Harmonic-Gaussian Double-Well Potential
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 . 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.[more]
The unperturbed Hamiltonian is . For convenience, we set . The unperturbed eigenfunctions are given by , where are Hermite polynomials. The unperturbed eigenvalues are then . The perturbation is the Gaussian function . To second-order in perturbation theory, we have . (In practice, the sum is truncated at some ).
When , the perturbation reduces to zero and the system reverts to a simple harmonic oscillator. As the central barrier becomes wider, the lower eigenvalues ( and ; also and ) approach degenerate pairs. The situation becomes similar to tunneling, giving two eigenstates of opposite parity, with their linear combinations approximating localized states.[less]
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
 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.
 K. T. Hecht, Quantum Mechanics, New York: Springer-Verlag, 2000 pp. 365–368.
 Wikipedia. "Perturbation Theory (Quantum Mechanics)." (Mar 11, 2013) en.wikipedia.org/wiki/Perturbation_theory_(quantum_mechanics).