are accurately described by the Dirac equation $$\label{corr_1}
\left[ \gamma^{\mu}\left( \hat{p}_{\mu} - \frac{e}{c}A_{\mu} \right) - mc \right] \psi_{D} = 0 \, ,$$ where $\hat{p}_{\mu}=  i \hbar \partial_{\mu}$ is the four-momentum operator, $A_{\mu}$ the electromagnetic potential, and the covariant $\gamma$ matrices satisfy the anticommutation relation $$\label{corr_2}
\left\lbrace \gamma^{\mu}, \gamma^{\nu} \right\rbrace_{+} = 2 g^{\mu \nu} \, ,$$ with $g^{\mu \nu}$ the metric tensor. We adopt the metric tensor with signature $\left( 1 , -1, -1, -1 \right)$ and consider the symmetric representation of the Dirac matrices: $$\begin{aligned}
\label{corr_3}
 & \gamma^{0} = \left( \begin{array}{cc}
0 &     \mathbf{1}_{2 \times 2}\\
\mathbf{1}_{2 \times 2} & 0 \\
\end{array} \right)      \, , \hspace{.025\linewidth}
\gamma^{i} = \left( \begin{array}{cc}
0 &     \sigma^{i}\\
-\sigma^{i} & 0 \\
\end{array} \right)  \, ,\end{aligned}$$ written in terms of the Pauli matrices $\sigma^{i}$.

Consider the scenario of a massless Dirac particle $(m=0)$ subject to constant and homogeneous electric, $\mathbf{E} = E {\mathbf{e}}_{1}$, and magnetic, $\mathbf{B} = - B {\mathbf{e}}_{3}$, fields. This field configuration is commonly used for measuring magnetoresistance and transport properties of electron gases; relevant examples are found in [@MacDonald_1983; @Lukose_2007] where the electron gas is effectively described by equation (\[corr\_1\]). The electromagnetic potential generating this field configuration is determined up to 