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.
r
r
The Hulthen potential has the following form:
V(r)=-=-
Z
a
-r/a
e
1-
-r/a
e
Z
a
-1
(-1)
r/a
e
you can adjust the parameter . In the limit as , this reduces to a Coulomb potential . For , the potential simulates a three-dimensional delta function .
a
a∞
V(r)=-
Z
r
a0
δ(r)
Consider the radial Schrödinger equation, in atomic units :
ℏ=m=e=1
-P''(r)-P(r)=EP(r)
1
2
Z
a
-r/a
e
1-
-r/a
e
in terms of the reduced radial function . The Schrödinger equation can be solved in closed form for -states (). The (unnormalized) solutions are given by
P(r)=r(r)
R
nl
s
l=0
P
n
-αr/a
e
-r/a
e
2
F
1
-r/a
e
where
α=-
n
2
Za
n
E
n
1
2
2
n
2a
Z
n
n=1,2,3,….
In the limit as , the energy approaches the Coulomb value =-.
a∞
E
n
2
Z
2
2
n
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 (r) in black and the corresponding Coulombic (hydrogen atom) functions (r) in red; they merge as is increased.
V(r)
-1/r
n=1
2
3
Z=1
P
n
H
P
n
a