Why can quantum tunnelling be handled as a static problem? [duplicate]

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?

marked as duplicate by sammy gerbil, Community♦Apr 12 '17 at 10:49

• @ACuriousMind Well, I can accept that quantum tunnelling can be handled as static problem. What I need in this post is some intrigue explanation or proof of the equivalence between the time-dependent solution and time-independent solution to Schrödinger equation. For example, in the time-independent case, we vie $e^{ipx}$ as incident wave, $e^{-ipx}$ as reflecting wave and outside the barrier, we take only the outgoing wave etc... – Wein Eld Apr 12 '17 at 10:38