# Hulthen potential

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