{\frac{-i
\cal{H} \tau}{\hbar}} \Big]} | x^{\prime} \rangle,$$ where $\theta(\tau)$ is the Heaviside step function and $|x\rangle$, $|x^{\prime}\rangle$ are the eingenkets of the position operator $X$ (in the Schrödinger picture) with eingenvalues $x$ and $x^{\prime}$, respectively. The extension for 3D systems is straightforward and will be done in the next section. For $\tau>0$ we have, from equation (\[DefinicaoPropagador\]), that $$\label{Eq2}
i\hbar\frac{\partial}{\partial\tau}K(x,x^{\prime};\tau) =
 \langle x |\mathcal{H} \exp{\Big[ {\frac{-i \mathcal{H} \tau}{\hbar}}\Big]} |
x^{\prime}\rangle.$$ Inserting the unity $\uma =\exp{[-(i/\hbar)\mathcal{H}\tau]}\exp{[(i/\hbar)\mathcal{H}\tau]}$ in the r.h.s. of the above expression and using the well known relation between operators in the Heisenberg and Schrödinger pictures, we get the equation for the Feynman propagator in the Heisenberg picture, $$i\hbar\frac{\partial}{\partial\tau}
 K(x,x^{\prime};\tau)=\langle x,\tau |\mathcal{H} (X(0),P(0))| x^{\prime},0\rangle, \label{Eq3}$$ where $|x,\tau\rangle$ and $|x^{\prime},0\rangle$ are the eingenvectors of operators $X(\tau)$ and $X(0)$, respectively, with the corresponding eingenvalues $x$ and $x^{\prime}$: $X(\tau)|x,\tau\rangle=x|x,\tau\rangle$ and $X(0)|x^{\prime},0\rangle=x^{\prime}|x^{\prime},0\rangle$, with $K(x,x^{\prime};\tau)=\langle x,\tau|x^{\prime},0\rangle$. Besides, $X(\tau)$ and $P(\tau)$ satisfy the Heisenberg equations, $$i\hbar \frac{dX}{d\tau}(\tau)=[X(\tau),\mathcal{H}] \hspace{0.2cm} ;
 \hspace{0.2cm}  i\hbar \frac{dP}{d\tau}(\tau)=[P(\tau),\mathcal{H}].
\label{Eq4}$$ Schwinger’s method consists in the following steps:

(i)

:   we solve the Heisenberg equations for $X(\tau)$ and $P(\tau)$, and write the solution for $P(0)$ only in terms of the operators $X(\tau)$ and $X(0)$;

(ii)

:   then, we substitute the results obtained in [**(i)**]{} 