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Analytic Solutions for Double Deltafunction Potential

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Among the small number of quantum-mechanical problems with known analytical solutions [1] is the double-well deltafunction potential [2], which can be considered a one-dimensional analog of the hydrogen molecule ion . The Schrödinger equation (in atomic units ) can be written

[more]

,

where the two attractive deltafunction potential wells, with effective "nuclear charges" , are located at and separated by an "internuclear distance" .

The solution of the single deltafunction problem suggests an ansatz for the double deltafunction in the form

,

for the even- and odd-parity solutions, respectively. A derivation of the exponential coefficients is given in the Details below.

The wavefunctions and energy curves are plotted for selected values of and , for both the even ground states (in black) and odd first-excited states (in red).

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Contributed by: S. M. Blinder (February 2021)
Open content licensed under CC BY-NC-SA

Snapshots

Details

The form given for satisfies the Schrödinger equation when

,

having noted that the deltafunctions arise from the derivative of absolute values. The exponential coefficients for the even and odd solutions are thus determined by the transcendental relations

.

These can be solved in terms of the Lambert function (or product logarithm) as

.

The normalized eigenfunctions for the ground and first-excited states are given by

,

with the corresponding energy eigenvalues .

The ground state for a single deltafunction potential is given by

,

with an energy , resembling a one-dimensional projection of the state of a hydrogen atom.

References

[1] Wikipedia. "List of Quantum-Mechanical Systems with Analytical Solutions." (Feb 3, 2021) en.wikipedia.org/wiki/List_of_quantum-mechanical_systems_with_analytical_solutions.

[2] Wikipedia. "Delta Potential." (Feb 3, 2021) en.wikipedia.org/wiki/Delta_potential.

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