The work continues systematic applications of the method developed by Di Martino and Facchi (2015) for obtaining, for systems with variable boundary conditions, an equivalent Schrdinger equation with constant boundary conditions. From the resulting equation with rapidly varying terms, an effective equation is then derived for the averaged (slowly varying) wave functions. In this work, we obtain this effective equation for rapidly driven systems in the interaction picture (in first order in reverse frequency of external driving). This is the most suitable picture for systems with external driving. As an example, an extended two-state fast driven system is considered. As a second example, we consider the one-dimensional motion of a particle in a potential box with high-frequency oscillations of its width. Oscillations of width lead to additional terms in the effective equation proportional not only to the second derivative of the potential, but to the first derivative as well. In the case of width oscillations, compared to the oscillations of the box as a whole, the changes are even more significant. In our examples, phenomena such as slow averaged oscillations, resonance, the appearance of volcano potentials are observed, as is often the case in driven systems.

这项工作继续系统地应用 Di Martino 和 Facchi (2015) 开发的方法，为具有可变边界条件的系统获得具有恒定边界条件的等效 Schrdinger 方程。从得到的具有快速变化项的方程，然后推导出平均（缓慢变化）波函数的有效方程。在这项工作中，我们获得了交互图中快速驱动系统的有效方程（外部驱动反向频率的一阶）。这是最适合外置驱动系统的画面。作为一个例子，考虑了一个扩展的两态快速驱动系统。作为第二个例子，我们考虑一个粒子在一个势盒中的一维运动，它的宽度有高频振荡。宽度的振荡导致有效方程中的附加项不仅与电位的二阶导数成正比，而且与一阶导数成正比。在宽度振荡的情况下，与盒子整体的振荡相比，变化更为显着。在我们的例子中，观察到了诸如缓慢平均振荡、共振、火山势出现等现象，这在驱动系统中经常发生。