Edwards Localization

TL;DR

This paper investigates localization phenomena in quantum stochastic mechanics using the Edwards model, demonstrating how coupling strength and random scattering center positions influence ground state localization.

Contribution

It introduces a detailed analysis of localization in the Edwards model, including effects of random scattering centers and affine quantization of the Lax model, advancing understanding of quantum localization mechanisms.

Findings

Localization increases with coupling strength g.

Random scattering center positions enhance average localization.

Affine quantization reduces the system to contiguous square wells.

Abstract

We study the localization problem in quantum stochastic mechanics. We start from the Edwards model for a particle in a bath of scattering centers and prove static localization of the ground state wavefunction of the particle in a one dimensional square well coupled to Dirac delta like scattering centers in arbitrary but fixed positions. We see how the localization increases for increasing coupling $g$. Then we choose the scattering centers positions as pseudo random numbers with a uniform probability distribution and observe an increase in the localization of the average of the ground state over the many positions realizations. We discuss how this averaging procedure is consistent with a picture of a particle in a Bose-Einstein condensate of of non interacting boson scattering centers interacting with the particle with Dirac delta functions pair potential. We then study the dynamics of the ground state wave function. We conclude with a discussion of the affine quantization version of the Lax model which reduces to a system of contiguous square wells with walls in arbitrary positions independently of the coupling constant $g$.