Quantum tunnelling is a process that can happen in quantum mechanics but is forbidden in classical mechanics. Roughly speaking, a particle can possibly escape from a potential well or penetrate into a potential barrier. Obviously, the probability for a trapped particle to escape from a potential well should be handled with the time-dependent Schrödinger equation. But in most cases, we simply solve the static Schrödinger equation with proper boundary conditions (and match conditions around the turning points).
I wonder, how can a time-dependent process be handled as a static problem?