J. Nig. Soc. Phys. Sci. 4 (2022) 242–250

Journal of the
Nigerian Society

of Physical
Sciences

Approximate Solutions of the Schrödinger Equation for a
Momentum-Dependent potential

C. A. Onatea,d,∗, I. B. Okonb, M. C. Onyeajuc, A. D. Antiab

aDepartment of Physical Sciences, Landmark University, Omu-Aran, Nigeria
bTheoretical Physics Group, Department of Physics, University of Uyo, Nigeria

cDepartment of Physics, University of Port Harcourt, Choba, Nigeria
dLandmark University SDG 4 (Quality Education)

Abstract

The solution of one-dimensional Schrödinger equation for a newly proposed potential called modified shifted Deng-Fan momentum-dependent
potential is obtained via supersymmetric approach. The expectation values of momentum and position were calculated using Hellmann Feynman
Theorem. The effects of momentum-dependent parameter on the solutions of the system as well as the expectation values were studied. Finally,
the special cases of the interacting potential were obtained.

DOI:10.46481/jnsps.2022.653

Keywords: Eigensolutions, wave equation, potential models, momentum dependent

Article History:
Received: 10 February 2022
Received in revised form: 13 April 2022
Accepted for publication: 14 April 2022
Published: 29 May 2022

c©2022 Journal of the Nigerian Society of Physical Sciences. All rights reserved.
Communicated by: C. O. Edet, P. Amadi & R. Khordad

1. Introduction

In both the relativistic and nonrelativistic mechanics, wave
equations play some significant roles. The wave functions em-
bedded information that describes a complete quantum system.
It is also noted that the solutions of a complicated physical sys-
tems are checked and improved through numerical method by
wave equations [1, 2]. However, the major objective of the
wave equations is the determination of the eigenvalue and its
wave function. The solution of this wave equation can be stud-
ied via H |Ψ〉 = E |Ψ〉, (in this case, H is Hamiltonian and E
is energy). The wave function Ψ is expanded as |Ψ(r, E)〉 =∑

n fn(E) |Ψn(r)〉, the term r stands for set of coordinates for
real space.

∗Corresponding author tel. no: +2348056985024
Email address: oaclems14@physicist.net (C. A. Onate )

In most of these cases, several potentials such as hyperbolic
potential [3, 4], modified Rosen- Morse [5], Hulthén potential
[6, 7], Manning-Rosen potential [8-10], Kratzer potential [11],
potential family [12, 13], and others, were used. In recent time,
a greater interest has been focused on the wave equation with
position dependent-potential [14], constant mass-dependent po-
tential [15], and energy-dependent potential for both relativistic
and nonrelativistic wave equations [16]. Motivated by these,
we want to study the non-relativistic equation with a modi-
fied shifted Deng- Fan momentum-dependent potential. In the
present study, the effect of the momentum-dependent parameter
on energy of the system will be examined under the proposed
potential and its subset potentials. The modified shifted Deng-
Fan potential is a combination of Deng-Fan and Hulthén like
potentials. The Deng-Fan potential also known as improved
Manning-Rosen potential, is an empirical and diatomic molec-

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Onate et al. / J. Nig. Soc. Phys. Sci. 4 (2022) 242–250 243

Table 1. Bound states of the modified shifted Deng-Fan momentum-dependent
potential model with µ = ~ = 1, α = 0.15 Å, re = 0.4 Å, De = 5 eV and
D0 = 0.2 eV for various n and l.

n l ρ = 1 ρ = 0
0 0 2.4925255 4.0257731
1 0 3.9418902 4.6881046

1 4.2651097 4.8366869
2 0 4.5080914 4.8958752

1 4.6571299 4.9527631
2 4.8134283 4.9944769

3 0 4.7739363 4.9739034
1 4.8496210 4.9946798
2 4.9305476 5.0055476
3 4.9863491 5.0015209

ular potential function proposed in 2012 [17]. This potential
was used to examine the molecular vibrations and the energy
eigenvalues under relativistic and nonrelativistic wave equation
[18]. In ref. [19], it was theoretically used to study HF and
information entropy. The Deng-Fan potential model is given by

V (r) = De

(
1 −

eαre − 1
eαr − 1

)2
. (1)

where, De is the dissociation energy, re is the equilibrium bond
length, α is the screening parameter and r is the internuclear
separation. The scheme of our presentation is as follows: In
section 2, we presented the bound state solutions, Results and
discussion are given in section 3 while conclusion is given in
section 4.

2. Bound State Solutions

In the nonrelativistic quantum mechanics, the radial Schröd−
inger equation with an interacting potential V (r) together with
a nonrelativistic energy En`, is given by

−
~2

2µ
d2Un,`(r)

dr2
+

[
V (r) − En,` +

~2`(` + 1)
2µr2

]
Un,`(r) = 0,

(2)

where, l is the angular quantum number, r is the internuclear
distance, n is the quantum number, ~ is the reduced Planck’s
constant and Un,`(r) is the wave function. In this study, we
shall solve the radial Schrödinger equation above for modified
shifted Deng-Fan momentum-dependent potential. Thus, the
potential function for momentum dependent is given as

V (r) =

De
(
1 −

eαre − 1
eαr − 1

)2
− D0

(
1

eαr − 1

) (1 + ρ). (3)
In the equation above, De is the dissociation energy with differ-
ent values for various diatomic molecules, D0 is the potential
strength whose value is arbitrary chosen and re is the bond sep-
aration. The Schrödinger equation in equation (2) has centrifu-
gal term which needs to be approximated. Several approxima-

tion scheme have been used for different potential [20]. In this
study, the centrifugal term will be approximated using

1
r2
≈ α2

(
C0 +

eαr

(eαr − 1)2

)
, (4a)

where C0 is a dimensionless constant obtained by using the fol-
lowing power series

α2
(
C0 +

eαr

(eαr − 1)2

)
= α2

[
C0 +

1
(αr)2

−
1

12

+
(αr)2

240
−

(αr)4

6048
+ O((αr)6)

]
. (4b)

This finally give the dimensionless constant as C0 = 1/12.
With equation (3) and equation (4), the radial Schrödinger equa-
tion in equation (2) becomes

d2Un,`(r)
dr2

=


(

2µDeβ2 b2 e−αr

~2 + `(` + 1)α
2
)

e−αr

(1 − e−αr )2

−

2µβ2 (2De b+D0 )e−αr

~2

1 − e−αr
+ E2

 Un,`(r), (5)
where,

β2 = 1 + ρ, (6)

E2 =
2µ(Deβ2 − En,`)

~2
+ `(` + 1)C0α

2, (7)

b = eαre − 1. (8)

In the present study, the authors adopt supersymmetric ap-
proach (SUSYQM) to solve equation (5). To proceed using the
basic concept and formalism of SUSYQM, we write the ground
state wave function as

U0,`(r) = ex p
(
−

∫
W(r)dr

)
, (9)

where W(r) is defined as a superpotential that is propose based
on the interacting potential [21]. On the basis of this work, the
superpotential is given as

W(r) = ρ0 +
ρ1

eαr − 1
, (10)

where the constants ρ0 and ρ1 will be determine later. To relate
the superpotential function in equation (10) to equation (5), we
establish a Reccati differential equation of the form

W 2(r) −
dW(r)

dr
=

(
2µDeβ2 b

2 e−αr

~2
+`(`+1)α2

)
e−αr

(1−e−αr )2 (11)

−

2µβ2 (2De b+D0 )e
−αr

~2
1−e−αr + Er.

Relating equation (11) with equation (5), we can now deter-
mine the two parametric constants in the superpotential func-
tion as

ρ20 = Er, (12)
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Onate et al. / J. Nig. Soc. Phys. Sci. 4 (2022) 242–250 244

Figure 1. Variation of energy En,`(eV ) against the screening parameter α(Å) with momentum dependent potential ρ1(a) and without momentum
dependent potential ρ0 (b) for three different quantum states with µ = ~ = l = 1, re = 0.2 Å, De = 3 eV and D0 = 6 eV .

Table 2. ro-vibrational energy spectra in (eV ) for 2p, 3p, 3d, 4p, 4d, and 4f for Deng-Fan and Hulthèn potential with three different values of the screening parameter
for various quantum and angular quantum states.

ρ = 0 ρ = 1
State α De = 5, D0 = 0 De = 0, D0 = 5 De = 5, D0 = 0 De = 0, D0 = 5
2p 0.05 3.7357639 -1252.5010 1.0959759 -5005.0010

0.15 3.8435208 -141.39826 1.2510768 -560.56493
0.25 3.9472499 -52.526042 1.4029712 -205.02604

3p 0.05 4.3380223 -558.05774 2.6082849 -2227.2244
0.15 4.4738042 -64.248083 2.8289266 -251.93327
0.25 4.5993040 -24.776910 3.0417528 -93.943576

4p 0.05 4.6792822 -315.00479 3.7798457 -1255.0048
0.15 4.8154953 -37.265347 4.0535781 -143.93201
0.25 4.9170844 -15.119792 4.2949652 -55.119792

4d 0.05 4.6480486 -315.00438 3.6133017 -1255.0044
0.15 4.7878709 -37.261597 3.8809305 -143.92826
0.25 4.9000869 -15.109375 4.1249749 -55.109375

4f 0.05 4.6313806 -315.00375 3.5085755 -1255.0038
0.15 4.7760037 -37.255972 3.7735201 -143.92264
0.25 4.8997892 -15.093750 4.0224272 -55.093750

ρ1 = −
α

2

1 ±
√

(1 + 2l)2 +
8µDeβ2b2

α2~2

 , (13)
ρ0 =

2µDeβ2 b(b+2)
~2

2µD0β2
~2 −ρ

2
0

2ρ1
. (14)

The bound state solution requires that the wave function sat-
isfies the boundary conditions Un,`(r)/r = 0 as r −→ ∞ and
Un,`(r)/r is finite at r = 0. The regularity conditions enable us
to determine ρ0 > 0 and ρ1 < 0 as can be justify in equation
(13) and Eq. (14). By using equation (10), a pair of partner

potentials V±(r) = W 2 ± dW(r)/dr is constructed as:

V+(r) = ρ
2
0 +

ρ1(2ρ0 −ρ1)e−αr

1 − e−αr
+
ρ1(ρ1 −α)e−αr

(1 − e−αr )2
, (15)

V−(r) = ρ
2
0 +

ρ1(2ρ0 −ρ1)e−αr

1 − e−αr
+
ρ1(ρ1 + α)e−αr

(1 − e−αr )2
, (16)

The two partner potentials satisfied the relationship

V+(r, a0) = V−(r, a1) + R(a1), (17)

where a0 is an old set of parameters and a1 is a new set of
parameters uniquely determined from a0 , the R(a1) is a re-
minder term that is independent of the variable r. However,

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Onate et al. / J. Nig. Soc. Phys. Sci. 4 (2022) 242–250 245

Figure 2. Variation of energy En,`(eV ) against the screening parameter
α(Å) with momentum dependent potential ρ1 (a) and without momen-
tum dependent potential ρ0 (b) for three different quantum states with
µ = ~ = l = 1, re = 0.2 Å, De = 3 eV and D0 = 6 eV .

as ρ1 −→ ρ1 + α for a0 = ρ1, we can write a recurrent relation
of the form ρ2 = a0 + 2α, ρ3 = a0 + 3α, ρ4 = a0 + 4α and
consequently, ρn = a0 + nα. For a proper relation, Eq. (17) can
be written using the above recurrence relation as

R(a1) =
(

2µDeβ2 b(b+2)−2µD0β2
~2

−a20
2a0

)2
− (18)(

2µDeβ2 b(b+2)−2µD0β2
~2

−a21
2a1

)2
,

Figure 3. Variation of energy En,`(eV ) against the screening parameter
α with momentum dependent potential ρ1 (a) and without momentum
dependent potential ρ0 (b) for three different quantum states with µ =
~ = l = 1, re = 0.2 Å, De = 0 eV and D0 = 8 eV .

R(a2) =
(

2µDeβ2 b(b+2)−2µD0β2
~2

−a21
2a1

)2
− (19)(

2µDeβ2 b(b+2)−2µD0β2
~2

−a22
2a2

)2
,

R(a3) =
(

2µDeβ2 b(b+2)−2µD0β2
~2

−a22
2a2

)2
− (20)(

2µDeβ2 b(b+2)−2µD0β2
~2

−a23
2a3

)2
,

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Onate et al. / J. Nig. Soc. Phys. Sci. 4 (2022) 242–250 246

Figure 4. Wave function and probability density for the modified shifted
Deng-Fan momentum-dependent potential.

R(an) =
(

2µDeβ2 b(b+2)−2µD0β2
~2

−a2n−1
2an−1

)2
− (21)(

2µDeβ2 b(b+2)−2µD0β2
~2

−a2n
2an

)2
.

Using all desirable results inconjuction with the negative part-
ner potential, we finally deduce the energy equation of the non-
relativistic equation for modified shifted Deng-Fan momentum

Figure 5. Wave function and probability density for the shifted Deng-
Fan momentum-dependent potential.

dependent potential as

En,` = De +
α2~2

2µ
[`(` + 1)C0−

2µβ2 [De b(b+2)−D0 ]
α2~2

1 + 2n +
√

(1 + 2l)2 + 8µDeβ2 b
2

α2~2

−
1 + 2n +

√
(1 + 2l)2 + 8µDeβ2 b

2

α2~2

4


2 . (22)

The radial wave function was obtained using the parametric
Nikiforov-Uvarov method. Defining a variable of the form y =
e−αr we obtain the wave function in terms of Jacobi polynomial
as

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Onate et al. / J. Nig. Soc. Phys. Sci. 4 (2022) 242–250 247

Figure 6. Wave function and probability density for the Hulthèn
momentum-dependent potential.

Un,`(y) = Ny
√

Er (1 − y)
1
2 +

1
2

√
(1+2l)2 +

8µDeβ2 b
2

α2~2 (23)

×P

2√Er,
√

(1+2l)2 +
8µDeβ2 b

2

α2~2


n (1 − 2y).

Detail procedure on how to calculate the wave function can be
found in ref. [22].

2.1. Expectation Values

In this section, we compute the expectation values using
Hellmann-Feynman theorem [23-27]. Given a formula

∂E(q)
∂q

=

〈
Ψ(q)

∣∣∣∣∣∂H(q)∂q
∣∣∣∣∣ Ψ(q)

〉
. (24)

Figure 7. Variation of the expectation values
〈
p2

〉
(eV ) and

〈
r−2

〉
(eV )

against the momentum dependent parameter ρ with µ = ~ = l = 1,
re = 0.2 Å, De = 3 eV , α = 0.8 Å and D0 = 2 eV .

Equation (24) only holds if the normalized eigenfunction is
continuous with respect to the parameter. Given the Hamilto-
nian of potential (1) as

H = −
~2

2µ
d2

dr2
+

De
(
1 −

eαre − 1
eαr − 1

)2
− D0

(
1

eαr − 1

) (1 + ρ),
(25)

the time and momentum expectation values can now be calcu-
lated by transforming the parameter q. To calculate the momen-
tum expectation value

〈
p2

〉
, we set q = De and have

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Onate et al. / J. Nig. Soc. Phys. Sci. 4 (2022) 242–250 248

Table 3. Comparison of bound states for Deng-Fan potential with re = 0.4 Å,
µ = ~ = 1 and De15 eV for 2p, 3p and 3d at various states.

State Present [18] [28]
2p 0.05 7.86080 7.86080 7.8628

0.10 7.95330 7.95330 7.95537
0.15 8.04510 8.04510 8.04724
0.20 8.13620 8.13620 8.13842
0.25 8.22663 8.22663 8.22892

3p 0.05 10.99776 10.9978 10.9998
0.10 11.16256 11.1626 11.1647
0.15 11.32425 11.3242 11.32647
0.20 11.48284 11.4828 11.48513
0.25 11.63834 11.6383 11.64068

3d 0.05 10.21598 10.21598 10.21651
0.10 10.35354 10.35354 10.35409
0.15 10.48935 10.48935 10.48992
0.20 10.62346 10.62346 10.62403
0.25 10.75591 10.75591 10.75645

Table 4. Comparison of bound states for Hulthèn potential with µ = ~ = 1 and
D0 = −α for 2p, 3p and 3d at various states.

State Present [29] [30]
2p 0.025 0.1128125 0.1127605 0.1127605

0.050 0.1012500 0.1010425 0.1010425
0.075 0.0903125 0.0898478 0.0898478
0.100 0.0800000 0.0791794 0.0791794
0.150 0.0612500 0.0594415 0.0594415

3p 0.025 0.0437587 0.0437069 0.0437069
0.050 0.0333681 0.0331645 0.0331645
0.075 0.0243837 0.0239397 0.0239397
0.100 0.0168056 0.0160537 0.0160537
0.150 0.0058681 0.0044663 0.0044663

3d 0.025 0.0437587 0.0436030 0.0436030
0.050 0.0333681 0.0327532 0.0327532
0.075 0.0243837 0.0230307 0.0230307
0.100 0.0168056 0.0144842 0.0144842
0.150 0.0018681 0.0013967 0.0013966

〈
p2

〉
= 1 −

1
µ

[
α2~2

2µ

(
2Λ0 − n −

1
2
−

Λ1

2

)
×

(
Λ0 − 2µD0β2 −

µβ2b2

α2~2Λ1

(
4Λ0

1 + 2n + Λ1

))]
, (26)

Λ0 =
2µβ2(Deb2 + 2Deb − D0)

α2~2(1 + 2n + Λ1)
,

Λ1 =

√
(1 + 2l)2 +

8µβ2 Deb2

α2~2


(27)

To calculate the position expectation value
〈
r−2

〉
we set q = l

and obtain

〈
r−2

〉
=

1
(2l + 1)~2

[
α2~2 ((1 + 2l)C0

−2
(

2Λ2
α2~2Λ3

−
1
4
−

n
2
−

Λ1

4

)
×

−4Λ2(1 + 2l)
α2~2Λ1Λ23

−
1 + 2l
2Λ1

 , (28)
Λ2 = µβ2(Deb(b + 2) − D0),
Λ3 = 1 + 2n + Λ1

 (29)
2.2. Special Cases of the potential

The two special cases of the potential are the Deng-Fan po-
tential and the Hulthén potential. When we put D0 = 0, the
potential becomes the Deng-Fan potential of the form

V (r) = De

(
1 −

eαre − 1
eαr − 1

)2
(1 + ρ), (30)

with the energy equation as

Enl = De +
α2~2

2µ
[`(` + 1)C0

−


2µβ2 De b(b+2)

α2~2

1 + 2n +
√

(1 + 2l)2 + 8µDeβ2 b
2

α2~2

−
1 + 2n +

√
(1 + 2l)2 + 8µDeβ2 b

2

α2~2

4


2 . (31)

If we put De = 0 the interacting potential reduces to Hulthén
potential of the form

V (r) = −D0

(
1

eαr − 1

)
(1 + ρ), (32)

with the energy equation of the form

En,` =
α2~2

2µ

`(` + 1)C0 −
−

2µβ2 D0
α2~2 −

(1+n+l)2

2

2(1 + n + l)


2 . (33)

3. Results and Discussion

In Figure 1, we examined the energy of the system with the
screening parameter for the modified shifted Deng-Fan poten-
tial. The energy of the system goes down while the screening
parameter increases. At some point, the energy of the system
has a turning point even when the screening parameter contin-
ues its linear increase. With momentum-dependent parameter,
the energy goes beyond -15 before the turning point while with-
out the momentum-dependent parameter, the energy has the
turning point before -13. The energy with momentum depen-
dent parameter for the ground state and first excited has their
turning point before the screening parameter equals 2 while

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Onate et al. / J. Nig. Soc. Phys. Sci. 4 (2022) 242–250 249

without the momentum parameter, all the turning points are ob-
tained above screening parameter equals 2. The turning point
physically shows that there is a revised in the direction in the
system. In Figure 2, we examined the energy of the system
with the screening parameter for the Deng-Fan potential. The
variation of energy against the screening parameter in the pres-
ence of the momentum dependent parameter at the ground state
and at the excited states differs. The energy at the excited states
decreases and then have a turning point while the screening pa-
rameter increases but for the ground state, the revise case is ob-
tained. This indicates that the effect of momentum-dependent
parameter on the energy at the ground state is insignificant but
highly significant for the excited states. In the absence of mo-
mentum dependent parameter, the energy at different quantum
states has the same variation with the screening parameter. For
both presence and absence of the momentum dependent param-
eter, the energy of the system at various state are equal for the
screening parameter less than 0.1. In Figure 3, we examined the
energy of the system with the screening parameter for Hulthén
potential. The energy of the system with and without momen-
tum dependent parameter at various quantum states decreases
monotonically as the screening parameter increases. However,
with momentum dependent parameter, the energy of the system
are lower than their counterpart without momentum dependent
parameter. It is noted that the energy of the system at different
quantum states without momentum dependent parameter are
closer compared to the energy of the system at different quan-
tum states with momentum dependent parameter. In Figures
4, 5 and 6, we plotted the wave function and probability den-
sity for modified shifted Deng-Fan momentum-dependent po-
tential, Deng-Fan momentum-dependent potential and Hulthén
momentum-dependent potential respectively. The wave func-
tion for the modified shifted Deng-Fan momentum-dependent
potential has the highest pick both at the negative and pos-
itive vertical component, followed by Deng-Fan momentum-
dependent potential and lastly, Hulthén momentum-dependent
potential. The probability densities are seen to be positive for
the three potentials. However, the same trend observed for the
wave function are also observed in the probability density for
the three potentials. Variation of the expectation values against
the momentum dependent parameter ρ with momentum depen-
dent potential are Figure 7. The position expectation value and
the momentum expectation value respectively rises as the mo-
mentum dependent parameter increases. A clear observation
shows that the position expectation value for various quantum
states are higher than their counterpart in the momentum expec-
tation value.

In Table 1, we presented the numerical results for modified
shifted Deng-Fan momentum-dependent potential and without
momentum-dependent potential. The numerical values without
momentum-dependent potential are higher than their counter-
part with momentum-dependent potential. This simply shows
that the present of momentum-dependent potential reduces the
energy of the system. The numerical values for Deng-Fan po-
tential (D0 = 0) and Hulthén potential (De = 0) in the pres-
ence and absence of the momentum dependent parameter are
presented in Table 2. In both the presence and absence of the

momentum dependent parameter, the variation of energy with
the screening parameter, quantum and angular quantum num-
bers are the same. However, the presence of the momentum
dependent parameter reduces the energy of the system for both
potentials. For Hulthén potential, there are similarities in the
numerical values for 3p and 3d as well as 4p, 4d and 4f. The
discrepancy for the numerical values rises as the screening pa-
rameter increases. In Table 3 and Table 4, we compared the re-
sults for Deng-Fan potential and Hulthén potential respectively
with existing results. The two results aligned with the previous
results.

4. Conclusion

The solutions for modified shifted Deng-Fan potential was
obtained with a momentum-dependent potential in the present
study. The results showed that the present of the momentum-
dependent parameter reduces the energy of the system. This
was also noticed for the special cases of the potential. The mo-
mentum dependent-parameter showed the same variation against
both the position and momentum expectation values.

Acknowledgments

We thank the referees for the positive enlightening com-
ments and suggestions, which have greatly helped us in making
improvements to this paper.

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