---
abstract: 'In order to popularize the so called Schwinger’s method we reconsider the Feynman propagator of two non-relativistic systems: a charged particle in a uniform magnetic field and a charged harmonic oscillator in a uniform magnetic field. Instead of solving the Heisenberg equations for the position and the canonical momentum operators, ${\bf R}$ and ${\bf P}$, we apply this 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. In our procedure we avoid fixing the gauge from the beginning and the result thus obtained shows explicitly the gauge dependence of the Feynman propagator.'
author:
- 'A. Aragão'
- 'F. A. Barone'
- 'H. Boschi-Filho'
- 'C. Farina'
title: 'Non-Relativistic Propagators via Schwinger’s Method'
---

Introduction
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In a recent paper published in this journal [@BaroneBoschiFarinaAJP2003], three methods were used to compute the Feynman propagators of a one-dimensional harmonic oscillator, with the purpose of allowing a student to compare the advantadges and disadvantadges of each method. The above mentioned methods were the following: the so called Schwinger’s method (SM), the algebraic method and the path integral one. Though extremely powerful and elegant, Schwinger’s 