Numerical Methods and Causality in Physics

TL;DR

This paper explores how numerical causality conditions in computational physics relate to physical causality, especially in the context of wave and Schrödinger equations, suggesting a fundamental link as grid size approaches zero.

Contribution

It proposes that numerical causality converges with physical causality in the continuum limit and derives a minimum spatial interval related to the reduced Compton wavelength for Schrödinger's equation.

Findings

Numerical causality merges with physical causality as grid size approaches zero.

A minimum spatial interval is required for Schrödinger's equation, proportional to the reduced Compton wavelength.

Implications for numerical analysis of wave phenomena and quantum mechanics.

Abstract

We discuss physical implications of the explicit method in numerical analysis. Numerical methods have there own condition for causality, known as the Courant-Friedrichs-Lewy condition. It is proposed that numerical causality merges with physical causality as the grid interval size approaches zero. We discuss the implications of this proposition on the numerical analysis of the wave equation. We also show that, insisting on physical causality, the numerical analysis of Schrodinger's equation implies that the minimum space interval should satisfy $\Delta x \ge a_0 \lambda_c$, where $\lambda_c$ is the reduced Compton wavelength and $a_0$ is a constant of the order unity.