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.

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.

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.