Delta Potential - Double Delta Potential

Double Delta Potential

The Double-well Dirac delta function model is described by the corresponding Schrödinger equation:

where the potential is now:

where is the "internuclear" distance with Dirac delta function (negative) peaks located at (shown in brown in the diagram). Keeping in mind the relationship of this model with its three-dimensional molecular counterpart, we use Atomic units and set . Here is a formally adjustable parameter. From the single well case, we can infer the "ansatz" for the solution to be:

Matching of the wavefunction at the Dirac delta function peaks yields the determinant:

Thus, is found to be governed by the pseudo-quadratic equation:

which has two solutions . For the case of equal charges (symmetric homonuclear case), and the pseudo-quadratic reduces to:

The "+" case corresponds to a wave function symmetric about the midpoint (shown in red in the diagram) where and is called gerade. Correspondingly, the "-" case is the wave function that is anti-symmetric about the midpoint where is called ungerade (shown in green in the diagram). They represent an approximation of the two lowest discrete energy states of the three-dimensional and are useful in its analysis. Analytical solutions for the energy eigenvalues for the case of symmetric charges are given by :

where W is the standard Lambert W function. Note that the lowest energy corresponds to the symmetric solution . In the case of unequal charges, and for that matter the three-dimensional molecular problem, the solutions are given by a generalization of the Lambert W function (see section on generalization of Lambert W function and references herein).

One of the most interesting cases is when which results in . Thus, we will have a non-trivial bound state solution that has . For these specific parameters, there are many interesting properties that occur, one of which is the unusual effect that the transmission coefficient is unity at zero energy.