The envelope theory as a pedagogical tool

Abstract

Abstract The envelope theory (ET) is a reliable and easy to implement method to solve time independent Schrödinger-like equations (eigenvalues and eigenvectors). It is particularly useful to solve many-body systems since the computational cost is independent from the number of particles. The purpose of this paper is twofold. First, we want to make known a method that is probably too little used. Second, we also want to show that this method can be used as a pedagogical tool, thanks to its simplicity and the reliable results that can be obtained. To reach these goals, the ET is applied to a simple problem in one dimension, the soft-Coulomb potential − k / x 2 + d 2 , characterised by a bias distance d. Such interaction is used for the study of excitons, electron–hole bound pairs where the two charges are kept separated in two different one-dimensional regions (quantum wires). In addition to its physical interest, this system has never been treated with the ET.