A Model of the Ground State Helium Atom

Let us model the Helium atom, at time t=0, as (a) two electrons at x=-r_e and x=r_e, (b) two protons at x=-r_p and x=r_p (with r_p<<r_e), and (c) two neutrons at z=r_n and z=-r_n. The diametrically opposed electrons are orbiting in a common circular orbit of radius r_e, at a common speed v_e. r_e and v_e are to be determined.

For brevity we write 1/4πε₀ as k:

k=1/4πε₀.    (1)

Then the net Coulomb force on each electron has the magnitude

F_coulomb=k2e²/r_e²-ke²/4r_e²    (2)

            =7ke²/4r_e².

There is also a Newtonian force on each electron:

F_newton=mv_e²/r_e.    (3)

Equating the Coulombic and Newtonian forces produces

mv_e²=7ke²/4r_e    (4)

or

v_e²r_e=7ke²/4m.    (5)

By deBroglie each electron is associated with a wavelength λ:

λ=h/mv_e.    (6)

In the ground state we assume that one of these wavelengths fits into the electron orbital:

λ=2πr_e.    (7)

This being the case,

h/mv_e=2πr_e,    (8)

and

v_er_e=h/2πm.    (9)

Dividing Eq. 5 by Eq. 9 produces

v_e=7πke²/2h.    (10)

Solving for r_e and v_e:

r_e=.3024e-10, (11)

v_e=3828460. (12)

Eq. 11 agrees well with the Internet value of .31e-10.