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Journal of Modern Physics, 2013, 4, 463-467
http://dx.doi.org/10.4236/jmp.2013.44065 Published Online April 2013 (http://www.scirp.org/journal/jmp)
Construction of Exactly Solvable
Ring-Shaped Potentials
Arup Bharali1*, Ngangkham Nimai Singh2
1Bajali College, Pathsala-781325, Assam, India
2Department of Physics, Gauhati University, Guwahati-781014, Assam, India
Email: *[email protected]
Received January 2, 2013; revised February 1, 2013; accepted February 10, 2013
Copyright © 2013 Arup Bharali, Ngangkham Nimai Singh. This is an open access article distributed under the Creative Commons
Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is
properly cited.
ABSTRACT
We propose a method for construction of exactly solvable ring-shaped potentials where the linear homogeneous second-
order differential equation satisfied by special function is subjected to the extended transformation comprising a coor-
dinate transformation and a functional transformation to retrieve the standard Schrödinger polar angle equation form in
non-relativistic quantum mechanics. By invoking plausible ansatze, exactly solvable ring-shaped potentials and corre-
sponding angular wave functions are constructed. The method is illustrated using Jacobi and hypergeometric polynomi-
als and the wave functions for the constructed ring-shaped potentials are normalized.
Keywords: Schrödinger Equation; Extended Transformation; Special Function; Ring-Shaped Potential; Normalization
1. Introduction formation and a functional transformation on second-
order differential equation for a particular special function
to mould the differential equation to the standard Schröd-
inger polar angle equation form. The coordinate transfor-
mation is the basic transformation required to change the
characteristics of the differential equation, while the func-
tional transformation is essential to retrieve the Schröd-
inger polar angle equation form. By invoking suitable
ansatze, exactly solvable ring-shaped potentials and their
angular wave functions are constructed. The special func-
tions are found to be the multiplicative factors in the an-
gular wave functions and the wave functions are normal-
ized. In standard literature, the Schrödinger equation for a
specific exactly solvable ring-shaped potential is solved
by transforming it to a differential equation satisfied by a
particular special function [1,2,9], but the ethics of the
present work is not to solve Schrödinger equation to ob-
tain wave functions for a particular ring-shaped potential,
but to construct exactly solvable ring-shaped potentials as
well as their angular wave functions starting from second-
order differential equations satisfied by special functions.
The fundamental nature of the adopted method for trans-
forming a solvable differential equation to a particular type
bearing some physical significance would have a wide
range of applicability not only in quantum mechanics but
in other branches of science also.
Generation/construction of exactly solvable quantum me-
chanical potentials is an important topic of fundamental
research; as such type of research always incorporates
new ideas and/or mathematical techniques to quantum
mechanics. Again, exactly solvable potentials are essen-
tial for the successful implementation of approximate
methods in the study of practical quantum systems. The
study of quantum systems with non-central potentials is
an upcoming field of research for the theoreticians. The
quantum systems with non-central potentials have been
studied extensively in quantum chemistry and nuclear
physics in the context of organic molecules and deformed
nuclei respectively in non-relativistic regime. Different
methods applied by various authors to obtain exact solu-
tions of Schrödinger equation for bound states with non-
central potentials, are the factorization method [1], the
standard approach [2], the path integral representation [3],
the Nikiforov-Uvarov method [4-6], the supersymmetric
approach [7], etc. Here, we apply a method for construc-
tion of exactly solvable ring-shaped potentials starting from
the linear homogeneous second-order differential equa-
tions satisfied by the special functions. We perform the ex-
tended transformation [8] consisting of a coordinate trans-
The plan of the paper is as follows: in Section 2, for-
*Corresponding author.
C
opyright © 2013 SciRes. JMP
A. BHARALI, N. N. SINGH
464
malism of the method is discussed, construction of exactly
solved ring-shaped potentials using the method is dem-
onstrated in Section 3 and concluding remarks are includ-
ed in Section 4.
2. Formalism
In the method, we start with a linear homogeneous sec-
ond-order differential equation satisfied by a particular
special function, on which we perform the extended trans-
formation to generate the Schrödinger polar angle equa-
tion form and by invoking a suitable ansatz, the Schrödin-
ger polar angle equation for a ring-shaped potential is re-
trieved from the generated equation. We now consider the
following linear homogeneous second-order differential
equation satisfied by a special function

F
z
 
0,QzFz
 
Qz

 
Fz PzFz
 
 (1)
where and are well defined for the special
function
Pz
F
z

zg
. We perform the extended transformation
[8] consisting of a coordinate transformation and a func-
tional transformation as follows
 
1
fFg
(2)
and




(3)
on the above differential Equation (1) yielding
  
 
0.
g
gg




g
2exp d
dln
d
exp d
dd
ln ln
dd
fPg
g
fP
fg



 





(4)
The transformation function
is a smooth differ-
entiable function of at least class and
2
C

1
f
is the
modulating function required to mold the above equation
to the standard Schrödinger polar angle equation form.
We make the coefficient of the first-order derivative equal
to cot
, fixing the functional form of

f
as
 
1d
2
sin exp
N
f
Cg

Pgg



, (5)
which changes Equation (4) to
 

  
2
11
cot,1 cs
24
11
24
g
gQgPgP
 
 

 
 
 

2
2
c
0g




 (6)
Where the Schwartzian derivative
 
3
2
, 2
gg g
 
g
 and
N
C in Equation (5)
will act as normalization constant.
To retrieve Schrödinger polar angle equation, the fol-
lowing identity must be prescribed
 
 
2
2
2
14
14 sin
111
,,
224
mV
ggQgPgPg

 
(7)

 
1ll
where
0,1, 2,, n
0, 1, 2,, l
and l and m are the orbital and mag-
netic quantum numbers. In presence of a central potential,
the admissible values for l are and for m are


Pg
where n is the principal quantum number.
But l needs to be redefined [9] for a quantum system with
central potential plus a polar angle dependent ring-shaped
potential.
By putting first the expressions for and
Qg

defining a special function
F
g in the above equation
(7) and then by invoking an ansatz that there should be at
least one constant term in right side of the above Equation
(7), the functional form of the transformation function
g

g
is specified. Putting
in the Equation (7)
again, one can in principle construct exactly solvable
ring-shaped potential
V
. The chosen term, of course,
should be integrable to obtain

g
which again to
be invert- ible to obtain the functional form of the trans-
formation function
g
. Using the expressions for
Pg ,
Qg and
g
in Equations (3) and (5), the
angular wave function for the constructed ring-shaped
potential is obtained as
 
 
12
sin
1
exp d.
2
N
Cg
Pgg F g
 







(8)
The normalization constant
N
C

is evaluated by using
the following normalization condition for
 
as
π
2
0
0, πsin dFiniteI
 




,
n
. (9)
3. Application
We choose Jacobi polynomials and hypergeometric func-
tion to construct exactly solvable ring-shaped potentials
using the method. Gegenbauer, Chebyshev and Legendre
polynomials can be obtained as special cases from Jacobi
polynomials. Again Jacobi polynomials can be obtained
from hypergeometric function as special cases and the
same holds for the generalized Laguerre, Hermite poly-
nomials and the confluent hypergeometric function [10].
3.1. Using Jacobi Polynomials
The differential equation [10] satisfied by Jacobi Polyno-
mial zP z

is
F
 

2
12
10,
zFz zFz
nnF z
 





  (10)
Copyright © 2013 SciRes. JMP
A. BHARALI, N. N. SINGH 465

for which
 
2
21z z
Pg
 

 

and
2
11Qg nnz

, the Equation (7) be-
comes
 
 






2
2
2
22
2
2
2
2
2
2
22
2
2
2
14
14 sin
11
,1
22
1
21
1
241
12.
41
mV
gnn
2
2
2
1
g
g
gg
g
g
g
gg
g



 

 




 






(11)
Introducing the ansatz, we choose the second term in
R.H.S. of the above equation as a constant independent of
θ. We suppose that

a constant

g
2
2
2
1
gC
g
,
(12)
which specifies the functional form of
as

sin Cg

. (13)
Selecting π2
and C1
to satisfy the local pro-
perty of the transformation function

π20g
, the above
Equation (11) gives us the exactly solvable Makarov ring-
shaped potential [7,9] as
 
11
2
cos
sin
BVV

(14)
and

2
1
11


1
24
ln
n

, where
2
1
mB
and 2
1
mB


1
V
.
Using Equations (13) and expressions of P and Q for
Equation (10) in (8), the angular wave functions corre-
sponding to
are



,cos
n
P

sin cos
22
N
C





 , (15)
while normalization constant
N
C

is evaluated by using
Equation (9) and orthogonality relation [10] for Jacobi
polynomials






1
,
1
1
11
21
21
n
xxP
nn







 
2
d
1
,
! 1
xx
nn
n




where 1
 and 1


and the constant is found to
be
1
22 1!1
.
11
N
nnn
Cnn

 


 


(16)
Again, if the fourth term in R.H.S. of the Equation (11)
is taken as a constant such that
2
2
2
2
1
gC
g


g
, (17)
the transformation function
becomes

icotgC

. (18)
To satisfy the local property

π20g
1C
, we choose
and
0
and the exactly solvable ring-shaped
potential [11] is found to be
 
22
cos
sin
VVB


(19)
with
124414mnn
 


and

2
2
12 ilB
 

V
.
The angular wave functions for 2
are obtained by
using Equation (18) and P and Q of Equation (10) in (8) as
 



12
,
cscexp i2
icot
N
n
C
P


 






(20)
and the normalization constant is calculated by using nor-
malization condition as Equation (9) and orthogonality
relation [11]







12
21
2
,
1
1expitan
id
11
,
22 1!1
n
xx
Pxx
nn
nnn





 
  

 




 

  




and is found to be
1
22 1!1
11
N
nnn
Cnn

 

 
 
 

(21)
Though
2
2
2
1
gg
g
in the third term is integrable to
g
have
, but it is not invertible and hence it cannot
be used to construct ring-shaped potential. Taking the last
Copyright © 2013 SciRes. JMP
A. BHARALI, N. N. SINGH
466
term as a constant with the choice

22
1
g
eq
and 2
C, the transformation ftions respectively come
2
2
g
gual to 2
C
unc
out a
 
s
1exp i2Cg

 


and
 
1exp 2gC

 


, yieldihaped po-
tentials not for physical interes
etric function
 ng ring-s
t.
3.2. Using Hypergeometric Function
The differential equation [10] for hypergeom
 
21 ,;;
F
Fz





0
zz
Fz
Fz





 
11zFz
 (22)
for which
 
11
P
zzzz 

and


 
1Qz

 comes z z and the equation (7) be
 






2
2
2
2
2
22
2
2
14 sin
11
,2
24 1
11
22
,
214
m
g
g
g
14 V
g
g
gg
g


 



 
(23)

1

 .
g the ansatz,e select the third term as a

where

ducinIntro w
constant such that
2
2
4
1
gC
gg
, (24)
which yields the transformnction

g
ation fu
as

2
cosgC

. (25)
oosing 0
Ch
and 1C so that

π20g
, the
constructed riaped potential become
 
ng-sh s
3
B
VV

32
cos
(26)
and the angular wave functions corresponding to
V3
are obtained by using expressions for P and Q and -
tion (25) in (8) as

Equa
 

1
2
2
cos
;cos ,
 
 
21
sin
,;
N
C
F

 


(27)
where

3
32 14Bl
 2mn ,

3
2 142Blm and 1
3
11 4Bm

  .
The normalization constant is determined by using
Equation (9) and orthogonality relation [9]
 
12
1
21
0
1,;;d


2
!1
2
x
xFnnxx



nn
nn n



 

and it is found to be
 

1
1!
nn
n
 

N
C
 
 . (28)
Again, choosing 0
and
12C in
so that
Equation (25),
π0g
, the constructed ring potential is found
tov rined pote be again Makarog-shapntial given by the
Equation (14) for which the angular wave functions be-
come

1
sin cosC
2
21
22
,;
;cos
2
N
F
 


 







(29)
where 12ln
 ,2l
 ,
1
12B

 and 242
1
2mmB
 and
tion constant
normaliza
 

1
1!
nn
n
 

N
C
 
 (30)
Picking the second and third terms sep
stants, we cannot construct any ring-shaped potentials that
w
We present a method for construction of exactly solvable
ith consideration of special func-
arately as con-
ill be functions of some trigonometric functions.
4. Conclusion
ring-shaped potentials w
tions in the framework of non-relativistic quantum me-
chanics. The method is realizable only for the implication
of the extended transformation which is a coordinate trans-
formation supplemented by a functional transformation.
The extended transformation is performed on the linear
homogeneous second-order differential equation satisfied
by a particular special function to retrieve the Schröd-
inger polar equation form and by invoking plausible an-
satze, exactly solvable ring-shaped potentials are con-
structed. For implementation of the method, we choose
Jacobi polynomial and hypergeometric function as spe-
cial functions to construct new (in Equation (26)) as well
as already known exactly solvable ring-shaped potentials.
The angular wave functions corresponding to the con-
structed potentials are normalized and also analytically
verified. Though orbital quantum number takes the val-
ues 0,1, 2,3, in presence of central potential, the same
Copyright © 2013 SciRes. JMP
A. BHARALI, N. N. SINGH
Copyright © 2013 SciRes. JMP
467
ed potenti
[1] J. Sadeghi and B. Pourhassan, “Exact Solution of the
Non-Central M Ring-Shaped Lik
Potential by th,” Electronic Jour-
cal Chemistry, Vol. 48, No. 4, 2010, pp. 876-882.
quantum number will depend on both magnetic quantum
number and characteristic constants in presence of ring-
shapals. Again, some unphysical potential are
also come up in the calculations and new technique is re-
quired to make them physical. Laguerre, Hermite, Roma-
novski polynomials, etc. can also be utilized in the me-
thod to construct ring-shaped potentials. The proposed
method has also the capability for construction of central
potentials in non-relativistic regime and the essence of
the method of converting a soluble differential equation
to another differential equation of practical interest can
be extrapolated to other branches of physics.
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