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 Wuhan Institute of Technology

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

Peng Yun-xiong

Published Jan 1, 2006• 0 citations
OpenAlex

Abstract

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.

Related papers

The Derivation of the Medium Wave Function in a One-Dimensional Finite Deep Symmetrical Square Potential Well

Zhao Peng-y

Three methods-a documental analysis, a textual method, a comprehensive analysis- are used in this article to analyze the energy levels of motional particles and their wave functions of one-dimensional finite deep symmetrical square potential wells, thus solving the eigen-values and eigen-functions at their corresponding energetic levels. Due to the three following advantages of a one-dimensional problem-its simpler mathematical processing, and the possible detailed-discussed results, and the display of the many features of quantum systems, so in the paper, the eigen-values and the exact formulas of the wave functions of moving particles in a one-dimensional finite deep symmetric square potential wells, which are not specified in the textbooks, are deviated according to the Schrodinger equation of stationary state through a case study of moving particles in a one-dimensional finite deep symmetric square potential wells.

集宁师范学院学报•Jan 1, 2015•0 citations

Solution of the particles bound in one-dimensional square potential well

Hao Zheng-tong

With the methods of the matrix representation,the wave functions and energy levels are carried out for the particle in one-dimensional square potential well,then the graphical representation of eigenvalue equation,eigenfunction and eigenvalue are presented by using Wolfram Mathematica.

College Physics•Jan 1, 2011•0 citations

BOUND STATES IN 2D AND 3D FINITE DEEP SPHERICAL SQUARE POTENTIAL WELLS

Leping LIU, Jiaze SUN, Xibin LI +1 more

By solving the Schrodinger equations in three-dimensional (3D) and two-dimensional (2D) finite depth center spherically symmetric square potential well, the eigenequations of bounded energy levels and their distributions are obtained, and the analytical solution for any bound state wave function is given. Both the eigenequations of the energy levels as well as the stationary state wave functions are closely related to the (spherical) Bessel functions. It is found that for the 3D case, there exists a bound solution when 2μa2V0/ħ2>π2/4 is present. While for the 2D case, there exists a bound state if 2μa2V0/ħ2>0.1993. Finally, numerical methods are used to analyze the characteristics of probability density distribution of different quantum states. It is found that in both three-dimensional and two-dimensional situations, the maximum number of probability density distributions inside the well corresponds exactly to the quantum number n. The larger n or the smaller α is, the higher the probability of discovering particles outside the well. In addition, the angular momentum quantum number l or m can characterize the degree to which particles deviate from the center or the uniform distribution.

Wuli yu gongcheng.•Apr 1, 2025•1 citation

The numerical methods and graphical displays of the particles bound in one-dimensional square potential wells

LI Da-nong

With the numerical methods and graphical displays,the energy levels and normalized wave functions are given for the particle in one-dimensional asymmetrical finite potential wells.The results are accordant with those of infinite wells when the potential tends to infinite.

College Physics•Jan 1, 2008•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

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
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.
Three methods-a documental analysis, a textual method, a comprehensive analysis- are used in this article to analyze the energy levels of motional particles and their wave functions of one-dimensional finite deep symmetrical square potential wells, thus solving the eigen-values and eigen-functions at their corresponding energetic levels. Due to the three following advantages of a one-dimensional problem-its simpler mathematical processing, and the possible detailed-discussed results, and the display of the many features of quantum systems, so in the paper, the eigen-values and the exact formulas of the wave functions of moving particles in a one-dimensional finite deep symmetric square potential wells, which are not specified in the textbooks, are deviated according to the Schrodinger equation of stationary state through a case study of moving particles in a one-dimensional finite deep symmetric square potential wells.
By solving the Schrodinger equations in three-dimensional (3D) and two-dimensional (2D) finite depth center spherically symmetric square potential well, the eigenequations of bounded energy levels and their distributions are obtained, and the analytical solution for any bound state wave function is given. Both the eigenequations of the energy levels as well as the stationary state wave functions are closely related to the (spherical) Bessel functions. It is found that for the 3D case, there exists a bound solution when 2μa2V0/ħ2>π2/4 is present. While for the 2D case, there exists a bound state if 2μa2V0/ħ2>0.1993. Finally, numerical methods are used to analyze the characteristics of probability density distribution of different quantum states. It is found that in both three-dimensional and two-dimensional situations, the maximum number of probability density distributions inside the well corresponds exactly to the quantum number n. The larger n or the smaller α is, the higher the probability of discovering particles outside the well. In addition, the angular momentum quantum number l or m can characterize the degree to which particles deviate from the center or the uniform distribution.
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 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.