Asymptotes of non-stationary solutions of the Schr\"{o}dinger equation for a particle interacting with a one-dimensional $\delta$-potential

Abstract

It is shown that non-stationary solutions of the Schr\"{o}dinger equation, which describes the quantum dynamics of a particle in the field of a one-dimensional delta potential (1DDP), are divided into two classes: some define pure states that have no free dynamics as $t\to\mp\infty$; others define states with asymptotically free dynamics but represent mixed states in whose space the asymptotic superselection rule holds. That is, according to the Schr\"{o}dinger equation, {\it pure scattering} states predicted by the conventional model of this scattering process do not exist. On mixed scattering states, the Hamiltonian with 1DDP is defined only in superselection sectors. The scattering process with one-sided incidence of a particle on 1DDP represents a decoherence process in a closed system.