{g_n\over A} = {eB\over 2\pi\hbar}\;\;\;\; (n=0,1,...)\;
     .\nonumber$$

Charged harmonic oscillator in a uniform magnetic field {#choumf}
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In this section we consider a particle with mass $m$ and charge $e$ in the presence of a constant and uniform magnetic field $\textbf{B}
= B{\bf e_3}$ and submitted to a 2-dimensional isotropic harmonic oscillator potential in the ${\cal OX}_1{\cal X}_2$ plane, with natural frequency $\omega_{0}$. Using the same notation as before, we can write the hamiltonian of the system in the form $$\mathcal{H} = \mathcal{H}_{\bot} + \frac{P_3^2}{2m}, \label{Eq48}$$ where $$\mathcal{H}_{\bot} = \frac{{\pi}_{1}^{2}+{\pi}_{2}^{2}}{2m} +
 \frac{1}{2}m\omega_0^2\left(X_1^2 + X_2^2\right). \label{Eq49}$$ As before, the Feynman propagator for this problem takes the form $K(\textbf{x},\textbf{x}^{\prime};\tau)=K_{\bot}
(\textbf{r},\textbf{r}^{\prime};\tau)K_{3}^{(0)}(x_3,x_3^{\prime};\tau)$, with $K_{3}^{(0)}(x_3,x_3^{\prime};\tau)$ given by equation (\[Eq14\]). The propagator in the ${\cal OX}_1{\cal X}_2$-plane satisfies the differential equation (\[EqDiferencialPropagador2D\]) and will be determined by the same used in the previous example.

Using hamiltonian (\[Eq49\]) and the usual commutation relations the Heisenberg equations are given by $$\begin{aligned}
\frac{d\textbf{R}(\tau)}{d\tau}&=&\frac{\mbox{\mathversion{bold}${\Pi}$}(\tau)}{m} \,,  \label{Eq51} \\
\frac{d\mbox{\mathversion{bold}${\Pi}$}(\tau)}{d\tau} &=& 2\omega
 \mathbb{C}  \mbox{\mathversion{bold}${\Pi}$}(\tau) - m
\omega_{0}^{2}\textbf{R}(\tau)\ ,  \label{Eq52}\end{aligned}$$ where we have used the matrix notation introduced in (\[Eq20\]) and (\[Eq21\]). Equation (\[Eq51\]) is the same as (\[EquacaoR\]), but equation (\[Eq52\]) contains an extra term when compared to (\[EquacaoPi\]). In order to 