J. Nig. Soc. Phys. Sci. 3 (2021) 165–171

Journal of the
Nigerian Society

of Physical
Sciences

Entropic system in the relativistic Klein-Gordon Particle

C. A. Onatea,∗, M. C. Onyeajub

aPhysics Programme, Department of Physical Sciences, Landmark University, Omu-Aran, Nigeria
bDepartment of Physics, Theoretical Physics Group, University of Port Harcourt, Choba, Nigeria

Abstract

The solutions of Kratzer potential plus Hellmann potential was obtained under the Klein-Gordon equation via the parametric Nikiforov-Uvarov
method. The relativistic energy and its corresponding normalized wave functions were fully calculated. The theoretic quantities in terms of
the entropic system under the relativistic Klein-Gordon equation (a spinless particle) for a Kratzer-Hellmann’s potential model were studied.
The effects of a and b respectively (the parameters in the potential that determine the strength of the potential) on each of the entropy were
fully examined. The maximum point of stability of a system under the three entropies was determined at the point of intersection between two
formulated expressions plotted against a as one of the parameters in the potential. Finally, the popular Shannon entropy uncertainty relation known
as Bialynick-Birula, Mycielski inequality was deduced by generating numerical results.

DOI:10.46481/jnsps.2021.209

Keywords: Eigensolutions, bound states, wave equation, theoretic quantity.

Article History :
Received: 25 April 2021
Received in revised form: 16 June 2021
Accepted for publication: 29 June 2021
Published: 29 August 2021

c©2021 Journal of the Nigerian Society of Physical Sciences. All rights reserved.
Communicated by: O. J. Oluwadare

1. Introduction

The understanding of correlations in quantum systems is
based on the analytic tools provided by the entropic measures.
These entropic measures are Shannon entropy, Rényi entropy,
and Tsallis entropy. The most outstanding of the entropic mea-
sures is the Shannon entropy introduced by Shannon [1]. The
Shannon entropy has several applications in various scientific
disciplines. In the concept of information, for instance, it presents
a discrete source without memory as a functional that quantifies
the uncertainty of a random variable at each discrete time. It
is the expected amount of information in a given event drawn
from a distribution that serves as a measure of uncertainty or
variability that is associated with random variables. Shannon

∗Corresponding author tel. no: +234(0)7036631325
Email address: oaclems14@physicist.net (C. A. Onate )

entropy has examined entropic uncertainty and has been tested
for different potential models. It serves as another form of
Heisenberg uncertainty relation. In physics, Shannon entropy
has been widely reported under the non-relativistic wave equa-
tion over the years for different potential models [2-25]. How-
ever, all the reports given in refs. [2-25] dwell under the non-
relativistic wave equation leaving out the relativistic wave equa-
tion. This motivates the present work. In the present study, the
authors want to examine the entropic system under the rela-
tivistic Klein-Gordon equation using Kratzer-Hellmann poten-
tial. The accuracy of Shannon entropy for any calculation can
be checked by the uncertainty relation of Shannon entropy that
relates position space and momentum space with the spatial di-
mension. This is otherwise called Bialynick-Birula, Mycielski
(BBM) inequality given as

S (ρ) + S (γ) = D(1 + ln π), (1)

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Onate & Onyeaju / J. Nig. Soc. Phys. Sci. 3 (2021) 165–171 166

where

S (ρ) = −
−4π
δ

∫ inf
0

ρ(r) log ρ(r) dr,

S (γ) = −
−4π
δ

∫ inf
0

γ( p) log γ( p) d p, (2)

ρ(r) and γ(p) are probability densities. D refers to the spatial
dimension, and ln π is a constant term. In this work, numeri-
cal results will be generated for equation (1) to verify whether
the results from the relativistic Klein-Gordon equation will sat-
isfy the BBM inequality. The Kratzer-Hellmann potential com-
prises of Kratzer potential and Hellmann potential. The physi-
cal form of Kratzer-Hellmann potential is

V (r) = De
( r − re

r

)2
−

(
a − be−δr

r

)
, (3)

where De is the dissociation energy, re is the equilibrium bond
length, a and b are the strengths of the Hellmann potential, r
is internuclear separation while δ (screening parameter) char-
acterized the range of the Hellmann potential. The two sub-sets
of the potential (3) have received attention in the bound states
and other areas of sciences. The Hellmann potential is known
to be suitable for the study of inner-shell ionization problems.
The potential was equally studied for alkali hydride molecules
by Varshni and Shukla [26]. Recently, the Hellmann potential
was used in ref. [27] as a tensor interaction for the breaking of
energy degenerate doublets in the Dirac equation. The Kratzer
potential, on the other hand, forms a potential pocket that is use-
ful for vibrational and rotational energy eigenvalues [28]. The
combination of these potentials is considered necessary because
of their applications.

2. Parametric Nikiforov-Uvarov method (PNUM)

This PNUM is a straight forward method that uses transfor-
mation of variable. The PNUM is short and accurate for solving
bound state problems. According to Tezcan and Sever [29], the
reference or standard equation for the PNUM is(

d2

d s2
+

c1 − c2
s(1 − c3 s)

d
d s

+
−ξ1 s2 + ξ2 s − ξ3

s2(1 − c3 s)2

)
ψ(s) = 0. (4)

According to ref. [29], the eigenvalues and eigenfunction re-
spectively can be obtained using [29, 30]

nc2 − (2n + 1) c5 + c7 + 2c3c8 + n (n − 1) c3

+ (2n + 1)
√

c9 +
(
2
√

c9 + c3 (2n + 1)
) √

c8 = 0, (5)

ψn,` (s) = Nn,` s
c12 (1 − c3 s)

−c12−
c13
c3 ×P

(
c10−1,

c11
c3
−c10−1

)
n (1 − 2c3 s) ,(6)

The parameters in equations (5) and (6) are deduced as follows

c4 =
1 − c1

2
, c5 =

c2 − 2c3
2

, c6 = c
2
5 + ξ1, c7 = 2c4c5 − ξ2, ,

c9 = c3 (c7 + c3c8) + c6, c10 = c1 + 2c4 + 2
√

c8, c8 = c
2
4 + ξ3

c11 = c2 − 2c5 + 2
(√

c9 + c3
√

c8
)
, c12 = c4 +

√
c8,

c13 = c5 −
(√

c9 + c3
√

c8
)

(7)

2.1. The Klein-Gordon equation (KGE) with Kratzer-Hellmann
potential

The KGE is use to describe spinless particles in the domain
of relativistic wave equation [31, 32, 33, 34, 35, 36, 37, 38]. The
Klein-Gordon equation for space-time scalar potential S (r) and
the time component of the Lorentz four-vector potential V (r)
arising from minimal coupling, in the relativistic unit (~ = c =
1), reads[

p̂2 + (M + S (r))2 − (E − V (r))2
]

R(r) = 0, (8)

where p̂ is the momentum operator, M is the particle’s mass,
E is the relativistic energy and R(r) is the wave function. The
KGE above has a potential 2V in which the nonrelativistic limit
cannot give the solutions of the Schrödinger equation. A criti-
cal investigation was done by Alhaidari et al. [39] who proved
that S = ±V . This is the nonrelativistic limit for the poten-
tial 2V . Thus, in the relativistic limit, the interacting potential
becomes V instead of 2V . Therefore, to obtain a solution of
the Klein- Gordon equation for any arbitrary `−state whose en-
ergy equation in the nonrelativistic limit equals the solution of
the Schrödinger equation, equation (8) becomes [39, 40, 41, 42,
43, 44, 45][

p̂2 − M2 + E2 − V (r)(M + E) −
`(` + 1)

r2

]
R(r) = 0, (9)

The solutions of the Klein-Gordon equation above and some di-
atomic molecular potential models have been obtained for dif-
ferent molecules [46, 47, 48, 49, 50], and the results compared
with experimental values. To get rid of the inverse squared
term in equation (9), we need to adopt a suitable approximation
scheme. In this work, we adopt the following approximation
that is valid for δ � 1,

1
r2
≈

δ2(
1 − e−δr

)2 . (10)
Plugging equations (3) and (10) into equation (9) and making a
simple transformation of the form y = e−δr , equation (9) turns
to be [

d2

dy2
+

1 − y
y − y2

d
dy

+
Ay2 + By + C

y2(1 − y)2

]
Rn,`(y) = 0, (11)

A =
Υ + bδβ
δ2

(12)

B =
2Υ − (b + a + 2Dere)δβ

δ2
, (13)

C =
E2 − M2 + βϑ− `(` + 1)δ2

δ2
, (14)

β = M + E, (15)

ϑ = 2Dereδ− De + aδ− Der
2
eδ

2, (16)

Υ = M2 − E2 + Deβ. (17)

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Onate & Onyeaju / J. Nig. Soc. Phys. Sci. 3 (2021) 165–171 167

Comparing equation (11) with equation (4), equation 7 numer-
ically becomes

c1 = c2 = c3 = 1, c4 = 0, c5 = −
1
2
, c6 =

1
4
− A, c7 = −B,

c8 = −C, c9 =
1
4
− A − B − C, c10 = 1 + 2

√
−C,

c11 = 2
(
1 +
√
−C

)
+

√
1 − 4(A + B + C), c12 =

√
−C,

c13 = −
1
2
−

1
2

√
1 − 4(A + B + C) −

√
−C (18)

Substituting c1 to c9 in equation (18) into equation (5), we have
energy equation for the Kratzer-Hellmann potential as

Υ −β(ϑ + De) + `(` + 1)δ2

δ2

=

 (ϑ− De − bδ)β− n(n + 1) −
1
2 − 2`(` + 1) −

(
n + 12

) √
1 − 4(A + B + C)

1 + 2n +
√

1 − 4(A + B + C)


2

.

(19)

The energy equation obtained for Kratzer-Hellmann potential in
equation (19) above has subset energy equations for Kratzer po-
tential, Yukawa potential, and Coulomb potential. However, the
wave function for the Kratzer-Hellmann potential is obtained by
substituting c10to c13in Eq. 18 into Eq. 6 to have

R(s) = N sη(1 − s)
1
2 +

λ
2 P(n2η,λ)(1 − 2s), (20)

where

η =

√
Υ − 2Dereδβ− aδβ

δ2
+ βDer2e + `(` + 1), (21)

λ =
√

(1 + 2`) + 4βDer2e. (22)

and N is a normalization factor which can easily be calculated
using normalization condition. Using∫

∞

0
|R(r)|2 dr = 1, (23)

the normalization factor can easily be obtained. Consider the
transformation y = e−δr and another transformation of the form
x = 2y − 1, with a relation of the form 1 − x = 1 −

(
1−x

2

)
,

when invoked on equation (23), using an appropriate integral,
we have the normalization factor as

N2n,` = −
n!2δηΓ(2η + λ + n + 1)
Γ(2η + 1)Γ(λ + n + 2)

. (24)

2.2. Kratzer-Hellmann potential and entropies

The three entropies mentioned in the introduction will be
calculated here using equation (20).

2.2.1. Shannon entropy
To obtain Shannon entropy, we plug equation (20) into equa-

tion (2). For S (ρ), we define a transformation of the form

s = 1 − y, and using the appropriate integral given in the ap-
pendix, we have

S (ρ) =
8πη(n!)Γ(2η + 1)Γ(λ + n + 3)Γ(2η + λ + n + 1)

Γ(2η + n)Γ(λ + n + 2)Γ(2η + λ + n + 3)

× log
[
(0.99)2η(0.01)1+λ

Γ(2η + n + 1)
Γ(2η + 1)

]
. (25)

To obtain S (γ), we define x = −1 + 2y and then, using integral
and formula in the appendix, we have

S (γ) = −
4πΓ(2η + n + 1)Γ(2η + λ + n + 1)

Γ(2η + n)Γ(2η + λ + n + 2)

× log
(0.99)2η (0.01)1+λ × [Γ(2η + n + 1)]2

n!
[
Γ(2η + 1)

]2
 . (26)

2.2.2. Rényi entropy
Rényi entropy is a generalization of Shannon entropy and is

defined as [51]

Rq(ρ) =
1

1 − q
log 4π

∫
∞

0
ρ(r)q dr. (27)

The q is called Tsallis index. Following the procedures used to
obtain Shannon entropy for position space, we have Rq(ρ) as

Rq(ρ) = −
2.5314δq−1

1 − q
(28)

×

[
2η(n!)Γ(2η + 1)Γ(λ + n + 3)Γ(2η + λ + n + 1)

Γ(2η + n)Γ(λ + n + 2)Γ(2η + λ + n + 3)

]q
.

For the momentum space, we follow step by step as in the
Shannon entropy for momentum space to have Rq(γ) as

Rq(γ) = −
1.2657δq−1

(1 − q)
(29)[

2Γ(2η + n + 1)Γ(2η + λ + n + 1)
Γ(2η + 1)Γ(2η + λ + n + 2)

]q
.

2.2.3. Tsallis entropy
The Tsallis entropy was introduced by Tsallis [52]. The

concept acts as a basis for generalizing the statistical mechan-
ics. This Tsallis entropy is defined as

Tq(ρ) =
1

q − 1

(
1 − 4π

∫
∞

0
ρ(r)q dr

)
, q , 1. (30)

The Tsallis entropy reduces to the usual Boltzmann-Gibbs en-
tropy as the Tsallis index q approaches one. With the wave
function in equation (20) and following previous procedures,
we have Tsallis entropy for position space as

Tq(ρ) =
1

q − 1
+

4πδq−1

q − 1
(31)(

2ηΓ(2η + 1)Γ(λ + n + 3)Γ(2η + λ + n + 1)(n!)
Γ(2η + n)Γ(λ + n + 2)Γ(2η + λ + n + 3)

)q
Following the procedures to obtain momentum space of Shan-
non entropy, the Tsallis entropy for momentum space is ob-
tained as

Tq(γ) =
1

q − 1
(32)

×

[
1 + 2πδq−1

(
2Γ(2η + n + 1)Γ(2η + λ + n + 1)

Γ(2η + 1)Γ(2η + λ + n + 2)

)q]
.

167



Onate & Onyeaju / J. Nig. Soc. Phys. Sci. 3 (2021) 165–171 168

3. Results and Discussion

Figure 1. S (ρ) against b

Figure 2. S (γ) against b

In Figure 1, we plotted S (ρ) against one of the potential
strength at the first excited state with ` = 1, re = 0.1, a = 1,
δ = 0.25, De = 2.5 and M = 2De. There is less concentration
of electron density and so, less concentration of the wave func-
tion which makes the system unstable as the potential strength
increases. In Figure 2, S (γ) is plotted against the potential
strength (b) at the first excited state with ` = 1, re = 0.1,
a = 1, δ = 0.25, De = 2.5 and M = 2De. There is a
more concentration of the spreading of electron density which
leads to more concentration of the wave function as the poten-
tial strength goes up. Thus, there is stability of the system as
b goes up. In Figures 3 and 4, we examined the variation of
the product of Rényi and Tsallis entropies respectively against
the potential strength (a) at the first excited state with ` = 1,
re = 0.1, b = 2, δ = 0.25, De = 2.5 and M = 2De and In
both cases, there is an inverse variation between the product of
the entropy and the potential strength. The point of intersection
for the entropies (Shannon, Rényi and Tsallis) is determined by
plotting S = RT /S T ; T = (RT /TT ) − 0.0a against a potential

Table 1. Shannon entropy relation with n = ` = 1, re = 0.1, b = 2, δ =
0.25, De = 2.5 and M = 2De.

a S (ρ) S (γ) S T = S (ρ) + S (γ)

1 -1.561445865 10.47908433 8.917638464
2 -0.884937778 10.33123825 9.446300475
3 -0.568858951 10.18281361 9.613954654
4 -0.396386446 10.05310205 9.656715603
5 -0.254891378 9.898107966 9.643216588

Table 2. Renyi entropy relation with n = ` = 1, re = 0.1, q = b = 2, δ =
0.25, De = 2.5 and M = 2De.

a R(ρ) R(γ) RT = R(ρ) + R(γ)

1 0.4291810608 2.803088803 3.232269864
2 0.3386588921 2.886297376 3.224956268
3 0.2792173453 2.936768150 3.215985495
4 0.2373724018 2.970645793 3.208018195
5 0.1937069142 3.004709576 3.198416490

Table 3. Tsallis entropy relation with n = ` = 1, re = 0.1, q = b = 2,
δ = 0.25, De = 2.5 and M = 2De.

a T2(ρ) T2(γ) TT = T2(ρ) + T2(γ)

1 0.077255921 8.181988939 8.259244860
2 0.048103385 8.674956863 8.723060249
3 0.032699075 8.980995892 9.013694967
4 0.023632579 9.189394955 9.213027534
5 0.015737688 9.401349043 9.417086731

Table 4. Shannon entropy S (ρ) for Kratzer, Coulomb and Yukawa potentials at
different excited states.

n Kratzer Coulomb Yukawa

0 -0.614945865 -0.36740543 -0.373228924
1 -0.489475768 -0.32198215 -0.330603267
2 -0.323785845 -0.19261467 -0.206643987
3 -0.191326241 -0.11715545 -0.130089665

Table 5. Shannon entropy S (γ) for Kratzer, Coulomb and Yukawa potentials at
different excited states.

n Kratzer Coulomb Yukawa

0 5.670453776 4.870762243 4.873986446
1 4.879443281 4.163817555 4.192208735
2 3.991974378 3.519430921 3.530045080
3 3.268931407 2.980665799 3.001043671

168



Onate & Onyeaju / J. Nig. Soc. Phys. Sci. 3 (2021) 165–171 169

Figure 3. R(ρ)R(γ) against a

Figure 4. T (ρ)T (γ) against a

strength as shown in Figure 5.

Table 1 presented the numerical results for Shannon en-
tropy. The results were numerically verified and confirmed the
Bialynick-Birula, Mycielski (BBM) inequality that gives a stan-
dard relation S (ρ) + S (γ) ≥ D(1 + log π). For D = 1, D(1 +
log π) = 1.497206180. However, the lower bound from Table 1
is 8.917638464. This verifies the accuracy of the present work.
In Tables 2 and 3, we numerically presented the Rényi entropy
and Tsallis entropy respectively for position space and momen-
tum space. In both entropies, the position space and momentum
space varies inversely with one another. In Tables 4 and 5, re-
sults of S (ρ) against S (γ) for the subset potentials were given.
The results of these subset potentials are similar to the results
of the mother potential. The result for Kratzer potential was ob-
tained by putting a = b = 0. The result for Coulomb potential
was obtained by putting b = De = 0. The result for Yukawa
potential was obtained by putting a = De = 0. The results in
Tables 1, 2, 3, 4 and 5 showed that a diffused density distri-
bution γ( p) in momentum space is associated with a localized
density distribution ρ(r) in the position space or configuration
space. The physical meaning is that a decrease in the position
space corresponds to an increase in the momentum space and

Figure 5. Entropies (S = RT /S T + 0.03a; T = RT /TT + 0.05) against the
potential strength a

vice visa.

4. Conclusions

We calculated the Shannon information, Rényi information
and Tsallis information for position and momentum entropies
of a relativistic Klein-Gordon equation. In the first excited
state i.e. the Shannon uncertainty yields the minimum value
of 8.917638464 which maintains the normal condition for the
entropic uncertainty relation with respect to BBM inequality.
The S (ρ), S (γ), R(ρ), R(γ), T (ρ) and T (γ) plotted against the
potential strengths determined the concentration of the electron
and wave density. The results obtained for the relativistic Klein-
Gordon equation were found to obey BBM inequality. The re-
sults for both the Rényi entropy and Tsallis entropy followed
the pattern of the results of the Shannon entropy.

Appendix

∫ 1
0

yα(1 − y)β2 F1(−n, n + 2(α + β); 2α + 1; y)
2dz

=
n!Γ(α + 1)2Γ(β + n + 2)

βΓ(α + n + 1)Γ(α + β + n + 2)

∫ 1
−1

(
1 − x

2

)η ( 1 + x
2

)ν
×

[
P(η,ν)n (x)

]2
d x

=
2Γ(η + n + 1)Γ(ν + n + 1)

n!ηΓ(η + ν + 2n + 1)Γ(η + ν + n + 1)

∫ 1
−1

(
1 − x

2

)s ( 1 + x
2

)ν
×

[
P(s,ν)n (x)

]2
d x

=
2Γ(s + n + 1)Γ(ν + n + 1)

n!sΓ(s + ν + 2n + 1)Γ(s + ν + n + 1)

169



Onate & Onyeaju / J. Nig. Soc. Phys. Sci. 3 (2021) 165–171 170

P(a,b)n (1 − 2x) =
Γ)a + n + 1
n!Γ(a + 1) 2

F1(−n, n + a + b + 1; a + 1; x)

2 F1(a, b; c; y) =
Γ(c)

Γ(a)Γ(b)

∞∑
n=0

Γ(a + n)Γ(b + n)
Γ(c + n)

yn

n!

∫ 1
−1

(1 − s)a−1 (1 + s)b
[
P(a,b)n (s)

]2
d s

=
Γ(a + n + 1)Γ(b + n + 1)

n!aΓ(a + b + n + 1)

2 F1
(
−n, n + v + u + 1; v + 1; 1−x2

)
=

Γ(n + v + 1)
Γ(v + 1)

P(v,u)n (x) = 2 F1

(
−n, n + v + u + 1; v + 1;

1 − x
2

)
.

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