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What is the physical relation, if any, that explains the factor of 4 in:

$\frac{\alpha}{4.001\pm 0.009} = \frac{r_{p_{Classical}}}{r_{p_{RMSCharge}}}$

where

$\alpha$ is the fine structure constant

$r_{p_{Classical}}$ is a proton's classical radius

$r_{p_{RMSCharge}}$ is a proton's charge radius as chosen by Mathematica: $(8.414\pm 0.019)\times10^{-16}m$

This can be calculated in Mathematica with:

codata[canonicalName_] := 
 Around @@ 
  Entity["PhysicalConstant", canonicalName][{"Value", 
    "StandardUncertainty"}]
codata["FineStructureConstant"]/(codata["ClassicalProtonRadius"]/
   codata["ProtonRMSChargeRadius"])

There is an apparently related question regarding the mass of the proton's relationship to its charge radius. While it is true that the proton's wavelength is physically related to its mass, and a proton's wavelength is related to some notion of its radius, it is unclear which of these notions apply to the present question. Therefore it is unclear that answering one of these related questions would imply the answer to the other.

If that implication can be explicitly stated, then it might be reasonable to close this question as "duplicate" if sufficiently obvious.

Otherwise, these two questions must be considered "related" only in the sense that they both involve properties of the proton being related, somehow, by a factor of almost exactly the number 4. The presumption that these two constants, being close to 4, are identical parameters of physics is reasonably characterized as "numerology". Therefore these questions must be presumed independent questions until this identity is given a theoretic basis.

James Bowery
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    Which value of the charge radius are you assuming? It's controversial, but presumably set primarily by an interaction between the strong and electromagnetic forces, and so won't have a simple theoretical reason for the ratio you propose. – J.G. Jan 07 '23 at 22:16
  • https://physics.stackexchange.com/q/254050/ – Mitchell Porter Jan 08 '23 at 02:33
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    Does this answer your question? The mass of the proton times its charge radius is very close to 4ħ/c. Is this a coincidence? – anna v Jan 08 '23 at 06:03
  • I don't understand why a question about the relationship between proton Mass and proton charge radius is the same as a question about the relationship between a proton classical radius and charge radius. Could someone explain it? – James Bowery Jan 09 '23 at 03:09
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    @JamesBowery The formula for the classical proton radius should be $$r_{p_{Classical}} = \frac{e^{2}}{4\pi\epsilon_{0}m_{p}c^{2}}$$ and the fine-structure constant is $$\alpha = \frac{e^{2}}{4\pi\epsilon_{0}\hbar c}.$$ The claim that $r_{p_{Classical}}/r_{p_{RMSCharge}} = \alpha/4$ is equivalent to $r_{p_{RMSCharge}} = 4r_{p_{Classical}}/\alpha$. When you plug in the two formulas, you find the claim to be equivalent to $r_{p_{RMSCharge}} = 4\hbar/(m_{p}c)$, or equivalently $r_{p_{RMSCharge}}\cdot m_{p} = 4\hbar/c$ which is what the other post is asking about. – Maximal Ideal Jan 09 '23 at 17:02
  • Thanks. There are two ways in which that answer should be considered nonobvious: 1) It is not obvious that the classical radius should be considered a function of the mass of the proton and 2) If you go searching for such a formula, you will find only the classical radius of the electron in terms of the electron mass and then reference to the proton RMS charge radius but not its classical radius. I'm reopening the question and will accept your answer if provided as such. – James Bowery Jan 09 '23 at 21:54
  • James, the "classical radius" of the proton is not a measured quantity, but is defined by the formula given by Maximal Ideal, so this question is mathematically equivalent to the question marked as a duplicate. If you can't find any good sources for this general definition of the classical radius, you could look at this answer or ask a question about why the classical radius is defined this way and why is was historically interesting. – David Bailey Jan 10 '23 at 02:34
  • After having added to the Wikipedia article on electron classical radius the above formula for proton classical radius, and it having survived for a day without being undone, I'm satisfied with it being marked as duplicate with one caveat:

    The question being given priority over this question really should have an answer that includes Maximal Ideal's derivation, rather than requiring people to click through and figure all this out.

     – James Bowery Jan 11 '23 at 20:53

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