Eigenstates for the Hulthen Potential

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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.
Contributed by: S. M. Blinder (July 2020)
Open content licensed under CC BY-NC-SA
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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
[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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