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WOLFRAM|DEMONSTRATIONS PROJECT

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eigenvalues
eigenfunctions
a
2.5
n
1
2
3
Energies for V(r)
E
n
= -
1
2
2
n
2a
-
1
n
E
1
= -0.32
E
2
= -0.005
E
3
= -0.0355556
Energies for -
1
r
E
n
= -
1
2
2
n
E
1
= -0.5
E
2
= -0.125
E
3
= -0.05556
The Hulthen potential is a short-range potential that behaves like a Coulomb potential for small values of
r
but decreases exponentially for large values of
r
. It has been applied to problems in nuclear, atomic and solid-state physics.
The Hulthen potential has the following form:
V(r)=-
Z
a
-r/a
e
1-
-r/a
e
=-
Z
a
-1
(
r/a
e
-1)
;
you can adjust the parameter
a
. In the limit as
a∞
, this reduces to a Coulomb potential
V(r)=-
Z
r
. For
a0
, the potential simulates a three-dimensional delta function
δ(r)
.
Consider the radial Schrödinger equation, in atomic units
ℏ=m=e=1
:
-
1
2
P''(r)-
Z
a
-r/a
e
1-
-r/a
e
P(r)=EP(r)
,
in terms of the reduced radial function
P(r)=r
R
nl
(r)
. The Schrödinger equation can be solved in closed form for
s
-states (
l=0
). The (unnormalized) solutions are given by
P
n
(r)=
-αr/a
e
(1-
-r/a
e
)
2
F
1
(2α+n+1,1-n;2α+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
a∞
, the energy approaches the Coulomb value
E
n
=-
2
Z
2
2
n
.
Choose "eigenvalues" to show the potential curve
V(r)
in black and the Coulomb potential
-1/r
in red. For each, the energy levels for
n=1
,
2
and
3
are shown as horizontal lines. Assume
Z=1
. Choose "eigenfunctions" to show plots of the radial functions
P
n
(r)
in black and the corresponding Coulombic (hydrogen atom) functions
H
P
n
(r)
in red; they merge as
a
is increased.
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