Numerical Solutions for a Two-dimensional Quantum Dot Model

Abstract

In this paper, a quantum dot mathematical model based on a twodimensional Schrödinger equation assuming the 1/r inter-electronic potential is revisited. Generally, it is argued that the solutions of this model obtained by solving a biconfluent Heun equation have some limitations. The known polynomial solutions are confronted with new numerical calculations based on the Numerov method. A good qualitative agreement between them emerges. The numerical method being more general gives rise to new solutions. In particular, we are now able to calculate the quantum dot eigenfunctions for a much larger spectrum of external harmonic frequencies as compared to previous results. Also the existence of bound state for such planar system, in the case = 0, is predicted and its respective eigenvalue is determined. Keywords Quantum dot model · Numerov numerical method · two-electron system · Schrödinger equation PACS PACS 81.07.Ta · 78.67.Hc · 36.10.-k 1 IntroductionModern technics in nanometer-scale semiconductor manufacturing enable the creation of quantum confinement of only a few electrons. These few-body systems are often called quantum dots [1]. They can be described by a model F. Caruso

A planar model for the muon-catalyzed fusion

“…We should remember that, over the recent past years, planar physics and truly two-dimensional systems have attracted a great deal of attention in connection with graphene [19][20][21], and with the Quantum Hall Effect [22,23]. All these planar quantum systems point to considering that the correct Coulomb inter-electronic potential should be ln(r) [24], rather than assuming the validity of the usual 3D dependence in planar systems, corresponding to the 1/r potential [25][26][27][28]. This brings us to our general motivation: why should one investigate the 2D μCF.…”

Experimental study of coupled oscillations on a Slinky Wilberforce pendulum

“…In a simple model for a quantum dot composed of two electrons, they can be described with an external harmonic oscillator potential of frequency Ω = 2ω. Following the steps of reference [10], we have the effective potential to be introduced into equation ( 2) for the quantum number = 0 is given by:…”

“…We will disclose here a powerful numerical calculation method originally developed by Boris Vasil'evich Numerov [1][2][3] -see also [4][5][6][7] -applying it to the timeindependent Schrödinger equation describing physical systems like the hydrogen atom, a diatomic molecule governed by the Morse potential and a particular model for the quantum dot atom [8][9][10]. These three examples will be solved and the parameters needed for each solution using Numerov's method will be shared in their respective sections.…”

An efficient approximation for accelerating convergence of the numerical power series. Results for the 1D Schrödinger's equation

“…It is used to determine presence of the energetics for short intramolecular O−H• • • O in enzyme and photo centers by nuclear motion of the involved hydrogen atom [8], to solve the Schödinger equation with complex potential boundaries for open multi−layer heterojunction systems [55] and to study energy spectra of mesons and hadrons [56]. Two−dimensional quantum dot eigen−functions for a much larger spectrum of external harmonic frequencies are calculated as well [57].…”