Ground State Potential Energy

From the home page, the velocity-to-light speed ratio for the ground state is:

Making the relativistic Lorentz factor, gamma:

Another property of the single-electron ion's ground state is that its shell circumference must be equal to the electron's de Broglie wavelength.

The ground state shell circumferences are, therefore:

So that the ratio of the Compton wavelength to shell circumference is:

From the definitions of alpha and the classical electron radius, the potential energy at a shell circumference of one Compton wavelength is:

Since potential energy is inversely proportional to shell radius, (and circumference), the potential at any allowed circumference is the product of the two previous equations.