Illustrating Electric Conductivity Using the Particle-in-a-Box Model: Quantum Superposition is the Key

TL;DR

This paper uses the particle-in-a-box quantum model with a tilting potential to illustrate electric conductivity, emphasizing quantum superposition's role and linking oscillatory dynamics to band gap dependence.

Contribution

It introduces a simple quantum model to visually demonstrate electric conductivity and highlights the importance of quantum superposition in this phenomenon.

Findings

Oscillatory particle dynamics mimic electric current.

Amplitude relates to the energy gap between states.

Conductivity depends on the valence-conduction band gap.

Abstract

Most of the textbooks explaining electric conductivity in the context of quantum mechanics provide either incomplete or semi-classical explanations that are not connected with the elementary concepts of quantum mechanics. We illustrate the conduction phenomena using the simplest model system in quantum dynamics, a particle in a box (PIB). To induce the particle dynamics, a linear potential tilting the bottom of the box is introduced, which is equivalent to imposing a constant electric field for a charged particle. Although the PIB model represents a closed system that cannot have a flow of electrons through the system, we consider the oscillatory dynamics of the particle probability density as the analogue of the electric current. Relating the amplitude and other parameters of the particle oscillatory dynamics with the gap between the ground and excited states of the PIB model allows us to demonstrate one of the most basic dependencies of electric conductivity on the valence-conduction band gap of the material.