# Unusual broadening of wave packets on lattices

## TL;DR

This paper investigates the unusual broadening behavior of wave packets on one-dimensional lattices, especially when their amplitude or phase varies on the scale of the lattice constant, revealing deviations from classical quantum mechanics expectations.

## Contribution

It provides analytical and numerical analysis of wave packet broadening on lattices, highlighting behaviors that differ from standard quantum mechanics textbook cases.

## Key findings

- Wave packets with amplitude or phase variations on the lattice scale behave differently.
- Narrow wave packets and those with high velocities show unique broadening patterns.
- The study offers new insights into wave packet dynamics on discrete lattices.

## Abstract

The broadening of one-dimensional Gaussian wave packets is presented in all textbooks on quantum mechanics. It is used to elucidate Heisenberg's uncertainty relation. The behaviour on a lattice is drastically different if the amplitude and/or phase of the wave packet varies on the scale of the lattice constant. Important examples are very narrow wave packets ot wave packets with an average velocity comparable to the maximum velocity on the lattice. Analytical and numerical results for the time dependence of wave packets on a one-dimensional lattice are presented.