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

Canadian Journal of Physics

Approximate solutions of the transcendental equation for the square quantum wells by finding the real root of the cubic equation

Jean-Marc Noël, Luc E. Levesque

Published Apr 14, 2023• 1 citation
Publisher

Abstract

The energy eigenvalues E n in finite square quantum wells (SQW) cannot be found using an analytic expression. As a result, numerical methods are normally used to find the eigenvalues from a transcendental equation. In this report, it will be shown that the eigenvalue solution for a given state consists in finding the only real positive root of a depressed trinomial polynomial of third order, which is as easy to solve as a quadratic equation. The method proposed can also be applied for semi-infinite, finite, and asymmetric SQW, which are often presented in quantum mechanics textbooks at the undergraduate level. The proposed method can be applied during an exam when programmable calculators are not allowed as the real root of the trinomial polynomial can be found using the formula for the cubic equation found nearly 500 years ago.

DOI
10.1139/cjp-2023-0004

Related papers

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

A simpler graphical solution and an approximate formula for energy eigenvalues in finite square quantum wells

F. M. S. Lima

The computation of allowed energy levels for a particle bounded to a finite square well potential is ubiquitous in modern physics and introductory quantum mechanics textbooks. For stationary bound states, the matching conditions for the wave functions lead to a pair of transcendental equations whose roots correspond to the energy eigenvalues. However, the graphical solutions available do not make clear the dependence of the energies on the well potential parameters. In this note, I present a simpler graphical solution involving only one dimensionless parameter that determines a straight-line crossing identical sinusoidal curves. I then reduce this solution to a single cosine curve, and from a three-point interpolation, I derive an approximate formula for all energy levels valid for any square quantum well and that demands only a pocket calculator.

American Journal of Physics•Oct 21, 2020•12 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

Particle a in One-Dimensional Finite and Semi-Infinite Well Revisited

Enrique Peacock-López

For the quantum particle in a finite and semi-infinite well, we scale space and energy to consider a dimensionless Schrdinger equation that can be solved for the allowed energy levels. In contrast to traditional approaches that determine the energies from dimensional transcendental equations associated with the boundary conditions, we solve for the square root of the dimensionless energies, which are also solutions of dimensionless transcendental equations. Our ability to define classes of wells depends on the value of the square root of the dimensionless well depth. We are also able to determine the maximum number of energy levels quite easily, and, besides deriving the relevant transcendental equations, we obtain asimple expression for the corresponding wave functions, which can be plotted using the numerical values of the allowed energies.

The Chemical Educator•Feb 1, 2011•0 citations

Exact and approximate energy spectrum for the finite square well and related potentials

O. F. de Alcantara Bonfim, David J. Griffiths

We investigate the problem of a quantum particle in a one-dimensional finite square well. In the standard approach the allowed energies are determined implicitly as the solutions to a transcendental equation. We obtain the spectrum analytically as the solution to a pair of parametric equations and algebraically using a remarkably accurate approximation to the cosine function. The approach is also applied to a variety of other quantum wells.

American Journal of Physics•Dec 21, 2005•41 citations

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

Barry I. Barker, Grayson H. Rayborn, Juliette W. Ioup +1 more

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.

American Journal of Physics•Nov 1, 1991•34 citations
The energy eigenvalues E n in finite square quantum wells (SQW) cannot be found using an analytic expression. As a result, numerical methods are normally used to find the eigenvalues from a transcendental equation. In this report, it will be shown that the eigenvalue solution for a given state consists in finding the only real positive root of a depressed trinomial polynomial of third order, which is as easy to solve as a quadratic equation. The method proposed can also be applied for semi-infinite, finite, and asymmetric SQW, which are often presented in quantum mechanics textbooks at the undergraduate level. The proposed method can be applied during an exam when programmable calculators are not allowed as the real root of the trinomial polynomial can be found using the formula for the cubic equation found nearly 500 years ago.
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 computation of allowed energy levels for a particle bounded to a finite square well potential is ubiquitous in modern physics and introductory quantum mechanics textbooks. For stationary bound states, the matching conditions for the wave functions lead to a pair of transcendental equations whose roots correspond to the energy eigenvalues. However, the graphical solutions available do not make clear the dependence of the energies on the well potential parameters. In this note, I present a simpler graphical solution involving only one dimensionless parameter that determines a straight-line crossing identical sinusoidal curves. I then reduce this solution to a single cosine curve, and from a three-point interpolation, I derive an approximate formula for all energy levels valid for any square quantum well and that demands only a pocket calculator.
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.
For the quantum particle in a finite and semi-infinite well, we scale space and energy to consider a dimensionless Schrdinger equation that can be solved for the allowed energy levels. In contrast to traditional approaches that determine the energies from dimensional transcendental equations associated with the boundary conditions, we solve for the square root of the dimensionless energies, which are also solutions of dimensionless transcendental equations. Our ability to define classes of wells depends on the value of the square root of the dimensionless well depth. We are also able to determine the maximum number of energy levels quite easily, and, besides deriving the relevant transcendental equations, we obtain asimple expression for the corresponding wave functions, which can be plotted using the numerical values of the allowed energies.
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.