scholarly journals Exact Solution of the Schrödinger Equation for Composition of Coulomb and Oscillator Potentials

2021 ◽  
Vol 24 (2) ◽  
pp. 203-206
Author(s):  
V. V. Kudryashov ◽  
A. V. Baran
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Hypergeometric Functions ◽  
Continuous Solution ◽  
Energy Levels ◽  
Discrete Energy ◽  
Radial Wave ◽  
Confluent Hypergeometric Functions ◽  
Radial Schrödinger Equation ◽  
Boundary Distance

The spherically symmetric potential is considered whose dependence on the distance r is described by the smooth composition of Coulomb at r < r0 and oscillator at r > r0 potentials. The boundary distance r0 is determined by the parameters of these potentials. The exact continuous solution of the radial Schrödinger equation is expressed in terms of the confluent hypergeometric functions. The discrete energy levels are obtained. The graphic illustrations for the energy spectrum and the radial wave functions are presented.

scholarly journals A Conditionally Integrable Bi-confluent Heun Potential Involving Inverse Square Root and Centrifugal Barrier Terms

2018 ◽  
Vol 73 (5) ◽  
pp. 407-414 ◽  
Author(s):  
Tigran A. Ishkhanyan ◽  
Vladimir P. Krainov ◽  
Artur M. Ishkhanyan
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Bound States ◽  
Hypergeometric Functions ◽  
Energy Levels ◽  
Hermite Functions ◽  
Exact Equation ◽  
Square Root ◽  
Centrifugal Barrier ◽  
Confluent Hypergeometric Functions

AbstractWe present a conditionally integrable potential, belonging to the bi-confluent Heun class, for which the Schrödinger equation is solved in terms of the confluent hypergeometric functions. The potential involves an attractive inverse square root term ~x−1/2 with arbitrary strength and a repulsive centrifugal barrier core ~x−2 with the strength fixed to a constant. This is a potential well defined on the half-axis. Each of the fundamental solutions composing the general solution of the Schrödinger equation is written as an irreducible linear combination, with non-constant coefficients, of two confluent hypergeometric functions. We present the explicit solution in terms of the non-integer order Hermite functions of scaled and shifted argument and discuss the bound states supported by the potential. We derive the exact equation for the energy spectrum and approximate that by a highly accurate transcendental equation involving trigonometric functions. Finally, we construct an accurate approximation for the bound-state energy levels.

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.

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.

A new determination of the parameters of the optimized Heine–Abarenkov potential for 27 elements

1976 ◽  
Vol 54 (23) ◽  
pp. 2348-2354 ◽  
Author(s):  
E. R. Cowley
Keyword(s):  
Schrödinger Equation ◽  
Numerical Integration ◽  
Coulomb Potential ◽  
Schrodinger Equation ◽  
Energy Levels ◽  
Inhomogeneity Correction ◽  
Radial Schrödinger Equation

We have calculated the energy levels of the truncated Coulomb potential using numerical integration of the radial Schrödinger equation, rather than interpolation in tables. The results are used to give the parameters of the optimized Heine–Abarenkov potential for 27 elements. Various methods of weighting other contributions to the potential in the solid are used, and the inhomogeneity correction introduced by Ballentine and Gupta is discussed.

scholarly journals Fourth derivative singularly P-stable method for the numerical solution of the Schrödinger equation

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Ali Shokri ◽  
Higinio Ramos ◽  
Mohammad Mehdizadeh Khalsaraei ◽  
Fikret A. Aliev ◽  
Martin Bohner
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Energy Levels ◽  
Stable Method ◽  
Implicit Methods ◽  
The Family ◽  
Radial Schrödinger Equation ◽  
Obrechkoff Methods ◽  
Fourth Derivative ◽  
Error Terms

AbstractIn this paper, we construct a method with eight steps that belongs to the family of Obrechkoff methods. Due to the explicit nature of the new method, not only does it not require another method as predictor, but it can also be considered as a suitable predictive technique to be used with implicit methods. Periodicity and error terms are studied when applied to solve the radial Schrödinger equation, considering different energy levels. We show its advantages in terms of accuracy, consistency, and convergence in comparison with other methods of the same order appearing in the literature.

The bound state solutions of the D-dimensional Schrödinger equation for the Hulthén potential within SUSY quantum mechanics

2017 ◽  
Vol 26 (05) ◽  
pp. 1750028 ◽  
Author(s):  
H. I. Ahmadov ◽  
M. V. Qocayeva ◽  
N. Sh. Huseynova
Keyword(s):  
Quantum Mechanics ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Dimensional Space ◽  
Energy Levels ◽  
Wave Functions ◽  
Hulthén Potential ◽  
Radial Wave ◽  
Energy Eigenvalues ◽  
Hulthen Potential

In this paper, the analytical solutions of the [Formula: see text]-dimensional hyper-radial Schrödinger equation are studied in great detail for the Hulthén potential. Within the framework, a novel improved scheme to surmount centrifugal term, the energy eigenvalues and corresponding radial wave functions are found for any [Formula: see text] orbital angular momentum case within the context of the Nikiforov–Uvarov (NU) and supersymmetric quantum mechanics (SUSY QM) methods. In this way, based on these methods, the same expressions are obtained for the energy eigenvalues, and the expression of radial wave functions transforming each other is demonstrated. The energy levels are worked out and the corresponding normalized eigenfunctions are obtained in terms of orthogonal polynomials for arbitrary [Formula: see text] states for [Formula: see text]-dimensional space.

scholarly journals METODE ELEMEN HINGGA UNTUK PENYELESAIAN PERSAMAAN SCHRÖDINGER ATOM HIDROGENIK

2006 ◽  
Vol 7 (1) ◽  
pp. 11-23
Author(s):  
Paken Pandiangan ◽  
Supriyadi Supriyadi ◽  
A Arkundato
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Hydrogen Atom ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Energy Levels ◽  
Wave Functions ◽  
Radial Wave ◽  
Action Integral ◽  
Method Approach

The research computed the energy levels and radial wave functions of the  Hydrogen Atom. The method used for computation was FEM (finite element method). Using the variational method approach, FEM was applied to the action integral of  Schrödinger equation. This lead to the eigenvalue equation in the form of  global matrix equation. The results of computation were depended on boundary of the action integral of Schrödinger equation and number of elements. For boundary 0 - 100a0 and 100 elements,  they were the realistic and best choice of computation to the closed  analytic results. The computation of first five energy levels resulted E1 = -0.99917211 R∞, E2 = -0.24984445 R∞, E3 = -0.11105532 R∞,           E4 = -0.06247405 R∞ and  E5 = -0.03998598 R∞ where 1 R∞ = 13.6 eV. They had relative error under 0.1% to the analytic results.  

scholarly journals A singular Lambert-W Schrödinger potential exactly solvable in terms of the confluent hypergeometric functions

2016 ◽  
Vol 31 (33) ◽  
pp. 1650177 ◽  
Author(s):  
A. M. Ishkhanyan
Keyword(s):  
Exact Solution ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Bound States ◽  
Hypergeometric Functions ◽  
Fundamental Solutions ◽  
Lambert W Function ◽  
Confluent Hypergeometric Functions ◽  
Heun Function ◽  
The One

We introduce two potentials explicitly given by the Lambert-W function for which the exact solution of the one-dimensional stationary Schrödinger equation is written through the first derivative of a double-confluent Heun function. One of these potentials is a singular potential that behaves as the inverse square root in the vicinity of the origin and vanishes exponentially at the infinity. The exact solution of the Schrödinger equation for this potential is given through fundamental solutions each of which presents an irreducible linear combination of two confluent hypergeometric functions. Since the potential is effectively a short-range one, it supports only a finite number of bound states.

scholarly journals WKB Energy Expression for the Radial Schrödinger Equation with a Generalized Pseudoharmonic Potential

2020 ◽  
pp. 13-20 ◽  
Author(s):  
E. Omugbe ◽  
O. E. Osafile ◽  
I. B. Okon
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Energy Levels ◽  
Turning Points ◽  
Wkb Method ◽  
Energy Expression ◽  
Pseudoharmonic Potential ◽  
Special Cases ◽  
Radial Schrödinger Equation ◽  
Exact Energy

In this paper, we applied the semi-classical quantization approximation method to solve the radial Schrödinger equation with a generalized Pseudoharmonic potential. The four turning points problem within the framework of the Wentzel-Kramers-Brillouin (WKB) method was transformed into two turning points and subsequently, the energy spectrum was obtained. Some special cases of the generalized Pseudoharmonic potential are presented. The WKB approximation approach reproduces the exact energy expression obtained with several analytical methods in the literature.  The values of the energy levels for some selected diatomic molecules (N2, CO, NO, CH) obtained numerically are in excellent agreement with those from previous works in the literature.

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