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.
The initial value problem method is formulated to calculate the propagator for time- dependent harmonic oscillators. The method is based on finding the initial position operator from Heisenberg equations. The investigated models in this paper are the damped harmonic oscillator, the harmonic oscillator with strongly pulsating mass, and the harmonic oscillator with mass growing with time. The comparison of the initial value problem method with Feynman path integral and Schwinger method is also described.