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
, ,
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.


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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.
[1] 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.
[2] 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.
[3] S. Flügge, Practical Quantum Mechanics, New York: Springer, 1999 pp. 175–178.
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