Proton Reference

 

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By Geek3 - Own work; created with hydrogen 1.0, CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=11293151

https://en.wikipedia.org/wiki/Hydrogen_atom#Theoretical_analysis

https://en.wikipedia.org/wiki/Hydrogen_atom#Bohr-Sommerfeld_Model

Reading over the history of the hydrogen atom and it's relationship to the foundation of quantum mechanics looks pretty sketchy compared to a more careful analysis.

More wholly:

$$F\left(x,\cdots,x_n\right)\equiv1\equiv{m_e}{e^4\over8c\epsilon_0^2h^3R_H}-{{\pi r_pcm_e}\over2h}$$

$x_6=r_p=2010\;and\;2013\;muonic\;hydrogen\;proton\;radius\;(Haramein's\;Equation)$

$x_5=R_H=Rydberg\;constant$

$x_4=h=$ Planck's constant

$x_3=\epsilon_0=$ Permittivity of vacuum

$x_2=c=$ Speed of light

$x_1=e=$ elementary charge

$x_0=m_e=mass\;of\;electron$

other useful constants and relationships:

$\alpha=fine\;structure\;constant$

$m_p=mass\;of\;proton$

$m_pr_p={2h\over{\pi c}}=4\ell m_{\ell}\;(Haramein's\;Equation)$

$\ell=Planck\;Length$

$m_{\ell}=Planck\;Mass$

A 4th order 7D polynomial, solved for roots using numerical method:

http://phxmarker.blogspot.com/2017/11/the-oracle-toppcg-beta-2-included-basic.html

The Surfer, OM-IV

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