# Eigenstates for the Hulthen Potential

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 ,
where , , 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.

### DETAILS

To solve the Schrödinger equation, make the variable transformation with and define the constants , .
The differential equation becomes ,
subject to the boundary conditions and . The solution is found to be .
The second boundary condition requires that , This leads to the forms of and given above.
References
 M. R. Setare and E. Karimi, "Algebraic Approach to the Hulthen Potential," International Journal of Theoretical Physics, 46(5), 2007 pp. 1381–1388. doi:10.1007/s10773-006-9276-z.
 J. Stanek, "The One-Dimensional Hulthén Potential in the Quantum Phase Space Representation," Central European Journal of Physics, 12(2), 2014 pp. 90–96. doi:10.2478/s11534-014-0433-3.
 S. Flügge, Practical Quantum Mechanics, New York: Springer, 1999 pp. 175–178.

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