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.

The second Exton potential for the Schrödinger equation

2019 ◽  
Vol 34 (24) ◽  
pp. 1950195
Author(s):  
Artur M. Ishkhanyan ◽  
Jacek Karwowski
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Bound States ◽  
Fractional Power ◽  
Fundamental Solutions ◽  
Transcendental Equation ◽  
Quantization Condition ◽  
Hermite Functions ◽  
Energy Quantization ◽  
Half Axis

Analytical solutions of the Schrödinger equation with a singular, fractional-power potential, referred to as the second Exton potential, are derived and analyzed. The potential is defined on the positive half-axis and supports an infinite number of bound states. It is conditionally integrable and belongs to a biconfluent Heun family. The fundamental solutions are expressed as irreducible linear combinations of two Hermite functions of a scaled and shifted argument. The energy quantization condition results from the boundary condition imposed at the origin. For the exact eigenvalues, which are solutions of a transcendental equation involving two Hermite functions, highly accurate approximation by simple closed-form expressions is derived. The potential is a good candidate for the description of quark–antiquark interaction.

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 Bound states of purely relativistic nature

2019 ◽  
Vol 204 ◽  
pp. 01014 ◽  
Author(s):  
V.A. Karmanov ◽  
J. Carbonell ◽  
H. Sazdjian
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Bound States ◽  
Energy Levels ◽  
Massive Particle ◽  
Structure Constant ◽  
Fine Structure Constant ◽  
Balmer Series ◽  
Photon Exchange ◽  
Exchange Form

Two particles interacting by photon exchange, form the bound states predicted by the non-relativistic Schrödinger equation with the Coulomb potential (Balmer series). More than 60 years ago, in the solutions of relativistic Bethe-Salpeter equation, in addition to the Balmer series, were found another series of energy levels. These new series, appearing when the fine structure constant α is large enough (α > π/4), are not predicted by the Schrödinger equation. However, this new (non-Balmer) states can hardly exist in nature, since in order to create a strong e.m. field with α > π/4 a point-like charge Z > 107 is needed. The nuclei having this charge, though exist starting with bohrium, are far from to be point-like. In the present paper, we analyze the more realistic case of a strong interaction created by exchange of a massive particle. It turns out that in the framework of the Bethe-Salpeter equation this interaction still generates a series of new relativistic states, which are similar to those of the massless exchange case, and which are absent in the Schrödinger equation. The properties of these solutions are studied. Their existence in nature seems possible.

Wave Function Properties and Shifted Energy Levels of Bound States in a Plasma

1991 ◽  
Vol 46 (7) ◽  
pp. 583-589 ◽  
Author(s):  
H. Lehmann ◽  
W. Ebeling
Keyword(s):  
Wave Function ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Bound States ◽  
Energy Levels ◽  
Plasma Composition ◽  
Energy Eigenvalues ◽  
Electron Positron ◽  
New Energy ◽  
Variation Procedure

On the basis of earlier work we show a simple way to estimate the properties of bound states in a plasma. The Bethe-Salpeter equation is approximated by an effective Schrodinger equation. The energy eigenvalues are found via a variation procedure. The treatment is applicated to helium-like bound states and excited hydrogen-like states. The effect of the new energy eigenvalues on the plasma composition is discussed for the symmetrical electron-positron plasma.

scholarly journals FOUR-PARTICLE ANYON EXCITON: BOSON APPROXIMATION

1995 ◽  
Vol 09 (02) ◽  
pp. 123-133 ◽  
Author(s):  
M. E. Portnoi ◽  
E. I. Rashba
Keyword(s):  
Angular Momentum ◽  
Electron Density ◽  
Ground State ◽  
Hypergeometric Functions ◽  
Energy Levels ◽  
Confluent Hypergeometric Functions ◽  
Full Symmetry ◽  
Hole Confinement ◽  
Boson Approximation

A theory of anyon excitons consisting of a valence hole and three quasielectrons with electric charges –e/3 is presented. A full symmetry classification of the k = 0 states is given, where k is the exciton momentum. The energy levels of these states are expressed by quadratures of confluent hypergeometric functions. It is shown that the angular momentum L of the exciton ground state depends on the distance between the electron and hole confinement planes and takes the values L = 3n, where n is an integer. With increasing k the electron density shows a spectacular splitting on bundles. At first a single anyon splits off of the two-anyon core, and finally all anyons become separated.

Analytical solution of the Shredinger equation for the linear combination of the Manning-Rosen and the class of Yukawa potential

2021 ◽  
pp. 157-169
Author(s):  
G.A. Bayramova ◽  
Keyword(s):  
Analytical Solution ◽  
Bound States ◽  
Hypergeometric Functions ◽  
Yukawa Potential ◽  
Energy Levels ◽  
Energy Eigenvalue ◽  
Radial Wave ◽  
Rosen Potential ◽  
Analytical Expressions ◽  
Potential Parameters

In the present work, an analytical solution for bound states of the modified Schrödinger equation is found for the new supposed combined Manning-Rosen potential plus the Yukawa class. To overcome the difficulties arising in the case l ≠ 0 in the centrifugal part of the Manning-Rosen potential plus the Yukawa class for bound states, we applied the developed approximation. Analytical expressions for the energy eigenvalue and the corresponding radial wave functions for an arbitrary value l ≠ 0 of the orbital quantum number are obtained. And also obtained eigenfunctions expressed in terms of hypergeometric functions. It is shown that energy levels and eigenfunctions are very sensitive to the choice of potential parameters.

Solution of the Schrödinger equation for bound states in closed form

1982 ◽  
Vol 26 (1) ◽  
pp. 662-664 ◽  
Author(s):  
Edgardo Gerck ◽  
Jason A. C. Gallas ◽  
Augusto B. d'Oliveira
Keyword(s):  
Schrödinger Equation ◽  
Closed Form ◽  
Schrodinger Equation ◽  
Bound States
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