A Local and Discrete Model Simulating Nonrelativistic Quantum Mechanical Systems

TL;DR

This paper introduces a discrete, local model that simulates nonrelativistic quantum mechanics results using simple integer operations and random walks on a lattice, avoiding complex wavefunctions and non-locality.

Contribution

The model uniquely reproduces quantum mechanical predictions through a purely local, discrete framework based on integer arithmetic and stochastic particle trajectories.

Findings

Successfully reproduces free particle and harmonic oscillator distributions

Models particle in a box and Delta potential scenarios accurately

Avoids complex wavefunctions and non-locality in quantum simulation

Abstract

This paper presents a simple model that mimics quantum mechanics (QM) results without using complex wavefunctions or non-localities. The proposed model only uses integer-valued quantities and arithmetic operations, in particular assuming a discrete spacetime under the form of a Euclidean lattice. The proposed approach describes individual particle trajectories as random walks. Transition probabilities are simple functions of a few quantities that are either randomly associated to the particles during their preparation, or stored in the lattice nodes they visit during the walk. Non-relativistic QM predictions are retrieved as probability distributions of similarly-prepared ensembles of particles. The scenarios considered to assess the model comprise of free particle, constant external force, harmonic oscillator, particle in a box, and the Delta potential.