same method was independently developed by M. Goldberger and M. GellMann in the autumn of 1951 in connection with an unpublished paper about density matrix in statistical mechanics [@Goldberger1951].

Our purpose in this paper is to provide the reader with two other examples of non-relativistic quantum propagators that can be computed in a straightforward way by Schwinger’s method, namely: the propagator for a charged particle in a uniform magnetic field and this same problem with an additional harmonic oscillator potential. Though these problems have already been treated in the context of the quantum action principle [@UrrutiaManterola1986], we decided to reconsider them for the following reasons: instead of solving the Heisenberg equations for the position and the canonical momentum operators, ${\bf R}$ and ${\bf P}$, as is done in [@UrrutiaManterola1986], we apply Schwinger’s method by solving the Heisenberg equations for the gauge invariant operators ${\bf R}$ and $\mbox{\mathversion{bold}${\pi}$} = {\bf P}-e{\bf A}$, the latter being the mechanical momentum operator. This is precisely the procedure followed by Schwinger in his seminal paper of gauge invariance and vacuum polarization [@Schwinger1951]. This procedures has some nice properties. For instance, we are not obligued to choose a particular gauge at the beginning of calculations. As 