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The Hulthen potential [[#References|[a1]]] is given by
 
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>
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$$ \tag{a1 }
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V ( r ) = - {
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\frac{z}{a}
 +
} \cdot {
 +
\frac{ { \mathop{\rm exp} } { {
 +
\frac{- r }{a}
 +
} } }{1 - { \mathop{\rm exp} } { {
 +
\frac{- r }{a}
 +
} } }
 +
} ,
 +
$$
  
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.
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where $  a $
 +
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]]].
+
The Hulthen potential is a short-range potential which behaves like a Coulomb potential for small values of $  r $
 +
and decreases exponentially for large values of $  r $.  
 +
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 $  s $
 +
waves. For $  l \neq 0 $,  
 +
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====
 
====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>
 
<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>

Latest revision as of 22:11, 5 June 2020


The Hulthen potential [a1] is given by

$$ \tag{a1 } V ( r ) = - { \frac{z}{a} } \cdot { \frac{ { \mathop{\rm exp} } { { \frac{- r }{a} } } }{1 - { \mathop{\rm exp} } { { \frac{- r }{a} } } } } , $$

where $ a $ 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 $ r $ and decreases exponentially for large values of $ r $. The Hulthen potential has been used in many branches of physics, such as nuclear physics [a2], atomic physics [a3], [a4], solid state physics [a5], and chemical physics [a6]. The model of the three-dimensional delta-function could well be considered as a Hulthen potential with the radius of the force going down to zero [a7]. The Schrödinger equation for this potential can be solved in a closed form for $ s $ waves. For $ l \neq 0 $, a number of methods have been employed to find approximate solutions for the Schrödinger equation with the Hulthen potential [a8], [a9], [a10], [a11]. The Dirac equation with the Hulthen potential has also been studied using an algebraic approach [a12].

References

[a1] L. Hulthen, Ark. Mat. Astron. Fys , 28A (1942) pp. 5 (Also: 29B, 1)
[a2] L. Hulthen, M. Sugawara, S. Flugge (ed.) , Handbuch der Physik , Springer (1957)
[a3] T. Tietz, J. Chem. Phys. , 35 (1961) pp. 1917
[a4] C.S. Lam, Y.P. Varshni, Phys. Rev. A , 4 (1971) pp. 1875
[a5] A.A. Berezin, Phys. Status. Solidi (b) , 50 (1972) pp. 71
[a6] P. Pyykko, J. Jokisaari, Chem. Phys. , 10 (1975) pp. 293
[a7] A.A. Berezin, Phys. Rev. B , 33 (1986) pp. 2122
[a8] C.S. Lai, W.C. Lin, Phys. Lett. A , 78 (1980) pp. 335
[a9] S.H. Patil, J. Phys. A , 17 (1984) pp. 575
[a10] V.S. Popov, V.M. Wienberg, Phys. Lett. A , 107 (1985) pp. 371
[a11] B. Roy, R. Roychoudhury, J. Phys. A , 20 (1987) pp. 3051
[a12] B. Roy, R. Roychoudhury, J. Phys. A , 23 (1990) pp. 5095
How to Cite This Entry:
Hulthen potential. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hulthen_potential&oldid=47279
This article was adapted from an original article by R. Roychoudhury (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article