1. Consider the case when \(U(\x) = -\frac{e^2}{4\pi\epsilon_0|\x|}\), which is the Coulomb potential. Show that the spin-orbit coupling we find from the Dirac equation agrees with the one we often use in undergraduate quantum mechanics.
    We compute \(\)\nabla U(\x) = \frac{e^2}{4\pi\epsilon_0 r^2}\rhat = \frac{e^2}{4\pi\epsilon_0 r^3}\x\(\) Then, can use the fact that \(\hat{\mathbf{L}} = \hat{\x}\times\hat{\p}\), and then get that $$\begin{align} \Hhat_\text{SOC} ={}& \frac{\hbar}{4m^2c^2}\boldsymbol{\sigma}\cdot\left(\frac{e^2}{4\pi\epsilon_0r^3}\hat{\mathbf{L}}\right) \\ ={}& \frac{e^2}{8\pi m^2c^2\epsilon_0r^3} \hat{\mathbf{S}}\cdot\hat{\mathbf{L}} \end{align}$$ where we have set \(\mathbf{S} = \hbar\boldsymbol{\sigma}/2\). This agrees with the standard expression from undergraduate quantum mechanics!