Semiclassical time-independent description of rapidly oscillating fields on a lattice in the Kapitza approximation.

We investigate a semiclassical dynamics driven by a high-frequency ($\omega )$ field, plus a static arbitrary potential on a one-dimensional tight-binding lattice. We find -in the spirit of the Kapitza pendulum- an effective, time-independent potential $V_{eff} (x)$ that describes the average motion to order $\omega ^{-2}$. This effective potential depends on the static external potential $V(x)$, on the lattice constant ``$a$'' and on the applied high-frequency field $f(x,t)$. One obtains that \[ \frac{V_{eff} (x)}{m}=\frac{a^2}{2}V^2(x)-a^2EV(x)+a^4\int {dx} (V(x)-E)^2\frac{\partial }{\partial x}\left[ {\sum\limits_{n=1}^\infty {\frac{f_n ^2(x)}{\omega ^2n^2}} } \right]. \] Where ``$m$'' and ``$E$'' are, respectively, the effective mass and unperturbed energy of the particle's average motion, and $f_n (x)$ is the n-th Fourier component of the driving field. Where appropriate, our results should be suitable for the description of semiclassical electronic motion in a crystal lattice and/or atomic motion in an optical one.