### The Quantum Electric Field

 The potential energy between two concentrations of charge is proportional to the product of the two charges, and inversely proportional to the distance between them. $E_{p}=\frac{q_{1}q_{2}}{r}k$This implies that the product of distance and potential energy is constant for any two charges.$rE_{p}=q_{1}q_{2}k$If the two charges are of opposite signs, the potential energy is negative, indicating that it requires positive energy to separate them.For the case where both charges are equal to one quantum charge, (as in the hydrogen atom), we have two well defined distances:A. The classical electron radius at which the potential energy, by definition, is the rest mass of the electron.B. The Bohr radius at which the potential energy is equal to the ground state potential energy of hydrogen.$r_{e}\left (- m_{e} c^{2}\right )=a_{0}E_{H}$So that the potential energy of hydrogen at one Bohr radius is:$E_{H}=-\frac{r_{e}}{a_{0}}m_{e}c^{2}$The ratio of the classical electron radius to the Bohr radius is the square of the fine structure constant, alpha.$E_{H}=-\alpha ^{2}m_{e}c^{2}\approx -\frac{511,000}{137^{2}}eV\approx- 27.2eV$The potential energy of a single electron is proportional to the number of protons, Z, and inversely proportional to radius.  With radius expressed in units of the Bohr radius, the general equation for the potential energy of an electron/nucleus pair becomes:$E_{p}=-\frac{Z}{r}\alpha ^{2}m_{e}c^{2}\approx- \frac{Z}{r}27.2eV$