Dynamics of Free Particle and Harmonic Oscillator Using Propagators
The time evolution of a one-dimensional quantum system from an initial state can be represented, in terms of the propagator, by [1, 2][more]
For the free particle,
while for the harmonic oscillator,
For compactness, we use units with . For the initial state, we consider the Gaussian wave packet and the rectangular pulse.
The plots, for selected cases, show the probability densities and .[less]
For the free particle with initial Gaussian wave packet , we find
For initial rectangular pulse ,
For the harmonic oscillator with initial Gaussian wave packet,
(For , this reduces to the time-dependent ground-state eigenfunction .)
For initial rectangular pulse,
 R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals, New York: McGraw-Hill, 1965.
 Wikipedia. "Propagator." (Feb 24, 2020) en.wikipedia.org/wiki/Propagator.