The Hulthen potential is a short-range potential that behaves like a Coulomb potential for small values of

but decreases exponentially for large values of

. It has been applied to problems in nuclear, atomic and solid-state physics.

The Hulthen potential has the following form:

;

you can adjust the parameter

. In the limit as

, this reduces to a Coulomb potential

. For

, the potential simulates a three-dimensional delta function

.

Consider the radial Schrödinger equation, in atomic units

:

,

in terms of the reduced radial function

. The Schrödinger equation can be solved in closed form for

-states (

). The (unnormalized) solutions are given by

,

In the limit as

, the energy approaches the Coulomb value

.

Choose "eigenvalues" to show the potential curve

in black and the Coulomb potential

in red. For each, the energy levels for

,

and

are shown as horizontal lines. Assume

. Choose "eigenfunctions" to show plots of the radial functions

in black and the corresponding Coulombic (hydrogen atom) functions

in red; they merge as

is increased.