- The document discusses using the Nikiforov-Uvarov method to find bound state solutions to the Schrodinger equation with Hulthen plus generalized exponential Coulomb potential.
- The method is applied to obtain the energy eigenvalues and total wave function for the potential.
- A C++ algorithm is used to numerically calculate the energy values for different quantum states and screening parameter values.
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.
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
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.
Some new exact Solutions for the nonlinear schrödinger equationinventy
Research Inventy : International Journal of Engineering and Science is published by the group of young academic and industrial researchers with 12 Issues per year. It is an online as well as print version open access journal that provides rapid publication (monthly) of articles in all areas of the subject such as: civil, mechanical, chemical, electronic and computer engineering as well as production and information technology. The Journal welcomes the submission of manuscripts that meet the general criteria of significance and scientific excellence. Papers will be published by rapid process within 20 days after acceptance and peer review process takes only 7 days. All articles published in Research Inventy will be peer-reviewed.
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 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.
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
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.
Some new exact Solutions for the nonlinear schrödinger equationinventy
Research Inventy : International Journal of Engineering and Science is published by the group of young academic and industrial researchers with 12 Issues per year. It is an online as well as print version open access journal that provides rapid publication (monthly) of articles in all areas of the subject such as: civil, mechanical, chemical, electronic and computer engineering as well as production and information technology. The Journal welcomes the submission of manuscripts that meet the general criteria of significance and scientific excellence. Papers will be published by rapid process within 20 days after acceptance and peer review process takes only 7 days. All articles published in Research Inventy will be peer-reviewed.
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 .
The module-algebra structures of quantum enveloping algebra Uq(sl(m+1)) on the coordinate algebra of quantum vector spaces (in other words, quantum actions/symmetries) are
investigated. We denote the coordinate algebra of quantum n-dimensional vector space by Aq(n). As our main result, first, we give a complete classification of module-algebra structures of Uq(sl(m+1)) on Aq(3), and with the same method, on Aq(2), all module-algebra structures of Uq(sl(m+1)) are characterized. The classical limit of the Uq(sl(3))-module algebra
structures on Aq(2) are presented, and, as opposite to Uq(sl(2)) case, there no cubic and fourth-power terms. Lastly, the module-algebra structures of Uq(sl(m+1)) on Aq(n) are obtained for any n ≥ 4, and are classified using the language of Dynkin diagrams.
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.
SCHRODINGER'S CAT PARADOX RESOLUTION USING GRW COLLAPSE MODELijrap
Possible solution of the Schrödinger's cat paradox is considered.We pointed out that: the collapsed
state of the cat always shows definite and predictable measurement outcomes even if Schrödinger's
cat consists of a superposition: cat=livecat+
deathcat
Anomalous Diffusion Through Homopolar Membrane: One-Dimensional Model by Guilherme Garcia Gimenez and Adélcio C Oliveira* in Evolutions in Mechanical Engineering
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.
optimal solution method of integro-differential equaitions under laplace tran...INFOGAIN PUBLICATION
In this paper, Laplace Transform method is developed to solve partial Integro-differential equations. Partial Integro-differential equations (PIDE) occur naturally in various fields of science. Engineering and Social Science. We propose a max general form of linear PIDE with a convolution Kernal. We convert the proposed PIDE to an ordinary differential equation (ODE) using the LT method. We applying inverse LT as exact solution of the problems obtained. It is observed that the LT is a simple and reliable technique for solving such equations. The proposed model illustrated by numerical examples.
A generalized bernoulli sub-ODE Method and Its applications for nonlinear evo...inventy
Research Inventy : International Journal of Engineering and Science is published by the group of young academic and industrial researchers with 12 Issues per year. It is an online as well as print version open access journal that provides rapid publication (monthly) of articles in all areas of the subject such as: civil, mechanical, chemical, electronic and computer engineering as well as production and information technology. The Journal welcomes the submission of manuscripts that meet the general criteria of significance and scientific excellence. Papers will be published by rapid process within 20 days after acceptance and peer review process takes only 7 days. All articles published in Research Inventy will be peer-reviewed.
Significance of Mathematical Analysis in Operational Methods [2014]SanjayKumar Patel
Dr Ajay Shukla from SVNIT came to Ahmedabad on 2nd August 2014,to deliver the lecture on Significance of Mathematical Analysis in Operational Methods ....
Lecture was held in St. Xavier's College,Ahmedabad under the Father Valles Lecture Series...
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.
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.
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 module-algebra structures of quantum enveloping algebra Uq(sl(m+1)) on the coordinate algebra of quantum vector spaces (in other words, quantum actions/symmetries) are
investigated. We denote the coordinate algebra of quantum n-dimensional vector space by Aq(n). As our main result, first, we give a complete classification of module-algebra structures of Uq(sl(m+1)) on Aq(3), and with the same method, on Aq(2), all module-algebra structures of Uq(sl(m+1)) are characterized. The classical limit of the Uq(sl(3))-module algebra
structures on Aq(2) are presented, and, as opposite to Uq(sl(2)) case, there no cubic and fourth-power terms. Lastly, the module-algebra structures of Uq(sl(m+1)) on Aq(n) are obtained for any n ≥ 4, and are classified using the language of Dynkin diagrams.
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.
SCHRODINGER'S CAT PARADOX RESOLUTION USING GRW COLLAPSE MODELijrap
Possible solution of the Schrödinger's cat paradox is considered.We pointed out that: the collapsed
state of the cat always shows definite and predictable measurement outcomes even if Schrödinger's
cat consists of a superposition: cat=livecat+
deathcat
Anomalous Diffusion Through Homopolar Membrane: One-Dimensional Model by Guilherme Garcia Gimenez and Adélcio C Oliveira* in Evolutions in Mechanical Engineering
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.
optimal solution method of integro-differential equaitions under laplace tran...INFOGAIN PUBLICATION
In this paper, Laplace Transform method is developed to solve partial Integro-differential equations. Partial Integro-differential equations (PIDE) occur naturally in various fields of science. Engineering and Social Science. We propose a max general form of linear PIDE with a convolution Kernal. We convert the proposed PIDE to an ordinary differential equation (ODE) using the LT method. We applying inverse LT as exact solution of the problems obtained. It is observed that the LT is a simple and reliable technique for solving such equations. The proposed model illustrated by numerical examples.
A generalized bernoulli sub-ODE Method and Its applications for nonlinear evo...inventy
Research Inventy : International Journal of Engineering and Science is published by the group of young academic and industrial researchers with 12 Issues per year. It is an online as well as print version open access journal that provides rapid publication (monthly) of articles in all areas of the subject such as: civil, mechanical, chemical, electronic and computer engineering as well as production and information technology. The Journal welcomes the submission of manuscripts that meet the general criteria of significance and scientific excellence. Papers will be published by rapid process within 20 days after acceptance and peer review process takes only 7 days. All articles published in Research Inventy will be peer-reviewed.
Significance of Mathematical Analysis in Operational Methods [2014]SanjayKumar Patel
Dr Ajay Shukla from SVNIT came to Ahmedabad on 2nd August 2014,to deliver the lecture on Significance of Mathematical Analysis in Operational Methods ....
Lecture was held in St. Xavier's College,Ahmedabad under the Father Valles Lecture Series...
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.
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.
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
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.
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.
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.
Conformable Chebyshev differential equation of first kindIJECEIAES
In this paper, the Chebyshev-I conformable differential equation is considered. A proper power series is examined; there are two solutions, the even solution and the odd solution. The Rodrigues’ type formula is also allocated for the conformable Chebyshev-I polynomials.
Strum Liouville Problems in Eigenvalues and Eigenfunctionsijtsrd
This paper we discusses with Strum Liouville problem of eigenvalues and eigenfunctions, within the standard equation where p,q and r are given functions of the independent variable x is an interval The is a parameter and is the dependent variable. The method of separation of variable applied to second order liner partial differential equations, the equation is known because the Strum Liouville differential equation. Which appear in the overall theory of eigenvalues and eigenfunctions and eigenfunctions expansions is one of the deepest and richest parts of recent mathematics. These problems are associate with work of J.C.F strum and J.Liouville. B. Kavitha | Dr. C. Vimala "Strum - Liouville Problems in Eigenvalues and Eigenfunctions" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-4 | Issue-5 , August 2020, URL: https://www.ijtsrd.com/papers/ijtsrd31721.pdf Paper Url :https://www.ijtsrd.com/mathemetics/other/31721/strum--liouville-problems-in-eigenvalues-and-eigenfunctions/b-kavitha
Crack problems concerning boundaries of convex lens like formsijtsrd
The singular stress problem of aperipheral edge crack around a cavity of spherical portion in an infinite elastic medium whenthe crack is subjected to a known pressure is investigated. The problem is solved byusing integral transforms and is reduced to the solution of a singularintegral equation of the first kind. The solution of this equation is obtainednumerically by the method due to Erdogan, Gupta , and Cook, and thestress intensity factors are displayed graphically.Also investigated in this paper is the penny-shaped crack situated symmetrically on the central plane of a convex lens shaped elastic material. Doo-Sung Lee"Crack problems concerning boundaries of convex lens like forms" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-2 | Issue-3 , April 2018, URL: http://www.ijtsrd.com/papers/ijtsrd11106.pdf http://www.ijtsrd.com/mathemetics/applied-mathamatics/11106/crack-problems-concerning-boundaries-of-convex-lens-like-forms/doo-sung-lee
FITTED OPERATOR FINITE DIFFERENCE METHOD FOR SINGULARLY PERTURBED PARABOLIC C...ieijjournal
In this paper, we study the numerical solution of singularly perturbed parabolic convection-diffusion type
with boundary layers at the right side. To solve this problem, the backward-Euler with Richardson
extrapolation method is applied on the time direction and the fitted operator finite difference method on the
spatial direction is used, on the uniform grids. The stability and consistency of the method were established
very well to guarantee the convergence of the method. Numerical experimentation is carried out on model
examples, and the results are presented both in tables and graphs. Further, the present method gives a more
accurate solution than some existing methods reported in the literature.
FITTED OPERATOR FINITE DIFFERENCE METHOD FOR SINGULARLY PERTURBED PARABOLIC C...ieijjournal
In this paper, we study the numerical solution of singularly perturbed parabolic convection-diffusion type
with boundary layers at the right side. To solve this problem, the backward-Euler with Richardson
extrapolation method is applied on the time direction and the fitted operator finite difference method on the
spatial direction is used, on the uniform grids. The stability and consistency of the method were established
very well to guarantee the convergence of the method. Numerical experimentation is carried out on model
examples, and the results are presented both in tables and graphs. Further, the present method gives a more
accurate solution than some existing methods reported in the literature.
Similar to Bound- State Solution of Schrodinger Equation with Hulthen Plus Generalized Exponential Coulomb Potential Using NIKIFOROV-UVAROV Method (20)
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)
is an open access peer-reviewed journal that provides an excellent international forum for sharing
knowledge and results in theory, methodology and applications of Artificial Intelligence, Soft
Computing. The Journal looks for significant contributions to all major fields of the Artificial
Intelligence, Soft Computing in theoretical and practical aspects. The aim of the Journal is to
provide a platform to the researchers and practitioners from both academia as well as industry to
meet and share cutting-edge development in the field.
Authors are solicited to contribute to the journal by submitting articles that illustrate research
results, projects, surveying works and industrial experiences that describe significant advances in
the areas of Database management systems.
SOME THEORETICAL ASPECTS OF HYDROGEN DIFFUSION IN BCC METALS AT LOW TEMPERATURESijrap
Purpose of the work is to discuss some theoretical aspects of the diffusion of hydrogen atoms in the crystal
lattice of BCC metals at low temperatures using the methods of statistical thermodynamics. The values of
the statistical model calculations of H diffusion coefficients in α-Fe, V, Ta, Nb, K are in good agreement
with the experimental data. The statistical model can also explain deviations from the Arrhenius equation
at temperatures 300-100 K in α-Fe, V, Nb and K. It was suggested that thermally activated fast tunnelling
transition of hydrogen atoms through the potential barrier at a temperature below 300 K provides an
almost free movement of H atoms in the α-Fe and V lattice at these temperatures. The results show that
quantum-statistical effects play a decisive role in the H diffusion in BCC metals at low temperatures. Using
the statistical model allows for the prediction of the diffusion coefficient for H in BCC metals at low
temperatures, where it’s necessary to consider quantum effects.
MASSIVE PHOTON HYPOTHESIS OPENS DOORS TO NEW FIELDS OF RESEARCHijrap
Mass, an inherent property of matter, is calculated directly for the photon particle from the very classical
principles of the kinetic theory of gases. It is not an end result with no perspective nor other outcome.
Quite the opposite, a single ponderable tiny photon frees the mind of old ways of thinking and opens up
new paths to a broad field of investigation where the very large can then be described and explained by the
very small. This reality of a non-zero mass suddenly shows up in the interpretation of many experiments
which become clear and simple to comprehend. Besides, that same key particle has the potential to unlock
and solve some long lasting major observational issues or enigmas. All this converges upon its
acknowledgement and acceptance.
PHENOMENOLOGICAL METHOD REGARDING A THIRD THEORY OF PHYSICS “THE EVENT:THE TH...ijrap
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.
3rd International Conference on Integrating Technology in Education (ITE 2022)ijrap
3rd International Conference on Integrating Technology in Education (ITE 2022) This forum also aims to provide a platform for exchanging ideas in new emerging trends that needs more focus and exposure and will attempt to publish proposals that strengthen our goals.
A SPECIAL RELATIONSHIP BETWEEN MATTER, ENERGY, INFORMATION, AND CONSCIOUSNESSijrap
This paper discusses the advantages of describing the universe, or nature, in terms of information and consciousness. Some problems encountered by theoretical physicists in the quest for the theory of everything stem from the limitations of trying to understand everything in terms of matter and energy only. However, if everything, including matter, energy, life, and mental processes, is described in terms of information and consciousness, much progress can be made in the search for the ultimate theory of the universe. As brilliant and successful as physics and chemistry have been over the last two centuries, it is important that nature is not viewed solely in terms of matter and energy. Two additional components are needed to unlock her secrets. While extensive writing exists that describes the connection between matter and energy and their physical basis, little work has been done to learn the special relationship between matter, energy, information, and consciousness.
This paper discusses the advantages of describing the universe, or nature, in terms of information and consciousness. Some problems encountered by theoretical physicists in the quest for the theory of everything stem from the limitations of trying to understand everything in terms of matter and energy only. However, if everything, including matter, energy, life, and mental processes, is described in terms of information and consciousness, much progress can be made in the search for the ultimate theory of the universe. As brilliant and successful as physics and chemistry have been over the last two centuries, it is important that nature is not viewed solely in terms of matter and energy. Two additional components are needed to unlock her secrets. While extensive writing exists that describes the connection between matter and energy and their physical basis, little work has been done to learn the special relationship between matter, energy, information, and
consciousness.
THE CONCEPT OF SPACE AND TIME: AN AFRICAN PERSPECTIVEijrap
Understanding the concept of space and time is critical, essential, and fundamental in searching for theall-encompassing theory or the theory of everything (ToE). Some physicists argue that time exists, whileothers posit that time is only a social or mental construct. The author presents an African thought systemon space and time conception, focusing on the African (Bantu) view of space and time. The author arguesthat before the advent of the Western linear view of space and time, Africans had their own visionregarding these two concepts. Their conception of time appears to be holistic, highly philosophical, non-linear, and thought-provoking. The author hopes that exploring these two concepts from an African perspective will provide a new and more in-depth insight into reality's nature. A scientific investigation of space and time from an African-centered perspective is a worthy and necessary endeavor in the quest forthe ToE
Learning to Pronounce as Measuring Cross Lingual Joint Orthography Phonology ...ijrap
Machine learning models allow us to compare languages by showing how hard a task in each language might be to learn and perform well on. Following this line of investigation, we explore what makes a language “hard to pronounce” by modelling the task of grapheme-to-phoneme (g2p) transliteration. By training a character-level transformer model on this task across 22 languages and measuring the model’s proficiency against its grapheme and phoneme inventories, we show that certain characteristics emerge that separate easier and harder languages with respect to learning to pronounce. Namely the complexity of a language's pronunciation from its orthography is due to the expressive or simplicity of its grapheme-to phoneme mapping. Further discussion illustrates how future studies should consider relative data sparsity per language to design fairer cross-lingual comparison tasks.
THE CONCEPT OF SPACE AND TIME: AN AFRICAN PERSPECTIVEijrap
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
that before the advent of the Western linear view of space and time, Africans had their own vision
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
perspective will provide a new and more in-depth insight into reality's nature. A scientific investigation of space and time from an African-centered perspective is a worthy and necessary endeavor in the quest for the ToE.
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.
The Concept of Space and Time: An African Perspectiveijrap
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 that before the advent of the Western linear view of space and time, Africans had their own vision 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 perspective will provide a new and more in-depth insight into reality's nature. A scientific investigation of space and time from an African-centered perspective is a worthy and necessary endeavor in the quest for the ToE.
The majority of physicists take it for granted that the universe is made up of matter. In turn, matter is composed of atoms; atoms are made up of particles such as electrons, protons, neutrons, etc. Also, protons
and neutrons are composed of quarks, etc. Furthermore, that everything in nature is governed by the known laws of physics and chemistry. The author only partially shares this view. He argues that many phenomena in the universe may depend on rules or factors as yet incorporated by the physical sciences.
The last few years have led him to reflect on the many unsolved physics problems, such as the quest for the theory of everything (ToE), the arrow of time, the interpretation of quantum mechanics, the fine-tuned
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
Cancer cell metabolism: special Reference to Lactate PathwayAADYARAJPANDEY1
Normal Cell Metabolism:
Cellular respiration describes the series of steps that cells use to break down sugar and other chemicals to get the energy we need to function.
Energy is stored in the bonds of glucose and when glucose is broken down, much of that energy is released.
Cell utilize energy in the form of ATP.
The first step of respiration is called glycolysis. In a series of steps, glycolysis breaks glucose into two smaller molecules - a chemical called pyruvate. A small amount of ATP is formed during this process.
Most healthy cells continue the breakdown in a second process, called the Kreb's cycle. The Kreb's cycle allows cells to “burn” the pyruvates made in glycolysis to get more ATP.
The last step in the breakdown of glucose is called oxidative phosphorylation (Ox-Phos).
It takes place in specialized cell structures called mitochondria. This process produces a large amount of ATP. Importantly, cells need oxygen to complete oxidative phosphorylation.
If a cell completes only glycolysis, only 2 molecules of ATP are made per glucose. However, if the cell completes the entire respiration process (glycolysis - Kreb's - oxidative phosphorylation), about 36 molecules of ATP are created, giving it much more energy to use.
IN CANCER CELL:
Unlike healthy cells that "burn" the entire molecule of sugar to capture a large amount of energy as ATP, cancer cells are wasteful.
Cancer cells only partially break down sugar molecules. They overuse the first step of respiration, glycolysis. They frequently do not complete the second step, oxidative phosphorylation.
This results in only 2 molecules of ATP per each glucose molecule instead of the 36 or so ATPs healthy cells gain. As a result, cancer cells need to use a lot more sugar molecules to get enough energy to survive.
Unlike healthy cells that "burn" the entire molecule of sugar to capture a large amount of energy as ATP, cancer cells are wasteful.
Cancer cells only partially break down sugar molecules. They overuse the first step of respiration, glycolysis. They frequently do not complete the second step, oxidative phosphorylation.
This results in only 2 molecules of ATP per each glucose molecule instead of the 36 or so ATPs healthy cells gain. As a result, cancer cells need to use a lot more sugar molecules to get enough energy to survive.
introduction to WARBERG PHENOMENA:
WARBURG EFFECT Usually, cancer cells are highly glycolytic (glucose addiction) and take up more glucose than do normal cells from outside.
Otto Heinrich Warburg (; 8 October 1883 – 1 August 1970) In 1931 was awarded the Nobel Prize in Physiology for his "discovery of the nature and mode of action of the respiratory enzyme.
WARNBURG EFFECT : cancer cells under aerobic (well-oxygenated) conditions to metabolize glucose to lactate (aerobic glycolysis) is known as the Warburg effect. Warburg made the observation that tumor slices consume glucose and secrete lactate at a higher rate than normal tissues.
Introduction:
RNA interference (RNAi) or Post-Transcriptional Gene Silencing (PTGS) is an important biological process for modulating eukaryotic gene expression.
It is highly conserved process of posttranscriptional gene silencing by which double stranded RNA (dsRNA) causes sequence-specific degradation of mRNA sequences.
dsRNA-induced gene silencing (RNAi) is reported in a wide range of eukaryotes ranging from worms, insects, mammals and plants.
This process mediates resistance to both endogenous parasitic and exogenous pathogenic nucleic acids, and regulates the expression of protein-coding genes.
What are small ncRNAs?
micro RNA (miRNA)
short interfering RNA (siRNA)
Properties of small non-coding RNA:
Involved in silencing mRNA transcripts.
Called “small” because they are usually only about 21-24 nucleotides long.
Synthesized by first cutting up longer precursor sequences (like the 61nt one that Lee discovered).
Silence an mRNA by base pairing with some sequence on the mRNA.
Discovery of siRNA?
The first small RNA:
In 1993 Rosalind Lee (Victor Ambros lab) was studying a non- coding gene in C. elegans, lin-4, that was involved in silencing of another gene, lin-14, at the appropriate time in the
development of the worm C. elegans.
Two small transcripts of lin-4 (22nt and 61nt) were found to be complementary to a sequence in the 3' UTR of lin-14.
Because lin-4 encoded no protein, she deduced that it must be these transcripts that are causing the silencing by RNA-RNA interactions.
Types of RNAi ( non coding RNA)
MiRNA
Length (23-25 nt)
Trans acting
Binds with target MRNA in mismatch
Translation inhibition
Si RNA
Length 21 nt.
Cis acting
Bind with target Mrna in perfect complementary sequence
Piwi-RNA
Length ; 25 to 36 nt.
Expressed in Germ Cells
Regulates trnasposomes activity
MECHANISM OF RNAI:
First the double-stranded RNA teams up with a protein complex named Dicer, which cuts the long RNA into short pieces.
Then another protein complex called RISC (RNA-induced silencing complex) discards one of the two RNA strands.
The RISC-docked, single-stranded RNA then pairs with the homologous mRNA and destroys it.
THE RISC COMPLEX:
RISC is large(>500kD) RNA multi- protein Binding complex which triggers MRNA degradation in response to MRNA
Unwinding of double stranded Si RNA by ATP independent Helicase
Active component of RISC is Ago proteins( ENDONUCLEASE) which cleave target MRNA.
DICER: endonuclease (RNase Family III)
Argonaute: Central Component of the RNA-Induced Silencing Complex (RISC)
One strand of the dsRNA produced by Dicer is retained in the RISC complex in association with Argonaute
ARGONAUTE PROTEIN :
1.PAZ(PIWI/Argonaute/ Zwille)- Recognition of target MRNA
2.PIWI (p-element induced wimpy Testis)- breaks Phosphodiester bond of mRNA.)RNAse H activity.
MiRNA:
The Double-stranded RNAs are naturally produced in eukaryotic cells during development, and they have a key role in regulating gene expression .
Richard's aventures in two entangled wonderlandsRichard Gill
Since the loophole-free Bell experiments of 2020 and the Nobel prizes in physics of 2022, critics of Bell's work have retreated to the fortress of super-determinism. Now, super-determinism is a derogatory word - it just means "determinism". Palmer, Hance and Hossenfelder argue that quantum mechanics and determinism are not incompatible, using a sophisticated mathematical construction based on a subtle thinning of allowed states and measurements in quantum mechanics, such that what is left appears to make Bell's argument fail, without altering the empirical predictions of quantum mechanics. I think however that it is a smoke screen, and the slogan "lost in math" comes to my mind. I will discuss some other recent disproofs of Bell's theorem using the language of causality based on causal graphs. Causal thinking is also central to law and justice. I will mention surprising connections to my work on serial killer nurse cases, in particular the Dutch case of Lucia de Berk and the current UK case of Lucy Letby.
Deep Behavioral Phenotyping in Systems Neuroscience for Functional Atlasing a...Ana Luísa Pinho
Functional Magnetic Resonance Imaging (fMRI) provides means to characterize brain activations in response to behavior. However, cognitive neuroscience has been limited to group-level effects referring to the performance of specific tasks. To obtain the functional profile of elementary cognitive mechanisms, the combination of brain responses to many tasks is required. Yet, to date, both structural atlases and parcellation-based activations do not fully account for cognitive function and still present several limitations. Further, they do not adapt overall to individual characteristics. In this talk, I will give an account of deep-behavioral phenotyping strategies, namely data-driven methods in large task-fMRI datasets, to optimize functional brain-data collection and improve inference of effects-of-interest related to mental processes. Key to this approach is the employment of fast multi-functional paradigms rich on features that can be well parametrized and, consequently, facilitate the creation of psycho-physiological constructs to be modelled with imaging data. Particular emphasis will be given to music stimuli when studying high-order cognitive mechanisms, due to their ecological nature and quality to enable complex behavior compounded by discrete entities. I will also discuss how deep-behavioral phenotyping and individualized models applied to neuroimaging data can better account for the subject-specific organization of domain-general cognitive systems in the human brain. Finally, the accumulation of functional brain signatures brings the possibility to clarify relationships among tasks and create a univocal link between brain systems and mental functions through: (1) the development of ontologies proposing an organization of cognitive processes; and (2) brain-network taxonomies describing functional specialization. To this end, tools to improve commensurability in cognitive science are necessary, such as public repositories, ontology-based platforms and automated meta-analysis tools. I will thus discuss some brain-atlasing resources currently under development, and their applicability in cognitive as well as clinical neuroscience.
Nutraceutical market, scope and growth: Herbal drug technologyLokesh Patil
As consumer awareness of health and wellness rises, the nutraceutical market—which includes goods like functional meals, drinks, and dietary supplements that provide health advantages beyond basic nutrition—is growing significantly. As healthcare expenses rise, the population ages, and people want natural and preventative health solutions more and more, this industry is increasing quickly. Further driving market expansion are product formulation innovations and the use of cutting-edge technology for customized nutrition. With its worldwide reach, the nutraceutical industry is expected to keep growing and provide significant chances for research and investment in a number of categories, including vitamins, minerals, probiotics, and herbal supplements.
Seminar of U.V. Spectroscopy by SAMIR PANDASAMIR PANDA
Spectroscopy is a branch of science dealing the study of interaction of electromagnetic radiation with matter.
Ultraviolet-visible spectroscopy refers to absorption spectroscopy or reflect spectroscopy in the UV-VIS spectral region.
Ultraviolet-visible spectroscopy is an analytical method that can measure the amount of light received by the analyte.
Multi-source connectivity as the driver of solar wind variability in the heli...Sérgio Sacani
The ambient solar wind that flls the heliosphere originates from multiple
sources in the solar corona and is highly structured. It is often described
as high-speed, relatively homogeneous, plasma streams from coronal
holes and slow-speed, highly variable, streams whose source regions are
under debate. A key goal of ESA/NASA’s Solar Orbiter mission is to identify
solar wind sources and understand what drives the complexity seen in the
heliosphere. By combining magnetic feld modelling and spectroscopic
techniques with high-resolution observations and measurements, we show
that the solar wind variability detected in situ by Solar Orbiter in March
2022 is driven by spatio-temporal changes in the magnetic connectivity to
multiple sources in the solar atmosphere. The magnetic feld footpoints
connected to the spacecraft moved from the boundaries of a coronal hole
to one active region (12961) and then across to another region (12957). This
is refected in the in situ measurements, which show the transition from fast
to highly Alfvénic then to slow solar wind that is disrupted by the arrival of
a coronal mass ejection. Our results describe solar wind variability at 0.5 au
but are applicable to near-Earth observatories.
This presentation explores a brief idea about the structural and functional attributes of nucleotides, the structure and function of genetic materials along with the impact of UV rays and pH upon them.
Observation of Io’s Resurfacing via Plume Deposition Using Ground-based Adapt...Sérgio Sacani
Since volcanic activity was first discovered on Io from Voyager images in 1979, changes
on Io’s surface have been monitored from both spacecraft and ground-based telescopes.
Here, we present the highest spatial resolution images of Io ever obtained from a groundbased telescope. These images, acquired by the SHARK-VIS instrument on the Large
Binocular Telescope, show evidence of a major resurfacing event on Io’s trailing hemisphere. When compared to the most recent spacecraft images, the SHARK-VIS images
show that a plume deposit from a powerful eruption at Pillan Patera has covered part
of the long-lived Pele plume deposit. Although this type of resurfacing event may be common on Io, few have been detected due to the rarity of spacecraft visits and the previously low spatial resolution available from Earth-based telescopes. The SHARK-VIS instrument ushers in a new era of high resolution imaging of Io’s surface using adaptive
optics at visible wavelengths.
PRESENTATION ABOUT PRINCIPLE OF COSMATIC EVALUATION
Bound- State Solution of Schrodinger Equation with Hulthen Plus Generalized Exponential Coulomb Potential Using NIKIFOROV-UVAROV Method
1. International Journal of Recent advances in Physics (IJRAP) Vol.4, No.3, August 2015
DOI : 10.14810/ijrap.2015.4301 1
BOUND- STATE SOLUTION OF SCHRODINGER
EQUATION WITH HULTHEN PLUS
GENERALIZED EXPONENTIAL COULOMB
POTENTIAL USING NIKIFOROV-UVAROV
METHOD
Ituen. B. Okon1
and Oyebola Popoola2
1
Department of Physics, University of Uyo, Nigeria.
2
Department of Physics, University of Ibadan, Nigeria.
Abstract
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.
Keywords
Nikiforov-Uvarov method, Schrodinger equation, Hulthen plus exponential Coulomb potential.
PACS: 03.65.Ge, 03.65. –w
1. Introduction
The exact and approximate bound state solution to Schrodinger wave equation has arouse the
interest of many authors for decades. Theses solutions are very important with enormous
applications in physical sciences and quantum field theory [1-9]. Finding the solution of the wave
equations whether relativistic or non relativistic help in quantum mechanical studies. Different
authors have used different techniques and methods for this purpose among them are: Analytical
Nikiforov-Uvarov method [10-12], factorization method [13], super-symmetric quantum
approach [14-18], shape invariant method [19], the ansatz solution approach [20-22], exact
quantization rule [23-24], tridiagonalisation program[25] , variational method [26] and many
others.
However, these techniques are applicable for selected potentials, few amongst them are : Rosen-
Morse, Hulthen, pseudoharmonic, Poschl-Teller, kratzer fues and Mie-Type potentials[27-30].
The study of the proposed potential is very essential in investigating the interaction existing
between diatomic molecules.
2. International Journal of Recent advances in Physics (IJRAP) Vol.4, No.3, August 2015
2
The main aim of this paper is to present and study Hulthen plus generalized exponential coulomb
potential from which other potentials are deduced as special cases. This article is organized as
follows: section 1 is the introduction, section 2 gives a brief discussion of Nikiforov-Uvarov
method. The method used in this article reproduces accurate analytical solutions for many
differential equations with significant application in physics, for example it can be used for
equation of Hermite, Laguerre and Jacobi. Section 3 gives the radial solution to Schrodinger
equation with the proposed potential, section 4 gives the result and discussion while section 5
gives the numerical computation of the resulting energy equation.
The proposed potential is given by
( )
2
0
2
( )
1
r r
r
V e Ae
V r
r
e
α α
α
− −
−
−
= −
−
(1)
Hulthen potential describes the interaction existing between diatomic molecules but addition of
the second term which is the exponential coulomb potential ascertain the attractive force that
holds the molecules together. However, this force exist as Coulomb force.
2. Review of Nikiforov-Uvarov method.
The NU method is based on reducing second order linear differential equation to a generalized
equation of hyper-geometric type .This method provides solution in terms of special orthogonal
functions as well as corresponding energy eigen value. With appropriate coordinate
transformation, ( )
s s x
= , this equation can be written as
2
( ) ( )
( ) ( ) ( ) 0
( ) ( )
s s
s s s
s s
τ σ
ψ ψ ψ
σ σ
′′ ′
+ + =
% %
(2)
In order to find the exact solution to equation (2), we set the wave function as
( ) ( ) ( )
s s s
ψ φ χ
= (3)
Substituting equation (3) into equation (2) gives the hyper-geometric equation
( ) ( ) ( ) ( ) ( ) 0
s s s s s
σ χ τ χ λχ
′′ ′
+ + = (4)
Where the wave function is defined as the logarithmic derivative
( ) ( )
( ) ( )
s s
s s
φ π
φ σ
′
= (5)
Where ( )
s
π is at most polynomial of first degree. However, the hyper-geometric type function
( )
s
χ in equation (4) for a fixed n can be express in Rodrigue relation
3. International Journal of Recent advances in Physics (IJRAP) Vol.4, No.3, August 2015
3
( )
( ) ( ) ( )
( )
n
n
n
n n
B s d
s s s
s ds
χ σ ρ
ρ
= (6)
n
B is the normalization constant and the weight function ( )
s
ρ satisfies the condition
( )
( ) ( ) ( ) ( )
d
s s s s
ds
σ ρ τ ρ
= (7)
Also ( ) ( ) 2 ( )
s s s
τ τ π
= +
% (8)
In order to accomplish the conditions imposed on the weight function ( )
s
ρ , it is necessary that
the classical orthogonal polynomial ( )
s
τ be equal to zero, so that its derivative be less than zero.
That is
( )
0
d s
ds
τ
< (9)
Therefore the function ( )
s
π and the parameter λ required for the NU method are defined as
follows:
2
( )
2 2
s k
σ τ σ τ
π σ σ
′ ′
− −
= ± − +
% %
% (10)
( )
k s
λ π′
= + (11)
k values in equation (10) are possible to evaluate if the expression under the square root must be
square of polynomials. This is possible if and only if its descriminant is zero. With this a new
eigen equation becomes
2
2
( ) ( 1)
2
n
nd s n n d
ds ds
τ σ
λ
−
= − (12)
n = 0, 1, 2-
Where ( )
s
τ is defined in equation (8) and on comparing equation (11) to (12), we can now
obtained the energy eigen value.
3. Solution of Schrodinger Equation
The Schrodinger wave equation is given by
2 2
2 2 2
( ) 2 ( 1)
( ) ( ) 0
2
d r l l
E V r r
dr r
ψ µ
ψ
µ
+
+ − − =
h
h
(13)
Substituting equation (1) into (13) gives
4. International Journal of Recent advances in Physics (IJRAP) Vol.4, No.3, August 2015
4
( )
2
2 2
0
2 2 2
2
( ) 2 ( 1)
( ) 0
2
1
r r
r
V e
d r Ae l l
E r
dr r r
e
α α
α
ψ µ
ψ
µ
− −
−
+
+ + + − =
−
h
h
(14)
Lets define some approximation scheme
( )
2 2
2
2 2
1 4
1
r
r
e
r e
α
α
α −
−
=
−
and
( )
2
1 2
1
r
r
e
r e
α
α
α −
−
=
−
(15)
Substituting equation (15) into equation (14) with the transformation 2 r
s e α
−
= gives
( )
( ) ( )
2
2 2 2 2 2 2
1 1
2
2 2
1
( ) ( ) 1
( ) (2 ( 1)) ( ) 0
1 1
s
d s d s
s l l s s
ds s s ds s s
ψ ψ
ε δ σ ε δ σ ε ψ
−
+ + − − − + + + − + − =
− −
(16)
Where
[ 2
2 2
2
4
E
µ
ε
α
= −
h
, 2 0
2 2
2
4
V
µ
δ
α
=
h
, 1 2
A
µ
σ
α
=
h
] (17)
Comparing equation (16) to equation (2) we obtained the followings:
( ) (1 )
s s
τ = −
% , ( ) (1 )
s s s
σ = − ,
2 2 2 2 2 2
1 1
( ) ( ) (2 ( 1))
s s l l s
σ ε δ σ ε δ σ ε
=− + + + + + − + −
%
(18)
Then, using equation (10), the polynomial equation ( )
s
π then become
2 2 2 2 2 2
1 1
1
( ) (2 ( 1) )
2 4
s
s k s l l k s
π ε δ σ ε δ σ ε
−
= ± + + + − − + + − + − +
(19)
To find the value of k we consider the discriminant such that 2
4 0
b ac
− =
Hence,
( )
2
1 1
( 1) 4 ( 1) 1
k l l l l
σ δ ε
= − + − − + + +
( )
2
2 1
( 1) 4 ( 1) 1
k l l l l
σ δ ε
= − + − − − + +
Substituting the values of 1
k and 2
k into equation (19), then the four values of ( )
s
π is given
below.
5. International Journal of Recent advances in Physics (IJRAP) Vol.4, No.3, August 2015
5
( )
( )
1 2
( 4 ( 1 ) 1 ) , ( 1 ) 4 ( 1 ) 1
1 1
2
1 2
( 4 ( 1 ) 1 ) , ( 1 ) 4 ( 1 ) 1
2 1
2
( )
2
l l s k l l l l
l l s k l l l l
s
s
ε ε σ δ ε
ε ε σ δ ε
π
+ + + − = − + − − + + +
+ + + + = − + − − − + +
−
= ±
(20)
( )
s
π has four solutions and one of these solutions satisfied bound state condition which is
1
( ) 4 ( 1) 1
2 2
s
s l l s
π ε ε
= − − + + + −
(21)
Using equation (8) ( )
( ) 1 2 4 ( 1) 1 2 2
s s l l s s
τ ε ε
= − − + + − + (22)
Which implies that ( )
( ) 2 4 ( 1) 1 2 0
s l l
τ ε
′ = − − + + − <
(23)
Equation (23) satisfies bound state condition. The energy eigen value can be calculated using
equation (11)
2
1
1 1
( 1) 4 ( 1) 1 4 ( 1) 1
2 2
l l l l l l
λ δ δ ε ε
= − + + + − + + − − + + −
(24)
Applying equation (12)
( )
2
2 3
2 (4 4 ( 1) 1) 2
n n n l l n
λ σ ε σ
= − − + + + − + (25)
Equating equation (24) and (25) gives the energy equation as follow
( )
2
1
1 1
2 4 ( 1) 1 ( 1)
2 2
2 4 ( 1) 1 1
n l l n l l
n l l
σ δ
ε
− + + + − + + − −
=
+ + + −
(26)
Substituting parameters in equation (17) into equation (26) gives
6. International Journal of Recent advances in Physics (IJRAP) Vol.4, No.3, August 2015
6
( )
2
0
2 2 2 2 2
2
1 1
2 4 ( 1) 1 ( 1)
2 2 2 4
2 4 ( 1) 1 1
nl
v
A
n l l n l l
E
n l l
µ
µ
α α α
µ
− + + + − + + − −
= −
+ + + −
h h h
(27)
Equation (27) is the energy of the proposed potential.
Calculation of the wave function
By using equation (5)
( )
( )
1 1
( ) ( ) 1 1
4 ( 1) 1
( ) ( ) 2(1 ) 2 (1 ) (1 ) (1 )
s
s s
l l
s s s s s s s s s
π
φ φ ε ε
φ σ φ
= ⇒ = − − + + + −
− − − −
(28)
Taking integral of equation (28) and engaging mathematical algebraic simplification gives
[ ]
1 1
4 ( 1) 1
2 2
( ) 1
l l
s s s ε
φ
+ + +
−
= − (29)
Equation (29) is the solution to the first part of the wave equation. To determine the second
solution of the wave function, it is necessary to compute the weight function .
Calculation of weight function
Using equation (7)
( ) ( )
( ) ( ) ( )
( )
( )
( ) ( )
( )
s s s
d
s s s s ds ds
ds s s
ρ τ σ
σ ρ τ ρ
ρ σ
′ ′
−
= ⇒ =
∫ ∫
Then the weight function can be given as
[ ]( )
4 ( 1) 1 2
( ) 1
l l
s s s ε
ρ
+ +
= − (30)
Rewriting equation(31) in its Rodrigue form by making use of equation(6) gives
[ ] ( )
[ ] ( )
4 ( 1) 1 4 ( 1) 1
2 2
( ) ( ) 1 1
n
l l n l l
n
n n n
d
Y s B s s s s s
ds
ε ε
− + + + + +
− +
= − −
(31)
Lets define standard associated Laguerre polynomial as
[ ]
2 ,
2 2
( ) ( ) (1 ) (1 ) ( ) (1 )
n
n n
n n n n
n
d
s B s s s s s B s P s
ds
µ υ µ
υ µ υ µ
χ +
− − + +
= − − = −
(32)
7. International Journal of Recent advances in Physics (IJRAP) Vol.4, No.3, August 2015
7
Then re-writing equation (31) in terms of equation (32) gives the second part of the wave function
as
( ) ( )
4 ( 1) 1 2 , 4 ( 1) 1
( ) ( ) (1 )
l l l l
n n n
Y s B s P s
ε
+ + + + +
= − (33)
Hence the total wave function is given by
[ ]( ) ( ) ( )
4 ( 1) 1 2 , 4 ( 1) 1
4 ( 1) 1 2
( ) ( ) ( ) ( ) 1 (1 )
l l l l
l l
n n n
s s Y s B s s s P s
ε
ε
ψ φ
+ + + + +
+ +
= = − − (35)
4. Result and Discussion
By setting some potential parameters into equation (1), we obtain some well known potentials.
4.1 Hulthen potential
If 0
A = in equation (1) then the potential reduces to Hulthen potential
( )
2
0
2
( )
1
r
r
V e
V r
e
α
α
−
−
−
=
−
(36)
and the resulting energy of this potential is given by
( )
2
0
2 2 2 2
2
1 1
2 4 ( 1) 1 ( 1)
2 2 2 4
2 4 ( 1) 1 1
nl
v
n l l n l l
E
n l l
µ
α α
µ
− + + + − + + −
= −
+ + + −
h h
(37)
4.2 Generalized exponential coulomb potential
If 0 0
v = in equation (1), then, the potential reduces to generalized exponential potential
r
Ae
r
α
−
− (38)
The resulting energy equation is given by
( )
2
2 2 2
1 1
2 4 ( 1) 1 ( 1)
2 2 2
2 4 ( 1) 1 1
nl
A
n l l n l l
E
n l l
µ
α α
µ
− + + + − + + −
= −
+ + + −
h h
(39)
5.0 Numerical result for Hulthen plus exponential coulomb potential for various value of α
9. International Journal of Recent advances in Physics (IJRAP) Vol.4, No.3, August 2015
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Table 4
Table5
Figure 1. A graph of Hulthen plus modified exponential Coulomb potential for 0.2
α =
n
l
( 0.8)
nl
E α = n l ( 0.8)
nl
E α = n l ( 0.8)
nl
E α = n l ( 0.8)
nl
E α =
1 0 -29.5936010 1 1 -2907.36620 1 2 -7011.27050 1 3 -11854.85400
2 0 -56.5504000 2 1 -1829.55790 2 2 -6108.98500 2 3 -11574.31800
3 0 -140.185610 3 1 -1383.83970 3 2 -5597.43400 3 3 -11389.15800
4 0 -195.999970 4 1 -1146.22860 4 2 -5268.82470 4 3 -11257.81600
5 0 -233.967640 5 1 -1000.24610 5 2 -5040.18850 5 3 -11159.81000
6 0 -261.145600 6 1 -902.058350 6 2 -4872.03960 6 3 -11083.88000
7 0 -281.472530 7 1 -831.745500 7 2 -4743.22950 7 3 -11023.32100
8 0 -297.217600 8 1 -779.030150 8 2 -4641.42530 8 3 -10973.89550
n l
( 1.0)
nl
E α = n l ( 1.0)
nl
E α = n l ( 1.0)
nl
E α = n l ( 1.0)
nl
E α =
1 0 -110.25000 1 1 -16965.059 1 2 -41412.250 1 3 -70291.250
2 0 -390.06244 2 1 -10712.250 2 2 -36147.510 2 3 -68696.400
3 0 -890.02770 3 1 -8122.5140 3 2 -33160.406 3 3 -67643.330
4 0 -1216.2654 4 1 -6740.4097 4 2 -31240.557 4 3 -66896.125
5 0 -1436.4098 5 1 -5890.5625 5 2 -29904.287 5 3 -66338.420
6 0 -1593.3401 6 1 -5318.5767 6 2 -28921.248 6 3 -65906.290
7 0 -1710.4133 7 1 -4908.7530 7 2 -28168.025 7 3 -65561.590
8 0 -1800.9410 8 1 -4601.3604 8 2 -27572.598 8 3 -65280.242
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Figure 2: A graph of Hulthen plus modified exponential Coulomb potential for 0.4
α =
5. CONCLUSION
In this paper, we have studied the bound state solution to Schrodinger equation using Hulthen
plus generalized exponential coulomb potential via conventional Nikiforov-Uvarov method. We
also obtained the energy of this potential and the total wave function in addition to energies of
special cases. The Numerical result shows that the energy of this potential decreases with an
increase in quantum state as shown in tables (1-5). However, the numerical result also shows that
these energies apart from decreasing with an increase in screening parameterα , have negative
values to ascertain the bound state of the solution. Graphical illustration s shows a unique stability
at different quantum state of the potential that describe the interaction existing in diatomic
molecules.
These energies exist at different discrete energy eigen state illustrating the reason for stability of
diatomic molecules.
ACKNOWLEDGEMENT
The authors are grateful to the anonymous reviewer for his useful comments which has
significantly help for the improvement of this article.
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