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Physica Scripta

Even-exponent potential wells solved by the finite difference method

Tisorn Na Phattalung, Wittaya Kanchanapusakit

Published Mar 19, 2025• 0 citations
Publisher

Abstract

Abstract This article examines a particle in a one-dimensional potential well V a (x) ∝ x a , where a ≥ 2 is an even integer. The system is referred to as a harmonic oscillator when a = 2 and as a particle in a box when a → ∞. The finite difference method is applied to solve the Schrödinger equation, determining the ground-state energies and wave functions. Giving accurate description in the cases of a = 2 and a → ∞, the numerical method allows us to explore the system for intermediate values of a, revealing how the energies and wave functions evolve between these two limits. The study includes discussions on the virial theorem, the uncertainty principle, excited states, and the limitation of the numerical method.

DOI
10.1088/1402-4896/adc2ba

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Abstract This article examines a particle in a one-dimensional potential well V a (x) ∝ x a , where a ≥ 2 is an even integer. The system is referred to as a harmonic oscillator when a = 2 and as a particle in a box when a → ∞. The finite difference method is applied to solve the Schrödinger equation, determining the ground-state energies and wave functions. Giving accurate description in the cases of a = 2 and a → ∞, the numerical method allows us to explore the system for intermediate values of a, revealing how the energies and wave functions evolve between these two limits. The study includes discussions on the virial theorem, the uncertainty principle, excited states, and the limitation of the numerical method.
This chapter examines some explicitly one-dimensional systems, including the finite square well, the harmonic oscillator, and the application of harmonic oscillator theory to molecular vibrations. It analyzes the computer solutions of the Erwin Schrodinger equation. Although there are no real one-dimensional square wells in nature, many real systems have potentials that resemble the square well. Near the center of force the different wave functions must differ from one another or else they would not satisfy the Schrodinger equation for different values of energy. In contemporary research almost all the manipulation of the Schrodinger equation is done not analytically but rather by computer using numerical methods. A major advantage of the dimensionless form of the Schrodinger equation is that a single solution for one kind of system is trivially easy to extend to all similar systems. The dimensionless form of the Schrödinger equation can be solved numerically if it is approximated by a so-called difference equation.
Abstract We consider the finite potential well defined by V x < a = − V o 1 − x n a n ,where n is an integer. The Schrödinger equation is recast into dimensionless form; the size of the well is characterized by a single parameter α = 2 m V o a / ℏ . We use three different mathematical techniques to solve for the energy eigenvalues and wave functions. Analytic solutions are given for the triangular well ( n = 1), the parabolic well ( n = 2), and the square well ( n → ∞). The finite element method is used to obtain accurate numerical results for 1 ⩽ n ⩽ ∞ over a full range of α . Finally, both the energy and wave function are expanded in a Taylor series in order to study the analytic behavior of the ground state energy (as a function of n ) as α → 0.
The ground-state energies and wave functions of a particle moving in one-dimensional potentials of the form V(x)=x2p, p=1,2,..., are obtained nearly exactly by iterative solution of the Schrödinger equation. These solutions are used as the ‘‘experimental benchmark’’ against which approximate solutions for the general case are tested. Both finite depth wells and infinitely deep wells are studied, as is the three-dimensional version (spherical symmetry, angular momentum quantum number l=0). The approach is aimed at student exploration of the physics of the family of potential wells, with emphasis upon connecting the familiar extreme cases, V(x)=x2 (simple harmonic oscillator) and the limiting case xk, k→∞ (square well).
Schrodinger equations are widely used in solving a variety of problems, one of them is particles in a box in especially of potential wells bound state. Particle in a box is a discrete circumstances, so we must examined whether or not the degeneration of these particles. The Schrodinger equation analysed to get a form of modeling of energy levels and wave functions in potential wells of the circumstance used numerical method. We also can show energy level of each state from the model. The numerical approaches is finnite difference method using  Matlab R2010 software. Numerical calculation shows k value. k 1 and k 3 gets from solution of cos equation, whereas k 2 and k 4 gets from solution of sin equation. The k value is used to know the energy level in potential wells. energy level model show wave fucntion. In potential well bound state, Energy level and wave functions be smaller when potential energy is high and it is high when potential when potential energy is small. When wide of potential well is smaller, energy level and wave function be smaller too. But there isn’t show energy level and wave function when wide of potential well is high. The shape of energy level and and wave function of particle in box.
The Schrödinger equation is a significant achievement on the development of quantum mechanics. By solving the Schrödinger equation, the fundamental behaviors and properties of a microscopic particle can be found in one- to three-dimensions. The article focuses on the derivations of quantum harmonic oscillators in one dimension by solving second order-differential equation. The wave functions, probability densities under different energy levels are presented. The results can be used to estimate different forms of continuous potential experienced by an oscillator. By calculating the uncertainty relation of the oscillator under one specific excited state, the general relation can be confirmed. Based on the output, by splitting the variables, the two-dimensional harmonic oscillator can also be derived. The degenerate energy levels are presented as . Numerical simulations are made to visualize those results and suggests the relation of energy levels and number of maxima of the probability densities.
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.