ANALYTICAL APPROXIMATIONS TO THE l-WAVE SOLUTIONS OF THE SCHRÖDINGER EQUATION WITH A HYPERBOLIC POTENTIAL

2008 ◽  
Vol 22 (07) ◽  
pp. 483-489 ◽  
Author(s):  
SHISHAN DONG ◽  
S. G. MIRANDA ◽  
F. M. ENRIQUEZ ◽  
SHI-HAI DONG
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Energy Levels ◽  
Bound State ◽  
Short Range Potential ◽  
Quantum Numbers ◽  
Analytical Approximations ◽  
Special Cases ◽  
Hyperbolic Potential ◽  
Arbitrary Quantum

The bound-state solutions of the Schrödinger equation for a hyperbolic potential with the centrifugal term are presented approximately. It is shown that the solutions can be expressed by the hypergeometric function 2F1(a, b; c; z). To show the accuracy of our results, we calculate the energy levels numerically for arbitrary quantum numbers n and l. It is found that the results are in good agreement with those obtained by other methods for short-range potential. Two special cases for l = 0 and σ = 1 are also studied briefly.

ANALYTICAL APPROXIMATIONS TO THE SCHRÖDINGER EQUATION FOR A SECOND PÖSCHL–TELLER-LIKE POTENTIAL WITH CENTRIFUGAL TERM

2008 ◽  
Vol 23 (10) ◽  
pp. 1537-1544 ◽  
Author(s):  
SHI-HAI DONG ◽  
WEN-CHAO QIANG ◽  
J. GARCÍA-RAVELO
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Bound State ◽  
Centrifugal Term ◽  
Short Range Potential ◽  
Quantum Numbers ◽  
Analytical Approximations ◽  
Special Cases ◽  
Arbitrary Quantum ◽  
Good Agreement

The bound state solutions of the Schrödinger equation for a second Pöschl–Teller-like potential with the centrifugal term are obtained approximately. It is found that the solutions can be expressed in terms of the hypergeometric functions 2F1(a, b; c; z). To show the accuracy of our results, we calculate the eigenvalues numerically for arbitrary quantum numbers n and l. It is found that the results are in good agreement with those obtained by other method for short-range potential. Two special cases for l = 0 and V1 = V2 are also studied briefly.

scholarly journals Solutions of Schrodinger equation and thermal properties of generalized trigonometric Poschl-Teller potential.

2020 ◽  
Vol 66 (6 Nov-Dec) ◽  
pp. 824
Author(s):  
C. O. Edet ◽  
P. O. Amadi ◽  
U. S. Okorie ◽  
A. Tas ◽  
A. N. Ikot ◽  
...  
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Hypergeometric Functions ◽  
Energy Equation ◽  
Bound State ◽  
Energy Spectra ◽  
Energy Eigenvalues ◽  
Quantum Numbers ◽  
Special Cases ◽  
Potential Parameters

Analytical solutions of the Schrödinger equation for the generalized trigonometric Pöschl–Teller potential by using an appropriate approximation to the centrifugal term within the framework of the Functional Analysis Approach have been considered. Using the energy equation obtained, the partition function was calculated and other relevant thermodynamic properties. More so, we use the concept of the superstatistics to also evaluate the thermodynamics properties of the system. It is noted that the well-known normal statistics results are recovered in the absence of the deformation parameter and this is displayed graphically for the clarity of our results. We also obtain analytic forms for the energy eigenvalues and the bound state eigenfunction solutions are obtained in terms of the hypergeometric functions. The numerical energy spectra for different values of the principal and orbital quantum numbers are obtained. To show the accuracy of our results, we discuss some special cases by adjusting some potential parameters and also compute the numerical eigenvalue of the trigonometric Pöschl–Teller potential for comparison sake. However, it was found out that our results agree excellently with the results obtained via other methods

Bound state solution of the Schrodinger equation for the Woods–Saxon potential plus coulomb interaction by Nikiforov–Uvarov and supersymmetric quantum mechanics methods

2021 ◽  
pp. 2150023
Author(s):  
Enayatolah Yazdankish
Keyword(s):  
Quantum Mechanics ◽  
Schrödinger Equation ◽  
Coulomb Interaction ◽  
Schrodinger Equation ◽  
Energy Eigenvalue ◽  
Bound State ◽  
Repulsive Interaction ◽  
Saxon Potential ◽  
Special Cases ◽  
Supersymmetry Quantum Mechanics

The generalized Woods–Saxon potential plus repulsive Coulomb interaction is considered in this work. The supersymmetry quantum mechanics method is used to get the energy spectrum of Schrodinger equation and also the Nikiforov–Uvarov approach is employed to solve analytically the Schrodinger equation in the framework of quantum mechanics. The potentials with centrifugal term include both exponential and radial terms, hence, the Pekeris approximation is considered to approximate the radial terms. By using the step-by-step Nikiforov–Uvarov method, the energy eigenvalue and wave function are obtained analytically. After that, the spectrum of energy is obtained by the supersymmetry quantum mechanics method. The energy eigenvalues obtained from each method are the same. Then in special cases, the results are compared with former result and a full agreement is observed. In the [Formula: see text]-state, the standard Woods–Saxon potential has no bound state, but with Coulomb repulsive interaction, it may have bound state for zero angular momentum.

Approximate analytical solution of continuous states for the l-wave Schrödinger equation with a diatomic molecule potential

Open Physics ◽  
2010 ◽  
Vol 8 (4) ◽  
Author(s):  
Gao-Feng Wei ◽  
Wen-Chao Qiang ◽  
Wen-Li Chen
Keyword(s):  
Schrödinger Equation ◽  
Diatomic Molecule ◽  
Schrodinger Equation ◽  
Scattering Amplitude ◽  
Energy Levels ◽  
Bound State ◽  
Radial Wave ◽  
Continuous States ◽  
Analytical Properties ◽  
Function Potential

AbstractThe continuous states of the l-wave Schrödinger equation for the diatomic molecule represented by the hyperbolical function potential are carried out by a proper approximation scheme to the centrifugal term. The normalized analytical radial wave functions of the l-wave Schrödinger equation for the hyperbolical function potential are presented and the corresponding calculation formula of phase shifts is derived. Also, we interestingly obtain the corresponding bound state energy levels by analyzing analytical properties of scattering amplitude.

Energy spectrum for trigonometric Pöschl–Teller potential

2012 ◽  
Vol 90 (12) ◽  
pp. 1259-1265 ◽  
Author(s):  
Babatunde James Falaye
Keyword(s):  
Bound State ◽  
Asymptotic Iteration Method ◽  
Generalized Hypergeometric Functions ◽  
Short Range Potential ◽  
Energy Eigenvalues ◽  
Molecular Potential ◽  
Quantum Numbers ◽  
Special Case ◽  
Arbitrary Quantum ◽  
Good Agreement

We present analytical solutions of the Schrödinger equation for the trigonometric Pöschl–Teller molecular potential by using a proper approximation to the centrifugal term within the framework of the asymptotic iteration method. We obtain analytic forms for the energy eigenvalues and the bound state eigenfunction solutions are obtained in terms of the generalized hypergeometric functions. Energy eigenvalues for a few diatomic molecules are calculated for arbitrary quantum numbers n and ℓ with various values of parameter α. We also studied special case ℓ = 0 and found that the results are in good agreement with findings of other methods for short-range potential.

scholarly journals The bound state solutions of the D-dimensional Schrödinger equation for the Woods–Saxon potential

2016 ◽  
Vol 25 (01) ◽  
pp. 1650002 ◽  
Author(s):  
V. H. Badalov
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Bound State ◽  
Wave Functions ◽  
Saxon Potential ◽  
Radial Wave ◽  
Energy Eigenvalues ◽  
Quantum Numbers ◽  
Radial Schrödinger Equation ◽  
Pekeris Approximation

In this work, the analytical solutions of the [Formula: see text]-dimensional radial Schrödinger equation are studied in great detail for the Wood–Saxon potential by taking advantage of the Pekeris approximation. Within a novel improved scheme to surmount centrifugal term, the energy eigenvalues and corresponding radial wave functions are found for any angular momentum case within the context of the Nikiforov–Uvarov (NU) and Supersymmetric quantum mechanics (SUSYQM) methods. In this way, based on these methods, the same expressions are obtained for the energy eigenvalues, and the expression of radial wave functions transformed each other is demonstrated. In addition, a finite number energy spectrum depending on the depth of the potential [Formula: see text], the radial [Formula: see text] and orbital [Formula: see text] quantum numbers and parameters [Formula: see text] are defined as well.

ARBITRARY l-WAVE SOLUTIONS OF THE SCHRÖDINGER EQUATION FOR THE SCREEN COULOMB POTENTIAL

2013 ◽  
Vol 22 (06) ◽  
pp. 1350036 ◽  
Author(s):  
SHISHAN DONG ◽  
GUO-HUA SUN ◽  
SHI-HAI DONG
Keyword(s):  
Schrödinger Equation ◽  
Coulomb Potential ◽  
Schrodinger Equation ◽  
State Energy ◽  
Energy Levels ◽  
Bound State ◽  
Screen Coulomb Potential ◽  
Normalization Constant ◽  
Angular Momentum State ◽  
Good Agreement

Using improved approximate schemes for centrifugal term and the singular factor 1/r appearing in potential itself, we solve the Schrödinger equation with the screen Coulomb potential for arbitrary angular momentum state l. The bound state energy levels are obtained. A closed form of normalization constant of the wave functions is also found. The numerical results show that our results are in good agreement with those obtained by other methods. The key issue is how to treat two singular points in this quantum system.

scholarly journals Bound State Solutions of the Hua Potential within the Framework of Two Approximations Scheme

2021 ◽  
Vol 3 (3) ◽  
pp. 38-41
Author(s):  
E. B. Ettah ◽  
P. O. Ushie ◽  
C. M. Ekpo
Keyword(s):  
Wave Function ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Energy Equation ◽  
Bound State ◽  
S Wave ◽  
Angular Momenta ◽  
Special Cases ◽  
Uvarov Method

In this paper, we solve analytically the Schrodinger equation for s-wave and arbitrary angular momenta with the Hua potential is investigated respectively. The wave function as well as energy equation are obtained in an exact analytical manner via the Nikiforov Uvarov method using two approximations scheme. Some special cases of this potentials are also studied.

scholarly journals Diatomic Molecules and Mass Spectrum of Heavy Quarkonium System with Kratzer- Screened Coulomb Potential (KSCP) through the Solutions of the Schrödinger Equation

2021 ◽  
Vol 3 (2) ◽  
pp. 48-55
Author(s):  
E. P. Inyang ◽  
E. P. Inyang ◽  
J. Karniliyus ◽  
J. E. Ntibi ◽  
E. S. William
Keyword(s):  
Schrödinger Equation ◽  
Coulomb Potential ◽  
Schrodinger Equation ◽  
Diatomic Molecules ◽  
Expansion Method ◽  
Bound State ◽  
Heavy Quarkonium ◽  
Energy Eigenvalues ◽  
Special Cases ◽  
Screened Coulomb Potential

In this work, we obtain solutions of the Schrödinger equation with Kratzer-screened Coulomb potential (KSCP) model using the series expansion method. Explicitly, we compute the bound state energy eigenvalues for selected diatomic molecules of N2, CO, NO, and CH, respectively, for the various vibrational and rotational quantum states and the numerical energy eigenvalues agree with the existing literature. Three special cases were considered. The energy eigenvalues are applied to obtain the mass spectra of heavy quarkonium system such as charmonium and bottomonium. The results agree with the experimental data and other recent theoretical studies.

Approximate and analytical bound state solutions of the Frost–Musulin potential

2014 ◽  
Vol 92 (1) ◽  
pp. 18-21 ◽  
Author(s):  
A.G. Adepoju ◽  
E.J. Eweh
Keyword(s):  
Functional Analysis ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Short Range ◽  
Bound State ◽  
State Solution ◽  
Centrifugal Term ◽  
Analysis Method ◽  
Bound State Solution ◽  
Short Range Potential

Despite all the attempts made by several authors to investigate the bound state solutions of the Schrödinger equation with various potentials, until now, such investigations have not been conducted for Frost–Musulin diatomic potential. In this study, we obtain the approximate bound state solution of this potential via the functional analysis method. We also numerically solved the Schrödinger equation without any approximation to centrifugal term for the same potential. The comparisons between the results reveal the accuracy of our approximate results for short-range potential.

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