Analytical Solution Of Schrödinger Equation With Mie–Type Potential Using Factorisation Method

International Journal of Recent advances in Physics (IJRAP) Vol.2, No.2, May 2013
1
Analytical Solution Of Schrödinger Equation With
Mie–Type Potential Using Factorisation Method
Ituen. B. Okon1
, Eno. E. Ituen1
, Oyebola Popoola2
and Akaninyene. D. Antia1
1
Department of Physics, University of Uyo ,Uyo, Nigeria.
aitismen@yahoo.com
1
Department of Physics, University of Uyo, Uyo, Nigeria.
enoituenus@gmail.com
2
Department of Physics, University of Ibadan, Ibadan, Nigeria.
oyebpopoola@yahoo.com
1
Department of Physics, University of Uyo, Uyo Nigeria.
antiacauchy@yahoo.com
ABSTRACT
we have obtained the analytical solution of Schrödinger wave equation with Mie – type potential
using factorization method. We have also obtained energy eigenvalues of our potential and the
corresponding wave function using an ansatz and then compare the result to standard Laguerre’s
differential equation. Under special cases our potential model reduces two well known potentials such as
Coulomb and the Kratzer Feus potentials.
KEYWORDS
Factorization method, Mie-type potential, Schrödinger equation. PACS numbers: 03.65.Ge, 03.65.Pm,
03.65-w
1. INTRODUCTION
It is well known in quantum mechanics that a total wave function provides implicitly all
relevant information about the behaviour of a physical system, hence, if it is exactly solvable for a
given potential, the wave function can describe such system completely [1-5]. The exact solution
to Schrodinger equation plays a major role in quantum mechanical systems, meanwhile, this
solution is only possible for several selected potentials and approximation methods are frequently
used to arrive at the solution [6]. Generally, there are few main methods use to study the solutions
of quantum mechanical systems. One of them is the analytical method which is basically the
variational method and perturbation theory, but because of some inadequacies of this method in
addressing some quantum mechanical problems since the problem of suitable choice of initial and
convergence with respect to perturbation theory, numerical methods have been introduced [3].
Correspondingly, various methods have been developed to find the approximate solution to
Schrödinger equation. Such methods include standard method, path integral approach, invariance
method [4], Nikiforov-Uvarov method [1] and others [5]. In this work, we use the factorization
method to obtain the exact solution to Schrödinger equation using Mie – type potential. However,
with appropriate choice of parameters, our potential model can be reduced to Kratzer –Feus
potential and coulomb potential . Meanwhile, the Kratzer potential is amongst the most attractive
physical potentials as it contains a degeneracy removing inverse square term besides the common

2
coulomb term [7-13]. Analytical and Exact solution of the Schrodinger equation are very
important because of the understanding of physics that can be brought out from such solutions
[14-18].
2. FACTORIZATION METHOD
The factorization method [19] is a way of solving differential equations by change of variables. In
spherical coordinate, the Schrödinger wave equation is given as:
−ħ2
2
∇2
ψ (r) + (r) (r) = (r) (1)
Where is the reduce mass and the Laplacian operation ∇2
in spherical coordinate is given by
∇2
=
1
2 + ( 2 ) +
1
2 sin
( ) +
1
2 2
2
2 (2)
Let the wave function be
In order to solve Eq. (1), we assume total wave function to be a product of three functions and
write it as
Ψ( ) = Ψ( , , ) = ( ) ⊝ ( )Φ( ) (3)
where ( ) is the radial part, ⊝ ( ) is the angular part and Φ( ) is the azimuthal part of the wave
function. Substituting Eqs. (2) and (3) into Eq. (1) we obtain the following equations
Substituting equation (4) into (3)
2
( )
2 + 2
= 0 (4)
1
( )
(sin
⊝( )
) + [ −
2
sin2
] ⊝ ( ) (5)
1
2
( 2 ) +
2 2
ħ2
[ − ] ( ) (6)
Where
= ( + 1), and l is the angular momentum quantum number
= ( ) −
( +1)ħ2
2 2 is the effective potential and
( +1)ħ2
2 2 is the centrifugal term. is the
magnetic quantum number.
The solutions of Eqs. (4) and (6) are well known. Eq. (5) results in Legendre polynomials. Their
product ( , ) called spherical harmonics or Condon-Shortley phase is which is documented
in literatures [20]
( , ) = Θ( )Φ( ) (7)
The solution of Eq. (5) depends mainly on V(r), the potential of the system. Some potential can
be solved exactly but in most cases only approximate solution can be obtained for cases for non-
varnishing centrifugal term. The energies of the system are also obtained from Eq. (6)

3
3. SOLUTION OF THE RADIAL PART
The potential of our study is a Mie-Type potential given by [21]
( ) = − + 2 + (8)
This potential is useful in describing diatomic molecules. The Mie type potential possesses the
general features of the true interaction energy [22, 23]. The Mie-Type and Pseudo-harmonic
potentials are the two special kinds of exactly solvable Power-law and inverse power law
potentials other than the Columbic and harmonic oscillator[22] The potential can reduce to two
special cases. The potential reduces to Coulomb-like potential if we set B=C=0
( ) = − (9)
and to Kratzer-Feus if we set C=0.
( ) = − + 2
(10)
Putting this potential of Eq.(8) into the Schrodinger wave equation of Eq.(6), we get
2
2 +
2
+
2
ħ2 [( + − 2 − ) −
( +1)ħ2
2 2
] ( ) = 0 (11)
Recently, Ikot et al. [14] used the ansatz
( ) = 0( ) [1 +
0( )
(1 + 0( ))] ( ) (12)
By the use of Taylor’s series expansion Eq. (12) can be reduced to
R( ) ≅ 2 0( ) ( ) (13)
Substituting Esq. (12) into Eq. (11) we obtain
′′( ) +
2
0+ (1+ 0) 0
′
+ 0
1+ 0
+ 2 ′( ) +
1
0+ (1+ 0) 0
′′
+ 0
′′
1+ 0
− 0
′ 2
(1+ 0)2 +
2 0+ 1+ 0 0′+ 0′1+ 0+2 ħ2 − + − 2− +12 2 =0 (14)
Where the primes are derivatives with respect to r and the argument of 0(r) has been
suppressed.
The standard Laguerre’s differential equation is defined as [20]
′′( ) + (1 + − ) ′( ) + (2 − ) − ( + ) ( ) =0 (15)
The ( ) are the Laguerre’s polynomial.

4
Comparing Eqs. (14) and (15) we obtain
( )
′
+
′
+ 2 = 1 + − (16)
Performing a simple integration on Eq. (16) we obtain
= − + (17)
Where D is a constant of integration.
This equation can be simplified further to obtain
( ) = (18)
The wave function of the Mie-Type potential can be obtained by
Substituting Eq. (18) into Eq. (13) as
( ) = 2 ( ) , (19)
Where  rLn is the Laquerre’s polynomial as given by the Rodrigues formula as [20]
 rn
mn
r
m
mn
er
dr
d
er
a
rL 


 










2
, ),(
)( (20)
The exact wave function is obtained by substituting Eq. (18) into Eq. (12) as
( ) = 1 + ln 1 + ( ) (21)
Where is the normalization constant.
The total wave function can be written from Eq. (3) as
Ψ = 1 + ln 1 + ( ) ( , ) (22)
3.1 Eigen values:
In order to obtain the eigenvalues, we shall use the approximate value of the eigenfunction. We
insert Eq. (19) into Eq. (13) to get
′′( ) + 2( − 1 − + 1) ′
( ) +
ħ
−
ħ
+ − 2 ( − 1) + 2 −
ħ
+
−1 −2+2 −1−2 ħ2− ( +1)1 =0 (23)
Comparing Eqs. (23) and (15) we obtain the following set of equations

5
ħ
−
ħ
+ = 0 (24)
ħ
− 2 ( − 1) − 2 = (2 − ) (25)
( − 1)( − 2) + 2( − 1) −
ħ
− ( + 1) = − ( + ) (26)
Combining equations (24) and (25) gives
= −
ħ [ ]
(28)
Equation (28) gives the energy spectrum. From Eq. (26) we obtain
=
( )± ( )( ) ( )
ℏ
(29)
3.2 Special Cases:
Kratzer-Fues Potential: if we set = 0 in equation (8) we obtain Kratzer-Fues potential and
using this parameter in Eq.(28), we obtain the energy eigenvalues of this system as
= −
ħ [ ]
(30)
Coulomb Potential: If we set = = 0 in equation (8),we obtain Coulomb potential and using
B=O in Eq. (29) and C=0 in Eq.(28), we obtain the energy spectrum of coulomb potential as
 2
1
2
2
22
2





mn
A
Ec
nlm

, (31)
where,
 
2
)1(4)31)(1(1
1


llmmm
 . (32)
3. CONCLUSION
In this paper, we have studied the analytic solution of the Schrödinger equation for Mie–
type potential using factorization method. We have obtained the wave function using an ansatz
and also obtain the corresponding energy spectrum. We have showed that under special cases our
results reduce to two well known potentials: Coulomb and Kratzer-Feus potentials by making
proper adjustments to constant A, B and C. However, the Mie-type potential as discuss by
Agboola [21] is a diatomic potential that can be studied using polynomial, ansartz wave function
and 1/N expansion methods. The result for energy eigen value is also consistent to that obtained
in this work.

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ACKNOWLEDGEMENT
The authors are deeply indebted to Dr. A.N.Ikot of department of Physics, University of Uyo, for
his useful comment and reading of this paper which has lead to the improvement of the work. We
also thank the reviewer for his positive comments which has lead to the improvement of this
paper.
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