The envelope theory as a pedagogical tool

TL;DR

This paper demonstrates the envelope theory as a simple, reliable, and computationally efficient pedagogical tool for solving Schrödinger-like equations, exemplified through a one-dimensional soft-Coulomb potential problem.

Contribution

It introduces the envelope theory as an underutilized method for solving quantum equations and highlights its pedagogical value for teaching and understanding many-body systems.

Findings

Envelope theory provides reliable eigenvalues and eigenvectors.

Method is computationally independent of particle number.

Applied successfully to a one-dimensional soft-Coulomb potential.

Abstract

The envelope theory is a reliable and easy to implement method to solve time independent Schr\"odinger-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 envelope theory is applied to a simple problem in one dimension, the soft-Coulomb potential $-k/\sqrt{x^2+d^2}$, 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 envelope theory.