The envelope theory as a pedagogical tool

Abstract

The envelope theory 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 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.

https://notebookarchive.org/the-envelope-theory-as-a-pedagogical-tool--2023-03-97cajed/

References in corpus (5)

• Auxiliary fields as a tool for computing analytical solutions of the Schrödinger equation
• A general comparison theorem
• Numerical tests of the envelope theory for few-boson systems
• Hyperspherical harmonics expansion on Lagrange meshes for bosonic systems in one dimension
• Many-quark interactions: Large-$N$ scaling and contribution to baryon masses

Cited by in corpus (3)

• Tests of the envelope theory for three-body forces
• Accuracy Tests of the Envelope Theory
• Upper bounds for critical coupling constants for binding some quantum many-body systems