a consequence, we end up with an expression for the propagator written in an arbitrary gauge. As a bonus, the transformation law for the propagator under gauge transformations can be readly obtained.

In order to prepare the students to attack more complex problems, we solve the Heisenberg equations in matrix form, which is well suited for generalizations involving Green functions of relativistic charged particles under the influence of electromagnetic fields (constant $F_{\mu\nu}$, a plane wave field or even combinations of both). For pedagogical reasons, at the end of each calculation, we show how to extract the corresponding energy spectrum from the Feynman propagator. Although the way Schwniger’s method must be applied to non-relativistic problems has already been explained in the literature [@UrrutiaManterola1986; @SchwingerBookEnglert2001; @BaroneBoschiFarinaAJP2003], it is not of common knowledge so that we start this paper by summarizing its main steps. The paper is organized as follows: in the next section we review Schwinger’s method, in section \[SectionExamples\] we present our examples and section \[SectionFinalRemarks\] is left for the final remarks.

Main steps of Schwinger’s method
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For simplicity, consider a one-dimensional time-independent Hamiltonian $\mathcal{H}$ and the corresponding non-relativistic Feynman propagator defined as $$\label{DefinicaoPropagador}
K(x,x^{\prime};\tau)=\theta(\tau) \langle x | \exp{ \Big[ 