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

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 $\mathrm{\Delta }x$, it has been shown that a finite standard deviation ${\sigma }_p<\mathrm{\infty }$ can only be ensured if the wave-function is zero at the edge of $\mathrm{\Delta }x$, otherwise it does not exist [8]. Under this circumstances the corresponding sharp inequality is ${\sigma }_p\mathrm{\Delta }x\ge \pi \mathrm{\hslash }$. 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 Fernández-Guasti (2022) [9].