Bound State Formation in Time Dependent Potentials

TL;DR

This paper investigates how quantum bound states form and evolve over time in one-dimensional potentials, analyzing the effects of various external perturbations and the applicability of perturbation theory.

Contribution

It introduces a detailed analysis of time-dependent bound state formation, including effects of noisy potentials and the relation between energy-time uncertainty and transition dynamics.

Findings

Bound states form following the pulse shape of external perturbations.

Formation times are not delayed by energy-time uncertainty.

First-order perturbation theory's applicability is examined.

Abstract

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