Consider the Schrödinger equation
$i \partial_t \psi(x,t) = -\frac{\hbar^2}{2m} \Delta \psi(x,t) + V(x) \psi(x,t) + W(t) \psi(x,t)$
where $\psi(x,t)$ is the wave function, $m$ the particle mass and $V(x)$ a time-Independent potential that is Zero if $x<0$ and has constant value $V_0$ on all other Points. Furthermore, we have a time-dependent, uniform driving potential
$W(t)=W \sin (\Omega t)$
that oscillates with Amplitude $W$ and frequency $\Omega$. This equation I want to study nonperturbatively. If the particle has average energy of
$\frac{\hbar^2 k^2}{2m}+W < V_0$,
while moving with velocity $v=\frac{\hbar k}{m}$
then time-Independent calculations (driving potential = const.) will give me an exponentially decreasing wave function in the Region $x \geq 0$. The particle will almost not tunnel over the step barrier. But when I have a time-dependent excitation, I will have some energy uncertainty
$\Delta E \ge \frac{\hbar}{2 \Delta t}$
with time uncertainty $\Delta t$ of order $\Omega^{-1}$. Will there be a short-time Tunneling over the step, more precisely: Will I observe a wave function that is not exponential decreasing with distance? Will the energy uncertainty give the particle a temporary push that we have wave Solutions for $x \geq 0$ over a short time?