instance: the Feynman propagator is obtained in a generic gauge; [*(ii)*]{} the gauge-dependent and gauge-independent parts of the propagator appear clearly separated and [*(iii)*]{} the transformation law for the propagator under gauge transformation can be readly obtained. Besides, we adopted a matrix notation which can be straightforwardly generalized to cases of relativistic charged particles in the presence of constant electromagnetic fields and a plane wave electromagnetic field, treated by Schwinger [@Schwinger1951]. For completeness, we showed explicitly how one can obtain the energy spectrum directly from que Feynman propagator. In the Landau problem, we obtained the (infinitely degenerated) Landau levels with the corresponding degeneracy per unit area. For the case where we included the bidimensional harmonic potential, we obtained the energy spectrum after identifying the poles of the corresponding energy Green function. We hope that this pedagogical paper may be useful for undergraduate as well as graduate students and that these two simple examples may enlarge the (up to now) small list of non-relativistic problems that have been treated by such a powerful and elegant method.

Acknowledgments {#acknowledgments .unnumbered}
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F.A. Barone, H. Boschi-Filho and C. Farina would like to thank Professor Marvin Goldberger for a private communication and for kindly 