We consider a model of an electron in a crystal moving under the influence ofan external electric field: Schroedinger's equation in one spatial dimensionwith a potential which is the sum of a periodic function $V$ and a smoothfunction $W$. We assume that the period of $V$ is much shorter than the scaleof variation of $W$ and denote the ratio of these scales by $\epsilon$. Weconsider the dynamics of $\textit{semiclassical wavepacket}$ asymptotic (in thelimit $\epsilon \downarrow 0$) solutions which are spectrally localized near toa $\textit{crossing}$ of two Bloch band dispersion functions of the periodicoperator $- \frac{1}{2} \partial_z^2 + V(z)$. We show that the dynamics isqualitatively different from the case where bands are well-separated: at thetime the wavepacket is incident on the band crossing, a second wavepacket is`excited' which has $\textit{opposite}$ group velocity to the incidentwavepacket. We then show that our result is consistent with the solution of a`Landau-Zener'-type model.
展开▼
