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The Chemical Educator

Thinking Outside the (Particle in a) Box

Nathaniel C. Blank, Kate Clemons, Ryan Crowdis, Carla E. Estridge, Monica Foster, Stacey Gash, Brandon Gish, Bradley Gollihue, Cameron Henzman, Deysi Hernandez, Talha Ijaz, Ashley Ivey, Jacob Jones, Andrea K. Loveless, Sloan Roberts, Tyla Sauley, Ellyn J. Velasco, Chase Wilson, Mary M. Blackburn, Henry E Montgomery Jr

Published Feb 1, 2010• 0 citations
Publisher

Abstract

We investigate the problem of a quantum particle confined in a finite one-dimensional square well. The eigenvalues and eigenfunctions are found using a computer algebra system to solve the equations that result from matching the logarithmic derivatives at the well boundaries. The wavefunctions are used to investigate the relation of well depth to tunneling and to the Heisenberg uncertainty relations. We demonstrate how dimensional analysis can be used to locate the minimum uncertainty for a given confining potential.

DOI
10.1333/s00897102266a

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American Journal of Physics•Dec 21, 2005•41 citations
We investigate the problem of a quantum particle confined in a finite one-dimensional square well. The eigenvalues and eigenfunctions are found using a computer algebra system to solve the equations that result from matching the logarithmic derivatives at the well boundaries. The wavefunctions are used to investigate the relation of well depth to tunneling and to the Heisenberg uncertainty relations. We demonstrate how dimensional analysis can be used to locate the minimum uncertainty for a given confining potential.
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.
Abstract In order to specify the uncertainty relation for a particle confined in a finite square well potential we derived the standard deviation of momentum Δ p and that of position Δ x . It is shown that Δ p has a peak as a function of the well width and both Δ x and Δ p ⋅ Δ x have a dip in contrast to the case of the particle confined in an infinite square well potential. We clarified that Δ p 0 ⋅ Δ x 0 of the particle occupying the ground state exists in the finite range as a function of the well width and the potential-barrier height, but a particle confined in an infinite square well potential has a constant Δ p 0 ⋅ Δ x 0 . In addition, we found that the values of Δ p n ⋅ Δ x n for excited states diverge in a certain range of well widths, where the energy level of the confined particle exceeds the potential barrier.
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 quantum model of finite potential well is a most subject in the book of quantum mechanics. The solutions maybe derive from one dimension Schrödinger’s equation for describe a bound state energy of confined particle in the finite potential well. Sometime, the problem of confined particle in the finite potential well does not have an exact solution yet, but there are in fact exact solution.In this work a numerical technique is adopted to find the exact solution for the quantum model of finite potential well. To achieve this numerical solution, I use Fortran code computer program by the Newton-Raphson method for calculating the Eigen state energy for a particle (electron) confined in the finite potential well. Five values of potential width and depth have studied, the results showed that large potential width and large potential depth led me to more bound states, while smaller potential width and depth led me to fewer bound states.A comparison between infinite potential and finite potential has been presented. The results bound state energy and penetration. The results of comparison showed that the wave function of particle confined in the infinite potential cannot penetrate in the well, while the wave function of particle in the finite potential was penetrate in the well; the energy levels of an infinite well are much higher than that the corresponding energy levels for finite potential well. For confined particle in 2 or 3-dimension, we can reduce from result of 1-dimension.
An in depth look to the quantum behavior of a particle in a unidimensional finite, square box of potential energy is presented.Wave function normalization constants are fully determined and are used to establish an expression to calculate the particle allowed energy eigenstates based in the confinement condition for each of the two possible wave function symmetries.These eigenstates turn out to be degenerate and it becomes possible to configure confined systems so that the particle confinement/near free duality condition, also known as entanglement is present.This condition, obtained by using the particle mean presence probability density function of the two wave function symmetries of the system degenerate states, is illustrated with a couple of calculation examples.