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.
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.