AMApplied Mathematics2152-7385Scientific Research Publishing10.4236/am.2017.83025AM-74880ArticlesPhysics&Mathematics The Uncertainty Principle in Terms of Isoperimetric Inequalities ThomasSchürmann1*Germaniastra?e 8, Düsseldorf, Germany * E-mail:t.schurmann@icloud.com030320170803307311July 14, 2016Accepted: March 21, March 24, 2017© Copyright 2014 by authors and Scientific Research Publishing Inc. 2014This work is licensed under the Creative Commons Attribution International License (CC BY).

Simultaneous measurements of position and momentum are considered in n dimensions. We find, that for a particle whose position is strictly localized in a compact domain (spatial uncertainty) with non-empty boundary, the standard deviation of its momentum is sharply bounded by , while is the first Dirichlet eigenvalue of the Laplacian on D.

Uncertainty Principle Dirichlet Eigenvalue Wirtinger Inequality
Cite this paper

Schürmann, T. (2017) The Uncertainty Principle in Terms of Isoperimetric Inequalities. Applied Mathematics, 8, 307-311.

ReferencesHeisenberg, W. (1927) über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik. Zeitschrift für Physik, 43, 172-198., W. (1930) The Physical Principles of the Quantum Theory. University of Chicago Press, Chicago. [Reprinted by Dover, New York (1949, 1967)].Kennard, E.H. (1927) Zur Quantenmechanik einfacher Bewegungstypen. Zeitschrift für Physik, 44, 326-352.ürmann, T. and Hoffmann, I. (2009) A Closer Look at the Uncertainty Relation of Position and Momentum. Foundations of Physics, 39, 958-963. Inequality States That for Any Bounded Interval Q=[a,b] and Any Continuously Differentiable f(x)∈C<sup>1</sup>(Q) with f(a)=f(b)=0. It Follows That ∫<sub>Q</sub>∣f∣<sup>2</sup> dx≤[(b-a)<sup>2</sup>/π<sup>2</sup>] ∫<sub>Q</sub> ∣f∣<sup>2</sup> dx. The Constant (b-a)<sup>2</sup>/π<sup>2</sup> Cannot Be Further Improved [6].Osserman, R. (1978) The Isoperimetric Inequality. Bulletin of the American Mathematical Society, 84, 1182-1238.ólya, G. and Szego, G. (1951) Isoperimetric Inequalities in Mathematical Physics. Annals of Mathematics Studies, 27, Princeton University Press.Payne, L.E. (1967) Isoperimetric Inequalities and Their Applications. SIAM Review, 9, 453-488., J.W.W. (1894/1896) The Theory of Sound. 2nd Edition, London, 339-340.Faber, G. (1923) Bayer. Akad. Wiss. München, Math.-Phys., 169.Krahn, E. (1927) über eine von Rayleigh formulierte Minimaleigenschaft des Kreises. Mathematische Annalen, 94, 97., J.R. and Sigillito, V.G. (1984) Eigenvalues of the Laplacian in Two Dimensions. SIAM Review, 26, 163-193., E. (1926) über Minimaleigenschaften der Kugel in drei und mehr Dimensionen. Acta Comm. Univ. Tartu (Dorpat), A9, 1. [English Translation: Lumiste, ü. and Peetre, J., Eds., Edgar Krahn, 1894-1961, A Centenary Volume, IOS Press, Amsterdam, 1994, Chapter 6.]