Quantitative analysis of the applicability limits of the model of a one-dimensional infinite square well

Vladimir Ivchenko

doi:10.1590/1806-9126-rbef-2019-0222

Abstract

Abstract We discuss the influence of several factors on the deviations from energy spectrum of an infinite square quantum well (QW) for real microscopic systems that can be approximately modelled using particle in a box. We introduce the “blurring” potential in the form of the modified Woods-Saxon potential and solve the corresponding Schrodinger equation. It is found that the increase of the degree of blurring δ of the QW leads to the increase of number of the energy levels inside it and to increase of deviations from the quadratic dependence e (n) (e is the particle energy, n is the energy level number) typical for the infinite square QW, especially, for the energy levels close to the QW “tops”. It is most surprising that for relatively “large” values of δ the difference between the levels energies of such well and the appropriate (with the same n) levels energies of the square QW with the same depth changes sign (from positive to negative) as number n increases. We also conclude that the asymmetry of the QW and non-equality min≠mout (where min and mout are the particle effective mass inside and outside the QW) play a significant role for the relatively “shallow” well near the QW top.

Highlights

As it is known, the study of model systems, for which there are simple analytic solutions of the time independent Schrödinger equation, makes it possible to understand the methods of quantum mechanics more comprehensively
The results obtained are of independent interest, since they reflect, in some approximation, the properties of the corresponding real systems. One of such idealized system is the particle in a onedimensional box model [1], [2] ( known as the infinite square quantum well (QW)) that describes a particle which can only move freely along a linear segment of finite length
We discuss the influence of several factors on the deviations from energy spectrum of an infinite square quantum well (QW) for real microscopic systems that can be approximately modelled using particle in a box

Summary

Introduction

The study of model systems, for which there are simple analytic solutions of the time independent Schrödinger equation, makes it possible to understand the methods of quantum mechanics more comprehensively. The results obtained are of independent interest, since they reflect, in some approximation, the properties of the corresponding real systems One of such idealized system is the particle in a onedimensional box model [1], [2] ( known as the infinite square quantum well (QW)) that describes a particle which can only move freely along a linear segment of finite length. The aim of this article is the consideration of the influence of such factors on the deviations from energy spectrum (1) in the real QW systems (heterostructures, quantum dots and atomic nuclei, etc.) that can be approximately modelled using particle in a box

The “blurring” of the walls and finite quantum well depth

The position-dependent effective mass

Existence of two additional mechanical degrees of freedom of the particle

Findings

Conclusions