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arXiv:2211.14724 · Physics Letters A

On the Experimental Verification of the Uncertainty Principle Of Position and Momentum

Thomas Schürmann, Ingo Hoffmann, Winfrid Görlich

Published Jan 1, 2022• 3 citations
arXivHTMLPDF
DOI
10.1016/j.physleta.2023.128787

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This experiment investigates the Heisenberg Uncertainty Principle (HUP) through an optical analogue using Single-slit diffraction of a monochromatic He–Ne laser. The spatial confinement of light through slits of varying width is compared with the resulting diffraction patterns, providing a means to examine the relationship between position uncertainty ∆x and transverse momentum uncertainty ∆px. Diffraction patterns were recorded using a photodiode detector, and the angular positions of the first minima were determined and used to estimate the spread in momentum via the de-Broglie relation. The results demonstrates and are consistent with the expected inverse relationship. The measured diffraction patterns were also found to adhere to the Fraunhofer diffraction relation, supporting the validity of the experimental model and the goodness of fit of the data.The calculated uncertainty products were further found to be consistent with the expected magnitude of Planck's constant. and therefore the the findings support the interpretation of diffraction as a classifcal manifestation of the Foruier relation between position and momentum space. Thus providing a physically intuituve demonstration of Heisenberg's Uncertainty Principle.
The uncertainty principle is considered to be one of the most striking features in quantum mechanics. In the textbook literature, uncertainty relations usually refer to the preparation uncertainty which imposes a limitation on the spread of measurement outcomes for a pair of noncommuting observables. In this work, we study the preparation uncertainty for the angular momentum, especially for spin-1/2. We derive uncertainty relations encompassing the triple components of angular momentum and show that, compared with the relations involving only two components, a triple constant 2/sqrt[3] often arises. Intriguingly, this constant is the same for the position and momentum case. Experimental verification is carried out on a single spin in diamond, and the results confirm the triple constant in a wide range of experimental parameters.
As a foundation of quantum physics, uncertainty relations describe ultimate limit for the measurement uncertainty of incompatible observables. Traditionally, uncertainty relations are formulated by mathematical bounds for a specific state. Here we present a method for geometrically characterizing uncertainty relations as an entire area of variances of the observables, ranging over all possible input states. We find that for the pair of position and momentum operators, Heisenberg’s uncertainty principle points exactly to the attainable area of the variances of position and momentum. Moreover, for finite-dimensional systems, we prove that the corresponding area is necessarily semialgebraic; in other words, this set can be represented via finite polynomial equations and inequalities, or any finite union of such sets. In particular, we give the analytical characterization of the areas of variances of (a) a pair of one-qubit observables and (b) a pair of projective observables for arbitrary dimension, and give the first experimental observation of such areas in a photonic system.