This document presents a solution to the Klein-Gordon equation for a modified screened Coulomb potential plus inversely quadratic Yukawa potential using the formula method. The formula method provides simple formulas to find bound state solutions to quantum mechanical wave equations. This method is applied to solve the radial part of the Klein-Gordon equation for the combined modified screened Coulomb plus inversely quadratic Yukawa potential. Solutions for the energy eigenvalues and wavefunctions are obtained using the formulas of the formula method.
Exact Bound State Solution of Qdeformed Woods-Saxon Plus Modified Coulomb Pot...ijrap
In this work, we obtained an exact solution to Schrodinger equation using q-deformed Woods-Saxon plus
modified Coulomb potential Using conventional Nikiforov-Uvarov method. We also obtained the energy
eigen value and its associated total wave function . This potential with some suitable conditions reduces to
two well known potentials namely: the Yukawa and coulomb potential. Finally, we obtained the numerical
results for energy eigen value with different values of q as dimensionless parameter. The result shows that
the values of the energies for different quantum number(n) is negative(bound state condition) and increases
with an increase in the value of the dimensionless parameter(arbitrary constant). The graph in figure (1)
shows the different energy levels for a particular quantum number.
Exact Bound State Solution of Qdeformed Woods-Saxon Plus Modified Coulomb Pot...ijrap
In this work, we obtained an exact solution to Schrodinger equation using q-deformed Woods-Saxon plus modified Coulomb potential Using conventional Nikiforov-Uvarov method. We also obtained the energy eigen value and its associated total wave function . This potential with some suitable conditions reduces to two well known potentials namely: the Yukawa and coulomb potential. Finally, we obtained the numerical results for energy eigen value with different values of q as dimensionless parameter. The result shows that the values of the energies for different quantum number(n) is negative(bound state condition) and increases with an increase in the value of the dimensionless parameter(arbitrary constant). The graph in figure (1) shows the different energy levels for a particular quantum number.
Bound- State Solution of Schrodinger Equation with Hulthen Plus Generalized E...ijrap
We apply an approximation to centrifugal term to find bound state solutions to Schrodinger equation with
Hulthen Plus generalized exponential Coulomb potential Using Nikiforov-Uvarov Method. Using this
method, we obtained the energy-eigen value and the total wave function. We implement C++ algorithm, to
obtained the numerical values of the energy for different quantum state starting from the first excited state
for different values of the screening parameter.
The approximate bound state of the nonrelativistic Schrӧdinger equation was
obtained with the modified trigonometric scarf type potential in the framework of
asymptotic iteration method for any arbitrary angular momentum quantum number l
using a suitable approximate scheme to the centrifugal term. The effect of the screening
parameter and potential depth on the eigenvalue was studied numerically. Finally, the
scattering phase shift of the nonrelativistic Schrӧdinger equation with the potential
under consideration was calculated.
Exact Solutions of the Klein-Gordon Equation for the Q-Deformed Morse Potenti...ijrap
In this work, we solve the Klein-Gordon (KG) equation for the general deformed Morse potential with
equal scalar and vector potentials by using the Nikiforov-Uvarov (NU) method, which is based on the
solutions of general second-order linear differential equation with special functions. The energy
eigenvalues and corresponding normalized eigenfunctions are obtained. It is found that the eigenfunctions
can be expressed by the Laguerre polynomials. Our solutions have a good agreement with earlier study.
Exact Bound State Solution of Qdeformed Woods-Saxon Plus Modified Coulomb Pot...ijrap
In this work, we obtained an exact solution to Schrodinger equation using q-deformed Woods-Saxon plus
modified Coulomb potential Using conventional Nikiforov-Uvarov method. We also obtained the energy
eigen value and its associated total wave function . This potential with some suitable conditions reduces to
two well known potentials namely: the Yukawa and coulomb potential. Finally, we obtained the numerical
results for energy eigen value with different values of q as dimensionless parameter. The result shows that
the values of the energies for different quantum number(n) is negative(bound state condition) and increases
with an increase in the value of the dimensionless parameter(arbitrary constant). The graph in figure (1)
shows the different energy levels for a particular quantum number.
Exact Bound State Solution of Qdeformed Woods-Saxon Plus Modified Coulomb Pot...ijrap
In this work, we obtained an exact solution to Schrodinger equation using q-deformed Woods-Saxon plus modified Coulomb potential Using conventional Nikiforov-Uvarov method. We also obtained the energy eigen value and its associated total wave function . This potential with some suitable conditions reduces to two well known potentials namely: the Yukawa and coulomb potential. Finally, we obtained the numerical results for energy eigen value with different values of q as dimensionless parameter. The result shows that the values of the energies for different quantum number(n) is negative(bound state condition) and increases with an increase in the value of the dimensionless parameter(arbitrary constant). The graph in figure (1) shows the different energy levels for a particular quantum number.
Bound- State Solution of Schrodinger Equation with Hulthen Plus Generalized E...ijrap
We apply an approximation to centrifugal term to find bound state solutions to Schrodinger equation with
Hulthen Plus generalized exponential Coulomb potential Using Nikiforov-Uvarov Method. Using this
method, we obtained the energy-eigen value and the total wave function. We implement C++ algorithm, to
obtained the numerical values of the energy for different quantum state starting from the first excited state
for different values of the screening parameter.
The approximate bound state of the nonrelativistic Schrӧdinger equation was
obtained with the modified trigonometric scarf type potential in the framework of
asymptotic iteration method for any arbitrary angular momentum quantum number l
using a suitable approximate scheme to the centrifugal term. The effect of the screening
parameter and potential depth on the eigenvalue was studied numerically. Finally, the
scattering phase shift of the nonrelativistic Schrӧdinger equation with the potential
under consideration was calculated.
Exact Solutions of the Klein-Gordon Equation for the Q-Deformed Morse Potenti...ijrap
In this work, we solve the Klein-Gordon (KG) equation for the general deformed Morse potential with
equal scalar and vector potentials by using the Nikiforov-Uvarov (NU) method, which is based on the
solutions of general second-order linear differential equation with special functions. The energy
eigenvalues and corresponding normalized eigenfunctions are obtained. It is found that the eigenfunctions
can be expressed by the Laguerre polynomials. Our solutions have a good agreement with earlier study.
Bound State Solution to Schrodinger Equation with Hulthen Plus Exponential Co...ijrap
In this work, we obtained an approximate bound state solution to Schrodinger with Hulthen plus
exponential Coulombic potential with centrifugal potential barrier using parametric Nikiforov-Uvarov
method. We obtained both the eigen energy and the wave functions to non -relativistic wave equations. We
implement Matlab algorithm to obtained the numerical bound state energies for various values of
adjustable screening parameter at various quantum state.. The developed potential reduces to Hulthen
potential and the numerical bound state energy conform to that of existing literature.
EXACT SOLUTIONS OF SCHRÖDINGER EQUATION WITH SOLVABLE POTENTIALS FOR NON PT/P...ijrap
We have obtained explicitly the exact solutions of the Schrodinger equation with Non PT/PT symmetric
Rosen Morse II, Scarf II and Coulomb potentials. Energy eigenvalues and the corresponding
unnormalized wave functions for these systems for both Non PT and PT symmetric are also obtained using
the Nikiforov-Uvarov (NU) method.
Analytical Solution Of Schrödinger Equation With Mie–Type Potential Using Fac...ijrap
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
Analytical Solution Of Schrödinger Equation With Mie–Type Potential Using Fac...ijrap
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.
ANALYTICAL SOLUTIONS OF THE MODIFIED COULOMB POTENTIAL USING THE FACTORIZATIO...ijrap
We have solved exactly Schrödinger equation with modified Coulomb Potential under the framework of
factorization method. Energy levels and the corresponding wave functions in terms of associated Laquerre
function are also obtained. For further guide to interested readers we have computed the energy
eigenvalue for some selected elements for various values of n and l .
Microscopic Mechanisms of Superconducting Flux Quantum and Superconducting an...Qiang LI
We have provided microscopic explanations to superconducting flux quantum and (superconducting and normal) persistent current. Flux quantum is generated by current carried by "deep electrons" at surface states. And values of the flux quantum differs according to the electronic states and coupling of the carrier electrons. Generation of persistent carrier electrons does not dissipate energy; instead there would be emission of real phonons and release of corresponding energy into the environment; but the normal carrier electrons involved still dissipate energy. Even for or persistent carriers,there should be a build-up of energy of the middle state and a build-up of the probability of virtual transition of electrons to the middle state, and the corresponding relaxation should exist accordingly.
Analytical Solution Of Schrodinger Equation With Mie-Type Potential Using Fac...ijrap
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.
Analytical Solutions of the Modified Coulomb Potential using the Factorizatio...ijrap
We have solved exactly Schrödinger equation with modified Coulomb Potential under the framework of factorization method. Energy levels and the corresponding wave functions in terms of associated Laquerre
function are also obtained. For further guide to interested readers we have computed the energy eigenvalue for some selected elements for various values of n and l .
Solutions of the Schrodinger Equation with Inversely Quadratic Hellmann Plus ...ijrap
The solutions of the Schrödinger equation with inversely quadratic Hellmann plus Mie-type potential for
any angular momentum quantum number have been presented using the Nikiforov-Uvarov method. The
bound state energy eigenvalues and the corresponding un-normalized eigenfunctions are obtained in terms
of the Laguerre polynomials. Several cases of the potential are also considered and their eigen values
obtained.
SOLUTIONS OF THE SCHRÖDINGER EQUATION WITH INVERSELY QUADRATIC HELLMANN PLUS ...ijrap
The solutions of the Schrödinger equation with inversely quadratic Hellmann plus Mie-type potential for
any angular momentum quantum number have been presented using the Nikiforov-Uvarov method. The
bound state energy eigenvalues and the corresponding un-normalized eigenfunctions are obtained in terms
of the Laguerre polynomials. Several cases of the potential are also considered and their eigen values obtained.
Bound State Solution to Schrodinger Equation with Hulthen Plus Exponential Co...ijrap
In this work, we obtained an approximate bound state solution to Schrodinger with Hulthen plus
exponential Coulombic potential with centrifugal potential barrier using parametric Nikiforov-Uvarov
method. We obtained both the eigen energy and the wave functions to non -relativistic wave equations. We
implement Matlab algorithm to obtained the numerical bound state energies for various values of
adjustable screening parameter at various quantum state.. The developed potential reduces to Hulthen
potential and the numerical bound state energy conform to that of existing literature.
EXACT SOLUTIONS OF SCHRÖDINGER EQUATION WITH SOLVABLE POTENTIALS FOR NON PT/P...ijrap
We have obtained explicitly the exact solutions of the Schrodinger equation with Non PT/PT symmetric
Rosen Morse II, Scarf II and Coulomb potentials. Energy eigenvalues and the corresponding
unnormalized wave functions for these systems for both Non PT and PT symmetric are also obtained using
the Nikiforov-Uvarov (NU) method.
Analytical Solution Of Schrödinger Equation With Mie–Type Potential Using Fac...ijrap
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
Analytical Solution Of Schrödinger Equation With Mie–Type Potential Using Fac...ijrap
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.
ANALYTICAL SOLUTIONS OF THE MODIFIED COULOMB POTENTIAL USING THE FACTORIZATIO...ijrap
We have solved exactly Schrödinger equation with modified Coulomb Potential under the framework of
factorization method. Energy levels and the corresponding wave functions in terms of associated Laquerre
function are also obtained. For further guide to interested readers we have computed the energy
eigenvalue for some selected elements for various values of n and l .
Microscopic Mechanisms of Superconducting Flux Quantum and Superconducting an...Qiang LI
We have provided microscopic explanations to superconducting flux quantum and (superconducting and normal) persistent current. Flux quantum is generated by current carried by "deep electrons" at surface states. And values of the flux quantum differs according to the electronic states and coupling of the carrier electrons. Generation of persistent carrier electrons does not dissipate energy; instead there would be emission of real phonons and release of corresponding energy into the environment; but the normal carrier electrons involved still dissipate energy. Even for or persistent carriers,there should be a build-up of energy of the middle state and a build-up of the probability of virtual transition of electrons to the middle state, and the corresponding relaxation should exist accordingly.
Analytical Solution Of Schrodinger Equation With Mie-Type Potential Using Fac...ijrap
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.
Analytical Solutions of the Modified Coulomb Potential using the Factorizatio...ijrap
We have solved exactly Schrödinger equation with modified Coulomb Potential under the framework of factorization method. Energy levels and the corresponding wave functions in terms of associated Laquerre
function are also obtained. For further guide to interested readers we have computed the energy eigenvalue for some selected elements for various values of n and l .
Solutions of the Schrodinger Equation with Inversely Quadratic Hellmann Plus ...ijrap
The solutions of the Schrödinger equation with inversely quadratic Hellmann plus Mie-type potential for
any angular momentum quantum number have been presented using the Nikiforov-Uvarov method. The
bound state energy eigenvalues and the corresponding un-normalized eigenfunctions are obtained in terms
of the Laguerre polynomials. Several cases of the potential are also considered and their eigen values
obtained.
SOLUTIONS OF THE SCHRÖDINGER EQUATION WITH INVERSELY QUADRATIC HELLMANN PLUS ...ijrap
The solutions of the Schrödinger equation with inversely quadratic Hellmann plus Mie-type potential for
any angular momentum quantum number have been presented using the Nikiforov-Uvarov method. The
bound state energy eigenvalues and the corresponding un-normalized eigenfunctions are obtained in terms
of the Laguerre polynomials. Several cases of the potential are also considered and their eigen values obtained.
Ethnobotany and Ethnopharmacology:
Ethnobotany in herbal drug evaluation,
Impact of Ethnobotany in traditional medicine,
New development in herbals,
Bio-prospecting tools for drug discovery,
Role of Ethnopharmacology in drug evaluation,
Reverse Pharmacology.
Operation “Blue Star” is the only event in the history of Independent India where the state went into war with its own people. Even after about 40 years it is not clear if it was culmination of states anger over people of the region, a political game of power or start of dictatorial chapter in the democratic setup.
The people of Punjab felt alienated from main stream due to denial of their just demands during a long democratic struggle since independence. As it happen all over the word, it led to militant struggle with great loss of lives of military, police and civilian personnel. Killing of Indira Gandhi and massacre of innocent Sikhs in Delhi and other India cities was also associated with this movement.
The Art Pastor's Guide to Sabbath | Steve ThomasonSteve Thomason
What is the purpose of the Sabbath Law in the Torah. It is interesting to compare how the context of the law shifts from Exodus to Deuteronomy. Who gets to rest, and why?
How to Create Map Views in the Odoo 17 ERPCeline George
The map views are useful for providing a geographical representation of data. They allow users to visualize and analyze the data in a more intuitive manner.
The Roman Empire A Historical Colossus.pdfkaushalkr1407
The Roman Empire, a vast and enduring power, stands as one of history's most remarkable civilizations, leaving an indelible imprint on the world. It emerged from the Roman Republic, transitioning into an imperial powerhouse under the leadership of Augustus Caesar in 27 BCE. This transformation marked the beginning of an era defined by unprecedented territorial expansion, architectural marvels, and profound cultural influence.
The empire's roots lie in the city of Rome, founded, according to legend, by Romulus in 753 BCE. Over centuries, Rome evolved from a small settlement to a formidable republic, characterized by a complex political system with elected officials and checks on power. However, internal strife, class conflicts, and military ambitions paved the way for the end of the Republic. Julius Caesar’s dictatorship and subsequent assassination in 44 BCE created a power vacuum, leading to a civil war. Octavian, later Augustus, emerged victorious, heralding the Roman Empire’s birth.
Under Augustus, the empire experienced the Pax Romana, a 200-year period of relative peace and stability. Augustus reformed the military, established efficient administrative systems, and initiated grand construction projects. The empire's borders expanded, encompassing territories from Britain to Egypt and from Spain to the Euphrates. Roman legions, renowned for their discipline and engineering prowess, secured and maintained these vast territories, building roads, fortifications, and cities that facilitated control and integration.
The Roman Empire’s society was hierarchical, with a rigid class system. At the top were the patricians, wealthy elites who held significant political power. Below them were the plebeians, free citizens with limited political influence, and the vast numbers of slaves who formed the backbone of the economy. The family unit was central, governed by the paterfamilias, the male head who held absolute authority.
Culturally, the Romans were eclectic, absorbing and adapting elements from the civilizations they encountered, particularly the Greeks. Roman art, literature, and philosophy reflected this synthesis, creating a rich cultural tapestry. Latin, the Roman language, became the lingua franca of the Western world, influencing numerous modern languages.
Roman architecture and engineering achievements were monumental. They perfected the arch, vault, and dome, constructing enduring structures like the Colosseum, Pantheon, and aqueducts. These engineering marvels not only showcased Roman ingenuity but also served practical purposes, from public entertainment to water supply.
2024.06.01 Introducing a competency framework for languag learning materials ...Sandy Millin
http://sandymillin.wordpress.com/iateflwebinar2024
Published classroom materials form the basis of syllabuses, drive teacher professional development, and have a potentially huge influence on learners, teachers and education systems. All teachers also create their own materials, whether a few sentences on a blackboard, a highly-structured fully-realised online course, or anything in between. Despite this, the knowledge and skills needed to create effective language learning materials are rarely part of teacher training, and are mostly learnt by trial and error.
Knowledge and skills frameworks, generally called competency frameworks, for ELT teachers, trainers and managers have existed for a few years now. However, until I created one for my MA dissertation, there wasn’t one drawing together what we need to know and do to be able to effectively produce language learning materials.
This webinar will introduce you to my framework, highlighting the key competencies I identified from my research. It will also show how anybody involved in language teaching (any language, not just English!), teacher training, managing schools or developing language learning materials can benefit from using the framework.
How to Make a Field invisible in Odoo 17Celine George
It is possible to hide or invisible some fields in odoo. Commonly using “invisible” attribute in the field definition to invisible the fields. This slide will show how to make a field invisible in odoo 17.
Students, digital devices and success - Andreas Schleicher - 27 May 2024..pptxEduSkills OECD
Andreas Schleicher presents at the OECD webinar ‘Digital devices in schools: detrimental distraction or secret to success?’ on 27 May 2024. The presentation was based on findings from PISA 2022 results and the webinar helped launch the PISA in Focus ‘Managing screen time: How to protect and equip students against distraction’ https://www.oecd-ilibrary.org/education/managing-screen-time_7c225af4-en and the OECD Education Policy Perspective ‘Students, digital devices and success’ can be found here - https://oe.cd/il/5yV
Unit 8 - Information and Communication Technology (Paper I).pdfThiyagu K
This slides describes the basic concepts of ICT, basics of Email, Emerging Technology and Digital Initiatives in Education. This presentations aligns with the UGC Paper I syllabus.
Instructions for Submissions thorugh G- Classroom.pptxJheel Barad
This presentation provides a briefing on how to upload submissions and documents in Google Classroom. It was prepared as part of an orientation for new Sainik School in-service teacher trainees. As a training officer, my goal is to ensure that you are comfortable and proficient with this essential tool for managing assignments and fostering student engagement.
Synthetic Fiber Construction in lab .pptxPavel ( NSTU)
Synthetic fiber production is a fascinating and complex field that blends chemistry, engineering, and environmental science. By understanding these aspects, students can gain a comprehensive view of synthetic fiber production, its impact on society and the environment, and the potential for future innovations. Synthetic fibers play a crucial role in modern society, impacting various aspects of daily life, industry, and the environment. ynthetic fibers are integral to modern life, offering a range of benefits from cost-effectiveness and versatility to innovative applications and performance characteristics. While they pose environmental challenges, ongoing research and development aim to create more sustainable and eco-friendly alternatives. Understanding the importance of synthetic fibers helps in appreciating their role in the economy, industry, and daily life, while also emphasizing the need for sustainable practices and innovation.
Read| The latest issue of The Challenger is here! We are thrilled to announce that our school paper has qualified for the NATIONAL SCHOOLS PRESS CONFERENCE (NSPC) 2024. Thank you for your unwavering support and trust. Dive into the stories that made us stand out!
1. www.ajms.com 30
ISSN 2581-3463
RESEARCH ARTICLE
Bound State Solution of the Klein–Gordon Equation for the Modified Screened
Coulomb Plus Inversely Quadratic Yukawa Potential through Formula Method
Onyemaobi Ifeanyichukwu Michael, Benedict Iserom Ita, Onyemaobi Obinna Eugene,
Alex Ikeuba, Imoh Effiong, Hitler Louis
Department of Pure and Applied Chemistry, University of Calabar, Calabar, Cross River State, Nigeria
Received: 25-02-2018; Revised: 30-03-2018; Accepted: 30-04-2018
ABSTRACT
We present solution of the Klein–Gordon equation for the modified screened Coulomb potential (Yukawa)
plus inversely quadratic Yukawa potential through formula method. The conventional formula method
which constitutes a simple formula for finding bound state solution of any quantum mechanical wave
equation, which is simplified to the form;
2
1 2
2 2
3 3
( ) ( )
''( ) '( ) ( ) 0
(1 ) (1 )
k k s As Bs c
s s s
s k s s k s
− + +
ψ + ψ + ψ =
− −
. The bound
state energy eigenvalues and its corresponding wave function obtained with its efficiency in spectroscopy.
Key words: Bound state, inversely quadratic Yukawa, Klein–Gordon, modified screened coulomb
(Yukawa), quantum wave equation
INTRODUCTION
Physicists, chemists, and other researchers in
science have shown much interest in searching for
the exponential-type potentials owing to reasons
that most of this exponential-type potentials play
an important role in physics, for example, Yukawa
potential a tool used in plasma, solid-state, and
atomic physics.[1]
It is expressed mathematically
as follows:
V r g
e
r
V
e
r
yukawa
kmr r
( ) = − ≡ −
− −
2
0
α
(1)
V r v
e
r
IQYP
r
( ) = −
−
0
2
2
α
(2)
Where α → 0 results in V r
V
r
IQP ( ) =
− 0
2
(3)
Respectively, where g is the magnitude sealing
constant, m is the mass of the affected particle,
r is the radial distance to the particles, and k is
another scaling constant. The inversely quadratic
potential has been used by Oyewumi et al.,[2]
in
combination with an isotropic harmonic oscillator
Address for correspondence:
Benedict Iserom Ita / Hitler Louis
E-mail: Iserom2001@yahoo.com /
louismuzong@gmail.com
in N-dimension spaces. Since then, several
papers on the potential have appeared in the
literature of Ita et al.,[3]
first proposed by Hideki
Yukawa, in 1935, on the paper titled “on the
interaction of elementary particles.” In his work,
he explained the effect of heavy nuclei interaction
on peons. According to Yukawa, he expanded
that the interactions of particles are not always
accompanied by the emission of light particles
when heavy particles are transmitted from neutron
state to proton state, but the liberated energy due
to the transmission is taken up sometimes by
another heavy particle which will be transformed
from proton state into neutron state. Hamzavi
et al. obtained bound state approximate analytical
solutions of the Yukawa potential for any l-state
through the Nikiforov-Uvarov (NU) method in
which their calculations to the energy eigenvalue
yielded reasonable result compared to other
numerical and analytical methods. Hitler et al.[5]
obtained the energy eigenvalue for the Yukawa
potential using the generalized scaling variation
method for a system with a spherically symmetric
potential, Coulombic at the origin. In a nutshell,
not much has been achieved on the application
or use of the Yukawa potential in both relativistic
and non-relativistic quantum mechanical cases.
The objective of this report is to obtain band state
2. Michael, et al.: Bound state solution of the Klein–Gordon equation for the modified screened coulomb plus inversely
quadratic yukawa potential through formula method
AJMS/Apr-Jun-2018/Vol 2/Issue 2 31
solutions of the Klein–Gordon equation with
the modified screened Coulomb potential plus
inversely quadratic Yukawa (MSC-IQY) potential
through formula method. Our work is arranged
as follows; in the following section, we obtain a
bound state solution, and we present discussion
and conclude remarks at the end of the article.
Theoretical approach
Formula method is based on finding bound
state solution of any quantum mechanical wave
equation which can be simplified to the form:
ψ ψ ψ
'' '
( )
( )
( )
( )
( )
( )
( )
s
k k s
s k s
s
As Bs c
s k s
s
+
−
−
+
+ +
−
=
1 2
3
2
2
3
2
1 1
0
(4)
The two cases where k3 0
= and 3
k ≠ are studied.
Furthermore, an expression for the energy
spectrumandwavefunctionintermsofgeneralized
hypergeometric functions 2 1 3
F k s
( . : : )
α β γ is
derived. In a study by Falaye et al.,[6] the accuracy
of the proposed formula has been proven in
obtaining bound state solutions for some existing
eigenvalue problems, which is seen to be accurate,
efficient, reliable, and particularly very easy to
manipulate when introduced to quite a good
number of quantum mechanical potential models.
In quantum mechanics, while solving relativistic
and nonrelativistic quantum wave equations in the
presence of central and non-central potential
models, we do often come across differential
equation of the form, as seen in equation (4) in
which several techniques have been formulated
for tackling equation (4). This includes the
Feynman integral formalism, proper quantization
rule, NU method, asymptotic iteration method
(AIM), exact quantization rule, and generalized
pseudospectral method. Falaye[7]
also proposed
that the energy eigenvalues and the corresponding
wave function can be obtained using the following
formulas with regard to this approach (formula
method).
k k
n
k
k k k A
4 5
3
2 3 2
2
1 2
2
1
2
4
+ =
−
− − − −
( )
( ) (5)
Or more explicitly as,
k k
n
k
k k k A
n
k
k k
4
2
5
2
3
2 3 2
2
2
3
2 3
1 2
2
1
2
4
2
1 2
2
1
2
− −
−
− − − −
( )
−
− −
( )
( −
− −
( )
− = ≠
k A
k k
2
2
5
2
3
4
0 0
)
,
(6)
and
ψ ( ) ( ) ( , ( )
; )
s N S k s F n n k k
k
k
k k k s
n
k k
= − − + +
+ − +
4 5
1 2
1 2
3 2 1 4 5
2
3
4 1 3
(7),
respectively, such that
k
k k C
k
k k
k
k k
k
4
1 1
2
5
1 2
3
1 2
3
2
1 1 4
2
1
2 2 2
1
2 2 2
=
− + − −
= + − +
+ −
−
( ) ( )
,
A
A
k
B
k
C
3
2
3
+ +
where Nn
is the normalization constant, also where
k3 0
→ the energy eigenvalue, and its
corresponding wave function is given as,
B nk k k
n k k
k k
− −
+ +
− = =
2 2 4
1 4
2
5
2
3
2 2
0 0
, (8)
ψ ( ) [ ; ;( ) ]
( )
s N S e F n k k k k s
n
k k s
= − + +
−
4 5
1 1 4 1 5 2
2 2
(9),
respectively.
Furthermore, for some other cases, where k3 0
= ,
3 0
k ≠ , and k5 = −A . In this approach, analysis
on the asymptotic behavior at the origin is first
noted and at infinity for the finiteness of the
solution. Considering the solution of equation (4)
where s → 0 , it is seen that
ψ ( )
s Sk
= 4
where k
k k C
4
1 1
2
1 1 4
2
=
− + − −
( ) ( )
(10)
Furthermore, when s
k
→ 1
3
the solution of
equation 4 is ψ ( ) ( )
s k s k
= −
1 3
5
,
3. Michael, et al.: Bound state solution of the Klein–Gordon equation for the modified screened coulomb plus inversely
quadratic yukawa potential through formula method
AJMS/Apr-Jun-2018/Vol 2/Issue 2 32
k
k k
k
k k
k
A
k
B
k
C
5
1 2
3
1 2
2
2
3
2
3
1
2 2 2
1
2 2 2
= + − + + −
− + +
(11)
Hence, the wave function in the intermediary
region, for this task, could be seen as follows:
ψ ( ) ( ) ( )
s s k s F s
k k
= −
4 5
1 3 (12)
On substituting equation (12) into equation (10),
gives a differential equation in its second-order
expressed as:
′′ + ′
+ − + +
−
F s F s
k k sk k k
k
k
s k s
( ) ( )
( )
( )
2 2 2
1
4 1 3 4 5
2
3
3
−
− + + + + −
−
2 1 2
1
3 4 4 5 3 4 1 4 1 3 2
3
k k k k k k k k k k k B
s k s
( ) ( ) ( )
( )
=
F s
( ) 0
(13)
′′ + ′
+ − + +
−
F s F s
k k sk k k
k
k
s k s
( ) ( )
( )
( )
2 2 2
1
4 1 3 4 5
2
3
3
−
+ + +
( ) − +
−
k k k k k k k
A
k
s k s
F s
3 4 5
2
4 5 2 3
3
3
1
( ) ( )
( )
( )
) = 0
(14)
It should be noted that equation (13) is equivalent
to equation (14), whereas (13) seems to be more
complex during the course of the calculation.
Therefore, we employ equation (14) to obtain
solutions, thereby invoking the traditional method
otherwise known as the functional analysis
approach and the AIM. In summary, it was noted
that on applying the AIM and functional analysis
method to solve equation (14) so as to obtain the
formula given in equation (6) yielded the same
results which imply that the result is in excellent
agreement.
The Klein–Gordon equation
Given the Klein–Gordon equation as:
2
2 2 2
2 2 2 2
2
2
( )
1
( ) 0
( 1)
( ( ) )
i V r c
t
U r
h c l l h C
S r M
r
∂
− − − ∇
∂
=
+
+ + −
n
(15)
Where M is the rest mass,
i
t
∂
∂
is energy eigenvalue,
and V(r) and S(r) are the vector and scalar
potentials, respectively.
The radial part of the Klein–Gordon equation
with vector V(r) potential = scalar S(r) potential
is given as:
( )
2
2 2 2
2 2
2 2 2
2
( ) 1
( 1)
2( ) ( )
( ) 0
d U r
dr h c
l l h c
E M c E Mc V r
r
U r
+
+
− − + −
=
(16)
In atomic units, where h c
= = 1
d U r
dr
E M E M V r
l l
r
U r
2
2
2 2
2
2
1 0
( )
( ) ( )
( ) ( )
+
−
( )− +
−
+
=
(17)
The Solution to the radial part of the Klein–
Gordon equation for the MSCIQY potential
using formula method
The sum of the MSCIQY potential is given thus,
V r V
e
r
V
e
r
r r
( ) = − −
− −
1 0
2
2
α α
(18)
Substituting equation (18) into equation (17), we
obtain:
d U r
dr
E M E M
V
e
r
V
e
r
l l
r
r r
2
2
2 2
1 0
2
2 2
2
1
( )
( )
( )
+
−
( )− +
− −
−
+
− −
=
U r
( ) 0
(19)
4. Michael, et al.: Bound state solution of the Klein–Gordon equation for the modified screened coulomb plus inversely
quadratic yukawa potential through formula method
AJMS/Apr-Jun-2018/Vol 2/Issue 2 33
Using an appropriate approximate scheme to deal
withthecentrifugal/inversesquaretermequation (19)
as proposed by Greene and Aldrich[8]
wherein,
1
1
1
1
2
2
2
r e r e
r r
=
−
( )
=
−
( )
− −
α α
α α
; (20)
Which is valid for α 1 for a short potential and
introducing a new variable of form s e r
= −α
;
d U s
ds s
s
s
dU s
ds
s
E M E M
V s
2
2
2 2
2 2
0
2 2
1 1
1
1
2
1
( ) ( )
( )
( )
( ) ( )
(
+
−
−
+
− − +
−
−
−
−
−
−
+
−
=
s
V s
s
l l
s
U s
) ( )
( )
( )
( )
2
1
2
2
1
1
1
0
(21)
d U s
ds s
s
s
dU s
ds
s
E M E M V s
s
2
2
2
2 2
2
0
2
2
1 1
1
1
2
1
( ) ( )
( )
( )
( )
( )
+
−
−
+
−
+
+
−
+
2
2
1
1
1
0
1
2
( )
( )
( )
( )
( )
E M V s
s
l l
s
U s
+
−
−
+
−
=
(22)
d U s
ds s
s
s
dU s
ds s s
E M
s s
2
2 2 2
2 2
2
2
1 1
1
1
1
1 2 2
( ) ( )
( )
( )
( )
( ) (
+
−
−
+
−
−
− + +
E
E M V s
E M
V s s l l
U s
+ +
+
− − +
=
)
( )
( ) ( )
( )
0
2
1
2
1 1
0
(23)
Where − =
−
=
+
β
α α
2
2 2
2 1
2
E M
B
E M
V (24)
d U s
ds s
s
s
dU s
ds s s
B V E M s
2
2 2 2
2
0
2
1 1
1
1
1
2
( ) ( )
( )
( )
( )
( )
+
−
−
+
−
− + − +
( ) +
2
2 1
0
2 2
+
( ) − + +
( )
=
B s l l
U s
( )
( ) (25)
Let
A B V E M
B B l l
= − + − +
( )
= + + +
( )
2
0
2 2
2
2 1
( )
( )
C = -
(26)
Now, comparing equation (25) with equation (4),
A, B, and C with k1 2 3
, ,
k k can easily be
determined. k4 5
and k can be obtained as
k C k l l
k l l V E M
4 4
2
5 0
1
1
2
1
4
1 2
= − = + +
= + + + − +
; ( );
( ) ( )
(27)
Thus, the energy eigenvalues can easily be
obtained using either equation 5 or 6; equation 5 is
more preferable. Hence, on substituting
k k k k k
1 2 3 4 5
, , , , into equation (5), the energy
eigenvalue is obtained thus,
2
0
0
2
2 1
1
2
1 1
4
2
2 1 2
=
+ − + + +
+ + + + − +
( )
+ +
V E M B l l
n l l V E M
n l l
( ) ( )
( ) ( )
( ) ( +
+ + − +
− +
1 1
4
2
1
0
2
) ( )
( )
V E M
l l
The corresponding wave functions can be
obtained from equation (7) by making the needful
substitutions.
R s N S
s
ne ne
l l
l l V E M
( )
( )
( )
( ) ( )
=
−
+ +
+ + − + +
2
0
1
1
4
1 2 1
2
1
2 1
2
0
2
2
1
1
2
1
4
1
2
2
F
n n
l l
l l
V E M
l
,
( )
( )
( )
;
(
+
+ + +
+
+ +
− +
+
l
l s
+ +
1 1
) ,
Where Nne is the normalization constant.
DISCUSSION
We have obtained the energy eigenvalues and the
corresponding wave function using the formula
method for the MSCIQY potential. As an analogy,
if we set parameters V V
0 1
0 0
= ≠
and
5. Michael, et al.: Bound state solution of the Klein–Gordon equation for the modified screened coulomb plus inversely
quadratic yukawa potential through formula method
AJMS/Apr-Jun-2018/Vol 2/Issue 2 34
2
2
1
1
2
1 1
4
2 1 2 1 1
4
=
− + + +
+ + + +
( )
+ + + +
B l l
n l l
n l l
( )
( )
( )
− +
≡
− + + + + +
( )+
+ + +
+ +
2
2
1
2 1 1
2
2 1 1 1
4
2 1 2
l l
B l l l l
n l l
n l
( )
( )
( ) ( )
(
( )
( )
l
l l
+ +
− +
1 1
4
1
2
The above gives the energy eigenvalue of the
Klein–Gordon equation for the Yukawa potential
or MSC potential.
Furthermore, if we set V V
1 0
0 0
= ≠
and
2
0
0
2
2 1
1
2
1
1
4
2
2 1
=
+ + +
+ + +
+
+ − +
+
( )+
V E M l l
n
l l
V E M
n
( ) ( )
( )
( )
2
2
1
1
4
2
1
0
2
l l
V E M
l l
( )
( )
( )
+
+ − +
− +
;
gives the energy eigenvalue of the K.GE for the
IQY potential.
CONCLUDING REMARKS
The analytical solutions of the Klein–Gordon
equation for the modified screened Coulomb
plus inversely quadratic Yukawa potential have
been presented through formula method in which
comparison to other methods used; previously,
this method has been seen to be efficient, easy to
manipulate, reliable, and self-explanatory. It is
energy eigenvalue, and wave function has been
obtained and can be employed in the analysis of
spectroscopy.
ACKNOWLEDGMENT
We thank God for the inspirational tips he has
bestowed on us, thereby making this research
article, reach a successful completion. Sequel to
this, OIM acknowledges author OOE and BJF.
REFERENCES
1. Yukawa H. On the interaction of elementary particles. I.
Proc Phys Math Soc Jap 1934;17:48-57.
2. Oyewumi KJ, Falaye BJ, Onate CA, Oluwadara OJ.
K state solutions for the fermionic massive spin-½
particles interacting with double ring-shaped kratzer and
oscillator potentials. Int J Mod Phys 2014;23:1450005.
3. Ita BI, Ikeuba AI, Louis H, Tchoua P. Wkb solution
for inversely quadratic Yukawa plus inversely
quadratic hellmann potential. J Theor Phys Cryptogr
2015;2:109-12.
4. Hamzavi M, Movahedi M, Thylwe KE, Rajabi AA.
Approximate analytical solution of the Yukawa
potential with arbitrary angular momenta. Chem Phys
Lett 2012;29:1-4.
5. Hitler L, Iserom IB, Tchoua P, Ettah AA. Boundstate
solutions of the Klein-Gordon equation for the more
general exponential screened Coulomb potential plus
Yukawa (MGESCY) potential using Nikiforov-uvawu
method. J Math Phys 2018;9:261.
6. Falaye BJ, Ikhdair SM, Hamzavi M. Formula method
for bound state problems. Few Body Syst 2014. DOI:
10.1007/s00601-014-0937-9.
7. Falaye BJ. The Klein-Gordon equation with ring-
shaped potentials: Asymptotic iteration method. J Math
Phys 2012;53:82107.
8. Greene RL, Aldrich C. Variational wave functions for a
screened Coulomb potential. Phys RevA1976;14:2363.