Bound-state formation in time-dependent potentials

We study the temporal formation of quantum-mechanical bound states within a one-dimensional attractive square-well potential by first solving time-independent Schr\"odinger equation and then studying time-dependent system with an external potential. For this we introduce Gaussian potentials different spatial extensions generalize description also for subsequent pulses random noisy potentials. Our main goal is to timescales in which state populated depopulated. Particularly, clarify likely connection between uncertainty relation energy time transition eigenstates. demonstrate that not delayed due but follows pulse shape perturbation. In addition investigate (non-)applicability first-order perturbation theory on considered quantum system.