The Fine Structure Constant and the Coulomb Potential

We will now grapple for the first time with the problem of which set of units to use. Advanced texts typically use CGS units in which the potential energy is

$$\bgroup\color{black}V(r)={-e^2\over r}\egroup$$

while the Standard International units

$$\bgroup\color{black}V(r)={1\over 4\pi\epsilon_0}{-e^2\over r}\egroup$$

We can circumvent the problem by defining the dimensionless fine structure constant $\alpha$.

$$\begin{eqnarray*} \alpha_{SI}&=&{1\over 4\pi\epsilon_0}{e^2\over \hbar c}\appro... ...37}\\ \alpha_{CGS}&=&{e^2\over \hbar c}\approx{1\over 137}\\ \end{eqnarray*}$$

So in either set of units the Hydrogen potential is

$$\bgroup\color{black}V(r)={-\alpha \hbar c\over r}\egroup$$