Analytic Solutions for Double Deltafunction Potential
Among the small number of quantum-mechanical problems with known analytical solutions  is the double-well deltafunction potential , 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).[less]
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.
 Wikipedia. "List of Quantum-Mechanical Systems with Analytical Solutions." (Feb 3, 2021) en.wikipedia.org/wiki/List_of_quantum-mechanical_systems_with_analytical_solutions.
 Wikipedia. "Delta Potential." (Feb 3, 2021) en.wikipedia.org/wiki/Delta_potential.