The Two Component Dirac Equation

First, we can write the two component equation that is equivalent to the Dirac equation. Assume that the solution has the usual time dependence $e^{-iEt/\hbar}$. We start from the equation in $\psi_A$ and $\psi_B$.

$$\begin{eqnarray*} \pmatrix{\hbar{\partial\over\partial x_4} & -i\hbar\vec{\sigma... ...{\psi_A \cr \psi_B \cr} +mc\pmatrix{\psi_A \cr \psi_B \cr}=0 \\ \end{eqnarray*}$$

Turn on the EM field by making the usual substitution $\vec{p}\rightarrow\vec{p}+{e\over c}\vec{A}$ and adding the scalar potential term.

$$\begin{eqnarray*} \pmatrix{-{1\over c}(E+eA_0-mc^2) & \vec{\sigma}\cdot\left(\ve... ...vec{\sigma}\cdot\left(\vec{p}+{e\over c}\vec{A}\right)\psi_A \\ \end{eqnarray*}$$

These two equations can be turned into one by eliminating $\psi_B$.

$$\begin{eqnarray*} {1\over c}(E+eA_0-mc^2)\psi_A = \vec{\sigma}\cdot\left(\vec{p}... ...vec{\sigma}\cdot\left(\vec{p}+{e\over c}\vec{A}\right)\psi_A \\ \end{eqnarray*}$$

This is the two component equation which is equivalent to the Dirac equation for energy eigenstates. The one difference from our understanding of the Dirac equation is in the normalization. We shall see below that the normalization difference is small for non-relativistic electron states but needs to be considered for atomic fine structure.