# Diffusion in the mean for a periodic Schr\"{o}dinger equation perturbed by a fluctuating potential

## TL;DR

This paper studies the diffusive behavior of a quantum particle on a lattice under a combined periodic and randomly fluctuating potential, demonstrating a diffusive limit and a heat equation for the wave packet's amplitude.

## Contribution

It establishes diffusive scaling and a central limit theorem for a quantum particle in a stochastic potential with a detailed analysis of the diffusion constant's dependence on disorder strength.

## Key findings

- Diffusive scaling for moments of position displacement.
- Diffusion constant inversely proportional to the square of disorder strength.
- Wave packet amplitude converges to a heat equation under diffusive rescaling.

## Abstract

We consider the evolution of a quantum particle hopping on a cubic lattice in any dimension and subject to a potential consisting of a periodic part and a random part that fluctuates stochastically in time. If the random potential evolves according to a stationary Markov process, we obtain diffusive scaling for moments of the position displacement, with a diffusion constant that grows as the inverse square of the disorder strength at weak coupling. More generally, we show that a central limit theorem holds such that the square amplitude of the wave packet converges, after diffusive rescaling, to a solution of a heat equation.