Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Journal of Physics A Mathematical and General

Analytical solution of the Schrodinger equation with linear confinement potential

Hirokazu Tezuka

Published Nov 21, 1991• 30 citations
Publisher

Abstract

It is shown that the Schrodinger equation with a linear confinement potential has an analytical solution. The existence of the analytical solution requires additional terms to the potential other than the linear one. The Coulomb-type potential is necessary to get non-zero energy eigenvalues. The author proposes an asymptotically linear potential, and shows that one can analytically solve the Schrodinger equation with this potential.

DOI
10.1088/0305-4470/24/22/010

Related papers

Analytical Solution Of Schrödinger Equation With Mie-Type Potential Using Factorisation Method

B. Okon, Eno E. Ituen, Oyebola O. Popoola +1 more

we have obtained the analytical solution of Schrodinger wave equation with Mie – type potential using factorization method. We have also obtained energy eigenvalues of our potential and the corresponding wave function using an ansatz and then compare the result to standard Laguerre’s differential equation. Under special cases our potential model reduces two well known potentials such as Coulomb and the Kratzer Feus potentials.

•Jan 1, 2013•6 citations

The Analytical Solution of the Schrödinger Particle in Multiparameter Potential

Taş, Ahmet, Ali Havare

In this study, we present analytical solutions of the Schrödinger equation with the Multiparameter potential containing the different types of physical potential via the asymptotic iteration method (AIM) by applying a Pekeris-type approximation to the centrifugal potential. For any n and l (states) quantum numbers, we get the bound state energy eigenvalues numerically and the corresponding eigenfunctions.Furthermore, we compare our results with the ones obtained in previous works and it is seen that our numerical results are in good agreement with the literature.

arXiv:1608.08571•Aug 29, 2016•0 citations

The Analytic Solution of Schrödinger Equation with Potential Function Superposed by Six Terms with Positive-power and Inverse-power Potentials

HU Xian-quan, Guang Luo, Cui Li-Peng +2 more

The analytic solution of the radial Schrodinger equation is studied by using the tight coupling condition of several positive-power and inverse-power potential functions in this article. Furthermore, the precisely analytic solutions and the conditions that decide the existence of analytic solution have been searched when the potential of the radial Schrodinger equation is V (r) = �1r 8 + �2r 3 + �3r 2 + �3r 1 + �2r 3 + �1r 4 . Generally speaking, there is only an approximate solution, but not analytic solution for Schrodinger equation with several potentials' superposition. However, the conditions that decide the existence of analytic solution have been found and the analytic solution and its energy level structure are obtained for the Schrodinger equation with the potential which is motioned above in this paper. According to the single-value, finite and continuous standard of wave function in a quantum system, the authors firstly solve the asymptotic solution through the radial coordinate r ! 1 and r ! 0; secondly, they make the asymptotic solutions combining with the series solutions nearby the neighborhood of irregular singularities; and then they compare the power series coefficients, deduce a series of analytic solutions ofthe stationary state wave function and corresponding energy level structure by tight coupling among the coefficients of potential functions for the radial Schrodinger equation; and lastly, they discuss the solutions and make conclusions.

Communications in Theoretical Physics•Mar 1, 2009•1 citation

Exact solutions of the Schrödinger equation with non-central potential by the Nikiforov–Uvarov method

F Yaşuk, C Berkdemir, A Berkdemir

The general solutions of the Schrödinger equation for a non-central potential are obtained by using the Nikiforov–Uvarov method. The Schrödinger equation with general non-central potential is separated into radial and angular parts, and energy eigenvalues and eigenfunctions are derived analytically. By making special selections, the non-central potential is reduced to Coulomb and Hartmann ring-shaped potentials, and the obtained results are compared with the solutions of Coulomb and Hartmann potentials given in the literature.

arXiv:quant-ph/0410144•Journal of Physics A Mathematical and General•Jul 6, 2005•92 citations

On Solutions of the Killingbeck Potential and Clarifying Comments on a Related Analytical Approach

Fatma Zohra Khaled, Mustafa Moumni, Mokhtar Falek

The work presents analytical solutions to the Schrodinger equation for the Killingbeck potential, a Hybrid model combining harmonic, linear and Coulomb terms, and which is also an approximate model of Yukawa-type potentials. The radial Schrodinger equation is solved by means of the series expansion method, thus yielding the exact expressions of both bound-states solutions and eigen-functions for the systems; these systems include quarkonium and confined hydrogen-like atoms in plasma environments. Furthermore, we offer a constructive commentary on the work of Obu et al. (East Eur. J. Phys. 3, 146-157, 2023), with the aim of clarifying a mathematical misstatement used in their analytical treatment of analogous systems.

quant-pharXiv:2607.19611•Jul 23, 2026

A Mathematical Method for Exact Analytical Solution of the Schrodinger Equation with Non-central Potential

Mohammad Reza Shojaei, ALI AKBAR RAJABI, Y. Momen +4 more

In this paper, we have studied the Schrödinger equation with central and non‐central potential. We have solved the Schrödinger equation in spherical coordinate analytically by using supersymmetric and shape invariance methods. Then we have found the radial and angular parts of wavefunction. Finally the energy eigenvalues of the Schrödinger equation with non‐central potential are obtained.

AIP conference proceedings•Jan 1, 2011•0 citations
It is shown that the Schrodinger equation with a linear confinement potential has an analytical solution. The existence of the analytical solution requires additional terms to the potential other than the linear one. The Coulomb-type potential is necessary to get non-zero energy eigenvalues. The author proposes an asymptotically linear potential, and shows that one can analytically solve the Schrodinger equation with this potential.
In this study, we present analytical solutions of the Schrödinger equation with the Multiparameter potential containing the different types of physical potential via the asymptotic iteration method (AIM) by applying a Pekeris-type approximation to the centrifugal potential. For any n and l (states) quantum numbers, we get the bound state energy eigenvalues numerically and the corresponding eigenfunctions.Furthermore, we compare our results with the ones obtained in previous works and it is seen that our numerical results are in good agreement with the literature.
The analytic solution of the radial Schrodinger equation is studied by using the tight coupling condition of several positive-power and inverse-power potential functions in this article. Furthermore, the precisely analytic solutions and the conditions that decide the existence of analytic solution have been searched when the potential of the radial Schrodinger equation is V (r) = �1r 8 + �2r 3 + �3r 2 + �3r 1 + �2r 3 + �1r 4 . Generally speaking, there is only an approximate solution, but not analytic solution for Schrodinger equation with several potentials' superposition. However, the conditions that decide the existence of analytic solution have been found and the analytic solution and its energy level structure are obtained for the Schrodinger equation with the potential which is motioned above in this paper. According to the single-value, finite and continuous standard of wave function in a quantum system, the authors firstly solve the asymptotic solution through the radial coordinate r ! 1 and r ! 0; secondly, they make the asymptotic solutions combining with the series solutions nearby the neighborhood of irregular singularities; and then they compare the power series coefficients, deduce a series of analytic solutions ofthe stationary state wave function and corresponding energy level structure by tight coupling among the coefficients of potential functions for the radial Schrodinger equation; and lastly, they discuss the solutions and make conclusions.
The general solutions of the Schrödinger equation for a non-central potential are obtained by using the Nikiforov–Uvarov method. The Schrödinger equation with general non-central potential is separated into radial and angular parts, and energy eigenvalues and eigenfunctions are derived analytically. By making special selections, the non-central potential is reduced to Coulomb and Hartmann ring-shaped potentials, and the obtained results are compared with the solutions of Coulomb and Hartmann potentials given in the literature.
The work presents analytical solutions to the Schrodinger equation for the Killingbeck potential, a Hybrid model combining harmonic, linear and Coulomb terms, and which is also an approximate model of Yukawa-type potentials. The radial Schrodinger equation is solved by means of the series expansion method, thus yielding the exact expressions of both bound-states solutions and eigen-functions for the systems; these systems include quarkonium and confined hydrogen-like atoms in plasma environments. Furthermore, we offer a constructive commentary on the work of Obu et al. (East Eur. J. Phys. 3, 146-157, 2023), with the aim of clarifying a mathematical misstatement used in their analytical treatment of analogous systems.