Proton Radius, the Rydberg Equation, and Fundamental Physics Constants
Mark Rohrbaugh - MSEE, ΩIV , Lyz Starwalker - Mµ* ©1991 - 2023 - ∞
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| n=1 Bohr_model |
Highlights
- Complete solution to proton radius puzzle
- Correction to Rydberg Equation (extends with proton to electron mass ratio)
- Roots of Full Rydberg Equation define fundamental physics constants
Abstract
A mathematical solution for the proton radius has been derived which adds a term to the Rydberg equation. By avoiding using the reduced mass approximation (as has been done in solid-state theory and the hydrogen atom's analytical solution to the Schrödinger Wave Equation) and including the proton to electron mass ratio which brings in the proton radius. A proton to electron mass ratio of 1836.15267 is calculated with a proton radius of 0.8412fm.
Keywords
Proton radius; proton to electron mass ratio; Rydberg equation
1.0 Introduction
This derivation makes use of, as a starting point,: $$R_M={R_\infty\over {1+{m_e\over M}}}[1]$$
This derivation makes use of, as a starting point,: $$R_M={R_\infty\over {1+{m_e\over M}}}[1]$$
1.1 Governing model
Schrödinger Equation (and Full Wave Equation)
(At lowest energies, n=1, Bohr model and Schrödinger model are equivalent for 1H hydrogen)
2.0 Derivation:
$R_M={R_\infty\over {1+{m_e\over M}}}$
$R_M(1+{m_e\over M})=R_\infty$
$(1+{m_e\over M})={R_\infty\over R_M}$
$1={R_\infty\over R_M}-{m_e\over M}$
For hydrogen, 1H @ 0°K:, $M=m_p$
$1={R_\infty\over R_H}-{m_e\over m_p}$
For the electron to proton mass ratio (inverse of proton to electron mass ratio), the derivation of the proton to electron mass ratio:
$$\mu={m_p\over m_e}=1836.15267\dots$$
$$m_e={2R_{H}h\over c\alpha^2}$$
( $m_e$ for the electron mass: Derivation of Rydberg equation for electron mass, also see [1] above)
$$m_e{\alpha^2\over R_{H}}={2h\over c}$$
$$m_e{\alpha^2\over \pi R_{H}}={2h\over \pi c}$$
By substitution, let $r_e={\alpha^2\over\pi R_{H}}$, 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:
$$m_er_e={2h\over\pi c}=m_pr_p$$
$$\therefore {m_p\over m_e}={r_e\over r_p}={\alpha^2\over\pi r_pR_{H}}=1836.15267$$
Where:
$$m_pr_p={2h\over\pi c}=4\ell m_{\ell}$$
$$r_p=0.841235640(294664)\;fm$$(Google Calculator for r_p)
$\mu={m_p\over m_e}={\alpha^2\over\pi r_pR_H}=1836.15267$ (Google Calculator for μ)
$1={R_\infty\over R_H}-{\pi r_pR_H\over \alpha^2}$
$R_\infty={m_ee^4\over 8{\epsilon_0}^2h^3c}$
$$1={{m_ee^4\over 8{\epsilon_0}^2h^3cR_H}}-{\pi r_pR_H\over \alpha^2}\;[2]$$
$m_p=$ proton mass
$m_e=$ electron mass
$e=$ elementary charge
$\epsilon_0=$ permittivity of free space
$r_p=$ proton radius
$\alpha=$ fine-structure constant
$h=$ Planck constant
$c=$ speed of light
$R_{H}=$ Rydberg constant for 1H hydrogen atom
$R_{\infty}=$ Rydberg constant for massive nucleus atom
$R_{M}=$ Rydberg constant for Multi-proton atom
$\ell=$ Planck length
$m_{\ell}=$ Planck mass
3. Other Solutions for proton mass (ratio)
Boundary Value Problem method of determining coefficients use with wave equation for both electron and proton; Geometrical assuming vacuum density and information theory of black hole event horizon - Haramein et al
4. Conclusions
This derivation provides a polynomial [2] that predicts future trends of fundamental physics constants coefficient values as the measurements are refined over the years to be more precise. The polynomial [2] may also actually DEFINE the constants - ongoing work with the polynomial investigating stability and convergence continues.
There are various algorithms one can envision setting some constants as known and iterating to find the others, and likely all constants in the equation. A timebase reference is also needed and likely why c is defined in mainstream via decree rather than derivation.
Credit authorship contribution statement
Mark's derivation with Lyz's input to equate/balance the proton and electron
Declaration of Competing Interest
n/a
Data availability
n/a
References
[standard]
*Mµ - Master of the µniverse, Macro and micro, inner and outer-dimensional
