Experiments with Schr\"odinger Cellular Automata

TL;DR

This paper introduces a class of Schr"odinger cellular automata derived from Hamiltonian splitting, providing insights into quantum phenomena like interference, refraction, and wave evolution through discrete automata simulations.

Contribution

It presents a novel framework for simulating Schr"odinger dynamics using cellular automata based on Hamiltonian splitting, enabling new experimental insights.

Findings

Quantitative analysis of phase and group velocities.

Observation of interference effects such as refraction and Aharonov-Bohm.

Insights into energy levels and waveform aliasing phenomena.

Abstract

We derive a class of cellular automata for the Schr\"odinger Hamiltonian, including scalar and vector potentials. It is based on a multi-split of the Hamiltonian, resulting in a multi-step unitary evolution operator in discrete time and space. Experiments with one-dimensional automata offer quantitative insight in phase and group velocities, energy levels, related approximation errors, and the evolution of a time-dependent harmonic oscilator. The apparent effects of spatial waveform aliasing are intriguing. Interference experiments with two-dimensional automata include refraction, Davisson-Germer, Mach-Zehnder, single & double slit, and Aharonov-Bohm.