非均匀周期介质中的波包：通过一维带交叉的传播
Wavepackets in inhomogeneous periodic media: propagation through a one-dimensional band crossing
摘要：
我们考虑了晶体中电子在外加电场作用下运动的模型：一维空间中的薛定谔方程，其势为周期函数$V$和光滑函数$W$之和。我们假设$V$的周期比$W$的变化尺度短得多，并将这些尺度的比值表示为$ε$。我们考虑$\textit{semiclassical wavepacket}$渐近（在极限$ε\downarrow 0$）解的动力学，这些解的谱局部化在周期算子$- \frac{1}{2} \partial_z^2 + V(z)$的两个Bloch带色散函数的$\textit{crossing}$附近。我们表明，动力学是定性不同的情况下，频带分离良好：在时间的波包是入射在带交叉，第二波包是“激发”的$\textit{opposite}$群速度的入射波包。然后，我们表明，我们的结果是一致的“Landau-Zener”型模型的解决方案。
英文摘要：
We consider a model of an electron in a crystal moving under the influence of an external electric field: Schroedinger's equation in one spatial dimension with a potential which is the sum of a periodic function $V$ and a smooth function $W$. We assume that the period of $V$ is much shorter than the scale of variation of $W$ and denote the ratio of these scales by $ε$. We consider the dynamics of $\textit{semiclassical wavepacket}$ asymptotic (in the limit $ε\downarrow 0$) solutions which are spectrally localized near to a $\textit{crossing}$ of two Bloch band dispersion functions of the periodic operator $- \frac{1}{2} \partial_z^2 + V(z)$. We show that the dynamics is qualitatively different from the case where bands are well-separated: at the time the wavepacket is incident on the band crossing, a second wavepacket is `excited' which has $\textit{opposite}$ group velocity to the incident wavepacket. We then show that our result is consistent with the solution of a `Landau-Zener'-type model.
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