A Model of the Hydrogen Atom, Any Level

Let us model the Hydrogen atom, at time t=0, as (a) an electron at x=-r_e, (b) a proton at x=r_p (with r_p<<r_e)_n. r_e and v_e are to be determined.

First determine the desired level, N (with N=1 being the ground level).

For brevity we write 1/4πε₀ as k:

k=1/4πε₀.    (1)

Then the net Coulomb force on the electron has the magnitude

F_coulomb=ke²/r_e²    (2)

There is also a Newtonian force on the electron:

F_newton=mv_e²/r_e.    (3)

Equating the Coulombic and Newtonian forces produces

mv_e²=ke²/r_e    (4)

or

v_e²r_e=ke²/m.    (5)

By deBroglie the electron is associated with a wavelength λ:

λ=h/mv_e.    (6)

In the N’th  state we assume that N of these wavelengths fits into the electron orbital:

λ=2πr_e/N.    (7)

This being the case,

h/mv_e=2πr_e/N,    (8)

and

v_er_e=Nh/2πm.    (9)

Dividing Eq. 5 by Eq. 9 produces

v_e=2πΝke²/Nh.    (10)

And,

r_e= Nh/2πmv_e.