In this paper, the propagator for a harmonic oscillator with mass obeying the function of m(t)=mtan2νt is derived by the Feynman path integral method. The wave function of this oscillator is calculated by expanding the obtained propagator. The propagator for a harmonic oscillator with strongly pulsating mass is evaluated by the Schwinger method. The propagator for a harmonic oscillator with mass rapidly growing with time is calculated by applying the integrals of the motion of quantum systems. The comparison between these methods are also discussed.
Using the Caldirola-Kanai Hamiltonian for the quantum dissipative system, the author expresses the propagator as the modified Feynman path integral in the configuration space, which can then be evaluated for the damped harmonic oscillator by Montroll's method. The propagator of the damped harmonic oscillator can also be calculated beyond and at caustics with the help of Horvathy-Feynman formula. The new results are confirmed by investigating the classical paths joining two fixed end-point positions, Finally the author obtains the time-dependent wavefunctions from the propagator of the dynamical system.