The Schrödinger Equation and the Confluent Hypergeometric Functions

2011 ◽  
pp. 69-75
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Hypergeometric Functions ◽  
Confluent Hypergeometric Functions

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.

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 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 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

scholarly journals Analytical Solution of the Schrödinger Equation with Spatially Varying Effective Mass for Generalised Hylleraas Potential

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Sanjib Meyur ◽  
Smarajit Maji ◽  
S. Debnath
Keyword(s):  
Schrödinger Equation ◽  
Effective Mass ◽  
Schrodinger Equation ◽  
Hypergeometric Functions ◽  
State Energy ◽  
Bound State ◽  
Exact Bound ◽  
Effective Mass Schrödinger Equation ◽  
Spatially Varying ◽  
Special Case

We have obtained exact solution of the effective mass Schrödinger equation for the generalised Hylleraas potential. The exact bound state energy eigenvalues and corresponding eigenfunctions are presented. The bound state eigenfunctions are obtained in terms of the hypergeometric functions. Results are also given for the special case of potential parameter.

scholarly journals Minimal Length Schrödinger Equation with Harmonic Potential in the Presence of a Magnetic Field

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
H. Hassanabadi ◽  
E. Maghsoodi ◽  
Akpan N. Ikot ◽  
S. Zarrinkamar
Keyword(s):  
Magnetic Field ◽  
Schrödinger Equation ◽  
Momentum Space ◽  
Schrodinger Equation ◽  
Hypergeometric Functions ◽  
Minimal Length ◽  
Wave Functions ◽  
Harmonic Potential ◽  
Energy Eigenvalues

Minimal length Schrödinger equation is investigated for harmonic potential in the presence of magnetic field and illustrates the wave functions in the momentum space. The energy eigenvalues are reported and the corresponding wave functions are calculated in terms of hypergeometric functions.

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

2010 ◽  
Vol 19 (01) ◽  
pp. 123-129 ◽  
Author(s):  
M. G. MIRANDA ◽  
GUO-HUA SUN ◽  
SHI-HAI DONG
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Bound States ◽  
Hypergeometric Functions ◽  
Jacobi Functions ◽  
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.

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

2016 ◽  
Vol 71 (1) ◽  
pp. 59-68 ◽  
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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