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

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 Schrödinger 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 ε(n) (ε 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. Keywords: infinite square quantum well, the "blurring" potential, the asymmetry of QW, the position-dependent effective mass. 1. The finite QW depth. 2. The small "blurring" of the QW walls. 3. The asymmetry of QW. 4. The position-dependent effective mass in solids.