method is by far the less popular among them. The main purpose of the present paper is to popularize Schwinger’s method providing the reader with two examples slightly more difficult than the harmonic oscillator case and whose solutions may serve as a preparation for attacking relativistic problems. In some sense, this paper is complementary to reference [@BaroneBoschiFarinaAJP2003].

The method we shall be concerned with was introduced by Schwinger in 1951 [@Schwinger1951] in a paper about QED entitled Gauge invariance and vacuum polarization". After introducing the proper time representation for computing effetive actions in QED, Schwinger was faced with a kind of non-relativistic propagator in one extra dimension. The way he solved this problem is what we mean by Schwinger’s method for computing quantum propagators. For relativistic Green functions of charged particles under external electromagnetic fields, the main steps of this method are summarized in Itzykson and Zuber’s textbook [@ItziksonZuberBook] (apart, of course, from Schwinger’s work [@Schwinger1951]). Since then, this method has been used mainly in relativistic quantum theory [@GitmanBook; @Dodonov75; @Dodonov76; @Lykken; @Ferrando:1994vt; @BFV96; @Gavrilov:1998hw; @McKeon:1998zx; @Chyi:1999fc; @Tsamis:2000ah; @Chaichian:2000eh; @Chung:2001mb; @BoschiFarinaVaidya1996].

However, as mentioned before, Schwinger’s method is also well suited for computing non-relativistic propagators, though it has rarely been 