In this paper, the propagators for a particle moving in a time-dependent linear potential and a free particle with linear damping are calculated by the application of the integrals of the motion of a quantum system. The propagator for a charged harmonic oscillator is derived from the Feynman path integrals and the propagator for a damped harmonic oscillator is evaluated by the Schwinger method. The relation between the integrals of the motion, Feynman path integrals, and Schwinger method are also described.
Based on the equivalence between momentum representation and coordinate representation,an alternative method is presented,where the momentum representation and the Feynman's path integral are employed to calculate the propagator function of a particle in a constant force field,then the result is transformed into the coordinate representation. We present a way to solve the Feynman's path integral,which exhibits the high value of momentum representation if the solution of one- dimensional motion of a particle in a constant force field.