# 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]

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Contributed by: S. M. Blinder (February 2020)

Open content licensed under CC BY-NC-SA

## Snapshots

## Details

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,

.

References

[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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