a gauge transformation, with the gauge commonly selected to match the boundary conditions of the problem at hand [@Champel_2007]. We select $$\label{corr_4}
A_{\mu} = \left(E x_{1}; 0 , - B x_{1}, 0 \right) \, ,$$ for which equation (\[corr\_1\]) becomes invariant under translations in the ${\mathbf{e}}_{2}$ and ${\mathbf{e}}_{3}$ directions giving rise to the conserved quantities $\hat{p}_{2}$ and $\hat{p}_{3}$. Solutions to equation (\[corr\_1\]) can be written in terms of $\phi^{+}$ and $\phi^{-}$ spinors through $$\label{corr_5}
\psi_{D} = \left( \begin{array}{c}
\phi^{+}_{D}\\
\phi^{-}_{D} \\
\end{array} \right)      \, ,$$ which, for $m=0$, satisfy the uncoupled equations $$\begin{aligned}
\left\lbrace i \hbar \partial_{t} - e E x_{1} + c \sigma_{1}\hat{p}_{1} + \sigma_{2}eBx_{1} + c \sigma_{j} \hat{p}_{j} \right\rbrace \phi^{+}_{D} =0 \, , \label{corr_6a} \\
\left\lbrace i \hbar \partial_{t} - e E x_{1} - c \sigma_{1}\hat{p}_{1} - \sigma_{2}eBx_{1} -c \sigma_{j} \hat{p}_{j} \right\rbrace \phi^{-}_{D} =0 \, , \label{corr_6b}\end{aligned}$$ with $j=2,3$ and the Einstein summation convention: $\sigma_{j} \hat{p}_    {j} = \sigma_{2} \hat{p}_{2} + \sigma_{3} \hat{p}_{3}$ is understood. The potential generated through the interaction with the magnetic field allows the position and momentum operators to be written in terms of bosonic operators, $$\label{corr_7}
\hat{a} = \frac{x_{1} + i(l^{2}_{\mathcal{B}}/\hbar)\hat{p}_{1}}{\sqrt{2} l_{\mathcal{B}}},\hspace{.025\linewidth} \hat{a}^{\dagger} = \frac{x_{1} - i(l^{2}_{\mathcal{B}}/\hbar)\hat{p}_{1}}{\sqrt{2} l_{\mathcal{B}}} \, ,$$ with the magnetic length 