K_{3}^{(0)}(x_3,x_3^{\prime};\tau),\;\;\;\;
(\tau>0)$$ where ${\bf r} = x_1{\bf e_1} + x_2{\bf e_2} $ and $K_{3}^{(0)}(x_3,x_3^{\prime};\tau)$ is the well known propagator of the free particle [@FeynmanHibbsBook], $$K_{3}^{(0)}(x_3,x_3^{\prime};\tau)=\sqrt{\frac{m}{2\pi i\hbar \tau}}
 \exp{ \Big[ \frac{im}{2\hbar}\frac{(x_3 - x_3^{\prime})^{2}}{\tau}\Big]}.
\label{Eq14}$$ In order to use Schwinger’s method to compute the two-dimensional propagator $K_{\bot}({\bf r},{\bf r}^{\prime};\tau) =
  \langle{\bf r},\tau|{\bf r}^{\prime},0\rangle$, we start by writing the differential equation $$\label{EqDiferencialPropagador2D}
i\hbar\frac{\partial}{\partial\tau}
 \langle{\bf r},\tau|{\bf r}^{\prime},0\rangle =
 \langle {\bf r},\tau|\mathcal{H}_{\bot}({\bf R}_\bot(0),
 \mbox{\mathversion{bold}${\pi}$}_\bot(0)  )| {\bf r}^{\prime},0\rangle\ ,$$ where ${\bf R}_\bot(\tau) =
 X_{1}(\tau){\bf e_1} + X_{2}(\tau){\bf e_2}$ and $\mbox{\mathversion{bold}${\pi}$}_\bot(\tau) =
  \pi_{1}(\tau){\bf e_1} + \pi_{2}(\tau){\bf e_2}$. In (\[EqDiferencialPropagador2D\]) $|{\bf r},\tau\rangle$ and $|{\bf r}^{\prime},0\rangle$ are the eigenvectors of position operators ${\bf R}(\tau)=X_{1}(\tau){\bf
e_1} + X_{2}(\tau){\bf e_2}$ and ${\bf R}(0)=X_{1}(0){\bf e_1} +
X_{2}(0){\bf e_2}$, respectively. More especifically, operators $X_{1}(0)$, $X_{1}(\tau)$, $X_{2}(0)$ and $X_{2}(\tau)$ have the eigenvalues $x_{1}^{\prime}$, $x_{1}$, $x_{2}^{\prime}$ and $x_{2}$, respectively. In order to solve the Heisenberg equations for operators ${\bf R}_\bot(\tau)$ and $\mbox{\mathversion{bold}${\pi}$}_\bot(\tau)$, we need the commutators $$\begin{aligned}
\label{ComutadorXcomPi2}
\Big[X_{i}(\tau) , \pi_{j}^{2}(\tau) \Big] &=&
 2i\hbar \pi_{i}(\tau)\, ,\cr
  \Big[\pi_{i}(\tau) , \pi_{j}^{2}(\tau)\Big] &=& 2i\hbar eB\epsilon_{ij3}\pi_{j}(\tau),
  \label{ComutadorPicomPi}\end{aligned}$$ where $\epsilon_{ij3}$ is the usual Levi-Civita symbol. Introducing the matrix notation $${\bf R}(\tau)=\left(
\begin{array}{c}
X_{1}(\tau)   \\ X_{2}(\tau)
\end{array}
\right)
\hspace{0.25cm};\hspace{0.25cm}
\mbox{\mathversion{bold}${\Pi}$}(\tau)=\left(
\begin{array}{c}
\pi_{1}(\tau)   \\ \pi_{2}(\tau)
\end{array}
\right)
\label{Eq20}\ ,$$ and using the 