On the experimental verification of the uncertainty principle of position and momentum

TL;DR

This paper revisits the uncertainty principle for position and momentum, analyzing the standard deviation as a measure, and verifies the theoretical bound through a laser diffraction experiment, comparing results with recent studies.

Contribution

It offers a mathematical reexamination of the uncertainty bound and provides experimental verification using a laser diffraction setup.

Findings

The uncertainty bound is reconsidered as a variational problem.

Experimental results support the theoretical inequality.

Comparison with recent laser experiments validates the analysis.

Abstract

Historically, Kennard was the first to choose the standard deviation as a quantitative measure of uncertainty, and neither he nor Heisenberg explicitly explained why this choice should be appropriate from the experimental physical point of view. If a particle is prepared by a single slit of spatial width $\Delta x$, it has been shown that a finite standard deviation $\sigma_p<\infty$ can only be ensured if the wave-function is zero at the edge of $\Delta x$, otherwise it does not exist [8]. Under this circumstances the corresponding sharp inequality is $\sigma_p \Delta x\geq \pi\hbar$. This bound will be reconsidered from the mathematical point of view in terms of a variational problem in Hilbert space and will furthermore be tested in a 4f-single slit diffraction experiment of a laser beam. Our results will be compared with a laser-experiment recently given by Fern\'andez-Guasti (2022) [9].