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New formula for Hydrogen Spectral Lines

  1. Aug 2, 2014 #1

    I have a new formula (or at least I think it's new) for predicting the wavelengths/frequencies of hydrogen's spectral lines. Please take a look and tell me if it is new. I am quite confident that it works because Bohr's original formula 'falls out' if one approximates cos(x) as 1 + x^2/2 for small x. But the formula was derived by deploying some relativistic ideas without actually knowing much about relativity! Here is the energy expression for 1s to 2s transition. Translate to frequency or wavelength with the usual E=hf=hc/λ.

    [cos(α/2) - cos(α)] * m * c^2


    α = fine structure constant
    m = mass of electron
    c = velocity of light

    More generally for level j to level k transitions:

    [cos(α/j) - cos(α/k)] * m * c^2 (j > k).

    I have checked this formula against a set of measured values I found here:


    Interesting that - like the Bohr formula - this formula seems to consistently overestimate transition energies thus leading to slightly higher than measured frequencies and conversely lower wavelengths.

    I learnt all about the 'fine structure constant' when developing this formula - it popped up again and again!
  2. jcsd
  3. Aug 2, 2014 #2


    Staff: Mentor

    New formulas should be published (along with their derivation and comparison to previous experiments) in the peer-reviewed literature. Not here on PF.
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