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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