State change via one-dimensional scattering in quantum mechanics

TL;DR

This paper develops a non-perturbative computational method to analyze state changes in a one-dimensional quantum scattering system involving a confined and a free particle, with potential applications in quantum wire and dot studies.

Contribution

It introduces a formulation and computational scheme that avoids perturbation theory for analyzing state changes in one-dimensional quantum scattering.

Findings

The method accurately predicts outcome probabilities based on initial states.

It demonstrates advantages over standard perturbative approaches.

Applicable to physical systems like quantum wires and quantum dots.

Abstract

This study aims to address the nature of state change, measurement, and probabilistic outcomes in non-relativistic quantum mechanics. We consider a pair of particles that interact in a one-dimensional setting via a delta-function potential. One of the particles is confined to a one-dimensional box, and the other particle is free. The free particle is incident from the left with specified energy, and it may cause changes in state of the confined particle before flying away to the left or to the right. We present a formulation and computational scheme that avoids the use of perturbation theory and determines the probability of any such outcome as a function of the initial state of the confined particle and the energy of the incident particle. As demonstrated by a direct comparison, this presented method holds multiple advantages over a standard perturbative method. The problem formulation and corresponding computational scheme may have applications in physical settings which admit one-dimensional scattering, e.g., in the study of quantum wires or quantum dots.