Binding energy from Dirac equation as a function of Z

Date: 2016-03-05

Here we shall plot the electron binding energy predicted by the Dirac equation for the hydrogen atom:

\[E_{nj}=mc^2\left\{\left[1+\left(\frac{\alpha Z}{n-(j+1/2)+\sqrt{(j+1/2)^2-(\alpha Z)^2}}\right)^2\right]^{\frac{-1}{2}}-1\right\}\]

We can see that the binding energy goes asymptotically to \(-\infty\) as \(Z\rightarrow 137\) (that is, as \(\alpha Z\rightarrow 1\)). In this regime, the coupling is so strong that we can have spontaneous production of electron-positron pairs from the vacuum, and we must turn to quantum field theory instead.