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1. Introduction
The Klein-Gordon and Dirac equations, or relativistic mod equations, can be solved
analytically or numerically to offer us a lot of information about a physical
systems. Various techniques are available in the literature to obtain their
solutions, such as the parametric Nikiforov-Uvarov (pNU) method, the exact
quantization rule, the Qiang-Dong proper quantization rule, the path integral
method, the asymptotic iteration method (AIM), the factorization method, the Laplace
transform approach, the supersymmetric quantum mechanics (SUSYQM), the ansatz
method, and the series expansion method. These equations play a vital role in
statistical physics, solid-state physics, quantum field theory, atomic and
sub-atomic physics, and molecular physics. Various research has been carried out on
these equations of many considerable potentials. Parmar and Vinodkumar (2021)
studied the combined modified Yukawa and Kratzer potential (MYKP) in the case of the
Klein-Gordon equation (KGE) for calculated numerical results of the energy spectrum
for CO, H2, LiH, and HCl molecules by pNU and SUSYQM methods using the
Greene-Aldrich approximation to handle 1/r and 1/r
2 terms in the effective potential [1]. Many researchers have dealt with either the modified Yukawa or
Kratzer potentials either individually or through the combination of one or both of
them with other potentials within the framework of the different fundamental
equations. Edet et al. analyzed the modified Kratzer potential plus
the screened Coulomb potential by the pNU method and obtained the energy spectrum
[2]. Ikot et al. obtained
the exact bound state energy spectrum of the Schrödinger equation with
energy-dependent molecular Kratzer potential using AIM [3]. Ikot et al. (2019) obtained the energy
eigenvalues and the corresponding normalized eigenfunctions of screened Kratzer
potential for lithium hydride (LiH) and hydrogen chloride (HCl) diatomic molecules
within the framework of non-relativistic quantum mechanics via the pNU method. Ikot
et al. (2019) obtained the energy eigenvalues and the
corresponding normalized eigenfunctions of the screened Kratzer potential for
lithium hydride (LiH) and hydrogen chloride (HCl) diatomic molecules within the
framework of non-relativistic quantum mechanics via the pNU method [4]. Ahmadov et al. [5] studied solutions of KGE with the
Manning-Rosen equation plus a class of Yukawa potentials using pNU and SUSYQM
methods and presented the energy spectrum for any l-state and the
corresponding radial wave functions in terms of the hypergeometric functions.
Purohit et al. (2021) solved the Schrodinger¨ equation in D
dimensions and obtained the eigenspectrum of the energy and momentum for
time-independent and timedependent Hulthen-screened cosines Kratzér potentials Using
the Qiang-Dong proper quantization rule and the supersymmetric quantum mechanics
approach [6]. Recently, Purohit et
al. (2022) obtained the energy spectrum for MYKP with the magnetic
field and Aharanov-Bohm flux field using the pNU approach and SEM in this study
[7]. The Kratzer potential (KP), which is
known first and foremost by Kratzer itself [8], is used in quantum atomic-molecular physics and has played a vital role
in the history of molecular and quantum chemistry [9]. Purohit et al. (2020) [10] obtained via the generalized Nikiforov-Uvarov method the
approximate bound-state solutions of the D-dimensional KGE for screened cosine
Kratzer potential (SCKP) using approximation suggested by Greene-Aldrich. Next year,
Purohit et al. also investigated SCKP under the influence of the
magnetic field and Aharanov-Bohm flux field and obtained energy eigenvalues and wave
functions with external fields via the pNU method using the approximation method
suggested by Greene-Aldrich for handling centrifugal barriers [11]. On the other hand, Yukawa potential, often called screened
Coulomb potential, is a short-range potential that has applications in particle,
high-energy, and molecular physics. It is used to study the interaction that occurs
between the atoms of diatomic molecules [12-14]. Recently, Purohit
et al. (2022) used the linear plus modified Yukawa potential as
the quarkantiquark interaction potential and obtained the energy eigenvalues and
associated wave function by solving the KGE analytically using the pNU method [15]. Based on this motivation, in this work, we
study new approximate bound state solutions of the deformed Klein-Gordon equation
(DKGE) in three-dimensional extended relativistic quantum mechanics (3D-ERQM)
symmetries of a newly proposed combined potential called the improved modified
exponential screened plus Yukawa potential (IMYKP) within the framework of
parametric Bopp’s shift method. This is a new potential model that has not yet been
studied to the best of our knowledge in 3D-ERQM symmetries. The main objective of
this study is to deepen the study of research [1] performed in the frameworks of symmetries known in quantum mechanics
known in the literature and research on new applications of this potential. It is
worth noting that the first foundations of quantum mechanics were based on
non-commutative axioms for all positions
where (V yk (r),S yk (r)) are the (vector, scalar) modified Yukawa-Kratzer potential, according to the view of 3D-RQM and 3D-NRQM symmetry, known in the literature [1]:
and
where potential strength parameters A
1 ≡ A
3 ≡ V
0, A
2 ≡ 2V
0, A
4 ≡ 2D
e
r
e
,
and
with
On the other hand, we notice that the RHS of Eq. (4) generates a novel uncertainty relation:
here h µν and G µν are equal to the average values:
The new subdivided three-uncertainties relations in Eq. (6) have no comparison in the
existing literature. Under the Lorentz transformation, which includes boosts and/or
rotations of the observer’s inertial frame, Eqs. (3) and (4) are covariant equations
(have the same behavior as
The second component in Eq. (7) provides a physical representation of the
consequences of space-space non-commutativity. The outline of the paper is as
follows: Section 2 presents an overview of the 3D-KGE under the modified
Yukawa-Kratzer potential models. Section 3 is devoted to investigating the 3D-DKGE
using the well-known Bopp’s shift method to obtain the effective potential for the
IMYKP model. Furthermore, using standard perturbation theory, we find the
expectation values of the radial terms (1/(1 −
z)4, z
2
/(1 − z)3, z
2
/(1 − z)4, z/(1 −
z)2, z/(1 −
z)3 and 1/(1 −
z)3) to calculate the corrected relativistic energy
generated by the effect of the perturbed effective potential
2. An overview of KGE under the modified Yukawa-Kratzer potential model in 3D-RQM symmetry
In order to construct a physical model describing a physical system that interacted with the improved modified Yukawa-Kratzer potential model in the 3D-ERQM symmetries, it is useful to recall the eigenvalues and the corresponding eigenfunctions under the influence of this system within the framework of relativistic quantum mechanics (RQM) known in the literature. In this case, the system is governed by the following Klein-Gordon equation:
The vector potential V
yk
(r) and space-time scalar potential S
yk
(r) are produced from the four-vector linear momentum
operator A
µ
(V
yk
(r), A = 0) and the reduced
mass M
yk
of (LiH, HCl, CO and, H2) molecules. While E
nl
is the relativistic eigenvalues, (n,l) represent the
principal and spin-orbit coupling terms. Since the modified Yukawa-Kratzer
potential model has spherical symmetry, it allows the wave function solution Ψ
nl
(r,θ,ϕ) of the known form
Parmar and Vinodkumar used the Alhaidari et al. [62] scheme to write the radial part of KGE in Eq. (9), by restyling the vector and scalar potentials (V yk (r),S yk (r)) → (V yk (r)/2,S yk (r)/2) under the non-relativistic limit. Using V yk (r) from Eq. (2) with V yk (r) = S yk (r) in Eq. (9), we obtain the following
with
The author of Ref. [1] used the supersymmetry quantum mechanics and factorization methods to obtain the expression of u nl (r) as a function of Gauss’s Hypergeometric function 2 F 1 (−n,n + 2ω nl + 2X nl ,2X nl + 1;z) in usual 3D-RQM symmetries as,
with
where z and
The corresponding relativistic energy eigenvalues for the modified Yukawa-Kratzer potential model for (LiH, HCl, CO and, H2) molecules in 3D-space, obtained the equation of energy [1]:
3. The new solutions of DKGE under the IMYKP models in 3D-ERQM symmetries:
3.1. Review of BS method
Let us begin in this subsection by finding the physical form of IMYKP models in three-dimensional extended relativistic quantum mechanics (3D-ERQM) symmetries. Our objective is achieved by applying the new principles that we have seen in the introduction (Eqs. (3), (4), and (7)), summarized in the new relationships of the modified equal-time noncommutative canonical commutation relations (METNCCCRs) and the notion of the Weyl-Moyal star product. These data allow us to rewrite the usual radial KG equations in Eq. (10) in the 3D-ERQM symmetries as follows:
There are two approaches to including non-commutativity in quantum field theory:
The first method is represented by rewriting the various NC physical fields
instead of the usual correspondence
This procedure is called Bopp’s shifts method (BSM) [72-77]. This
quantization procedure is called Bopp quantization [76]. The Weyl-Moyal star product
h(x,p)∗g(x,p)
induces BSM in the respect that it is replaced by
The Bopp’s shift method has achieved great success when applied by specialized researchers to the four basic equations correspond to the relativistic deformed Schrödinger equation (see, e.g., [32-34,77-82]) and the other three relativistic equations represented by the deformed Klein-Gordon equation (see, e.g., [83-88]), deformed Dirac equation (see, e.g., [47,93,94]) and the deformed Duffin-Kemmer-Petiau equation (see, e.g., [86-89]). It is worth motioning that that Bopp’s shift method permutes us to rewrite Eq. (16.1) without star product to the simplest form:
The modified algebraic structure of covariant canonical commutation relations with the notion of the Weyl-Moyal star product in Eqs. (3) and (4), which become new METNCCCRs with ordinary known products in literature, are as follows (see, e.g., [72-77]):
In 3D-ERQM symmetries, one possible way of implementing the algebra defined by
Eq. (18) is to construct the noncommutative set of variables
This allows us to find the operator
The Taylor expansion of effective potential
and
Substituting Eqs. (20) and (21) into Eq. (17), we obtain the following as the radial Schrödinger equation:
with
If we compare Eq. (23) and Eq. (10), we observe an additive potential
Furthermore, by using the unit step function (also known as the viside step
function θ (x) or simply the theta function),
it is possible to rewrite the global induced potential
where the step function θ(x) is given by:
Eq. (23) cannot be solved analytically for any state l ≠ 0
because of the centrifugal term and the studied potential itself. In fact, since
the effective perturbative potential
This allows us, after direct calculations, to find the following results:
This gives the perturbative effective potential as follows:
with
The IMYKP model is extended by including new radial terms 1/(1 −
z)4, z
2
/(1 − z)3, z
2
/(1 − z)4, z/(1 −
z)2, z/(1 −
z)3 and 1/(1 −
z)3 to become the IMYKP model in 3D-ERQM
symmetries. The new additive part
3.2. The expectation values under the IMYKP models in the 3D-ERQM symmetries
In this subsection, we want to apply the perturbative theory, in the case of
3D-ERQM symmetries, to find the 6-expectation values
and
We have used useful abbreviations
and
We calculate the integrals in Eqs.
here Γ(ξ) denoting the usual Gamma function. By identifying
Eqs.
and
with D nl = 2ω nl + 2X nl .
3.3. The corrected energy for the IMYKP models in 3D-ERQM symmetries
What draws attention here is the application of our physical method resulting
from the principle of superposition for the purpose of determining the total
values of the relativistic energy, in 3D-RNCQM symmetries. The global effective
potential
This degree of freedom results from the arbitrary nature of the infinitesimal NC vector Θ. We have oriented the spin-s of the (LiH, HCl, CO and, H2) molecules to become parallels to the vector Θ which interacted with the IMYKP model. Additionally, we use the following transformation which is well known in 3D-RQM symmetries:
In 3D-ERQM symmetry, the operators
with |l − s| ≤ j ≤
|l + s| for (LiH, HCl, CO and,
H2) molecules. As a direct consequence, the square partially
corrected energies
The global expectation values
The second principal physical contribution is due to the effect of the magnetic
perturbative effective potential, which generates the perturbed potential
with the additive physical condition
here (ℵ and τ) are the intensity of the magnetic field induced by the effect of the deformation of space-space geometry and a new infinitesimal non-commutativity parameter. This choice, which comes from the fact that the vector Θ is arbitrary, or that the magnetic field is directed according to the (O z ) axis serves to simplify quantitative calculations without affecting the nature of the physical point of view, we also need to apply:
with −|l| ≤ m ≤ +|l| for the
(LiH, HCl, CO and, H2) molecules. All of these data allow for the
discovery of the new square improved energy shift
After we have completed the first and second stages of the self-production of
energy, we are going to discover another very important case under the IMYKP
model in 3D-ERQM symmetries. This physical new phenomenon is produced
automatically from the influence of perturbed effective potential
Here χ is just an infinitesimal real proportional constant. The
effective potentials
We chose a rotational velocity Ω parallel to the (Oz) axis (Ω= Ωe z ) to simplify the calculations, this, of course, does not change the physical characteristics of the examined problem as much as it simplifies the calculations. The pertubed generated spin-orbit coupling is then transformed into new physical phenomena as follows:
All of this data allow for the discovery of the new corrected square improved
energy
It’s worth noting that the authors of ref. [99] were studied rotating isotropic and anisotropic harmonically
confined ultracold Fermi gases in two and 3-dimensional space at zero
temperature, but in this case, the rotational term was added to the Hamiltonian
operator manually, whereas, in our study, the rotation operator
where E nl are usual relativistic energies under the IMYKP model obtained from equations of energy in Eq. (15). It should be noted that the corrected relativistic energy in Eq. (46) can be generalized to include negative energy (the bosonic antiparticle) and the positive relativistic energy (the bosonic particle) as follows:
which can be written explicitly using the step
It is important to point out that because we have only used corrections of the first order of infinitesimal noncommutative parameters (Θ,τ,χ), perturbation theory cannot be used to find corrections of the second order (Θ2 ,τ 2 ,χ 2).
4. The SE with IMYKP models in 3D-ENRQM symmetries
The main purpose of this section is to analyze the non-relativistic limit, in
three-dimensional extended non-relativistic QM (3D-ENRQM) symmetries, for the IMYKP
model. Two steps must be applied. The first one corresponds to the non-relativistic
limit, in usual three-dimensional non-relativistic QM (3D-NRQM) symmetries. This is
achieved by transferring the following values (
with
Now, under the non-relativistic limit, the relativistic expectation values
and
with
while the relativistic factors
As a direct consequence, the new non-relativistic improved energy
where
5. Study of important particular cases in 3D-ERQM and 3D-ENRQM symmetries
We will look at some specific examples involving the new bound state energy eigenvalues in Eqs. (46) and (54) in this section. By adjusting relevant parameters of the IMYKP model in 3D-ERQM and three-dimensional extended non-relativistic quantum mechanics (3D-ENRQM) symmetries, we could derive some specific potentials useful for other physical systems for much concern the specialist reaches. It should be noted that these special cases were treated within the framework of relativistic and non-relativistic quantum mechanics known in the literature in ref. [1], and we are now in the process of generalizing them to include extended relativistic and non-relativistic quantum mechanics symmetries.
(1) If we choose, V 0 = 0, we obtain improved modified Kratzer potential (IMKP), and α → 0, from Eqs. (46) and (54), we deduced eigenvalues correspond to IMKP for 3D-ERQM and 3D-ENRQM symmetries as [33]:
and
where the relativistic eigenvalues
while the new relativistic and non-relativistic expectations values
(2) If we choose, A 1 = A 2 = 0 and A 3 = −λ, we obtain improved generalized Kratzer potential (IGKP), and α → 0, from Eqs. (46) and (54), we deduced eigenvalues correspond to IGKP for 3D-ERQM and 3D-ENRQM symmetries as:
and
where the relativistic eigenvalues
while the new relativistic and non-relativistic expectations values
(3) If we choose, A 1 = A 2 = 0 and A 3 = −D e , we obtain improved Kratzer potential, and α → 0, from Eqs. (46) and (54), we deduced eigenvalues correspond to improved Kratzer potential for 3D-ERQM and 3D-ENRQM symmetries as:
and
where the relativistic eigenvalues
while the new relativistic and non-relativistic expectations values
(4) If we choose, A 1 = A 3 = 0 and A 2 = −A, we obtain improved modified Kratzer plus screened Coulomb potential (IMKSCP). From Eqs. (46) and (54), we deduced eigenvalues correspond to IMKSCP for 3D-ERQM and 3D-ENRQM symmetries as:
and
where the relativistic eigenvalues
while the new relativistic and non-relativistic expectations values
(5) If we choose, A 1 = A 5 = 0 , A 2 = B and A 4 = C, we obtain improved Hellmann potential (IHP). From Eqs. (46) and (54), we deduced eigenvalues correspond to IMKSCP for 3D-ERQM and 3D-ENRQM symmetries as [39]:
and
where the relativistic eigenvalues
while the new relativistic and non-relativistic expectations values
(6) If we choose, A 1 = A 3 = D e = 0 and A 2 = −A , we obtain an improved screened Coulomb potential or improved Yukawa potential (IYP). From Eqs. (46) and (54), we deduced eigenvalues correspond to IMKSCP for 3D-ERQM and 3DENRQM symmetries as [38,88,103]:
and
where the relativistic eigenvalues
while the new relativistic and non-relativistic expectations values
(7) If we choose, A 2 = A 4 = A 5 = 0 , A 1 = A and A 2 = D 2, we obtain improved inversely square Yukawa potential. From Eqs. (46) and (54), we deduced eigenvalues correspond to improved inversely square Yukawa potential for 3D-ERQM and 3D-ENRQM symmetries as [37,105]:
and
where the relativistic eigenvalues
while the new relativistic and non-relativistic expectations values
6. Spin-averaged mass spectra of HLM under IMKSCP models in 3D-ENRQM symmetries
The quark-antiquark interaction potentials, are spherically symmetrical and
provide a good description of the heavy-light mesons (HLM) such as
cc and bb. This would give us a strong
incentive to dedicate this section to the purpose to determine the modified
spin-averaged mass spectra of the heavy quarkonium system such as
where
here m
q
is the quark mass which equals the antiquark mass
Permuted us to obtain the new energy eigenvalues
1. For the case:
2. For the case:
3. For the case: j = l − 1and s = 1, can be express on the new energy eigenvalue E nl yk−l by the following formula:
4. For the case (j = l, s =
0), we can be express on the new energy eigenvalue
By substituting Eqs. (86), (87) and (88) into Eq. (89), the new mass spectrum of the meson systems in 3D-ENRQM symmetries under the IMYKP model for any arbitrary radial and angular momentum quantum numbers becomes:
Thus the spin-averaged mass spectra
is extended to include
Which is sensitive to the atomic quantum numbers (n,j,l,s,m), potential depths (V 0 ,α), and the non-commutativity parameters (Θ,τ,χ) under the deformed properties of space-space. This allows us to realize logical physical limits:
to be achieved.
6.1. Composite systems
In this section, and in the context of NC algebra, consider composite systems such as molecules comprised of N = 2 particles of mass m α (α = 1,2). In the non-relativistic context, it is important to consider the characteristics of the system descriptions. It was discovered that those composite systems with various mass descriptions need various NC parameters [51,52,109]:
where the non-commutativity parameter
with Λ
c
can be present Θ
c
,τ
c
and χ
c
. As mentioned above, in the case of a system of two particles with the
same mass µ
1 = µ
2, we have
It is worth noting that for the three-simultaneous limits (Θ,τ,χ) → (0,0,0) and (Θ c ,τ c ,χ c ) → (0,0,0), we recover the energy equations for both the KGE and SE with modified Yukawa-Kratzer potential in 3D-RQM and 3DNRQM symmetries, which are obtained in main Ref. [1].
7. Conclusions
In summary, this paper presents an approximate analytical solution of the 3-dimensional deformed Klein-Gordon and deformed equations in 3D-ERQM symmetries with the improved modified Yukawa-Kratzer potential model models using the parametric Bopp’s shift method and standard perturbation theory. Under the deformed features of spacespace, we found new bound-state energies that appear sensitive to quantum numbers (n,j,l,s,m), the mixed potential depths (V 0 ,D e ,r e ), the screening parameter’s inverse α and the non-commutativity parameters (Θ,τ,χ). Moreover, the non-relativistic limit of the studied potential in 3DENRQM symmetries has been investigated. The modified spin-averaged mass spectra of heavy and heavy-light mesons such as cc and bb in both 3D-NRQM (commutative space) and 3D-ENRQM symmetries were determined by applying our results of the new non-relativistic energies that represent the binding energy between the quark and anti-quark.
In the context of 3D-ERQM and 3D-ENRQM symmetry, we have treated certain significant particular instances that we hope will be valuable to the specialized researcher, such as the improved modified Kratzer potential, improved generalized Kratzer potential, improved Kratzer potential, improved modified Kratzer plus screened Coulomb potential, improved screened Coulomb potential and improved inversely square Yukawa potential. It is shown that the IMYKP model in a 3D-ERQM and 3D-ENRQM symmetry has similar behavior to the dynamics of a bosonic particle and bosonic antiparticle with equal scalar and vector potential for the modified Yukawa-Kratzer potential model models in a 3D-RQM symmetry (commutative space) influenced by the effect of a constant magnetic field, a self-rotational and a perturbed spin-orbit interaction. We recover the energy equations for the KGE and SE with modified Yukawa-Kratzer potential in 3D-RQM and 3D-NRQM symmetries, which were found in the main reference [1], for the three-simultaneous limits (Θ,τ,χ) → (0,0,0) and (Θ c ,τ c ,χ c ) → (0,0,0).
