View source for Hulthen potential

The Hulthen potential [[#References|[a1]]] is given by

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110350/h1103501.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110350/h1103502.png" /> is the screening parameter and z is a constant which is identified with the atomic number when the potential is used for atomic phenomena.

The Hulthen potential is a short-range potential which behaves like a Coulomb potential for small values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110350/h1103503.png" /> and decreases exponentially for large values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110350/h1103504.png" />. The Hulthen potential has been used in many branches of physics, such as nuclear physics [[#References|[a2]]], atomic physics [[#References|[a3]]], [[#References|[a4]]], solid state physics [[#References|[a5]]], and chemical physics [[#References|[a6]]]. The model of the three-dimensional [[Delta-function|delta-function]] could well be considered as a Hulthen potential with the radius of the force going down to zero [[#References|[a7]]]. The [[Schrödinger equation|Schrödinger equation]] for this potential can be solved in a closed form for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110350/h1103505.png" /> waves. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110350/h1103506.png" />, a number of methods have been employed to find approximate solutions for the Schrödinger equation with the Hulthen potential [[#References|[a8]]], [[#References|[a9]]], [[#References|[a10]]], [[#References|[a11]]]. The Dirac equation with the Hulthen potential has also been studied using an algebraic approach [[#References|[a12]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  L. Hulthen,   ''Ark. Mat. Astron. Fys'' , '''28A'''  (1942)  pp. 5  (Also: 29B, 1)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  L. Hulthen,   M. Sugawara,   S. Flugge (ed.) , ''Handbuch der Physik'' , Springer  (1957)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  T. Tietz,   ''J. Chem. Phys.'' , '''35'''  (1961)  pp. 1917</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  C.S. Lam,   Y.P. Varshni,   ''Phys. Rev. A'' , '''4'''  (1971)  pp. 1875</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  A.A. Berezin,   ''Phys. Status. Solidi (b)'' , '''50'''  (1972)  pp. 71</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  P. Pyykko,   J. Jokisaari,   ''Chem. Phys.'' , '''10'''  (1975)  pp. 293</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  A.A. Berezin,   ''Phys. Rev. B'' , '''33'''  (1986)  pp. 2122</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  C.S. Lai,   W.C. Lin,   ''Phys. Lett. A'' , '''78'''  (1980)  pp. 335</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  S.H. Patil,   ''J. Phys. A'' , '''17'''  (1984)  pp. 575</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  V.S. Popov,   V.M. Wienberg,   ''Phys. Lett. A'' , '''107'''  (1985)  pp. 371</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top">  B. Roy,   R. Roychoudhury,   ''J. Phys. A'' , '''20'''  (1987)  pp. 3051</TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top">  B. Roy,   R. Roychoudhury,   ''J. Phys. A'' , '''23'''  (1990)  pp. 5095</TD></TR></table>