Why Does the Proton-Electron Mass Ratio Diverge from Direct Division?

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Discussion Overview

The discussion revolves around the discrepancy between the direct division of the proton and electron masses and the proton-electron mass ratio as provided in the CODATA table. Participants explore the implications of this divergence for calculating the reduced mass of the electron, particularly in the context of atomic transition energy calculations.

Discussion Character

  • Technical explanation
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant notes that dividing the mass of the proton by the mass of the electron does not yield the same value as the proton-electron mass ratio, diverging at about the 8th significant figure.
  • Another participant suggests that different units and correlated uncertainties can affect the accuracy of ratios, recommending caution in uncertainty estimates.
  • A participant mentions that the reduced mass of the electron is used in atomic transition energy calculations and expresses the need for a precise value.
  • There is a discussion about the correlation of uncertainties in the electron mass and Planck's constant, with one participant stating that the product of the reduced mass and other constants is better known than the individual components.
  • One participant raises a question about the validity of the Rydberg formula in the presence of QED effects, noting discrepancies between their regression analysis results and the CODATA value of the Rydberg constant.
  • A later reply corrects an earlier statement about the regression gradient yielding the Rydberg constant, providing a specific value and expressing a desire for CODATA to provide additional data for reduced mass calculations.
  • One participant expresses satisfaction with a formula they are using, which was suggested earlier in the discussion.

Areas of Agreement / Disagreement

Participants express differing views on the implications of the proton-electron mass ratio and the calculations involving reduced mass. There is no consensus on the best approach to resolve the discrepancies noted, and multiple competing views remain regarding the calculations and their accuracy.

Contextual Notes

Participants acknowledge the potential for correlated uncertainties in the parameters involved, which may affect the reliability of the derived values. The discussion also highlights the need for precise definitions and values from CODATA to clarify the calculations.

neilparker62
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In the CODATA table of physical constants, there are very precise numbers given for mass of electron and mass of proton. And an even more precise number for proton electron mass ratio. But when you divide the mass of proton by the mass of electron, you don't get the same number as the proton electron mass ratio The numbers start diverging at about the 8th significant figure.

I'm asking because I need a very precise value of reduced electron mass. So I'm not sure whether to use the standard formula:

me * mp / (me + mp)

or one with the proton electron mass ratio:

mp / (1 + mp/me)

They should be the same but they're not and it makes quite a big difference in calculated results.
 
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Which units do you use for the calculation, and which units do you need for the result? The different parameters have different, often correlated sources of uncertainties. For nonlinear combinations (like ratios), this can lead to a best value that is not the ratio of the individual best values. Unless you get information about those correlations, you'll have to be conservative with the uncertainty estimate.

I would check "me / (me/mp + 1)", but this is just a guess.

You can also ask CODATA if they have the reduced mass somewhere, or check the corresponding publications - it could even be an input to the CODATA fit.
 
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It's an atomic transition energy calculation. So output unit would be energy. Convert to frequency via Planck's constant. Inputs: reduced mass of electron x c^2 x terms with fine structure constant.

Sounds like a good idea to contact CODATA for a value of the electron's reduced mass.

Thanks for the suggestion.
 
neilparker62 said:
Inputs: reduced mass of electron x c^2 x terms with fine structure constant.
This product is much better known than the individual components. 6*10-12 uncertainty - expressed as 1/m, but as the speed of light is fixed the frequency has the same relative uncertainty.

Electron mass in MeV and Planck constant in eV*s are nearly 100% correlated: Comparison
 
Are we saying that if QED effects in certain energy levels somehow cancel out, then the Rydberg formula with 1/n^2 - 1/(n+1)^2 is still valid? I see a lot of literature where scientists seem to be doing something like this in order to calculate ever more accurate values of the Rydberg constant. Correlation coefficient on Balmer series frequencies (H I outermost level of each primary quantum number) vs 1/n^2 - 1/(n+1)^2 is 1 at the level of accuracy my PC can manage. But the resultant gradient from the regression analysis does not yield the CODATA value of the Rydberg constant. It is miles out.
 
Have to correct myself. The gradient of the above regression yields the Rydberg constant Rh as:

10 967 776.9

The 'super accurate' value given by CODATA is

10 973 731.568 508(65)

but that is for R (infinity).

It's a pity they don't give Rh as well and then we could easily determine reduced mass. I'll ask them for it!
 
mfb said:
I would check "me / (me/mp + 1)", but this is just a guess.
Thanks, I'm using that formula and it seems to work fine.
 

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