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

Scientia Plena

Partícula em um Poço Infinito Suave

Andrei Smirnov, Antonio Jorge Dantas Faria

Published Mar 6, 2015• 0 citations
OpenAlex

Abstract

In this paper it is considered a particle in an infinite potential well of special smooth form. To obtain eigenfunctions and eigenvalues of the problem, a method of Darboux transformations is applied. An orthonormal set of the eigenfunctions is presented. Some important properties of the eigenfunctions are shown. Principal characteristics of the particle in the well are discussed. A function of distribution of the probability density is analyzed. Conditions for determination of points of extrema of the probability density are formulated and corresponding transcendental equations are presented. An approximated method for determination of points of extrema is also proposed. Expected values of position and momentum of the particle are calculated explicitly. For some lower energy states, distributions of probability density are shown graphically in comparison with distributions for the infinite square well. As an additional useful result, some integrals of special combinations of trigonometric functions are presented.

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

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

Improved simple graphical solution for the eigenvalues of the finite square well potential

Edward J. Burge

The three principal graphical methods for obtaining the energy eigenvalues of the finite square well potential are presented. The simplest method, with sine, cosine and straight line, is provided with an easily applied rule, based on a physically obvious argument, for eliminating the unacceptable roots. The forms of the wavefunctions within the well, and the corresponding linear probability densities, are derived directly from the method. A simple extension of the method allows the energy level spectrum to be obtained directly on a linear energy scale. The variations of the energy eigenvalues with well depth and width are separately and jointly displayed, and explicit corresponding functional relationships are derived. Finally, two universal graphs are deduced, each of which allows the rapid appreciation and calculation of the dependence of the energy levels on the depth and width of the well and on the mass of the particle.

European Journal of Physics•Jul 1, 1985•7 citations

The effect of a $δ$ distribution potential on a quantum mechanical particle in a box

Pedro M. Girão, João P. Nunes

We study the effect of a $δ$ distribution potential placed at $x_0\geq 0$ and multiplied by a parameter $α$ on a quantum mechanical particle in an infinite square well over the segment $\left[-\,\frac{L}{2},\frac{L}{2}\right]$. We obtain the limit of the eigenfunctions of the time independent Schrödinger equation as $α\nearrow+\infty$ and as $α\searrow-\infty$. We see how each solution of the Schrödinger equation corresponding to $α=0$ changes as $α$ runs through the real line. When $x_0$ is a rational multiple of $L$, there exist solutions of the Schrödinger equation which vanish at $x_0$ and are unaffected by the value of $α$. We show that each one of these has an energy that coincides with the energy of a certain limiting eigenfunction obtained by taking $|α|\to\infty$. The expectation value of the position of a particle with wave function equal to the limiting eigenfunction is $x_0$.

arXiv:2311.02611•Nov 5, 2023•0 citations

Using the graphic method to study two-dimensional square potential well

Peng Yun-xiong

In this paper, the energy levels, wave function and probability density of a particle moving in a two-dimensional square potential well have been studied with the stationary state Schr[AK]dinger equation. The wave form of wave function and probability density are drawn and presented with TC or MATLAB. Some special examples concerning about wave form of wave function have been also discussed.

Journal of Wuhan Institute of Technology•Jan 1, 2006•0 citations

Exact Treatment of the Infinite Square Well in One Dimension with λδ^' (x) Potential

Abstract: This work considered the infinite square well in one dimension with a contact potential. The Dirac delta derivative function potential λδ^' where λ is a coupling constant was used to represent the contact potential. Using Green’s function technique, exact implicit expressions of the energy eigenvalues and eigenfunctions were obtained. The energy eigenvalues were expressed using a transcendental equation. The energy eigenfunctions satisfy the Schrödinger equation and the infinite square well boundary conditions. Also, the eigenfunctions and their first derivative were shown to be discontinuous. The values of these discontinuity jumps agreed with the required conditions for a self-adjoint extension Hamiltonian. In the weak coupling region, the energy eigenvalues are close to that of the even parity solution before adding the contact potential. The energy eigenvalues in the strong coupling regime reveal the energy eigenvalues of the odd parity solution. Keywords: Point interactions, Infinite square well, Green’s function technique. PACS numbers: 03.65.-w, 03.65.Db, 03.65.Ge

Jordan Journal of Physics•Jun 30, 2023•0 citations
In this paper it is considered a particle in an infinite potential well of special smooth form. To obtain eigenfunctions and eigenvalues of the problem, a method of Darboux transformations is applied. An orthonormal set of the eigenfunctions is presented. Some important properties of the eigenfunctions are shown. Principal characteristics of the particle in the well are discussed. A function of distribution of the probability density is analyzed. Conditions for determination of points of extrema of the probability density are formulated and corresponding transcendental equations are presented. An approximated method for determination of points of extrema is also proposed. Expected values of position and momentum of the particle are calculated explicitly. For some lower energy states, distributions of probability density are shown graphically in comparison with distributions for the infinite square well. As an additional useful result, some integrals of special combinations of trigonometric functions are presented.
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.
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.
The three principal graphical methods for obtaining the energy eigenvalues of the finite square well potential are presented. The simplest method, with sine, cosine and straight line, is provided with an easily applied rule, based on a physically obvious argument, for eliminating the unacceptable roots. The forms of the wavefunctions within the well, and the corresponding linear probability densities, are derived directly from the method. A simple extension of the method allows the energy level spectrum to be obtained directly on a linear energy scale. The variations of the energy eigenvalues with well depth and width are separately and jointly displayed, and explicit corresponding functional relationships are derived. Finally, two universal graphs are deduced, each of which allows the rapid appreciation and calculation of the dependence of the energy levels on the depth and width of the well and on the mass of the particle.
The effect of a δδδ distribution potential on a quantum mechanical particle in a box
We study the effect of a δδδ distribution potential placed at x0≥0x_0\geq 0x0​≥0 and multiplied by a parameter ααα on a quantum mechanical particle in an infinite square well over the segment [− L2,L2]\left[-\,\frac{L}{2},\frac{L}{2}\right][−2L​,2L​]. We obtain the limit of the eigenfunctions of the time independent Schrödinger equation as α↗+∞α\nearrow+\inftyα↗+∞ and as α↘−∞α\searrow-\inftyα↘−∞. We see how each solution of the Schrödinger equation corresponding to α=0α=0α=0 changes as ααα runs through the real line. When x0x_0x0​ is a rational multiple of LLL, there exist solutions of the Schrödinger equation which vanish at x0x_0x0​ and are unaffected by the value of ααα. We show that each one of these has an energy that coincides with the energy of a certain limiting eigenfunction obtained by taking ∣α∣→∞|α|\to\infty∣α∣→∞. The expectation value of the position of a particle with wave function equal to the limiting eigenfunction is x0x_0x0​.
Abstract: This work considered the infinite square well in one dimension with a contact potential. The Dirac delta derivative function potential λδ^' where λ is a coupling constant was used to represent the contact potential. Using Green’s function technique, exact implicit expressions of the energy eigenvalues and eigenfunctions were obtained. The energy eigenvalues were expressed using a transcendental equation. The energy eigenfunctions satisfy the Schrödinger equation and the infinite square well boundary conditions. Also, the eigenfunctions and their first derivative were shown to be discontinuous. The values of these discontinuity jumps agreed with the required conditions for a self-adjoint extension Hamiltonian. In the weak coupling region, the energy eigenvalues are close to that of the even parity solution before adding the contact potential. The energy eigenvalues in the strong coupling regime reveal the energy eigenvalues of the odd parity solution. Keywords: Point interactions, Infinite square well, Green’s function technique. PACS numbers: 03.65.-w, 03.65.Db, 03.65.Ge