SOLUTION OF SCHRÖDINGER EQUATION FOR POSCHL-TELLER PLUS SCARF NON-CENTRAL POTENTIAL USING
SUPERSYMMETRY QUANTUM MECHANICS APROACH
Antomi Saregar
Physics Education Department, Lampung Raden intan Institute, Indonesia; E-mail : [email protected]
Abstract: In this paper, we show that the exact energy eigenvalues and eigenfunctions of the Schrödinger equation for charged particles moving in certain class of noncentral potentials can be easily calculated analytically in a simple and elegant manner by using Supersymmetric method (SUSYQM). We discuss the Poschl-Teller plus Scarf non-central potential systems. Then, by operating the lowering operator we get the ground state wave function, and the excited state wave functions are obtained by operating raising operator repeatedly. The energy eigenvalue is expressed in the closed form obtained using the shape invariant properties.
The results are in exact agreement with other methods.
Keyword: supersymmetry, non-central potentials, poschl teller plus scarf.
INTRODUCTION
One of the important work in theoretical physics is to obtain exact solution of the Schrödinger equation for special potentials (Ballentine, 1999). It is well known that exact solution of Schrödinger equation are only possible for certain cases. The exact solution of Schrödinger equation for a class of non- central potentials already studied in quantum chemistry (Gonul and Kocak, 2005), With the advent of supersymmetric quantum mechanics SUSYQM (Gonul and Zorba, 2000), and the idea of shape invariance (Gendenshtein1983), study of potential problems in non-relativistic quantum theory has received renewed interest. SUSYQM allows one to determine eigenvalues and eigenstates of known analytically solvable potentials using algebra operator formalism without ever having to solve the Schrödinger differential equation by standard series method.
However, the operator method has so far been applied only to one dimensional and spherically symmetric theree dimensional problems. Supersymmetri is, by definition (Ranabir at all, 1988), a symmetry between fermions and boson. A supersymmetric field theoretical model consists of a set of
quantum fields and of a lagrangian for them which exhibit such a symmetry. The Lagrangian determines, through the action principle, the equations of motion and hence the dynamical behaviour of the particle.
Recently, some authors have investigated the energy spectra and eigenfunction with Non-central potential (Meyur at all, 2009), Trigonometric Poschl- Teller plus Rosen-Morse using SUSY (Antomi at all, 2013), Hulthén plus Manning-Rosen potential, and Scarf poential plus Poschl-Teller using NU (Cari at all, 2012). In this paper, we investigate the energy eigenvalues and eigenfunction of the Poschl-Teller plus Scarf non-central potential system using SUSYQM aproach.
The trigonometric Scarf potential is also called as generalized Poschl-Teller potential (Xu at all, 2010), The trigonometric Poschl-Teller play the essential roles in electrodynamics interatomic and intermolecular forces and can be used to describe molecular vibrations. Some of these trigonometric potential are exactly solvable or quasi – exactly solvable and their bound state solutions have been reported(El Kineni, 2001).
REVIEW OF FORMULA FOR SUPERSYMMETRY QUANTUM MECHANICS
A. Supersymmetry Quantum Mechanics (SUSY QM)
Witten defined the algebra of a supersymmetry quantum system, there are super charge operatorsQiwhich commute with the Hamiltonian Hss[21]
Qi,Hss
0 with, i = 1, 2, 3, …N (1) and they obey to algebra
Qi,Qj
ijHsswith Hss is called Supersymmetric Hamiltonian. Witten stated that the simplest quantum mechanical system has N=2, it was later shown that the case where N = 1, if it is supersymmetric, it is equivalent to an N = 2 supersymmetric quantum system (Ranabir, 1988), In the case where N = 2 we can define,
( )
2 2 1
2 1
1 x
m
Q p and
( )
2 2 1
1 2
2 x
m
Q p (2)
Here the are the usual Pauli spin matrices, and
p x
1 is the usual momentum operator. For example two component, we shall write Hss as H. using equation (1) and (2) we get,
H H
dx x x d m dx
d m dx x
x d dx m d Hss m
0 0
) ) ( ( 2 2
0
0 )
) ( ( 2 2
2 2
2 2 2 2
2 2
(3) with,
) 2 2 (
2 2
x dx V
d
Hm with
) ( 2 ' ) ( )
( 2 x
x m x
V (4a) and,
) 2 2 (
2 2
x dx V
d
Hm ,with
) ( 2 ' ) ( )
( 2 x
x m x
V (4b)
with and , is defined as supersymmetry partner in the Hamiltonian.
) (x
V and V(x) are the supersymmetry partner each other.
Thus, solving equation (4a) and (4b), Hamiltonian equation can be faktorizated,
A A x
H( ) , and H(x) AA (5)
were, ( )
2 x
dx d
A m and
) (
2 x
dx d m
A (6) with, A as raising operator, and A as lowering operator.
Shape Invariance
Gendenshteın [6] discovered another symmetry which if the supersymmetric system satisfies it will be an exactly solvable system, this symmetry is known as shape invariance. If our potential satisfies shape invariance we can readily write down its bound state spectrum, and with the help of the charge operators we can find the bound state wave functions. It turned out that all the potentials which were known to be exactly solvable until then have the shape invariance symmetry. If the supersymmetric partner potentials have the same dependence on x but differ in a parameter, in such a way that they are related to each other by a change of of that parameter, then they are said to be shape invariant. Gendenshteın stated this condition in this way,
) ( )
; ( )
;
( 1 1
x aj V x aj R aj
V (7)
with, '( ; )
) 2
; ( )
;
( j 2 j xaj
a m x a
x
V (8a)
'( ; )
) 2
; ( )
;
( j 2 j x aj
a m x a
x
V (8b)
where j = 0,1,2,.., and a is a parameter in our original potential whose ground state energy is zero. aj1 f(aj) where f is assumed to be an arbitrary function for the time being. The remainder R(aj) can be dependent on the parametrization variable a
but never on x. In this case Vis said to be shape invariant, and we can readily find its spectrum, take a look at H,
0 2 0
2 2
0 ( ; )
2 V x a E
dx d E m
H
H
(9) Acoording to equation.(9) a further equation is obtained between V(x)and V(x) we get,
0 0 0
2 0
0 '( ; )
2 )
; ( )
; ( )
( x a E
m a
x E a x V x
V
(10) where V(x) is often stated os effective potential Veff. While (x)is determined hypothetically based on the shape of effective potential from the associated system.
The hamiltonian equation can be generalized,
k ki i
k V x a R a
dx d H m
2 1 2 2
) ( )
; 2 (
,
dengan k = 0, 1, 2,… (11)
By comparing equation (8) and (9), it is found that
ki R ai
E0 1 ( ). So that, in eigen energy spectra, the value of , can be generalized as follows,
n
k k
n R a
E 1
)
( ( ) (12)
Furthermore, we get the total energy spectra,
0 )
( E
E
En n (13)
with as ground state energy in a Hamiltonian lowering partner potential.
Based on the characteristics of lowering operator, then the equation of ground state wave function can be obtained from the following equation,
) 0
(
0
A (14)
Meanwhile, the excited wave function, one and so forth ( ) can be obtained by using raising operator and ground state wave function ( ). In general, the equation of wave function can be stated as follow,
)
; ( )
; ( )
;
( 0 0 ( 1) 1
)
( x a A x a n x a
n
(15)
SOLUTION OF SCHRÖDINGER EQUATION FOR POSCHL-TELLER PLUS TRIGONOMETRIC SCARF NON-CENTRAL POTENTIAL USING SUPERSYMMETRY
Schrödinger equation Poschl-Teller plus Scarf Non-central potential is the potentials present simulataneusly in the quantum system. This non-central potential is expressed as,
2 2 1 2
2 2 2 2 2 2 2
sin cos ) ( 2 sin
) 1 ( 2
cos ) 1 ( sin
) 1 ( 2 ) ,
( b aa ba
mr b b a a m r
V r r
(16)
The three dimensional Schrödinger equation for Poschl-Teller plus trigonometric Scarf non-central potential is written as,
2 2 2 2 2 2 22
sin sin 1
sin 1 1
2
r r
r r r r m
b aa ba E
mr b
b a a
m r r
22 2 2 22 2 2 221
sin cos ) ( 2 sin
) 1 ( 2 cos
) 1 ( sin
) 1 ( 2
(17) If equation (17) multiplied by factor, (
2
2 2
mr ), and the result is solve using separation variable method since the non- central potential is separable. By setting
) ( ) ( ) ( ) , ,
(
r R r P , with
sin ) ( ) ) ( , , (
( r
e Q r r X
RP
im
, we obtain,
sin 0 cos ) ( 2 sin
) 1 ( sin
sin 1 sin
1
2 cos
) 1 ( sin
) 1 ( 1
2 2 1 2
2 2 2 2
2 2 2 2 2 2 2
a a b a b P
P
mr E b b a a r r r R r
R r r
(18) from equation (18) we obtain radial and angular Schrödinger equation as,
) 1 sin (
sin 1 sin
1 sin
cos ) ( 2 sin
) 1 (
2 cos
) 1 ( sin
) 1 ( 1
2 2 2 2
2 1 2
2
2 2 2 2 2 2 2
P P
a a b a b
mr E b b a a r r r R r
R r r
(19)
with ( ) is constanta variabel separable, where as orbital momentum number.
From equation (19) we get radial and angular wave function Schrödinger equation with single variable as following,
cos( 1) 2 ( 1)
sin ) 1 ( 1
2 2 2
2 2 2
2
E
mr b
b a a r r r R r
R r r
(20)
or equation (20) multiplied by ( Rr ), with
r
r r
R( ) ( ), so using symple algebra, we get,
mE X r b
b a a
r2 2 2 r 2 r 2 2
2 ( 1) 2
cos ) 1 ( sin
) 1 ( 1
(21) and than, for solve radial Schrödinger equation, we use approximation for centrifugal term,
d r
r2 2 0 sin2 1 1
1 for r 1, with
12 1
0
d , mE
2
2 2
, we get,
2 0 2
2 2
2 2 2 2
sin 1 )1
1 cos (
) 1 ( sin
) 1 (
1
r r
r bb d
a a
r
(22) from equation (22) simplied by (
m 2
2
) we
get radial Schrödinger equation,
2 2 2 0
2 2
2 2 2 2 2 2
) 2 1 2 ( cos
) 1 ( sin
) 1 ( 2
2
d m m
b b a a m r
m r r
(23) and we have the angular and Schrödinger equation as,
sin ( 1)
sin 1 sin
1 cos
) 1 ( sin
) 1 (
2 2 2 2
2
P P
b b a a
r r
(24) and we have set 2
2
1 2
m
that give azimuthalwave function as
,...
2 , 1 , 0 2 ,
1
eim m
(25)
equation (24) will be,
) 1 sin (
sin sin 1 sin
cos ) ( 2 sin
) 1 (
2 2 2
2 1 2
2
m P P
a a b a b
(26) with m2as variable separation and we get angular Schrödinger equation one dimentional,
m H a H
b m a a b m d
d
m (( 1) )
2 sin
cos ) ( 2 sin
) 1 ( 2
2 4
1 2 2
2 1 2
4 1 2 2 2
2 2 2
(27) The solution of Radial Scrodinger equation Poschl Teller plus trigonometric Scarf potential
Factor R in equation (20) is defined as wave function , then the Schrödinger equation for Poschl-Teller plus Scarf non- central potential in radial with the assumption of '2m2 2 can be rewritten as follow,
' ) 1 2 ( cos
) 1 ( sin
) 1 ( ) 1 ( 2
2 2 0
2 2
2 2 2 2 2 2
d
m b
b a
a m r
m r r
(28) Based on equation (28), the effective potential of radial SE Poschl- Teller plus trigonometric Scarf non-central potential can be rewritten as follow,
2 0
2 2
2 2
2
) 1 2 (
cos ) 1 ( sin
) 1 ( ) 1 (
2 d
m b
b a
a
Veff m r r
(29) or,
2 02 2
2 2 2
) 1 2 (
cos ) 1 ( sin
) 1 ' ( '
2 d
m b
b a a V m
r
eff r
(30)
with a' a(a1)(1)41 21
By inserting effective potential in equation (30) into equation (10), its obtained
2 0
2 2 2 2 2
2 ( 1)
2 cos
) 1 ( sin
) 1 ' ( ' ) 2
( 2 ' )
( d
m b b a a x m
x m
r
r
(31) By using incisive hypothesis, it is assumed that superpotential in equation (30) is,
) cot(
) tan(
)
(
x A r B r (32)
Where A and B are indefinite constantans that will be calculated. From equation (32), we can determine the value of '(x) and
)
2(
x , then the result is distributed into equation (31), then the following is obtained,
r r r r
B A m AB B B
A A
2 2 2
2 2 2 2
2
sin 2 cos
sin 2 cos
2 2 2 2 2 2 ( 1) 0
2 cos
) 1 ( sin
) 1 ' ( '
2 d
m b
b a a
m r r
(33)
By analysing the similar concept between left flank and right flank, from equation (33), it is obtained,
) ( cos ) 1 1 ( ) 2
( cos
1
2 2 2
2 2
2
m A r m bb r
A
;
r r
a a B m
m
B 2 2
2 2
2
sin ) 1 ' ( ' 2
) ( sin
1 2
;
and 2 0 0
2 2
2 ( 1)
2 2 d E
AB m B
A
(34)
from the three equation in equation (34), it is obtained,
mb
A 2
or 1
2
b
m
A
(35a)
) 1 '
2 (
a
B m
or '
2 a
B m
(35b)
0
2
0 ( ') ( 1)
2 b a d
E m
(35c)
The value of A and B are determined in certain way so the value of ( ) is equal to zero, so,
) cot(
2 ' ) 2 tan(
)
(
r a r
b m
x m (36)
By using equation. (6) and (36), we get
) cot(
' 2 ) tan(
2 2
) (
2 r a r
m b
dr m d m dr r
d m
A
(37) and
) cot(
' 2 ) tan(
2 2
) (
2 r a r
m b
dr m d m dr r
d m
A
(38) The ground state wave function can be obtained from equation (14) and (38), which are,
0 ) cot(
2 ' ) 2 tan(
2 0
r
r a
b m dr m
d m
Then the ground state wave function of rosen more potential is as follow,
) ( ) tan(
) ( ) cot(
' ) ,
( 0
0 0
r r r r
d b
d a
a d r
a b
Ca
r r r
( , ) 'lnsin( ) lncos( )
ln0 0
r r
CK
lncsc cot
'0
0(r,a ) C cos(r) b sin(r) a
(39)
By using equation (15) we can obtain excited wave function on the first level as follow,
)
; ( )
; ( )
;
( 0 0 0( ) 1
) (
1 r a A x a r a
(40)
wherea0 a'a1 a'1, ……,an a'n is the independent parametre to variable
“r”. By inserting the value of the parametre to equation (39) and (37) and by using equation (40), the following we get,
) cot(
' 2 ) tan(
2 2
)
;
( 0
) (
1
r a r
m b
dr m d m a
r
C cos(r) (b1) sin(r) (a'1)
1 cos( ) sin( )('2) ( ' 1)
cos( )
( 2)sin( )
'
2
r a r b
r a
r b a
m b C
'cot( )
2 )
tan(
2
r
r a
m b
m
C cos(r) (b1) sin(r) (a'1)
2 1 sin( )2 (2 ' 1)
cos( )
2
2
m b r a r
C cos(r) b sin(r) a'
(41) The breakdown in equation (41) can be continued to find wave function)
; ( ),
;
( 0 3( ) 0
) (
2 r a x a
, ... and so on.
The determination of the potential partner which have shape invariant, by using equation (8a) and (8b) results,
2 2 2 2 2
2 2
0 '
2 ) ( sin
1 ' ' ) ( cos
1 ) 2
,
( a b
m a
a b b a m
r
V r r
(42a)
and,
2 2 2 2 2
2 2
0 '
2 ) ( sin
1 )
( cos
1 ' ' ) 2
,
( a b
m b
b a a a m
r
V r r
