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.


Since,


,



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