BOUND STATE SOLUTION TO SCHRODINGER EQUATION WITH MODIFIED HYLLERAAS PLUS INVERSELY QUADRATIC POTENTIAL USING SUPERSYMMETRIC QUANTUM MECHANICS APPROACH
In this work, we obtained an approximate bound state solution to Schrodinger equation with modified
Hylleraass plus inversely quadratic potential using Supersymmetric quantum mechanics approach.
Applying perkeris approximation to the centrifugal term, we obtained the eigen-energy and the normalized
wave function using Gauss and confluent hypergeometric functions. We implement Fortran algorithm to
obtained the numerical result of the energy for the screening parameter α = 0.1,0.2,0.3, 0.4 0.5 and .
The result shows that the energy increases with an increase in the quantum state. The energy spectrum
shows increase in angular quantum state spacing as the screening parameter increases.
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 .
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.
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.
These slides are especially made to understand the postulates of quantum mechanics or chemistry better. easily simplified and at one place you will find each of relevant details about the 5 postulates. so go through it & trust me it will help you a lot if you are chemistry or a science student.
well done
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 .
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.
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.
These slides are especially made to understand the postulates of quantum mechanics or chemistry better. easily simplified and at one place you will find each of relevant details about the 5 postulates. so go through it & trust me it will help you a lot if you are chemistry or a science student.
well done
this slide is introduce the postulates of quantum mechanics in which has all important definable objects is defined. so that presentation is helpful for the undergraduate students
International Journal of Computational Engineering Research (IJCER) is dedicated to protecting personal information and will make every reasonable effort to handle collected information appropriately. All information collected, as well as related requests, will be handled as carefully and efficiently as possible in accordance with IJCER standards for integrity and objectivity.
Bound State Solution of the Klein–Gordon Equation for the Modified Screened C...BRNSS Publication Hub
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; 2122233()()''()'()()0(1)(1)kksAsBscsssskssks−++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
Bound State Solution to Schrodinger Equation With Modified Hylleraas Plus Inv...ijrap
In this work, we obtained an approximate bound state solution to Schrodinger equation with modified
Hylleraass plus inversely quadratic potential using Supersymmetric quantum mechanics approach.
Applying perkeris approximation to the centrifugal term, we obtained the eigen-energy and the normalized
wave function using Gauss and confluent hypergeometric functions. We implement Fortran algorithm to
obtained the numerical result of the energy for the screening parameter α = 0.1,0.2,0.3, 0.4 0.5 and .
The result shows that the energy increases with an increase in the quantum state. The energy spectrum
shows increase in angular quantum state spacing as the screening parameter increases.
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 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 .
this slide is introduce the postulates of quantum mechanics in which has all important definable objects is defined. so that presentation is helpful for the undergraduate students
International Journal of Computational Engineering Research (IJCER) is dedicated to protecting personal information and will make every reasonable effort to handle collected information appropriately. All information collected, as well as related requests, will be handled as carefully and efficiently as possible in accordance with IJCER standards for integrity and objectivity.
Bound State Solution of the Klein–Gordon Equation for the Modified Screened C...BRNSS Publication Hub
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; 2122233()()''()'()()0(1)(1)kksAsBscsssskssks−++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
Similar to BOUND STATE SOLUTION TO SCHRODINGER EQUATION WITH MODIFIED HYLLERAAS PLUS INVERSELY QUADRATIC POTENTIAL USING SUPERSYMMETRIC QUANTUM MECHANICS APPROACH
Bound State Solution to Schrodinger Equation With Modified Hylleraas Plus Inv...ijrap
In this work, we obtained an approximate bound state solution to Schrodinger equation with modified
Hylleraass plus inversely quadratic potential using Supersymmetric quantum mechanics approach.
Applying perkeris approximation to the centrifugal term, we obtained the eigen-energy and the normalized
wave function using Gauss and confluent hypergeometric functions. We implement Fortran algorithm to
obtained the numerical result of the energy for the screening parameter α = 0.1,0.2,0.3, 0.4 0.5 and .
The result shows that the energy increases with an increase in the quantum state. The energy spectrum
shows increase in angular quantum state spacing as the screening parameter increases.
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 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.
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
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.
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.
On Approach of Estimation Time Scales of Relaxation of Concentration of Charg...Zac Darcy
In this paper we generalized recently introduced approach for estimation of time scales of mass transport.
The approach have been illustrated by estimation of time scales of relaxation of concentrations of charge
carriers in high-doped semiconductor. Diffusion coefficients and mobility of charge carriers and electric
field strength in semiconductor could be arbitrary functions of coordinate.
Schrodinger equation and its applications: Chapter 2Dr.Pankaj Khirade
Wave function and its physical significance, Schrodinger time dependent equation, Separation in time dependent and time independent parts, Operators in quantum Mechanics, Eigen functions and Eigen values, Particle in one dimensional and three dimensional box (Energy eigen values). Qualitative analysis of potential barrier Tunneling effect). Simple Harmonic Oscillator (Qualitative analysis of Zero point energy)
IOSR Journal of Applied Physics (IOSR-JAP) is an open access international journal that provides rapid publication (within a month) of articles in all areas of physics and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications in applied physics. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
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.
Quantum-Gravity Thermodynamics, Incorporating the Theory of Exactly Soluble Active Stochastic Processes, with Applications
by Daley, K.
Published in IJTP in 2009. http://adsabs.harvard.edu/abs/2009IJTP..tmp...67D
IOSR Journal of Engineering (IOSR-JEN) Volume 5 Issue 7 Version 3
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On the Unification of Physic and the Elimination of Unbound Quantitiesijrap
This paper supports Descartes' idea of a constant quantity of motion, modernized by Leibniz. Unlike Leibniz, the paper emphasizes that the idea is not realized by forms of energy, but by energy itself. It remains constant regardless of the form, type, or speed of motion, even that of light. Through force, energy is only transformed. Here it is proved that force is its derivative. It exists even at rest, representing the object's minimal energy state. With speed, we achieve its multiplication up to the maximum energy state, from which a maximum force is derived from the object. From this point, corresponding to Planck's Length, we find the value of the force wherever we want. Achieving this removes the differences between various natural forces. The new idea eliminates infinite magnitudes. The process allows the laws to transition from simple to complex forms and vice versa, through differentiation-integration. For this paper, this means achieving the Unification Theory.
Gravity Also Redshifts Light – the Missing Phenomenon That Could Resolve Most...ijrap
In this paper I discover that gravity also redshifts light like the velocity of its source does. When light travels towards a supermassive object, its waves (or photons) undergo continuous stretching, thereby shifting towards lower frequencies. Gravity redshifts light irrespective of whether its source is in motion or static with respect to its observer. An equation is derived for gravitational redshift, and a formula for combined redshift is presented by considering both the velocity, and gravity redshifts. Also explained is how frequencies of electromagnetic spectrum continuously downgrade as a light beam of mix frequencies passes towards a black hole. Further, a clear methodology is provided to figure out whether expansion of the universe is accelerating or decelerating, or alternatively, the universe is contracting.
In this paper I present a new theory that explains as to when and how dark energy is created as mass is destroyed. The theory extends Einstein’s mass energy equation to a more generic form in order to make it work even in high gravity conditions. It also explains why dark energy is created. Further, it is proved Einstein’s mass energy equation holds good only when the destroyed mass has no supermassive object in its close vicinity. The relationship between dark energy and dark matter is unveiled. An extended mathematical form of Einstein’s mass energy equation is derived, based on which the conditions leading to dark energy creation are explained. Three new physical parameters called dark energy discriminant, dark energy radius and dark energy boundary are introduced to facilitate easy understanding of the theory. It is explained in detail that an extremely superdense object has two dark energy boundaries, outer and inner. Mass destroyed only between these two boundaries creates dark energy. Dark energy space, the space between the two aforementioned boundaries, shrouds visible matter in obscurity from optical and electromagnetic telescopes. This theory identifies Gargantuan as a superdense black hole currently creating fresh dark energy, which could be the subject of interest for the astronomical research community having access to sophisticated telescopes, and working on dark energy. It also upholds dark energy and denies the existence of dark matter. Dark matter is nothing but the well-known visible matter positioned in dark energy space. An important relationship is derived between a photon’s frequency and its distance from a black hole to demonstrate the effect of gravity on light. Another important fact revealed by this theory is gravity stretches out light, thereby causing redshift, which is unaccounted in the computation of velocities of outer galaxies. Whether the universe is undergoing accelerated or decelerated expansion, or accelerated contraction can precisely be determined only after accounting for the redshift caused by gravity
International Journal on Soft Computing, Artificial Intelligence and Applicat...ijrap
International Journal on Soft Computing, Artificial Intelligence and Applications (IJSCAI)
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The quest for a third theory uniting macro-cosmos (relativity) and micro-cosmos (quantum mechanics) has coexisted with the denial of feminine/subjective polarity to masculine/objective. The dismissal of electromagnetism as the tension of opposites in quest of inner/outer unity is sourced in the denial of the feminine qualia -- the negative force field attributed to dark energy/dark matter. However, a conversion philosophy sourced in the hieros gamos and signified by the Mobius strip has formulated an integral consciousness methodology producing quantum objects by means of embracing the shadow haunting contemporary physics. This Self-reflecting process integrating subject/object comprises an ontology of kairos as the “quantum leap.” An interdisciplinary quest to create a phenomenological narrative is disclosed via a holistic apparatus of hermeneutics manifesting image/text of a contemporary grail journey. Reflected in this Third space is the sacred reality of autonomous number unifying polarities of feminine/subjective (quality) and objective/masculine (quantity) as new measurement apparatus for the quantum wave collapse.
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Understanding the concept of space and time is critical, essential, and fundamental in searching for the all-encompassing theory or the theory of everything (ToE). Some physicists argue that time exists, while others posit that time is only a social or mental construct. The author presents an African thought system on space and time conception, focusing on the African (Bantu) view of space and time. The author argues
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regarding these two concepts. Their conception of time appears to be holistic, highly philosophical, nonlinear, and thought-provoking. The author hopes that exploring these two concepts from an African
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universe, etc. to mention just a few. The author posits that a field carries information, performs various mathematical and computational operations, and behaves as an intelligent entity embedded with consciousness.
Call For Papers - International Journal of Recent advances in Physics (IJRAP)ijrap
International Journal of Recent advances in Physics (IJRAP) is a peer-reviewed, open access journal, addresses the impacts and challenges of Physics. The journal documents practical and theoretical results which make a fundamental contribution for the development of Physics.
Call For Papers - International Journal of Recent advances in Physics (IJRAP)ijrap
International Journal of Recent advances in Physics (IJRAP) is a peer-reviewed, open access journal, addresses the impacts and challenges of Physics. The journal documents practical and theoretical results which make a fundamental contribution for the development of Physics.
Call For Papers - International Journal of Recent advances in Physics (IJRAP)ijrap
International Journal of Recent advances in Physics (IJRAP) is a peer-reviewed, open access journal, addresses the impacts and challenges of Physics. The journal documents practical and theoretical results which make a fundamental contribution for the development of Physics.
Call For Papers - International Journal of Recent advances in Physics (IJRAP)ijrap
International Journal of Recent advances in Physics (IJRAP) is a peer-reviewed, open access journal, addresses the impacts and challenges of Physics. The journal documents practical and theoretical results which make a fundamental contribution for the development of Physics
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BOUND STATE SOLUTION TO SCHRODINGER EQUATION WITH MODIFIED HYLLERAAS PLUS INVERSELY QUADRATIC POTENTIAL USING SUPERSYMMETRIC QUANTUM MECHANICS APPROACH
1. International Journal of Recent advances in Physics (IJRAP) Vol.4, No.4, November 2015
DOI : 10.14810/ijrap.2015.4403 27
BOUND STATE SOLUTION TO SCHRODINGER
EQUATION WITH MODIFIED HYLLERAAS PLUS
INVERSELY QUADRATIC POTENTIAL USING
SUPERSYMMETRIC QUANTUM MECHANICS
APPROACH
Ituen .B.Okon1
, Oyebola Popoola2
and Eno.E. Ituen1
1
Department of Physics, University of Uyo, Uyo, Nigeria.
2
Department of Physics, University of Ibadan, Ibadan, Nigeria.
1
Department of Physics, University of Uyo, Uyo, Nigeria.
ABSTRACT
In this work, we obtained an approximate bound state solution to Schrodinger equation with modified
Hylleraass plus inversely quadratic potential using Supersymmetric quantum mechanics approach.
Applying perkeris approximation to the centrifugal term, we obtained the eigen-energy and the normalized
wave function using Gauss and confluent hypergeometric functions. We implement Fortran algorithm to
obtained the numerical result of the energy for the screening parameter 0.1,0.2,0.3,0.4 0.5andα = .
The result shows that the energy increases with an increase in the quantum state. The energy spectrum
shows increase in angular quantum state spacing as the screening parameter increases.
KEYWORDS
Schrodinger, Supersymmetric Quantum Mechanics Approach, Modified Hylleraass plus Inversely
Quadratic potential.
1. INTRODUCTION
Schrodinger wave equation belongs to non-relativistic wave equation. The total wave function of
any quantum mechanical system basically provides implicitly the relevant information about the
physical behavior of the system. Bound state solutions most time provides negative energies
because oftenly, the energy of the particle is less than the maximum potential energy therefore,
causing the particle to be trapped within the potential well. However, in a well that is infinitely
long, the particles can have positive energies and are still trapped within the potential well, hence
we can conclude that for infinitely long potential well, bound state energy of a particle is either
less than the potential at negative infinity( )E < −∞ or less than the potential at positive infinity
( )E < ∞ which provides the reason for obtaining both negative and positive bound state energies
predominantly in Klein-Gordon equation. A lot of authors developed interest in studying bound
state solutions majorly due to its scientific applications in both physical and chemical sciences in
particle, high energy Physics and molecular dynamics . [1-3].
2. International Journal of Recent adv
Different analytical techniques have been adopted by different authors in providing solutions to
relativistic and non- relativistic wave equations. These are: Nikifor
quantisation, asymptotic iteration method, supersymmetric quantum mechanics approach,
factorization method, tridiagonalisation method etc.[4
consideration are: woods-Saxon plus modified exponential
generalized exponential coulomb potential,
Teller, kratzer fues and Mie-Type potential, Eckart potential and
etc [11-20]. This paper is organized
introduced the concept of supersymmetric quantum mechanics approach. In section 3, we apply
the concept of supersymmetry to provide the solution to Schrodinger equation using the proposed
potential and obtained the energy eigen value and the wave function.. In section 4, we implement
an algorithm to obtained numerical computation for the resulting energy.
The modified Hylleraas plus inversely quadratic potential is given by
2
0 31
2 2
2
( )
1
r
r
v e
V r
e r
α
α
χχ
χ
−
−
−
= +
−
Where 3χ is a constant , 1χ and
against the inter-nuclear distance with various values of the screening parameter
1.0,2.0,3.0 4.0andα = is shown below. The chosen
graph which is quite different from the one for numerical computation.
2.THE CONCEPT OF SUPERYSYMMETRIC QUANTUM
MECHANICS(SUSYQM)
The supersymmetric approach deals with the partner Hamiltonian of the form
2
( )
2
p
H V x
m
± = +
Where P is the momentum and V(x) is the effective potential which can be expressed in terms of
super-potential as
2
( ) ( ) ( )effV x x xφ φ±
′= ±
The ground state energy is obtained as
International Journal of Recent advances in Physics (IJRAP) Vol.4, No.4, November 2015
Different analytical techniques have been adopted by different authors in providing solutions to
relativistic wave equations. These are: Nikiforov-Uvarov method, exact
quantisation, asymptotic iteration method, supersymmetric quantum mechanics approach,
factorization method, tridiagonalisation method etc.[4-10] . Some of the potentials under
Saxon plus modified exponential coulomb potential, Hulthen plus
generalized exponential coulomb potential, Rosen-Morse, Hulthen, pseudo harmonic, Poschl
pe potential, Eckart potential and P-T symmetric Hulthen potential
20]. This paper is organized as follows: section 1 is the introduction. In section 2, we
introduced the concept of supersymmetric quantum mechanics approach. In section 3, we apply
the concept of supersymmetry to provide the solution to Schrodinger equation using the proposed
al and obtained the energy eigen value and the wave function.. In section 4, we implement
an algorithm to obtained numerical computation for the resulting energy.
The modified Hylleraas plus inversely quadratic potential is given by
and 2χ are Hylleraas parameter. The graph of this potential
nuclear distance with various values of the screening parameter
is shown below. The chosen α is to enable one sees the nature of the
graph which is quite different from the one for numerical computation.
Figure a.
CEPT OF SUPERYSYMMETRIC QUANTUM
MECHANICS(SUSYQM)
The supersymmetric approach deals with the partner Hamiltonian of the form
Where P is the momentum and V(x) is the effective potential which can be expressed in terms of
The ground state energy is obtained as
ances in Physics (IJRAP) Vol.4, No.4, November 2015
28
Different analytical techniques have been adopted by different authors in providing solutions to
Uvarov method, exact
quantisation, asymptotic iteration method, supersymmetric quantum mechanics approach,
10] . Some of the potentials under
coulomb potential, Hulthen plus
Morse, Hulthen, pseudo harmonic, Poschl-
T symmetric Hulthen potential
as follows: section 1 is the introduction. In section 2, we
introduced the concept of supersymmetric quantum mechanics approach. In section 3, we apply
the concept of supersymmetry to provide the solution to Schrodinger equation using the proposed
al and obtained the energy eigen value and the wave function.. In section 4, we implement
(1)
are Hylleraas parameter. The graph of this potential
nuclear distance with various values of the screening parameter
one sees the nature of the
CEPT OF SUPERYSYMMETRIC QUANTUM
(2)
Where P is the momentum and V(x) is the effective potential which can be expressed in terms of
(3)
3. International Journal of Recent advances in Physics (IJRAP) Vol.4, No.4, November 2015
29
1
0 ( ) N
x Ceφ− −
= (4)
where N is the normalization constant which for very simple cases can be determined using the
expression
0
( ) ( )
x
x
N x r drφ= ∫
(5)
However, the superpotential satisfies shape invariant condition of the form
0 1 1( , ) ( , ) ( )V a x V a x R a+ −= + (6)
Where 1a is a new set of parameter uniquely determined from the old set 0a through the
mapping 0 1 0: ( )f a a f a→ = (7)
The supersymmetric energy is determined as
1
( )
n
n s
s
E R a
=
= ∑ (8)
While higher order state solutions are obtained through the expression
( )
†1
0 01
0 2
( )
( , ) ( , )
n
s
n n
s
n s
A a
a x a x
E E
φ φ
−
− −
=
=
−
∏ (9)
Where †
( )sA a is a raising ladder operator expressed as
†
( , )s sA a x
x
φ
∂
= − +
∂
(10)
3. RADIAL SOLUTION OF SCHRODINGER EQUATION
Schrodinger equation is given by
( )
2
2 2 2
2 ( 1)
( ) ( ) 0nl
d R l l
E V r R r
dr r
µ +
+ − − = h
(11)
Substituting equation (1) into (11) gives
22
0 31
2 2 2 2 2
2
2 ( 1)
( ) 0
1
r
nl r
v ed R l l
E R r
dr e r r
α
α
χχµ
χ
−
−
+
+ + − − =
− h
(12)
Let’s define suitable approximation to the centrifugal term as
( )
2 1
02 2
1
1 r
D
D
r e α
α −
= +
−
(13)
Substituting equation (13) into (12) and re-arranging gives
4. International Journal of Recent advances in Physics (IJRAP) Vol.4, No.4, November 2015
30
( )
2 22
20 1 0 3 1
12 2 22
2
2
2 0 3 0
02 2
2 21
( 1) ( )
1
22
( 1) ( )
r
r
v e v Dd R
l l D R r
dr e
v DE
l l D R r
α
α
µ χ µ χ α
α
χ
µ χ αµ
α
−
−
+ − − +
−
= − − + −
h h
h h
(14)
This can also be represented as
( )
2 22
20 3 1 0 1
12 2 22
2
2
2 0 3 0
02 2
2 21
( 1) ( )
1
22
( 1) ( )
r
r
v D v ed R
l l D R r
dr e
v DE
l l D R r
α
α
µ χ α µ χ
α
χ
µ χ αµ
α
−
−
− + + −
−
= − − + −
h h
h h
(15)
Let’s define second order differential equation containing effective potential as
2
2
( )
( ) ( ) ( )nl
eff nl nl nl
d R r
V r R r E R r
dr
− + = % (16)
In order to represent equation (15) in the form of equation (16) , then equation (15) is multiply by
-1.
( )
2 22
20 3 1 0 1
12 2 22
2
2
2 0 3 0
02 2
2 21
( 1) ( )
1
22
( 1) ( )
r
r
v D v ed R
l l D R r
dr e
v DE
l l D R r
α
α
µ χ α µ χ
α
χ
µ χ αµ
α
−
−
− + + + −
−
= − + −
h h
h h
(17)
Comparing equation (17) to (16), we obtained the following:
( )
2 2
20 3 1 0 1
12 22
2
2
2 0 3 0
02 2
2 21
( ) ( 1)
1
22
( 1) (18 )
r
eff r
nl
v D v e
V r l l D
e
v DE
E l l D b
α
α
µ χ α µ χ
α
χ
µ χ αµ
α
−
−
= + + −
−
= − + −
h h
%
h h
(18).
Let
2
20 3 1
12
2
( 1)
v D
A l l D
µ χ α
α= + +
h
and 0 1
2
2
2 v
B
µ χ
χ
= −
h
(19)
Then, the effective potential reduced to
( )
2
2
1
( )
1
r
eff r
V r A Be
e
α
α
−
−
= + −
(20)
The super-potential suitable for the effective potential is given as
( )
1
22
( )
1 r
q
r q
e α
φ −
−
= +
−
(21)
In order to construct partner potential, we apply equation (6) by first taking the square and first
derivative of equation (21)
5. International Journal of Recent advances in Physics (IJRAP) Vol.4, No.4, November 2015
31
( )
2
1
22
2
( )
1
r
r
q e
r
e
α
α
α
φ
−
−
′ =
−
(22)
( ) ( )
2
2 2 1 2 1
2 22 2
2
( )
1 1
r r
q q q
r q
e e
α α
φ − −
= − +
− −
(23)
The partner potentials are
( ) ( ) ( )
2 2
2 2 1 2 1 2
2 2 22 2 2
2 2
( ) ( ) ( )
1 1 1
r
eff r r r
q q q q e
V r r r q
e e e
α
α α α
α
φ φ
−
+
− − −
′= + = − + +
− − −
(24)
( ) ( ) ( )
2 2
2 2 1 2 1 2
2 2 22 2 2
2 2
( ) ( ) ( )
1 1 1
r
eff r r r
q q q q e
V r r r q
e e e
α
α α α
α
φ φ
−
−
− − −
′= − = − + −
− − −
(25)
Equation (24) and (25) satisfies shape invariant condition.
3.1 CALCULATION OF GROUND STATE ENERGY
The ground state energy can be calculated by solving associated Riccati equation. This equation
is given as
2
0( ) ( ) ( )eff lr r V r Eφ φ′− = − % (26)
Where 0lE% is the ground state energy.
Substituting equation (20) and (25) into (26) and simplifying in decreasing order of exponent
gives rise to three pairs of simultaneous equations
1
2 2
0 1 2 22lA E q q q q+ = + −% (27)
1 1
2
0 2 22 2 2 2lB A E q q q qα− − = − + −% (28)
2
0 2lE B q− =% (29)
Solving the following simultaneous equations then
2
1 2 2 ( )q q q A B= ± + + (30)
2
21 1
2 2
1 1
2 2
2 2
A B q A B q
q q
q q
α α + + + +
= − ⇒ =
(31)
The ground state energy is calculated using equation (29)
2 2
1 1
0 0
1 1
2 2
2 2
l l
A B q A B q
E B E B
q q
α α + + + +
= + ⇒ − = − −
% % (32)
6. International Journal of Recent advances in Physics (IJRAP) Vol.4, No.4, November 2015
32
3.2 CALULATION OF HIGHER ORDER SUPERSYMMETRIC ENERGY
Using the condition of shape invariant, higher order supersymmetric energy can be calculated as
using
1
( )nl k
k
E R a
∞
−
=
= ∑ (33)
Equation (33) satisfying shape invariant condition can be evaluated as follows
1 0 1
2 0 2
3 0 3
1
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )n n n
R a R a R a
R a R a R a
R a R a R a
R a R a R a−
= −
= −
= −
= −
(34)
Thus, there is one to one corresponding mapping as shown below.
1 1
0 0
1 0
1
:
:
:
:
:
n n
n
q q
a a
a a n
q a
q a
α
α
α
+
+
+
⇒
⇒
a
a
a
a
a
(35)
Therefore, using (35), equation (31) can be written as
22
2 2 01
2 2
1 0
22
2 2
A B qA B q
q q
q q
αα + ++ +
= ⇒ =
(36)
Using equation (34), the following evaluations is carried out
2 2
0 1
1
0 1
2 2
1 2
2
1 2
22
32
3
2 3
2 2
0
1 0
2 2
( )
2 2
2 2
( )
2 2
22
( )
2 2
2 2
( )
2 2
n
nl k
k n
A B q A B q
R a
q q
A B q A B q
R a
q q
A B qA B q
R a
q q
A B a A B a
E R a
a a
α α
α α
αα
α α∞
−
=
+ + + +
= −
+ + + +
= −
+ ++ +
= −
+ + + +
= = −
∑
(37)
7. International Journal of Recent advances in Physics (IJRAP) Vol.4, No.4, November 2015
33
3.3 CALCULATION OF TOTAL ENERGY
The total energy is the sum of the ground state energy and higher order supersymmetric energy.
This is given as
0nl nl lE E E= +% % (38)
Sometimes, the ground state energy can be negative (bound State condition) like in hydrogen
atom which is about -13.6eV.
Hence, equation (38) can in this manner be expressed as
0( )nl nl lE E E= + −% % (39)
Substituting equation (32) and (37) into (39) gives
2
2
n
nl
n
A B a
E B
a
α + +
= − +
% (40)
Let’s recall equation (30)
2 2
1 2 2 1 2( ) ( ) , :q q q A B q A B qα α α= ± + + ⇒ = ± + + a
If 1: , :n n na a n for q aα+a a
Then ( )2 2
( ) 1 ( )n na n A B a n A Bα α α α α = + ± + + ⇒ = + ± + +
(41)
Substituting equation (18b) and equation (41) into (40) and simplifying gives the total energy as
2 2
2 2 23 1 0 1 3 1 0 1
1 12 2 2
2
2 2 0 1
22
223 1 0 1
12 2
2
2 2
2
3 0
2 2 2 2
( 1) 2 1 ( 1)
2
2 2 2
2 1 ( 1)
( 1)
nl
D v D v
l l D n l l D
v
E
D v
n l l D
l l
D
µχα µ χ µχα µ χ
α α α
χ χ µ χ
µ χµχα µ χ
α α
χ
α
χα
+ + − + + ± + + − =− −
+ ± + + −
+
+ +
h h hh
h
h h
h 0
(42)
2
D
µ
Equation (42) is the energy equation for Hylleraas plus inversely quadratic potential. However,
because of the plus and minus accaompanied by the square root sign, equation (42) can be
expressed as follows:
2 2
2 2 23 1 0 1 3 1 0 1
1 12 2 2
2
2 2 0 1
22
223 1 0 1
12 2
2
2 2
2
3 0
2 2 2 2
( 1) 2 1 ( 1)
2
2 2 2
2 1 ( 1)
( 1)
nl
D v D v
l l D n l l D
v
E
D v
n l l D
l l
D
µχα µ χ µχα µ χ
α α α
χ χ µ χ
µ χµχα µ χ
α α
χ
α
χα
+ + − + + + + + − =− −
+ + + + −
+
+ +
h h hh
h
h h
h 0
(43)
2
D
µ
8. International Journal of Recent advances in Physics (IJRAP) Vol.4, No.4, November 2015
34
2 2
2 2 23 1 0 1 3 1 0 1
1 12 2 2
2
2 2 0 1
22
223 1 0 1
12 2
2
2 2
2
3 0
2 2 2 2
( 1) 2 1 ( 1)
2
2 2 2
2 1 ( 1)
( 1)
nl
D v D v
l l D n l l D
v
E
D v
n l l D
l l
D
µχα µ χ µχα µ χ
α α α
χ χ µ χ
µ χµχα µ χ
α α
χ
α
χα
+ + − + + − + + − =− −
+ − + + −
+
+ +
h h hh
h
h h
h 0
(44)
2
D
µ
3.4 CALCULATION OF THE WAVE FUNCTION
Furthermore, in order to calculate the radial wave function, we used the coordinate transformation
2 r
s e α−
= into equation (16) and obtained the following
( )
20 1 0 3 0
2 2 2
2
2
0 1 3 0 3 1 0 1
2 2 2 2 2 2
2
3 0 3 1 0 1
2 2
( 1) 2
4 4 4 4
1 2 2 ( 1) ( 1)1
(45)
(1 ) (1 ) 4 4 4 4 4
( 1) ( 1) 2
4 4 4 4 4
v l l D D E
s
s v D D l l D l l Dd R dR E
s
dr s s ds s s
D D l l D l l D E
χ χ µ
α χ α
χ χ χ µ
α χ α
χ χ µ
α
+
− + + −
− + +
+ + + + + + + −
− −
+ +
− + + + −
h
h
h
The corresponding radial wave function is then given by
( )
3 0 3 1 0 13 0 3 1 0 1
2 22 2
3 0 3 1 0 3 0 3 1 01 1
2 2
( 1) ( 1) 2( 1) ( 1) 2 1
2 2 4 4 4 4 44 4 4 4 4
( 1) ( 1)( 1) ( 1)2 2
1 2 , 3 2
4 4 4 4 4 4 4 44 4
( ) (1 )
n
D D l l D l l D ED D l l D l l D E
r r
nl n
D D l l D D D l l Dl l D l l DE E
R r N e e
P
χ χ µχ χ µ
α α αα
χ χ χ χµ µ
α
+ ++ + − − + + + −+ + + −− −
+ ++ +
+ + + + − + + + + −
= −
×
hh
h
( )
2 2
2
1 2 r
e
α α
−
−
h
(46)
4. NUMERICAL COMPUTATION OF THE ENERGY EQUATION
Using equation (43), we implement Fortran algorithm to compute for the energy of the equation
in electron volt with various values of the screening parameter α . We use the following values
for our computation. 0 0 1 1 2 2
1
1.0, , 1.0, 1, 2, 1
12
V D Dµ χ χ χ= = = = = = − = =h .
9. International Journal of Recent advances in Physics (IJRAP) Vol.4, No.4, November 2015
35
Bound State energy with 0.1α = for different quantum state
n l ( 0.1)nE eVα = n l ( 0.1)nE eVα = n l ( 0.1)nE eVα = n l ( 0.1)nE eVα =
0 0 -1.81557810 0 1 -1.83414400 0 2 -1.87091950 0 3 -1.92523000
1 0 -1.39580210 1 1 -1.40959080 1 2 -1.43698190 1 3 -1.47761940
2 0 -1.18513580 2 1 -1.19601170 2 2 -1.21765080 2 3 -1.24983630
3 0 -1.05848570 3 1 -1.06744430 3 2 -1.08528610 3 3 -1.11186640
4 0 -0.97394794 4 1 -0.98155713 4 2 -0.99672174 4 3 -1.01933810
5 0 -0.91351247 5 1 -0.92012280 5 2 -0.93330324 5 3 -0.95297605
6 0 -0.86815786 6 1 -0.87399983 6 2 -0.88565266 6 3 -0.90305570
7 0 -0.83286536 7 1 -0.83809840 7 2 -0.84853950 7 3 -0.86414030
8 0 -0.80462086 8 1 -0.80935950 8 2 -0.81881640 8 3 -0.83295200
Table 1
Bound State energy with 0.2α = for different quantum state
( 0.2)nE eVα = ( 0.2)nE eVα = ( 0.2)nE eVα = ( 0.2)nE eVα =
0 0 -1.25588080 0 1 -1.29197690 0 2 -1.36170600 0 3 -1.46096090
1 0 -1.04142170 1 1 -1.06843020 1 2 -1.12113100 1 3 -1.19725980
2 0 -0.93265150 2 1 -0.95405185 2 2 -0.99604530 2 3 -1.05721940
3 0 -0.86688805 3 1 -0.88456786 3 2 -0.91938436 3 3 -0.97037660
4 0 -0.82283480 4 1 -0.83788250 4 2 -0.86758830 4 3 -0.91125740
5 0 -0.79126420 5 1 -0.80435646 5 2 -0.83024790 5 3 -0.86841345
6 0 -0.76752913 6 1 -0.77911305 6 2 -0.80205260 6 3 -0.83593720
7 0 -0.74903460 7 1 -0.75942045 7 2 -0.78000940 7 3 -0.81047153
8 0 -0.73421750 8 1 -0.74362910 8 2 -0.76230270 8 3 -0.78996754
Table 2
Bound State energy with 0.3α = for different quantum state
( 0.3)nE eVα = ( 0.3)nE eVα = ( 0.3)nE eVα = ( 0.3)nE eVα =
0 0 -1.11232240 0 1 -1.16417930 0 2 -1.26113510 0 3 -1.39352740
1 0 -0.96447120 1 1 -1.00370690 1 2 -1.07845380 1 3 -1.18307350
2 0 -0.88827163 2 1 -0.91957426 2 2 -0.97985350 2 3 -1.06547440
3 0 -0.84179723 3 1 -0.86777450 3 2 -0.91814520 3 3 -0.99038374
4 0 -0.81049335 4 1 -0.83267320 4 2 -0.87588680 4 3 -0.93828130
5 0 -0.78797410 5 1 -0.80731654 5 2 -0.84513450 5 3 -0.90001200
6 0 -0.77099670 6 1 -0.78814130 6 2 -0.82175210 6 3 -0.87071250
7 0 -0.75773966 7 1 -0.77313274 7 2 -0.80337375 7 3 -0.84755990
8 0 -0.74710070 8 1 -0.76106584 8 2 -0.78854835 8 3 -0.8288036
Table 3
n l n l n l n l
n l n l n l n l
10. International Journal of Recent adv
Bound State energy with
( 0.4)nE eVα =
0 0 -1.07219960 0 1
1 0 -0.95648575 1 1
2 0 -0.89565750 2 1
3 0 -0.85815040 3 1
4 0 -0.83271027 4 1
5 0 -0.81432050 5 1
6 0 -0.80040705 6 1
7 0 -0.78951290 7 1
8 0 -0.78075130 8 1
Bound State energy with
( 0.5)nE eVα =
0 0 -1.07287570 0 1
1 0 -0.97570810 1 1
2 0 -0.92349565 2 1
3 0 -0.89090130 3 1
4 0 -0.86861694 4 1
5 0 -0.85241880 5 1
6 0 -0.84011304 6 1
7 0 -0.83044730 7 1
8 0 -0.82265410 8 1
Figure 1: The graph of energy against the distance for different quantum state with
n l n l
n l n l
International Journal of Recent advances in Physics (IJRAP) Vol.4, No.4, November 2015
Bound State energy with 0.4α = for different quantum state
( 0.4)nE eVα = ( 0.4)nE eVα = ( 0.4)nE eVα
-1.13772730 0 2 -1.25619530 0 3 -1.41204810
-1.00673560 1 2 -1.10005060 1 3 -1.22680750
-0.93608590 2 2 -1.01235960 2 3 -1.11805080
-0.89188860 3 2 -0.95620050 3 3 -1.04651460
-0.86163010 4 2 -0.91715777 4 3 -0.99588360
-0.83961457 5 2 -0.88844097 5 3 -0.95816210
-0.82287750 6 2 -0.86643210 6 3 -0.92897120
-0.80972373 7 2 -0.84902660 7 3 -0.90571094
-0.79911387 8 2 -0.83491695 8 3 -0.88674080
Table 4
Bound State energy with 0.5α = for different quantum state
( 0.5)nE eVα = ( 0.5)nE eVα = ( 0.5)nE eVα
-1.15000000 0 2 -1.28518370 0 3 -1.45766960
-1.03571430 1 2 -1.14444880 1 3 -1.28834840
-0.97222220 2 2 -1.06231940 2 3 -1.18440910
-0.93181820 3 2 -1.00849700 3 3 -1.11411000
-0.90384614 4 2 -0.97049820 4 3 -1.06339450
-0.88333330 5 2 -0.94223930 5 3 -1.02507970
-0.86764705 6 2 -0.92040080 6 3 -0.99511280
-0.85526310 7 2 -0.90301790 7 3 -0.97103360
-0.84523810 8 2 -0.88885320 8 3 -0.95126210
Table 5
Figure 1: The graph of energy against the distance for different quantum state with α =
n l n l
n l n l
ances in Physics (IJRAP) Vol.4, No.4, November 2015
36
( 0.4)E eVα =
1.41204810
1.22680750
1.11805080
1.04651460
0.99588360
0.95816210
0.92897120
0.90571094
0.88674080
( 0.5)E eVα =
1.45766960
1.28834840
1.18440910
1.11411000
1.06339450
1.02507970
0.99511280
0.97103360
0.95126210
0.1= .
11. International Journal of Recent adv
Figure 2:The graph of energy against the distance for different quantum state with
Figure 3: The graph of energy
Figure 4: The graph of energy against the distance for different quantum state with
International Journal of Recent advances in Physics (IJRAP) Vol.4, No.4, November 2015
Figure 2:The graph of energy against the distance for different quantum state withα =
Figure 3: The graph of energy against the distance for different quantum state withα =
Figure 4: The graph of energy against the distance for different quantum state withα =
ances in Physics (IJRAP) Vol.4, No.4, November 2015
37
0.2= .
0.3= .
0.4= .
12. International Journal of Recent adv
Figure 5. : The graph of energy against the distance for
CONCLUSION
We used supersymmetric quantum mechanics approach to obtained bound state solution to
Hylleraas plus inversely quadratic potential using Schrodinger equation. We also obtained eigen
energy and the wave function. The result shows that the energy is negativ
state condition and increases with an increase in quantum as shown in tables (1
graphical spectrum shows that the interspacing within the quantum state is more pronounce the
value of α increases. This can be shown from figures (1
ACKNOWLEDGEMENT
We are deeply grateful to the referees for their useful comments which we have significantly use
to improve the article.
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International Journal of Recent advances in Physics (IJRAP) Vol.4, No.4, November 2015
Figure 5. : The graph of energy against the distance for different quantum state withα
We used supersymmetric quantum mechanics approach to obtained bound state solution to
Hylleraas plus inversely quadratic potential using Schrodinger equation. We also obtained eigen
energy and the wave function. The result shows that the energy is negative to ascertain bound
state condition and increases with an increase in quantum as shown in tables (1-5). The energy
graphical spectrum shows that the interspacing within the quantum state is more pronounce the
This can be shown from figures (1-5).
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We are deeply grateful to the referees for their useful comments which we have significantly use
OkonI.B , Ituen E.E, .Popoola O.O and Antia A.D.(2013), “Analytical Solutions of Schrodinger
Type Potential Using Factorisation Method”, International Journal of Recent
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0.5α = .
We used supersymmetric quantum mechanics approach to obtained bound state solution to
Hylleraas plus inversely quadratic potential using Schrodinger equation. We also obtained eigen
e to ascertain bound
5). The energy
graphical spectrum shows that the interspacing within the quantum state is more pronounce the
We are deeply grateful to the referees for their useful comments which we have significantly use
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Type Potential Using Factorisation Method”, International Journal of Recent
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Rosen Potential Using Supersymmetric Quantum Mechanics
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Uvarov Method”,
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