Non-relativistic propagators via Schwinger's method

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, R and P, we apply this method by solving the Heisenberg equations for the gauge invariant operators R and pi = P-eA, the latter being the mechanical momentum operator. In our procedure we avoid fixing the gauge from the beginning and the result thus obtained shows ...