(a) Express the Bohr energies in terms of the fine structure constant and the rest energy (mc2) of the electron.
(b) Calculate the fine structure constant from first principles (i.e., without recourse to the empirical values of ϵ0, e, ℏ, and c). *Comment:* The fine structure constant is undoubtedly the most fundamental pure (dimensionless) number in all of physics. It relates the basic constants of electromagnetism (the charge of the electron), relativity (the speed of light), and quantum mechanics (Planck's constant). If you can solve part (b), you have the most certain Nobel Prize in history waiting for you. But I wouldn't recommend spending a lot of time on it right now; many smart people have tried, and all (so far) have failed.