Project 12: - Rectangular Quantum Well Filled with Electrons – The Idea of Self-Consistent Calculations

Summary

This chapter covers the problem of a single quantum well filled with electrons, specifically, the changes that occur when electrons are introduced into an empty quantum well. Utilising the ’jellium’ model, the chapter commences by identifying the energy levels of an empty quantum well composed of infinite dipole planes filled with positively charged jellium. The subsequent introduction of electrons leads to a significant change in the well’s shape due to interactions within a non-uniform electron gas. However, the complexity of this problem is simplified by considering only Hartree interactions, allowing for self-consistent calculations. The density of the negative charge introduced is determined, then added to the uniform density of the jellium, resulting in a new potential energy shape of the well. The iterative process of determining new energy levels, populating them with electrons, and re-evaluating continues until a chosen property converges. The chapter concludes by demonstrating how to model the potential of an empty well and how this potential changes in a self-consistent procedure when the well is filled with electrons.