THE SOLUTION OF THE SECOND PÖSCHL–TELLER LIKE POTENTIAL BY NIKIFOROV–UVAROV METHOD

The bound states of the Schrödinger equation for a second Pöschl–Teller like potential are obtained exactly using the Nikiforov–Uvarov method. It is found that the solutions can be explicitly expressed in terms of the Jacobi functions or hypergeometric functions. The complicated normalized wavefunctions are found.

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

Zeitschrift für Naturforschung A ◽

10.1515/zna-2017-0314 ◽

2018 ◽

Vol 73 (5) ◽

pp. 407-414 ◽

Cited By ~ 1

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.

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EXACT SOLUTIONS OF THE MODIFIED KRATZER POTENTIAL PLUS RING-SHAPED POTENTIAL IN THE D-DIMENSIONAL SCHRÖDINGER EQUATION BY THE NIKIFOROV–UVAROV METHOD

International Journal of Modern Physics C ◽

10.1142/s0129183108012030 ◽

2008 ◽

Vol 19 (02) ◽

pp. 221-235 ◽

Cited By ~ 62

Author(s):

SAMEER M. IKHDAIR ◽

RAMAZAN SEVER

Keyword(s):

Schrödinger Equation ◽

Schrodinger Equation ◽

Bound States ◽

Laguerre Polynomials ◽

Three Dimensional ◽

Wave Functions ◽

Modified Kratzer Potential ◽

Energy Bound ◽

Uvarov Method ◽

Exact Energy

We present analytically the exact energy bound-states solutions of the Schrödinger equation in D dimensions for a recently proposed modified Kratzer plus ring-shaped potential by means of the Nikiforov–Uvarov method. We obtain an explicit solution of the wave functions in terms of hyper-geometric functions (Laguerre polynomials). The results obtained in this work are more general and true for any dimension which can be reduced to the well-known three-dimensional forms given by other works.

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

Modern Physics Letters A ◽

10.1142/s0217732316501777 ◽

2016 ◽

Vol 31 (33) ◽

pp. 1650177 ◽

Cited By ~ 17

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.

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Closed Analytical Solutions of the D-Dimensional Schrödinger Equation with Deformed Woods–Saxon Potential Plus Double Ring-Shaped Potential

Zeitschrift für Naturforschung A ◽

10.1515/zna-2015-0136 ◽

2016 ◽

Vol 71 (1) ◽

pp. 59-68 ◽

Cited By ~ 7

Author(s):

Mohamed Chabab ◽

Abdelwahed El Batoul ◽

Mustapha Oulne

Keyword(s):

Schrödinger Equation ◽

Schrodinger Equation ◽

Hypergeometric Functions ◽

Asymptotic Iteration Method ◽

Saxon Potential ◽

Double Ring ◽

Ring Shape ◽

Rule Approach ◽

Uvarov Method ◽

Pekeris Approximation

AbstractBy employing the Pekeris approximation, the D-dimensional Schrödinger equation is solved for the nuclear deformed Woods–Saxon potential plus double ring-shaped potential within the framework of the asymptotic iteration method (AIM). The energy eigenvalues are given in a closed form, and the corresponding normalised eigenfunctions are obtained in terms of hypergeometric functions. Our general results reproduce many predictions obtained in the literature, using the Nikiforov–Uvarov method (NU) and the improved quantisation rule approach, particularly those derived by considering Woods–Saxon potential without deformation and/or without ring shape interaction.

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Exact solutions of the D-dimensional Schrödinger equation for a ring-shaped pseudoharmonic potential

Open Physics ◽

10.2478/s11534-008-0024-2 ◽

2008 ◽

Vol 6 (3) ◽

Cited By ~ 21

Author(s):

Sameer Ikhdair ◽

Ramazan Sever

Keyword(s):

Schrödinger Equation ◽

Schrodinger Equation ◽

Bound States ◽

Jacobi Polynomials ◽

Wave Functions ◽

Three Dimensions ◽

Eigenvalues And Eigenfunctions ◽

Energy Eigenvalues ◽

Pseudoharmonic Potential ◽

Uvarov Method

AbstractA new non-central potential, consisting of a pseudoharmonic potential plus another recently proposed ring-shaped potential, is solved. It has the form $$ V(r,\theta ) = \tfrac{1} {8}\kappa r_e^2 \left( {\tfrac{r} {{r_e }} - \tfrac{{r_e }} {r}} \right)^2 + \tfrac{{\beta cos^2 \theta }} {{r^2 sin^2 \theta }} $$. The energy eigenvalues and eigenfunctions of the bound-states for the Schrödinger equation in D-dimensions for this potential are obtained analytically by using the Nikiforov-Uvarov method. The radial and angular parts of the wave functions are obtained in terms of orthogonal Laguerre and Jacobi polynomials. We also find that the energy of the particle and the wave functions reduce to the energy and the wave functions of the bound-states in three dimensions.

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