A continuum limit for the Kronig-Penney model

TL;DR

This paper studies the behavior of a quantum one-dimensional periodic system with many barriers and wells as the number of barriers increases, revealing it acts like a single effective barrier in the continuum limit.

Contribution

It introduces a continuum limit for the Kronig-Penney model, showing the multi-barrier system converges to a single effective barrier with specific properties.

Findings

The transmission coefficient approaches that of a single barrier as N increases.

Forbidden energy bands persist at any finite number of barriers.

The effective barrier height is determined by the ratio of barrier and well widths.

Abstract

We investigate the transmission properties of a quantum one-dimensional periodic system of fixed length $L$, with $N$ barriers of constant height $V$ and width $\lambda$, and $N$ wells of width $\delta$. In particular, we study the behaviour of the transmission coefficient in the limit $N\to \infty$, with $L$ fixed. This is achieved by letting $\delta$ and $\lambda$ both scale as $1/N$, in such a way that their ratio $\gamma= \lambda/\delta$ is a fixed parameter characterizing the model. In this continuum limit the multi-barrier system behaves as it were constituted by a unique barrier of constant height $E_o=(\gamma V)/(1+\gamma)$. The analysis of the dispersion relation of the model shows the presence of forbidden energy bands at any finite $N$.