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Thermodynamics properties study of diatomic molecules with q-deformed modified Poschl-

Teller plus Manning Rosen non-central potential in D dimensions using SUSYQM approach

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2016 J. Phys.: Conf. Ser. 710 012026

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ScieTech 2016 IOP Publishing
Journal of Physics: Conference Series 710 (2016) 012026 doi:10.1088/1742-6596/710/1/012026

Thermodynamics properties study of diatomic molecules with

q-deformed modified Poschl-Teller plus Manning Rosen non-
central potential in D dimensions using SUSYQM approach
A Suparmi,C Cari and B N Pratiwi
Physics Department, Faculty of Mathematics and Science, Sebelas Maret University
Jl. Ir. Sutami 36A Kentingan Surakarta 57126, Indonesia

Email: soeparmi@staff.uns.ac.id

Abstract. D-dimensional Dirac equation of q-deformed modified Poschl-Teller plus Manning

Rosen non-central potential was solved using supersymmetric quantum mechanics (SUSY
QM). The relativistic energy spectra were analyzed by using SUSY QM and shape invariant
properties from radial part of D dimensional Dirac equation and the angular quantum numbers
were obtained from angular part of D dimensional Dirac equation. The SUSY operators was
used to generate the D dimensional relativistic wave functions both for radial and angular parts.
In the non-relativistic limit, the relativistic energy equation was reduced to the non-relativistic
energy. In the classical limit, the partition function of vibrational, the specific heat of
vibrational, and the mean energy of vibrational of some diatomic molecules were calculated
from the equation of non-relativistic energy with the help of error function and Mat-lab 2011.

1. Introduction
In some area of physics, relativistic quantum mechanics play important roles. Finding an accurate
exact solution of Dirac equation for a certain potential is one of its important roles. Various methods
have been applied to solve the Dirac equation for some potentials, central and non-central potentials,
with or without tensor coupling potentials, such as NU method, [1-6] SUSY QM method, [7-11] and
Romanovski polynomial method. [12-16]
For very limited potentials, three dimensional radial Dirac equations are exactly solved only for s-
wave (l = 0). However, the three dimensional radial Dirac equations for the spherically symmetric
potentials can only be solved approximately for 𝑙 ≠ 0 states due tothe approximation scheme of the
centrifugal term ~𝑟 −2. [17-23]
Furthermore, the extension in higher dimensional spaces for some physical problems is very
important in some physics area. The D-dimensional non-relativistic and relativistic physical systems
have been investigated by many authors, such as ring-shaped pseudoharmonic potential, [21] the
isotropic harmonic oscillator and inverse quadratic potential, [22] Pseudoharmonic potential,[23]
Kratzer-Fues potential, [24-25]hydrogen atom, [26]modified Poschl-Teller potential, [27] linierly
energy dependent quadratic potential, [28] trigonometric scarf potential, [29] ring-shaped Kratzer
potential. [30]
The Dirac equation for a charged particle that moves in a field governed by q-deformed Poschl-
Teller potential[31] in D dimension is investigated using supersymmetric quantum mechanic (SUSY
QM) with idea of shape invariance. SUSY QM method is developed based on Witten proposal [32]
and the idea of shape invariant potential is proposed by Gendenshtein [33]. SUSY QM is a powerful
tool to determine energy spectrum and wave function of shape invariant potentials for one dimensional
Schrodinger equation. Thus the relativistic energy spectrum is obtainable by using the idea of shape
invariance and the wave functions are achieved by using lowering and raising SUSY operators. Some
of hyperbolic and trigonometric potentials are exactly solvable within the approximation of centrifugal
term and their solutions have been reported in the previous papers [10-11].The q-deformed modified
Poschl-Teller plus Manning Rosen non-central potential that govern the diatomic molecules vibration
is expressed as
 
V (r , )  t 2  a(a  1) sinh 2q tr    b(b  1) cosh 2q tr   1/ r 2   p( p  1) sin 2    2s cot  
 
(1)

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ScieTech 2016 IOP Publishing
Journal of Physics: Conference Series 710 (2016) 012026 doi:10.1088/1742-6596/710/1/012026

with a  1, b  0;   0; p  0, s  0 , a, b, p and s are positive constants that control the depth of

the potential, t is positive constant which controls the width of the potential, 0  r   , q causes the
deformation of the potential shape, 0  q  1 . In the non-relativistic limit, the relativistic energy
equation reduces the non-relativistic energy equation. In the non relativistic limit is the condition when
the energy is subtracted by the mass equals to the non-relativistic energy which is usually obtained
from Schrodinger equation solution of the system, and the condition where the sum of the energy and
the mass equals to twice of masses. The non-relativistic energy is used in calculation of thermal
properties in classical limit. In classical regime, the thermal properties including vibrational partition
function Z, mean energy U, and specific heat C [34,35] are determined by applying the non-
relativistic energy equation.
This paper is organized as follows. Brief review of SUSY quantum mechanics is presented in
section 2, solution of Dirac equations and its application to study thermodynamical properties are
presented in section 3 and conclusion is presented in section 4.

2. Review of Supersymmetric Quantum Mechanics Approach Using Operator

2.1. Supersymmetry Quantum Mechanics (SUSY QM)
According to the definition proposed by Witten, in a SUSY QM there are super charge operators
Qi which commute with the Hamiltonian H ss and obey to anti commutation algebra [32]
Qi , H ss   0 ; Qi , Q j    ij H ss with, i = 1, 2, 3, … (2)
with H ss is called supersymmetric Hamiltonian. Witten proposed that the SUSY QM is the one
dimensional model of SUSY field theory and he stated that the simplest SUSY QM system has
N=2[32] where the two charge operators are given as

Q1  1/ 2 1 p     
2m   2 ( x) ; Q2  1/ 2  2 p   
2m  1 ( x)  (3)
where𝜎𝑖 are the usual Pauli spin matrices, p  i   x  is the usual one dimensional momentum
operator, and  (x) is superpotential. By inserting equation (3) into the second equation of equation (2)
we get,
 2 d2 d ( x ) 2 
    ( x) 0  (4)
 ; H   H  0 
2
2 m dx 2 m dx
H ss  
 2
d 2
d ( x ) 2 
ss
 0 H 
 0  2
   ( x ) 
 2m dx 2m dx 
From equation (4) we have
V ( x)   2 ( x)   '( x); V ( x)   2 ( x)   '( x) (5)
2m 2m
Here 𝐻− and 𝐻+ , are supersymmetry partner of the Hamiltonian, V (x) and V (x) are the
supersymmetry partner potential. To simplify the determination of the energy spectrum and the wave
functions, the new operators, raising and lowering operators, are introduced as
 d  d
A     ( x) and A    ( x) (6)
2m dx 2m dx
By inserting equation (6) into equation (5) we get the SUSY Hamiltonian as
H  ( x)  A A and H  ( x)  AA (7)
to factorize the usual Hamiltonian as
2 d 2 (8)
H  H   E0    V ( x; a0 )  E0
2m dx 2
By using equation (5) and equation (8) it is obtained that

V ( x)  V ( x; a0 )  E0   2 ( x; a0 )   ' ( x; a0 )  E0 (9)
2m

2
ScieTech 2016 IOP Publishing
Journal of Physics: Conference Series 710 (2016) 012026 doi:10.1088/1742-6596/710/1/012026

where V (x ) is the effective potential, while  (x) is determined hypothetically from equation (9)
based on the shape of effective potential from the associated system.

2.2. Shape Invariance

It is observed that the supersymmetry only gives the relationship between the eigenvalues and
eigenfunctionsbetween the two Hamiltonian partners but does not provide the actual spectrum.
[36].The energy spectrum is only obtainable by implementing SUSY charged operators properties and
the condition of shape invariant proposed by Gendenshtein.[33]. If a pair of potentials V (x) defined
in equation (5) are similar in shape but different in the parameters, then they are called to be shape
invariant. More specifically, if V ( x, a0 ) satisfy the condition that
V ( x; a j )  V ( x; a j 1 )  R (a j 1 ) (10)
with V ( x; a j )   2 ( x; a j )   '( x; a j );V ( x; a j 1 )   2 ( x; a j 1 )   '( x; a j 1 ) (11)
2m 2m
where j = 0,1,2,.., and a is a parameter in our original potential, V-, whose ground state energy is zero,
a j  f j (a0 ) where f jis a function applied j times, the remainder R (a j ) is a’s dependence but is
independent of x, then V ( x, a0 ) is said to be shaped invariant. The energy eigenvalue of the
Hamiltonian H  is given by [33]
 k 1 R(ak )
( ) n
En (12)
and by using equation (8) and equation (12) we get the energy spectra of the system given as,
( )
En  En  E0 (13)
Based on the characteristics of lowering operator, the ground state wave function of H-, whose ground
state energy is zero, is obtained from condition that,
( )
 0  A 0 0
( )
H  0 (14)
( ) ( )
Subsequently, the excited wave function,  ( x, a0 ),... n ( x, a0 ) of H  are obtained by using
1
raising operator operated to the lower wave function, [37] given as
 n (  ) ( x; a0 )  A ( x; a0 ) A ( x.a1 )....A ( x, an 1 ) 0(  ) ( x; an ) (15)
In the Dirac equation we have applied the SUSY potential partner and the SUSY operator equations
(5-6) as
V ( x)   2 ( x)   ' ( x) ; V ( x)   2 ( x)   ' ( x) (16)
d d
A   ( x); A     ( x) (17)
dx dx
The SUSY QM and the idea of shape invariance potential are suitable to be used in solving one
dimensional Schrodinger equation problems. One dimensional Dirac equation reduces to one
dimensional Schrodinger type equation by suitable change in parameters which are the coefficient of
the potential function and the energy term.
By obtaining the super-potential, the potential partners, V ( x, a0 ) and V ( x, a0 ) , and the SUSY
operators, A  and A are obtained and so the energy spectrum and the wave function.

3. Solution of Dirac Equation for Non-central Potential in D dimension

3.1. Solution of radial part of D dimensional Dirac Equation
 
The Dirac equation with the scalar potential S (r ) and magnitude of vector potential V (r ) is
given as in Hu et al. [38]
    

. p(MS(r) )(r) EV(r) (r)   
(18)

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ScieTech 2016 IOP Publishing
Journal of Physics: Conference Series 710 (2016) 012026 doi:10.1088/1742-6596/710/1/012026


where M is the relativistic mass of the particle, E is the total relativistic energy, and p is the three-
dimensional momentum operator,  i 
0  I 0 
  ;     (19)
  0   0  I

with  are the three-dimensional Pauli matrices and I is the2 × 2 identity matrix. The potential in
equation (18) is spherically symmetric potential,it does not only depend on the radial coordinate

r  r , and we have taken   1 and c  1 . The Dirac equation expressed in equation (18) is invariant
under spatial inversion and therefore its eigenstates have definite parity. By writing the spinor in D
dimension as
  r   (20)
 (r )   
 r 
and if we insert equations (19) and (20) into equation (18) and use matrices multiplication, we achieve
    
.p(r) MS(r) EV(r)(r) (21)
    
.p(r) MS(r) EV(r)(r) (22)
In the exact spin symmetric case, when the scalar potential is equal to the magnitude of vector
 
potential S ( r )  V ( r ) , then the upper Dirac spinor obtained from equations (21) and (22) are
 
  .p   
.p  (r )   M  E  2V (r ) (r )
M E (23)
   
By applying the Pauli matrices, it is simply shown that if  . p  . p   p 2 , then equation (23)
becomes

p 2 (r )  2V  r  M  E   (r )  E 2  M 2  (r )  (24)
Since p 2   D   2D with the hypersphericalLaplacian  2D is given by [39-40]
  D 1   1  1 d  d  L2D  2 
 2D  r1 D r  2   sin D 2
    
r  r  r  sin D  2  D 1 d D 1  d D 1  sin 2  D 1  (25)
D 1

with L2D  2  lD  2 (lD  2  D  3)

.
To reduce equation (24) into Schrodinger like equation we set V  (1 / 2)V and if V is modified
Poschl-Teller plus trigonometric Manning-Rosennon-central potential then equation (24) becomes
  D 1   d  d  (r )   
 (r   2
1 1 L2D  2
r1 D r   sin
D 2
 D 1   (r )  
r  r  r  sin D2
 D 1 d D 1  d D 1  sin 2
 D 1 
  a (a  1) b(b  1)  1  p( p  1)  
t2    2   2 s cot    M  E     (r      E 2  M 2    (r  
  sinh tr cosh tr  r  sin  
2 2 2
 (26)
The upper component of Dirac spinor in equation (20) is given as
  r   D 1 Fl (r )Y 1 ... D2 . ( xˆ  1 , 2 ,..., D 1 ) and Y 1... D2 . ( xˆ  1 ,...,D1 )   1    H 2 ,..., D1 
1
(27)
2
r
where x is a D dimensional position vector in hyper-spherical Cartesian coordinate[40,41], the unit
  
vector along x vector is denoted as x  x / r .The hyper-spherical Cartesian coordinate components
x1 , x2 , x3 , . . . . are given as
x1  r cos 1 sin 2 sin 3 ...sin  D1; x2  r sin 1 sin 2 sin 3...sin  D1;
(28)
x j  r cos  j 1 sin  j sin  j 1...sin  D1 3  j  D  1
xD1  r cos  D2 sin  D1; xD  r cos  D1 (29)

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ScieTech 2016 IOP Publishing
Journal of Physics: Conference Series 710 (2016) 012026 doi:10.1088/1742-6596/710/1/012026

The simultaneous eigenfunctions are given as

L12Y (1 ...) D2
 m2Y (1 ...) D2
;m  1 ; L2D1Y (1 ...) D2
 D1  D1  D  2 Y (1 ...) D2
(30)
L2jY (1 ...) D 2
 j  j  j  1 Y (1 ...) D 2
;1  j  D  1 (31)
 1
k 1
d  k 1 d  L2k 1 
 ab  sin k 1  d 
L2k 
L2
  sin  k  ;2  k  D  2
d k  sin 2  k 
(32)
a b  2  k k 

By applying the variable separation method we have radial and angular parts of D dimensional Dirac
equation which are given as
 D 1 D  3 
a(a  1)( E  M )  l  ( )( )
d 2 F (r ) 2  2 2 ( E  M )b(b  1)  2 D 1 D  3 
t    F (r )  ( E  M  t  l  ( )  d 0 ) F (r )  0
2 2
)(
dr 2  2 
2 2
 sinh q tr cosh q tr  2 (33)
 

 d  D2 d   p( p  1)  
 D2
1
 sin  D 1   2
1
 L2D  2    2  2s cot  D 1  ( M  E )  H ( D 1 )  0 (34)
 sin  D 1 d D 1  d D 1  sin  D  2  sin  D 1  
In order to solve the radial Dirac equation in Eq.(33), we use the approximation value for the
centrifugal term as in Greene and Aldirch, and in Ikdhair [20-21], 1 r  t
2 2
d 0 
 1 sinh 2q tr , for
tr  1 and d 0  1 / 12 . By setting
 a(a  1)  E  M   l  (
D 1 D  3
)( )  c(c  1) ; b(b  1)( E  M )  b' b'1 (35)
2 2
 D 1 D  3 
E 2  M 2  t 2  l  ( )( )  d0  E ' (36)
 2 2 
equation (33) becomes
d 2 F (r ) 2  c(c  1) b' b'1 
 t  F (r )  E ' F (r ) (37)
dr 2  sinh 2 tr cosh 2 tr 
 q q 
Equation (37) is solved using SUSY QM and by introducing the hypothetical super-potential as in [26-
27]
 (r )  tP coth q tr  tB tanh q tr (38)
using equation (9), equation (37) and equation (38) we get
 c(c  1) b '  b ' 1  2 2
t2    t ( P  P)(q csc hq2tr )  t 2 ( B 2  B )(q sec hq2tr )  E '0   2 BP  P 2  B 2  t 2 (39)
 sinh 2 tr cosh 2 tr 
 q q 
that gives
P  (1/ 2)  (c(c  1) / q)  1/ 4 ; B  (1/ 2)  (b'(b' 1) / q)  1/ 4 ; E'0  P  B t 2 (40)
2

The superpotential,super-partner potentials, ground state energy and raising-lowering operators

obtained from equations (5), (6),(12), (13), and (38) are
 1 c(c  1) 1   1 b '(b ' 1) 1  (41)
 (r )    t  t   coth tr    t  t   tanh tr
4   4 
q q
 2 q  2 q

V (r, a0 )  q t 2 P  P  1 csc hq2tr  qt 2 B  B  1 sech 2q tr  t 2  P  B 

2
(42)
V (r, a0 )  q t 2 P  P  1 csc hq2tr  qt 2 B  B  1 sech 2q tr  t 2  P  B 
2
(43)
d d
A  
 tP coth q tr  tB tanh q tr ; A   tP coth q tr  tB tanh q tr (44)
dr dr
By comparing the super-partner potential V (r , a0 ) and V (r , a0 ) we get the translation parameters as

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Journal of Physics: Conference Series 710 (2016) 012026 doi:10.1088/1742-6596/710/1/012026

a0  P, a1  P  1,.... an  P  n, and b0  B, b1  B  1,.... bn  B  n, (45)

By implementing equations (10-13) and (42-43) and (45) we get the equation of relativistic energy
spectra given as

    
2
En'  t 2  c(c  1) q  1/ 4  b '(b ' 1) q  1/ 4  1  2n
(46)
The relativistic energy equation obtained from equations (35-36) and (46) is given as
2
    
 E  M   l  ( D21)( D2 3) t 2d0   t 2    c(cq1)  14    b '(bq' 1)  14   1  2n 
2 2 (47)
      
with c(c  1), b '  b ' 1 is expressed in equation (35) and
s '2
l   ( D  2) / 2  ( p ' n)2  ; p '  1/ 2   D2  ( D  3) / 2   ( E  M ) p( p  1) (48)
2 2

( p ' n)2
which are obtained from angular solution in the next section.From the relativistic energy equation in
equation (47), we can obtain the numerical value of relativistic energy by using Mat-Lab in Table 1.
Table 1. Relativistic energy for various dimensions level
t  0.8, s  0.3, d0  1. /12, a  6, b  2, p  4, s  2, q  0.75, M  5 1/ fm , lD2  2
𝐷 𝐸0 𝐸1 𝐸2 𝐸3 𝐸4
3 -5.9455 -7.1363 -7.9716 -9.2079 -9.2079
4 -5.8748 -7.0824 -7.9306 -8.6091 -9.1800
5 -5.8075 -7.0212 -7.8823 -8.5695 -9.1467

By using equations (14-15) and (44) we obtain the un-normalized relativistic ground state and first
exited state wave functions for upper Dirac spinnor
F0  C (sinh q tr )  P (cosh q tr )  B ; F1  2  tP coth q tr  tB tanh q tr  (sinh q tr )  P (cosh q tr )  B (49)
with the values of P and B are expressed in equation (40). By using raising operator in equation (44)
we obtain all exited states of wave functions in exact spin symmetric case.

(a) (b)
Figure 1.Ground state (solid line) and first exited (dash line) radial wave functions for (a)𝐷 =
3,(b)𝐷 = 4

3.2. Solution of Anggular part of D dimensional Dirac Equation

The angular part of D dimensional Dirac equation for Manning Rosen angular potential are
 1 d  j 1 d  j 1 ( j 1  j  2)

 j 1  sin  j   j ( j  j  1)   H ( j )  0; j  2, D  2 (50)
 sin  j d j  d j  sin 2
 j 

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
 D2
1 d  D2

d 
  
 L2D2   ( E  M )  p( p  1)  2s cot   H ( )  0 (51)
 sin D 1   j
 D 1
 sin  D 1 d D 1  d D 1  sin 2  D 1  sin  j
2

These two equations are solvable by using SUSY QM approach by reducing it into one dimensional
Schrodinger like equation by suitable substitution of wave functions in equations (50) and (51) as
H ( j )  Q( j ) sin ( j 1)/2  j and H ( D 1 )  Q( D 1 ) sin ( D  2)/2  D 1 (52)
By inserting equation (52) into equations (50) and (51) we obtain
d Q( j ) / d j    j ( j  j  1)   ( j  1) / 2   j 1 ( j 1  j  2)  ( j  1) / 2( j  3) / 2 sin  j  Q( j )  0
2 2  2 2  (53)
and

d 2Q( D1 ) / d D2 1    ( D  2) / 2 
2  L2D2   (D  2) / 2(D  4) / 2  (E  M )  p( p  1)  2s cot   Q( )  0 (54)
 2 D 1  D 1
 sin 2  D1  sin  D 1 
By setting the parameters in equation (53) as
( j  j  1)   ( j  1) / 2    ; j 1 ( j 1  j  2)  ( j 1) / 2( j  3) / 2  o(o  1)
2
j (55)
then we have
 d 2Q( j ) / d j2     o(o  1) sin 2  j  Q( j )  0; (56)
where  is assumed to stand for the energy.The hypotheticalsuperpotential for equation (56) is set as
 ( j )  I cot  j (57)
By using equations (9), (56), and (57) we get the values

  j 1  ( j  2) / 2  
2
I  (o  1) and  0  (o  1)2  1/ 2  (58)
The superpotential,super-partner potentials, and raising-lowering operators obtained from equations
(5), (6),(12), (13), and (57) are
V (a0 ,  )  o(o  1) sin 2    o  1 ;V (a0 ,  )  (o  2)(o  1) sin 2    o  1 (59)
2 2

A  d d j  (o  1) cot  j ; A   d d j  (o  1)cot  j (60)

By shifting o  o  1 in equation (59) and by applying equations (10-13) with equations (59-60) we
obtain
  j  1)   ( j  1) / 2    o  n  1   (1   n)
2 2
j ( j j j 1 (61)

The ground state and first excited state wave functions are obtained by using equations (14-15), and
(60) as

Q0 ( j )  C  sin  j  ; Q1 ( j )  (o  1) cos  j )  sin  j 

o 1 o

(62)
The angular wave function for the highest level of angular component is obtained from the solution of
equation (54) by setting

 L   ( D  2) / 2( D  4) / 2  ( E  M ) p( p  1)  p '( p ' 1)

2
D 2 (63)


( E  M ) s  s ' ; E '    ( D  2) / 2
2
 (64)
such that equation (54) become
d 2Q( D 1 ) / d D2 1   p '( p ' 1) sin 2  D 1  2s 'cot  D 1  Q( D 1 )   E ' Q( D 1 ) (65)

By setting the hypothetical superpotential for equation (65) as

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 (D1 )  N cot D 1  B / N (66)

and by using equations (9) and (65-66) we have
N  1 / 2  ( p'1 / 2)  p' ;  s'  B ; p '2  s '2 p '2  E '0 (67)

By combining equations (5-7) and (65-67) we have the super-potential, super partner potential and
SUSY operator as given as
 ( D 1 )  p 'cot  D 1  s ' p ' (68)
p' ( p'1) s' p' ( p'1)
2
s' 2
V ( D 1 , a0 )   2s' cot  D 1  p'2  2 ; V ( , a0 )   2s' cot  D 1  p'2  2 (69)
sin  D 1
2
p' sin  D 1
2
p'
A  d d D1  p 'cot  D1  s ' p ' ; A   d d D 1  p 'cot  D 1  s ' p ' (70)
By shifting the parameter p '  p '1 in equation (69) and by applying equation (10-13) with
equations (68-69) we obtain relativistic energy equation
E 'n  ( p ' n) 2  s '2 ( p ' n) 2 (71)
with p ' in equation (63).
By using equations (63-64), and (71) we get orbital quantum number equation given as
l    ( D  2) / 2  ( p ' n) 2  s '2 ( p ' n) 2
2
(72)
By using equations (8), (17), (18) and (89) we get the relativistic angular wavefunctions for ground
state and first exited state which are obtained using equations (14-15) and (70) are ,
s' s'
 D1  D1
  sin  D 1  Q1  2  p 'cot  D 1  s ' p '  sin  D 1 
()  p' p'  p' p'
Q 0 e ; e (73)

The total relativistic ground state and first excited state wave function for any D dimension is obtained
by combining equations (49), (62), and (73).

3.3. Thermodynamical Properties

In non-relativistic condition, the relativistic energy equation expressed in equation (47) reduces

into non-relativstic energy by taking M  E  2 where  is the non-relativistic mass, 
E  M   ENR , ENR is the non-relativistic energy, and if d0 is small then equation (47)
E    t 2     c (c  1) q  1/ 4    b '(b ' 1) q  1/ 4  1  2n 
2

nr
2
nr nr nr nr
(74)

with  a(a  1)  2    ( D 1 2)( D  3 2)  c (c 1) ; b(b  1)2 )  bnr ' bnr '1
l nr nr
(75)
In classical regimes [43], the vibrational partition function, vibrational mean energy, and specific heat
are obtained from the non-relativistic energy equation in equation (74).The vibrational partition
function is defined as

1
Z ( ,  )   e  E ,   (76)
nl

n 0 kT
k is Boltzman constant, Enr is non-relativistic energy spectrum of the system. When the temperature,
T, is high enough, then the value of  is high ,  is small, and if 1  2  2t 2   then equation (76)
becomes
 
 
  
Z ( ,  )   e y    e y dy  
2 2
erfi   (77)
n 0 2  
 
where ENR   2t 2    n  ;   1/ 2  cnr (cnr  1) q  1/ 4 
2
   bnr '(bnr ' 1) q  1/ 4 1   (78)

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with y   n     and erfi is the imaginary error function [44].

The vibrational mean energy and the vibrational specific heat are defined as
U ( ,  )       ln Z ( ,  ); C     T U  k  2    U (79)
By using equations (77-79) we obtain the vibrational mean energy and specific heat equation given as
 
U (  ,  )   1/    1/ 2   1/   t 
2  /  exp   2t 2 2  /    
erf  t 2  /   (80)

  1  t 2 
   2t 2 2     2t 2 2    2t 2 2   
 exp      exp  2 
1        1  t 2
2 2
  2      
C ( ,  )  k   
and (82)
    
2   t 2  2      t 2  
2

 erf    erf    
       
   

(a) (b)
Figure 3.Graph of (a) mean energy U as a function of  ,(b) specific heat as a function of  (for
𝐷 = 4, 𝑛 = 5, 𝑙 = 4, 𝑎 = 6, 𝑏 = 3, 𝑝 = 4, 𝑠 = 2, 𝑞 = 0.75)
The graphs of mean energy and specific heat as a function of  are shown in figure 3 (a and b).
From figure 3 (a) and (b) we see that for larger values of  the values of vibrational mean energy and
specific heat are constant. The specific heat for system whose is governed by q-deformed modified
Poschl-Teller plus trigonometric Manning-Rosen non-central potential are negatives. The negative
specific heat may occur for the astronomical objects [45].

4. Conclusion
The Dirac equation in D dimensions of q-deformed modified Poschl-Teller potential plus
trigonometric Manning-Rosen non-central potential is solved using SUSY QM. The radial part of D-
dimensions of the Dirac equation reduces to one dimensional Schrodinger type equation in centrifugal
approximation scheme. There are two solutions of angular Dirac equations, the first one is for hyper-
spherical harmonic and the other one is for angular potential functions.
In the exact spin symmetric case, the relativistic energy equation reduces to the non-relativistic
energy in the non-relativistic condition. In the classical regime , some thermodynamics properties are
derived from the non-relativistic energy equation. The mean energy and specific heat are numerically
calculated from non-relativistic energy equation by using Mat-Lab.

Acknowledgement
This research was partly supported by Higher Education Project Grant with contract no.
698/UN27/PN/2015.

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