# Harmonic-Gaussian Double-Well Potential

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A variant of a double-well potential is a harmonic oscillator perturbed by a Gaussian, represented by the potential . A similar function was used to model the inversion of the ammonia molecule [1]. The problem can be treated very efficiently using second-order perturbation theory based on the unperturbed harmonic oscillator. The first six energy levels are computed here.

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Contributed by: S. M. Blinder (April 2013)

Open content licensed under CC BY-NC-SA

## Snapshots

## Details

Snapshot 1: with perturbation turned off, simple harmonic oscillator with energy levels

Snapshot 2: relatively small perturbation, showing convergence of levels and

Snapshot 3: larger perturbation showing approach to degeneracy of two pairs of levels

References

[1] J. D. Swalen and J. A. Ibers, "Potential Function for the Inversion of Ammonia," *Journal of Chemical Physics,* 36(7), 1962 pp. 1914–1918. doi:10.1063/1.1701290.

[2] K. T. Hecht, *Quantum Mechanics*, New York: Springer-Verlag, 2000 pp. 365–368.

[3] Wikipedia. "Perturbation Theory (Quantum Mechanics)." (Mar 11, 2013) en.wikipedia.org/wiki/Perturbation_theory_(quantum_mechanics).

## Permanent Citation

"Harmonic-Gaussian Double-Well Potential"

http://demonstrations.wolfram.com/HarmonicGaussianDoubleWellPotential/

Wolfram Demonstrations Project

Published: April 4 2013