How do wave packets spread? Time evolution on Ehrenfest time scales
How do wave packets spread? Time evolution on Ehrenfest time scales
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DOI：
10.1088/1751-8113/45/21/215307
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发表时间：
2011-12
期刊：
影响因子：
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通讯作者：
R. Schubert;R. O. Vallejos;F. Toscano
中科院分区：
文献类型：
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作者：
R. Schubert;R. O. Vallejos;F. Toscano
我们得到了标准含时WKB理论的一个推广，它可以应用于长时间传播相干态和其他强局域态。它特别使我们能够给出一个统一的描述从本地相干态到离域拉格朗日状态，这发生在Escherichfest时间的转换。主要的新成分是一个metaplectic算子，用于修改初始状态的方式，标准的时间相关的WKB理论，然后可以应用于传播。我们给出了一个详细的分析相空间几何结构的基础上，这种建设，并使用它来确定新方法的有效性范围。通过几个例子来说明和测试该方案，并讨论了两个应用。(i)对于波包在临界能量附近的势垒上的散射，我们可以导出从反射到透射的过渡的统一近似。(ii)一个波包沿沿着双曲线轨迹传播，在埃克塞特时刻变成一个与不稳定流形相关的拉格朗日状态;这用反冲谐振子来说明。
We derive an extension of the standard time-dependent WKB theory, which can be applied to propagate coherent states and other strongly localized states for long times. It in particular allows us to give a uniform description of the transformation from a localized coherent state to a delocalized Lagrangian state, which takes place at the Ehrenfest time. The main new ingredient is a metaplectic operator that is used to modify the initial state in a way that the standard time-dependent WKB theory can then be applied for the propagation. We give a detailed analysis of the phase space geometry underlying this construction and use this to determine the range of validity of the new method. Several examples are used to illustrate and test the scheme and two applications are discussed. (i) For scattering of a wave packet on a barrier near the critical energy, we can derive uniform approximations for the transition from reflection to transmission. (ii) A wave packet propagated along a hyperbolic trajectory becomes a Lagrangian state associated with the unstable manifold at the Ehrenfest time; this is illustrated with the kicked harmonic oscillator.