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arXiv · 1409.3453

A continuum limit for the Kronig-Penney model

Matteo Colangeli·Sokol Ndreca·Aldo Procacci
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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 $N$.

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Matteo Colangeli, Sokol Ndreca, Aldo Procacci. 2014-09-11. A continuum limit for the Kronig-Penney model. https://arxiv.org/abs/1409.3453

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