Single-slit diffraction and the Heisenberg principle for position and momentum

TL;DR

This paper explores the relationship between single-slit diffraction patterns and the Heisenberg uncertainty principle, deriving the probability distribution of momentum and confirming the asymptotic law linking slit width and momentum uncertainty.

Contribution

It provides a detailed analysis connecting diffraction patterns with the Heisenberg principle, deriving the probability density of momentum from optical diffraction.

Findings

Verification of the asymptotic law P ~ ΔxΔp/h for small Δp

Derivation of the probability density of momentum from diffraction patterns

Confirmation of the fundamental link between slit width and momentum uncertainty

Abstract

Monochromatic light is prepared by a single slit of spatial width {\Delta}x. The diffraction pattern is considered to obtain the corresponding probability density of the momentum. Let P be the probability weight to measure a momentum in a finite window of width {\Delta}p. For small {\Delta}p the asymptotic law P ~ {\Delta}x{\Delta}p/h is verified, while h is Planck's constant of action.