Dynamics of Free Particle and Harmonic Oscillator Using Propagators

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The time evolution of a one-dimensional quantum system from an initial state can be represented, in terms of the propagator, by [1, 2]



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 .


Contributed by: S. M. Blinder (February 2020)
Open content licensed under CC BY-NC-SA



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,



[1] R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals, New York: McGraw-Hill, 1965.

[2] Wikipedia. "Propagator." (Feb 24, 2020) en.wikipedia.org/wiki/Propagator.

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