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

The Infinite Tilted-Well: An Example of Elementary Quantum Mechanics with Applications toward Current Research

John N. Churchill, Floyd O. Arntz

Published Jul 1, 1969• 10 citations
Publisher

Abstract

The quantum-mechanical problem of a particle under the simultaneous influence of a linear potential and one-dimensional infinite square-well is considered. A graphical technique is described whereby approximate solutions are obtained for the general case, and some analytical methods are discussed by which exact solutions may be obtained for various special cases. The results should provide insight into a problem of current theoretical and experimental interest: the behavior of electrons in solids under the influence of an applied electric field.

DOI
10.1119/1.1975775

Related papers

Position Expectation Values for an Electron in an Infinite Tilted Well

Noble M. Johnson, John N. Churchill

A numerical technique is used to determine the manner in which the expectation values of position for an electron in an infinite tilted well depend on the potential gradient and quantum state of the system. The results are compared to the classical case and to the exact analytic expression for the special case in which the total energy of the electron is equal to the maximum value of the linear potential.

American Journal of Physics•Apr 1, 1970•1 citation

Graphical Approach to Eigenvalue Problems of The One-Dimensional Finite Deep Square Potential Well

Shu Long

The one-dimensional finite-depth potential well problem under certain boundary conditions discussed in this paper is mainly the eigenvalue problem in quantum mechanics. The basis of this article is the solution of Schrodinger equation and the treatment of transcendental equation. In many quantum mechanics textbooks, the one-dimensional infinite potential well problem is regarded as the foundation of quantum mechanics because it is easy to handle and understand. On this basis, this paper discusses the more challenging finite-depth potential well situation. In this paper, the transcendental equation is obtained by analogy between the wave function with time and the continuity of the boundary conditions. In order to solve the problem that transcendental equation is difficult to solve in this process, this paper adopts the method of solving numerical solutions by using images, which includes two different cases of odd and even functions. This paper reveals the possibility of particles appearing outside the well in this problem, and on this basis, some discussions and analysis are made on the one-dimensional asymmetric finite-depth potential well problem.

Highlights in Science Engineering and Technology•Aug 20, 2024•0 citations

Non-Relativistic Scenario of a Free Particle in an Infinite Cylindrical Potential Well

Pratik Adarsh, Sabyasachi Ghosh

There are various types of infinite potential well problems occurring in elementary quantum mechanics formalism. The infinite square well (one dimensional), cubical box and, spherical well are quite common in textbooks. In this paper, we consider a rather uncommon potential well, an infinite cylindrical well, and try to find its energy eigenvalues and eigenfunctions using Schrödinger equation. We also plot some radial wavefunctions and density plots.

arXiv:2105.08283•May 18, 2021•0 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

Potential Problems

P. Strange

I believe that physicists gain much of their physics intuition from solving simple model problems explicitly. Apart from the hydrogen atom, this is something that is not often done in relativistic quantum theory books (except the book by Greiner (1990)). In this chapter we set up and solve five simple models that have exact analytical solutions. Furthermore, they are all related to well-known non-relativistic counterparts, and most find application in many areas of physics, particularly solid state physics. These relationships will be pointed out and where appropriate we will also mention the applications and consequences of the models. There are very few models in quantum mechanics that yield exact solutions, hence any that do are of fundamental interest, the hydrogen atom being, perhaps, the most famous example. The hydrogen atom with the potential V(r) = — Ze 2 /4πε 0 r is unique in that it can be solved analytically classically and in both non-relativistic and relativistic quantum theory, with and without spin. This has been discussed in detail in chapters 3 and 8. The five examples we choose to consider here are the following. Firstly we will solve the Dirac equation for an electron in a one-dimensional well, a relativistic generalization of the non-relativistic particle in a box problem.

Cambridge University Press eBooks•Sep 17, 1998•0 citations

The Harmonic Oscillator

Abstract The problem of a particle moving under the influence of a linear restoring force, F (x) = −Kx, or equivalently a quadratic potential, V (x) = Kx 2/2, is a problem which is studied at all levels of theoretical physics, from elementary classical mechanics through quantum field theory.É One of its most useful features is that, at every stage of development, it is exactly soluble and so can be easily used as a closed-form, analytic example. If it had no connection to real physical systems, however, that fact would be of only academic interest.

•Apr 13, 2006•0 citations
The quantum-mechanical problem of a particle under the simultaneous influence of a linear potential and one-dimensional infinite square-well is considered. A graphical technique is described whereby approximate solutions are obtained for the general case, and some analytical methods are discussed by which exact solutions may be obtained for various special cases. The results should provide insight into a problem of current theoretical and experimental interest: the behavior of electrons in solids under the influence of an applied electric field.
The one-dimensional finite-depth potential well problem under certain boundary conditions discussed in this paper is mainly the eigenvalue problem in quantum mechanics. The basis of this article is the solution of Schrodinger equation and the treatment of transcendental equation. In many quantum mechanics textbooks, the one-dimensional infinite potential well problem is regarded as the foundation of quantum mechanics because it is easy to handle and understand. On this basis, this paper discusses the more challenging finite-depth potential well situation. In this paper, the transcendental equation is obtained by analogy between the wave function with time and the continuity of the boundary conditions. In order to solve the problem that transcendental equation is difficult to solve in this process, this paper adopts the method of solving numerical solutions by using images, which includes two different cases of odd and even functions. This paper reveals the possibility of particles appearing outside the well in this problem, and on this basis, some discussions and analysis are made on the one-dimensional asymmetric finite-depth potential well 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.
I believe that physicists gain much of their physics intuition from solving simple model problems explicitly. Apart from the hydrogen atom, this is something that is not often done in relativistic quantum theory books (except the book by Greiner (1990)). In this chapter we set up and solve five simple models that have exact analytical solutions. Furthermore, they are all related to well-known non-relativistic counterparts, and most find application in many areas of physics, particularly solid state physics. These relationships will be pointed out and where appropriate we will also mention the applications and consequences of the models. There are very few models in quantum mechanics that yield exact solutions, hence any that do are of fundamental interest, the hydrogen atom being, perhaps, the most famous example. The hydrogen atom with the potential V(r) = — Ze 2 /4πε 0 r is unique in that it can be solved analytically classically and in both non-relativistic and relativistic quantum theory, with and without spin. This has been discussed in detail in chapters 3 and 8. The five examples we choose to consider here are the following. Firstly we will solve the Dirac equation for an electron in a one-dimensional well, a relativistic generalization of the non-relativistic particle in a box problem.
Abstract The problem of a particle moving under the influence of a linear restoring force, F (x) = −Kx, or equivalently a quadratic potential, V (x) = Kx 2/2, is a problem which is studied at all levels of theoretical physics, from elementary classical mechanics through quantum field theory.É One of its most useful features is that, at every stage of development, it is exactly soluble and so can be easily used as a closed-form, analytic example. If it had no connection to real physical systems, however, that fact would be of only academic interest.