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Gyula Szász

Greatest possible mass density and closest approach between elementary particles

According to the formulae for Lagrange multipliers

h = e^2/2c ∙(m’∙c^2 /2∙E(bound))1/2

and for the radii of two-particle system

r= h^2/(4π^2m’e^2).

With the reduced mass m(P,e)’ = mP∙me/(mP+me) and the bound energy E(bound) =13.8 eV the “ground state of hydrogen atom” is characterized with the Planck constant as Lagrange multiplier

h(Planck) = 6.62607004∙10^-27 cm^2 g/s,

and the Bohr radius

r(Bohr) = 0.529177∙10^-8 cm.

Indeed, the energetic lowest ground system of the proton-electron system occurs at the bound energy

E((P,e)-ground state, bound) = (mp+me)∙c^2.

With this bound energy and with the reduces mass m(P,e)’ = mP∙me/(mP+me) the Lagrange multiplier is

h’ = h(Planck)∙0.0011831 = h(Planck)/845.2

and the radius is

r’ = r(Bohr)∙1.4∙10-8= 0.748∙10-16 cm.

This radius leads to a greatest mass density of matter

ρmax = (mP+me)/(4/3πr’^3) = 1.75∙10^+24 g/cm^3.

As the sizes, d, of the two particle systems in the ground state of the electron-positron (e,p) is

d(e,p) = 0.703∙10^-13 cm

and for the proton-elton (P,E) system

d(P,E) = 0.383∙10^-16 cm

one must notice that two elementary particles cannot approach each other under the influences of their mutual interaction nearer than 10^-17 cm.