Tuesday, March 1, 2016

Derivation of Proton to Electron Mass Ratio Equation from the Rydberg Equation



$$\mu={m_p\over m_e}=1836.15267\dots$$
$$m_e={2R_{\infty}h\over c\alpha^2}$$
$$m_e{\alpha^2\over R_{\infty}}={2h\over c}$$
$$m_e{\alpha^2\over \pi R_{\infty}}={2h\over \pi c}$$
Let $r_e={\alpha^2\over\pi R_{\infty}}$, then:
$$m_er_e={2h\over\pi c}$$
Now, because for every action there is an equal and opposite reaction, for every force there is an equal and opposite force, for every torque, there is an equal and opposite torque, equate $m_pr_p$ to $m_er_e$ to balance torque/spin between proton and electron:
$$m_er_e={2h\over\pi c}=m_pr_p$$
$$\therefore {m_p\over m_e}={r_e\over r_p}={\alpha^2\over\pi r_pR_{\infty}}=1836.15267$$
Where:
$$m_pr_p={2h\over\pi c}=4\ell m_{\ell}$$
$$r_p=0.841235640294664\;fm$$
Compare to TOP_PCG calculated $r_p$:
$r_p=0.841235640294664\;fm$
$r_p=0.841235640479985\;fm$ <~~ TOP_PCG


$m_p=$ proton mass
$m_e=$ electron mass
$r_p=$ proton radius
$r_e=$ effective torque arm radius for electron
$\alpha=$ fine-structure constant
$h=$ Planck constant
$c=$ speed of light
$R_{\infty}=$ Rydberg constant
$\ell=$ Planck length
$m_{\ell}=$ Planck mass
QED.

The Surfer, OM-IV

5 comments:

  1. https://en.wikipedia.org/wiki/Q.E.D.

    ReplyDelete
  2. Years are passing by as the mainstream FLOUNDERS!
    Ha ha
    🀑🀑🀑 <--- Rare Triple CLown!!!

    ReplyDelete
  3. NASA's having trouble with the Hubble constant.
    Ha ha
    🀑🀑🀑 <--- Rare Triple CLown!!!
    (usually comes in pairs)

    ReplyDelete
  4. What is "TOP_PCG"?

    PS: By using your formula for rp I get:

    8.41235641(34 ±26)e-16 m

    I obtained the calculated result's uncertainty by using Mathematica's "Around" function to do error propagation of the uncertainties in the constants retrieved from Mathematica's CODATA database.

    This is of course within Mathematica's CODATA rp ("ProtonRMSChargeRadius") of:

    8.414 ± 0.019 e-16 m

    ReplyDelete

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