Kronig-Penney模型的一个连续极限
A continuum limit for the Kronig-Penney model
摘要：
研究了一维周期性量子系统的传输特性，该系统具有固定长度的$L$，$N$势垒为固定高度的$V$和宽度的$λ$，$N$阱为固定宽度的$δ$ .特别是，我们研究的行为的传输系数的限制$N\to \infty$，与$L$固定。这是通过让$δ$和$λ$都作为$1/N$进行缩放来实现的，以这种方式，它们的比率$γ= λ/δ$是表征模型的固定参数。在这个连续极限中，多势垒系统表现为由恒定高度$E_o=(γV)/(1+γ)$的唯一势垒构成。色散关系的分析表明，在任何有限的$N$的禁带的存在。
英文摘要：
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 $λ$, and $N$ wells of width $δ$. In particular, we study the behaviour of the transmission coefficient in the limit $N\to \infty$, with $L$ fixed. This is achieved by letting $δ$ and $λ$ both scale as $1/N$, in such a way that their ratio $γ= λ/δ$ 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 $E_o=(γV)/(1+γ)$. The analysis of the dispersion relation of the model shows the presence of forbidden energy bands at any finite $N$.
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