because it seemed to me that if one could derive the magnetic field experienced by an electron as it runs around the nucleus one could get the same result. Bu my attempts to do so so far have met with little success-- I feel like I am perhaps overcomplicating things. (I tried starting with the old equation for B field in a current loop and going from there, using the Bohr radius, but I ended up with stuff that stopped making sense).

Anyhow, to recap: first, justify the answer, second, is there a link to a derivation or something that I am missing.

THanks folks.

 

[Matterwave]

The fine structure of the Hydrogen atom arises due to three effects, the kinetic energy effect, spin-orbit coupling (basically what you're thinking about with regards to the magnetic field), and the Darwinian term. Summing the three effects gives you the ~$\alpha^2$ dependence of the energy splitting.

See: http://en.wikipedia.org/wiki/Fine_structure

 

[Emspak]

thanks, i knew of the wiki page, the issue is that in the answer sheet to the test this appeared on I am jut tryng to figure out where the 242 came from. It's like it comes out of nowhere.