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NUCLEAR PHYSICS GROUP
DEPARTMENT OF
ENGINEERING PHYSICS
Unıversity of Gaziantep 27310, Gaziantep, Türkiye
NPG Web Page : http://www1.gantep.edu.tr/~ozer
E-Mail Addresses: [email protected]
Unified treatment of screening Coulomb
and
anharmonic oscillator potentials
in arbitrary dimensions
Okan Özer, Bülent Gönül
Department of Engineering Physics, University of Gaziantep,
27310, Gaziantep, Türkiye
UNIFIED TREATMENT OF
SCREENING COULOMB AND
ANHARMONIC OSCILLATOR POTENTIALS
IN ARBITRARY DIMENSIONS
Abstract
A mapping is obtained relating radial screened Coulomb
systems with low screening parameters to radial
anharmonic oscillators in N-dimensional space. Using the
formalism of supersymmetric quantum mechanics, it is
shown that exact solutions of these potentials exist when
the parameters satisfy certain constraints.
p.1
MAPPINGS BETWEEN THE TWO DISTINCT
SYSTEMS
1  d 2 R N  1 dR  (  N  2)
 

R  E  V ( r )R

2
2
2  dr
r dr 
2r
(1)
Eq. (1) is transformed to
d 2
 ( M  1)( M  3)



 2V (r )   2 E
2
2


dr
4r
where
(2)
(r )  r ( N 1) / 2 R(r ) and M  N  2 .
If it is substituted r  2 and R  F (  ) /  
1  d 2 F N   1 dF  L( L  N   2)
 


F  Eˆ  Vˆ (  ) F (3)

2  d 2
 d 
2 2

where
N   2 N  2  2
,
L  2  

p.2
And from Eq. (3), it is obtained that
Eˆ  Vˆ (  )  E 2  2   2  2V (2 / 2)
(4)
The static-screened Coulomb potential is given as
VSC ( r )  e
2 e
r
r
(5)
Then, within the frame of low screening parameter,  , it
becomes as
e2
e2 2
e2 3 2 e2 4 3 e2 5 4
2
VSC (r )  
e  
r
r 
r 
r
r
2
6
24
120
A1
(6)

 A2  A3r  A4r 2  A5r 3  A6r 4
r
p.3
NOW, Eq. (6) is transformed to the anharmonic oscillator
using the procedure as mentioned above (with the choice
of  2  1 / En 0 )

A2
ˆ
V (  )  1 
En 0

 2
A3
A4
4
6
 




3/ 2
2
2 En 0
4 En 0

A5
A6
8
10




5/ 2
3
8 En 0
16 En 0
(7)
with the eigenvalue
Eˆ n 0 
 2 A1
En 0
1/ 2
Thus the system of Eq. (5) is reduced to another system
defined by Eq. (7) !!!
(8)
p.4
Supersymmetric treatment for the ground state
Using the SUSYQM, we set the superpotential term as
a
W (r )  1  a2  a3r  a4r 2 , a4  0
r
(9)
for the potential given in Eq. (6). Then the SUSY-partner
potential is found as
V (r )  W 2 (r )  W (r )


2a1a2

 a22  a3 (2a1  1)  2(a1a4  a4  a2a3 )r
r
a (a  1)
 2(a2a4  a32 )r 2  2a3a4r 3  a42r 4  1 1
r2
 2 A1


 2 A2  2 A3r  2 A4r 2  2 A5r 3  2 A6r 4  
 r

( M  1)( M  3)
 2 En 0 (10)
4r 2
p.5
where
a1 
M 1
2 A1
, a2 
, a3  
2
M 1
A5
, a4   2 A6
2 A6
(11)
The physically observables for the interested potential under the
constraints
2
8 A6 A4  2 A5
A1  ( M  1)
,
16 A6 2 A6
 ( M  1)
A1
A5 
A3   2 A6 


2
(
M

1
)
A
6

(12)
are found as
a
a


n 0 (r )  N0 r a1 exp a2r  3 r 2  4 r 3 
2
3


1  4 A12
En 0  A2  

2
2  ( M  1)

A5
M
2 A6

(13)
p.6
For the anharmonic oscillator potential, we set
W (  )  a  b 
5
3
c
 d ,
a0,d 0
(14)
b
d
a

n 0 (  )  C0  c exp  6   4   2 
4
2
6

(15)

which leads to
and leads to physically meaningful eigenvalue
2
d
8
A
A

2
A
M
6
4
5
ˆ
En 0   (2c  1) 
2
16 A6 2 A6 En 0 1 / 2
where M   N   2 L .
(16)
p.7
Significance of mapping parameter
To make clear the significance of the mapping parameter, ,
we consider Eq. (13) and Eq. (16) together with and arrive at
M
A1
ˆ
En 0  
1/ 2
M 1 E
n 0
To be consistent with Eq. (8), it is imposed that
such that
0   1
M
2( N  1   )  2(2   )

2
M 1
N  2  1
Numerical results for the interested potentials are tabulated for
different values of screening parameter, angular momentum
quantum number in arbitrary dimensions in Table 1 and 2.
p.8
Table 1. The first four eigenvalues of the screening Coulomb
potential as a function of the screening parameter in atomic
units.
p.9
Table 2. Ground-state eigenvalues of the anharmonic potential
8
p.10
CONCLUDING REMARKS
As the objective of this presentation we have highlighted a
different facet of these studies and established a very general
connection between the screened Coulomb and anharmonic
oscillator potentials in higher dimensional space through the
application of a suitable transformation.
The purpose being the emphasize the pedagogical value
residing in this interrelationship between two of the most
practical applications of quantum mechanics.
The exact ground state solutions for the potentials considered
are obtained analytically within the framework of
supersymmetric quantum mechanics.