NUCLEAR PHYSICS GROUP DEPARTMENT OF ENGINEERING PHYSICS Unıversity of Gaziantep 27310, Gaziantep,...
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Transcript of NUCLEAR PHYSICS GROUP DEPARTMENT OF ENGINEERING PHYSICS Unıversity of Gaziantep 27310, Gaziantep,...
NUCLEAR PHYSICS GROUP
DEPARTMENT OFENGINEERING 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 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