Question

A pre-quantum model of the hydrogen atom treats the electron as a point particle floating in a spherical ball of uniform positive charge out to some radius V. Show that the electron in this model exhibits simple harmonic motion, and find the angular frequency of oscillation. If the radius V is the size of an atom, say 0.5 nm, what is the frequency of oscillation?

Answer 1

The proton or the positive charge is assumed to be uniformly distributed over the sphere of radius r₀ . Let at one instant, the electron is at a point inside the positive charge distribution. Let this distance be the radius v.
The Gauss's law tells that the force on the electron is proportional to the positive charge contained in the sphere of radius v, where the electron is.
The charge contained in the sphere of radius r is
q = e (4 v³ / 4 r₀³) = e (v³ / r₀³)
Because the charge is distributed uniformly over the sphere.
The force exerted by this positive charge on the electron is
F = k e q / v²
F = k e² (v / r₀³)
Thus the force is proportional to the distance v. The electron executes simple harmonic oscillation.
The angular frequency of oscillation can be found
F = ² v
The angular frequency
= sqrt (k e² / r₀³ )
c) The size of the atom r₀ = 0.5 nm
= sqrt (9 x 10⁹ x (1.6 x 10^-19)² / (0.5 x 10^-9 )³ )
= 1.843 rad/s²