NUCLEAR PHYSICS GROUP

DEPARTMENT OFENGINEERING PHYSICS

Unıversity of Gaziantep 27310, Gaziantep, Türkiye NPG Web Page : http://www1.gantep.edu.tr/~ozer

E-Mail Addresses: ozer@gantep.edu.tr

Unified treatment of screening CoulombUnified treatment of screening Coulomband and

anharmonic oscillator potentialsanharmonic oscillator potentialsin arbitrary dimensionsin arbitrary dimensions

Okan Özer, Bülent Gönül

Department of Engineering Physics, University of Gaziantep, 27310, Gaziantep, Türkiye

AbstractAbstract

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.

UUNIFINIFIED TREATMENT OFED TREATMENT OFSCREENSCREENIING COULOMBNG COULOMB AND AND

ANHARMONANHARMONIIC OSCC OSCIILLATOR POTENTLLATOR POTENTIIALSALSIIN ARBN ARBIITRARY DTRARY DIIMENSMENSIIONSONS

MAPPINGS BETWEEN THE TWO DISTINCT MAPPINGS BETWEEN THE TWO DISTINCT SYSTEMSSYSTEMS

RrVERr

N

dr

dR

r

N

dr

Rd)(

2

)2(1

2

122

2

(1)

(2)

Eq. (1) is transformed to

ErV

r

MM

dr

d2)(2

4

)3)(1(22

2

)()( 2/)1( rRrr N where and 2NM

/)(2 FRandr If it is substituted

FVEFNLL

d

dFN

d

Fd)(ˆˆ

2

)2(1

2

122

2

(3)

.

where 2,222 LNN

p.1p.1

)2/()(ˆˆ 22222 VEVE (4)

r

eerV

r

SC

2)( (5)

452

342

23222

22

1202462)( r

er

er

er

ee

r

erVSC

(6)46

35

2432

1 rArArArAAr

A

The static-screened Coulomb potential is given as

Then, within the frame of lowlow screening parameter, , it becomes as

And from Eq. (3), it is obtained that

p.2p.2

103

0

682/5

0

5

62

0

442/3

0

32

0

2

168

421)(ˆ

nn

nnn

E

A

E

A

E

A

E

A

E

AV

2/10

10

2ˆ

nn

E

AE

(7)

NOW, Eq. (6) is transformed to the anharmonic oscillator using the procedure as mentioned above (with the choice of )

(8)

with the eigenvalue

02 /1 nE

Thus the system of Eq. (5) is reduced to another system defined by Eq. (7) !!!

p.3p.3

Supersymmetric treatment for the ground stateSupersymmetric treatment for the ground state

Using the SUSYQM, we set the superpotential term as

0,)( 42

4321 araraar

arW

for the potential given in Eq. (6). Then the SUSY-partner potential is found as

(9)

21142

43

4322

342

324411322

21

2

)1(2)(2

)(2)12(2

)()()(

r

aararaaraaa

raaaaaaaar

aa

rWrWrV

02

46

35

2432

1

24

)3)(1(

222222

nEr

MM

rArArArAAr

A

(10)

p.4p.4

646

53

121 2,

2,

1

2,

2

1Aa

A

Aa

M

Aa

Ma

where

(11)

3423200 32

exp)( 1 ra

ra

rarNr an

M

A

A

M

AAEn

6

52

21

20 2)1(

4

2

1

6

5163

66

2546

1

)1(2

)1(2

,216

28)1(

A

A

M

AMAA

AA

AAAMA

The physically observables for the interested potential under the constraints

(12)

are found as

(13)

p.5p.5

For the anharmonic oscillator potential, we set

0,0,)( 35 dadc

baW

(14)

which leads to

24600 246

exp)( dbaC c

n (15)

2/1066

2546

0 216

28)12(

2ˆ

nn

E

M

AA

AAAc

dE

and leads to physically meaningful eigenvalue

where .LNM 2

(16)

p.6p.6

Significance of mapping parameterSignificance of mapping parameter

2/10

10 1

ˆ

nn

E

A

M

ME

212

)2(2)1(2

1

N

N

M

M

To make clear the significance of the mapping parameter, ,

we consider Eq. (13) and Eq. (16) together with and arrive at

To be consistent with Eq. (8), it is imposed that

such that

10

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.7p.7

Table 1. The first four eigenvalues of the screening Coulomb potential as a function of the screening parameter in atomic units.

p.8p.8

Table 2. Ground-state eigenvalues of the anharmonic potentialp.9p.9

8

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.

CONCLUDING REMARKSCONCLUDING REMARKS

p.10p.10