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Energy Spectrum of a Particle in an Infinitely Deep Potential Well with a Non-flat Bottom

S. I. Denisov, Stanislav Ivanovych Denysov, A.V. Husachenko, O.S. Ryzhkov

Published Jan 1, 2018• 0 citations
OpenAlex

Abstract

To model the quantum behavior of current carriers in heterogeneous nanostructures, different potential wells and barriers are widely used. Of particular interest are those (including artificial ones) for which the corresponding Schrödinger equation can be solved exactly. One of them is an infinitely deep potential well with a flat bottom defined as 𝑈(𝑥) = ∞ for |𝑥| ≥ 𝑎 and 𝑈(𝑥) = 0 for |𝑥| < 𝑎. Here, we report on exact solution of the stationary Schrödinger equation in the case, when the potential bottom is non-flat, i.e., 𝑈(𝑥) = −𝑓𝑥 (𝑓 > 0) if |𝑥| < 𝑎. Physically, this means that the particle in the well is subjected to an external force 𝑓 in the 𝑥-direction.

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To model the quantum behavior of current carriers in heterogeneous nanostructures, different potential wells and barriers are widely used. Of particular interest are those (including artificial ones) for which the corresponding Schrödinger equation can be solved exactly. One of them is an infinitely deep potential well with a flat bottom defined as 𝑈(𝑥) = ∞ for |𝑥| ≥ 𝑎 and 𝑈(𝑥) = 0 for |𝑥| < 𝑎. Here, we report on exact solution of the stationary Schrödinger equation in the case, when the potential bottom is non-flat, i.e., 𝑈(𝑥) = −𝑓𝑥 (𝑓 > 0) if |𝑥| < 𝑎. Physically, this means that the particle in the well is subjected to an external force 𝑓 in the 𝑥-direction.
The main goal of this work is to solve the nonrelativistic wave equation for a new potential configuration that describes the quantum states of a particle that lies within a onedimensional infinite well of width L using the Asymptotic Iteration Method (AIM). This potential was introduced recently by Alhaidari to be added to the class of exactly solvable potentials in the Tridiagonal Representation Approach (TRA). We have obtained the energy eigenvalues for different choices of the potential parameters. A good match between our results and the ones obtained by the TRA are shown in Table 1. Moreover, new results have been presented in Table 2 and Table 3 for the energy spectrum.
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.
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).
A new method for consideration of an electron stationary motion in the field of an arbitrary one-dimensional potential. It is shown that for a case of an electron infinite motion, the problem of determination of wave functions can be represented as Cauchy problem for the one-dimensional Schrodinger equation. For the case of finite motion the equation determining the energy spectrum is found. It is proved, when the spectrum of bound states is know, the problem of building the wave functions of the discrete spectrum can be formulated as Cauchy problem for wave equation as well.
Abstract: We present an investigation for a particle confined in an infinite well with sinusoidal bottom, using the perturbation theory and numerical solution for the Schrödinger equation to obtain the eigen energies and wavefunctions. Potential strength and potential oscillation dependence of the state are examined and analyzed. It is shown that the particle in a box with sinusoidal bottom does not show up the Klauder phenomenon when the perturbations are gradually reduced to zero. The research results show that the potential oscillation significantly affects certain quantum states and, therefore, the ability to manipulate the energy difference between the states. In addition, our results for the present system converge to their corresponding values for the unperturbed one in the high-potential oscillation limit. Keywords: Infinite well, Perturbation theory, Sinusoidal potential, Numerical calculations, Klauder phenomenon.
Scanning tunneling microscopy is introduced as an analytic tool for characterizing nanostructures and dynamics on the nanoscale. To analyze the underlying tunneling phenomenon, the Schrödinger equation is solved for a particle confined between finite potential energy steps. For a single potential energy well, the proper solutions to the Schrödinger equation are similar to those of a particle in an infinite box, except for the emerging “tails” of the stationary wave functions in the “classically forbidden” regions of the external confining potentials. For a symmetric double well potential, wave function penetration of the stationary solutions manifests in their even distribution among the wells. An attempt to localize the particle in one of them would lead to periodic oscillations between the wells, as if the particle can “tunnel” under the separating energy barrier. The wave function penetration and tunneling phenomena are discussed also in relation to energy band formation in periodic lattices.