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

American Journal of Physics

Approximating the finite square well with an infinite well: Energies and eigenfunctions

Barry I. Barker, Grayson H. Rayborn, Juliette W. Ioup, George E. Ioup

Published Nov 1, 1991• 34 citations
Publisher

Abstract

Polynomial expansions are used to approximate the equations of the eigenvalues of the Schrödinger equation for a finite square potential well. The technique results in discrete, approximate eigenvalues which, it is shown, are identical to the corresponding eigenvalues of a wider, infinite well. The width of this infinite well is easy to calculate; indeed, the increase in width over that of the finite well is simply the original width divided by the well strength. The eigenfunctions of this wider, infinite well, which to first order has the same width for the ground state and all excited states, are also good approximations to the exact eigenfunctions of the finite well. These approximate eigenfunctions and eigenvalues are compared to accurate numeric calculations and to other approximations from the literature.

DOI
10.1119/1.16644

Related papers

Solving Schrödinger Equation for Finite Potential Well Using the Iterative Method

Laith Abdul Aziz Al-Ani, Russul K. Abid

The quantum Finite Square Well (FSW) model is a well-known topic in most quantum mechanics (QM) books.A couple of equations can be derived from one dimensional Schrodinger equation for a finite potential well for describing the bound Eigen states within the well.Sometimes the FSW problem does not have an exact solution, yet, there are in fact exact solutions.In this work a computational techniques is adopted to find the exact solution for FSW.To achieve this computational solution, a computer program has been written in Basic language for calculating the Eigen state energy for a particle confined in finite potential well using the iterative method (IM).Six values of potential width have been studied with potential depth.The results showed that wider potential width led us to more bound states, while narrower potential width led us to less bound states.Six values of potential depth have been studied with potential width.The results showed that larger potential depth led us to more bound states, while smaller potential width led us to less bound states.A Comparison Between Finite and Infinite Potential Well has been also presented.The result of comparison showed that energy levels of an infinite well are much higher than that the corresponding energy levels for finite potential well.In general, the matching between the results of the iterative method and the graphical method (GM) proving that the iteration method can be regarded as a useful tool for describing the solutions of the 1-dimensional FSW problem.

Journal of Al-Nahrain University-Science•Dec 1, 2019•2 citations

Bound states in a semi-infinite square potential well

Nivaldo A. Lemos

The finite square potential well is a staple problem in introductory quantum mechanics. There is an extensive literature on the determination of the allowed energies, which requires the solution of a transcendental equation by numerical, graphical or approximate analytic methods. Here we investigate the less explored problem of a particle in a semi-infinite potential well. The energy eigenvalues, which are also determined by a transcendental equation, are found by a standard graphical method, and a simple rule that yields the number of stationary states is provided. Next a simplification of the aforementioned transcendental equation is attempted. During the process pitfalls are encountered and a purportedly simpler graphical treatment of the problem given in the solutions manual to a fine textbook is shown to be flawed. A more careful analysis brings forth the correct simplification, which is shown to be particularly suitable for finding highly accurate approximations to the energy levels. Finally, a class of exact solutions is produced, the associated normalized eigenfunctions are constructed and the probability of finding the particle inside the well is computed.

quant-pharXiv:2603.08394•Mar 10, 2026•0 citations

Modeling Time-Independent Schrödinger Equation for an Infinite Potential Well Using Numerov and Matrix Methods

Dalal Y. Saad, Fawzi A. Ikraiam

The objective of this study is to numerically solve and apply two approximate methods to investigate the Time Independent Schrödinger Equation (TISE) in one dimension for an infinite potential square well. These two numerical methods are the Numerov Method (NM) and Matrix Method (MM). As a simulation tool, MATLAB, a high-level programming language and an efficient simulation tool, is used for modeling and solving TISE in one dimension. Exact analytical solutions for these potential functions are obtained and compared with numerical solutions and computational techniques. The energy eigenvalues and Eigen functions of a particle (such as an electron) restricted to move inside this potential are discussed as an illustration. The numerically calculated energies of several states with increasing numbers of points were obtained from both methods and compared with the simulation results of the exact solution. As an exemplary case, the first five wave functions are accurately determined numerically where the discreteness is found since the wave function vanishes at the boundary. The obtained results show very good agreement and the similarity is clearly confirmed between the three cases. This agreement confirms that this approach was highly accurate and efficient. The accuracy and the convergence of the numerical obtained results were easily checked. The stability of these methods is due to the fact that there are no restrictions on the time steps to be taken. The merits of these numerical methods are to avoid a huge expense in time when solving Schrödinger equation.

Libyan Journal of Science &Technology•Mar 1, 2025•0 citations

Bound states in a semi-infinite square potential well

Nivaldo Agostinho Lemos

Abstract The finite square potential well is a staple problem in introductory quantum mechanics. There is an extensive literature on the determination of the allowed energies, which requires the solution of a transcendental equation by numerical, graphical or approximate analytic methods. Here we investigate the less explored problem of a particle in a semi-infinite potential well. The energy eigenvalues, which are also determined by a transcendental equation, are found by a standard graphical method, and a simple rule that yields the number of stationary states is provided. Next a simplification of the aforementioned transcendental equation is attempted. During the process pitfalls are encountered and a purportedly simpler graphical treatment of the problem given in the solutions manual to a fine textbook is shown to be flawed. A more careful analysis brings forth the correct simplification, which is shown to be particularly suitable for finding highly accurate approximations to the energy levels. Finally, a class of exact solutions is produced, the associated normalized eigenfunctions are constructed and the probability of finding the particle inside the well is computed.

Revista Brasileira de Ensino de Física•Jan 1, 2026•0 citations

Simulation of Time Independent Schrödinger Equation for Finite Potential Well Using the Graphical Solution Method

Dalal Y. Saad, Fawzi A. Ikraiam

This paper investigates applying the approximate method; Graphical Solution Method (GSM), to theoretically solve the Time Independent Schrödinger Equation (TISE) in one dimension for a finite square well using MATLAB. With just few lines of MATLAB coding, calculating and plotting accurate eigenvalues (energy), eigenvectors (wave functions) and the bound eigenstates are possible for the finite square well of a negative potential (depth of the well) of -400 eV and a well width of 0.1 nm for an electron confined to this quantum well. These eigenvalues, eigenvectors and eigenstates are obtained and discussed. The found energy eigenvalues and states are discrete and yield physical acceptable solutions. The even and odd solutions of the TISE are also considered. The graphical solutions for the finite potential well are shown. The locations of discrete eigenvalues for even and odd solutions are also presented. These eigenvalues are tested confirming the correct eigenfunctions. The precision of these solutions depend on well width L and on the interval dx used to integrate the equation. Exact analytical solutions for this case are obtained and compared with results from the GSM. The accuracy and the convergence of the numerical results are easily checked. The results showed that the GSM can be considered as a suitable mean for determining the one dimensional solutions for the finite square well.

Journal of Pure & Applied Sciences•Oct 2, 2023•0 citations

1D potential wells of the form V x < a = − V o 1 − x n a n

Abby Corrigan, Janine Shertzer

Abstract We consider the finite potential well defined by V x < a = − V o 1 − x n a n ,where n is an integer. The Schrödinger equation is recast into dimensionless form; the size of the well is characterized by a single parameter α = 2 m V o a / ℏ . We use three different mathematical techniques to solve for the energy eigenvalues and wave functions. Analytic solutions are given for the triangular well ( n = 1), the parabolic well ( n = 2), and the square well ( n → ∞). The finite element method is used to obtain accurate numerical results for 1 ⩽ n ⩽ ∞ over a full range of α . Finally, both the energy and wave function are expanded in a Taylor series in order to study the analytic behavior of the ground state energy (as a function of n ) as α → 0.

European Journal of Physics•Dec 4, 2020•1 citation
Polynomial expansions are used to approximate the equations of the eigenvalues of the Schrödinger equation for a finite square potential well. The technique results in discrete, approximate eigenvalues which, it is shown, are identical to the corresponding eigenvalues of a wider, infinite well. The width of this infinite well is easy to calculate; indeed, the increase in width over that of the finite well is simply the original width divided by the well strength. The eigenfunctions of this wider, infinite well, which to first order has the same width for the ground state and all excited states, are also good approximations to the exact eigenfunctions of the finite well. These approximate eigenfunctions and eigenvalues are compared to accurate numeric calculations and to other approximations from the literature.
The quantum Finite Square Well (FSW) model is a well-known topic in most quantum mechanics (QM) books.A couple of equations can be derived from one dimensional Schrodinger equation for a finite potential well for describing the bound Eigen states within the well.Sometimes the FSW problem does not have an exact solution, yet, there are in fact exact solutions.In this work a computational techniques is adopted to find the exact solution for FSW.To achieve this computational solution, a computer program has been written in Basic language for calculating the Eigen state energy for a particle confined in finite potential well using the iterative method (IM).Six values of potential width have been studied with potential depth.The results showed that wider potential width led us to more bound states, while narrower potential width led us to less bound states.Six values of potential depth have been studied with potential width.The results showed that larger potential depth led us to more bound states, while smaller potential width led us to less bound states.A Comparison Between Finite and Infinite Potential Well has been also presented.The result of comparison showed that energy levels of an infinite well are much higher than that the corresponding energy levels for finite potential well.In general, the matching between the results of the iterative method and the graphical method (GM) proving that the iteration method can be regarded as a useful tool for describing the solutions of the 1-dimensional FSW problem.
The finite square potential well is a staple problem in introductory quantum mechanics. There is an extensive literature on the determination of the allowed energies, which requires the solution of a transcendental equation by numerical, graphical or approximate analytic methods. Here we investigate the less explored problem of a particle in a semi-infinite potential well. The energy eigenvalues, which are also determined by a transcendental equation, are found by a standard graphical method, and a simple rule that yields the number of stationary states is provided. Next a simplification of the aforementioned transcendental equation is attempted. During the process pitfalls are encountered and a purportedly simpler graphical treatment of the problem given in the solutions manual to a fine textbook is shown to be flawed. A more careful analysis brings forth the correct simplification, which is shown to be particularly suitable for finding highly accurate approximations to the energy levels. Finally, a class of exact solutions is produced, the associated normalized eigenfunctions are constructed and the probability of finding the particle inside the well is computed.
The objective of this study is to numerically solve and apply two approximate methods to investigate the Time Independent Schrödinger Equation (TISE) in one dimension for an infinite potential square well. These two numerical methods are the Numerov Method (NM) and Matrix Method (MM). As a simulation tool, MATLAB, a high-level programming language and an efficient simulation tool, is used for modeling and solving TISE in one dimension. Exact analytical solutions for these potential functions are obtained and compared with numerical solutions and computational techniques. The energy eigenvalues and Eigen functions of a particle (such as an electron) restricted to move inside this potential are discussed as an illustration. The numerically calculated energies of several states with increasing numbers of points were obtained from both methods and compared with the simulation results of the exact solution. As an exemplary case, the first five wave functions are accurately determined numerically where the discreteness is found since the wave function vanishes at the boundary. The obtained results show very good agreement and the similarity is clearly confirmed between the three cases. This agreement confirms that this approach was highly accurate and efficient. The accuracy and the convergence of the numerical obtained results were easily checked. The stability of these methods is due to the fact that there are no restrictions on the time steps to be taken. The merits of these numerical methods are to avoid a huge expense in time when solving Schrödinger equation.
Abstract The finite square potential well is a staple problem in introductory quantum mechanics. There is an extensive literature on the determination of the allowed energies, which requires the solution of a transcendental equation by numerical, graphical or approximate analytic methods. Here we investigate the less explored problem of a particle in a semi-infinite potential well. The energy eigenvalues, which are also determined by a transcendental equation, are found by a standard graphical method, and a simple rule that yields the number of stationary states is provided. Next a simplification of the aforementioned transcendental equation is attempted. During the process pitfalls are encountered and a purportedly simpler graphical treatment of the problem given in the solutions manual to a fine textbook is shown to be flawed. A more careful analysis brings forth the correct simplification, which is shown to be particularly suitable for finding highly accurate approximations to the energy levels. Finally, a class of exact solutions is produced, the associated normalized eigenfunctions are constructed and the probability of finding the particle inside the well is computed.
This paper investigates applying the approximate method; Graphical Solution Method (GSM), to theoretically solve the Time Independent Schrödinger Equation (TISE) in one dimension for a finite square well using MATLAB. With just few lines of MATLAB coding, calculating and plotting accurate eigenvalues (energy), eigenvectors (wave functions) and the bound eigenstates are possible for the finite square well of a negative potential (depth of the well) of -400 eV and a well width of 0.1 nm for an electron confined to this quantum well. These eigenvalues, eigenvectors and eigenstates are obtained and discussed. The found energy eigenvalues and states are discrete and yield physical acceptable solutions. The even and odd solutions of the TISE are also considered. The graphical solutions for the finite potential well are shown. The locations of discrete eigenvalues for even and odd solutions are also presented. These eigenvalues are tested confirming the correct eigenfunctions. The precision of these solutions depend on well width L and on the interval dx used to integrate the equation. Exact analytical solutions for this case are obtained and compared with results from the GSM. The accuracy and the convergence of the numerical results are easily checked. The results showed that the GSM can be considered as a suitable mean for determining the one dimensional solutions for the finite square well.
Abstract We consider the finite potential well defined by V x < a = − V o 1 − x n a n ,where n is an integer. The Schrödinger equation is recast into dimensionless form; the size of the well is characterized by a single parameter α = 2 m V o a / ℏ . We use three different mathematical techniques to solve for the energy eigenvalues and wave functions. Analytic solutions are given for the triangular well ( n = 1), the parabolic well ( n = 2), and the square well ( n → ∞). The finite element method is used to obtain accurate numerical results for 1 ⩽ n ⩽ ∞ over a full range of α . Finally, both the energy and wave function are expanded in a Taylor series in order to study the analytic behavior of the ground state energy (as a function of n ) as α → 0.