The Uncertainty Principle in Terms of Isoperimetric Inequalities

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DOI: 10.4236/am.2017.83025    230 Downloads   323 Views  

ABSTRACT

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.

Cite this paper

Schürmann, T. (2017) The Uncertainty Principle in Terms of Isoperimetric Inequalities. Applied Mathematics, 8, 307-311. doi: 10.4236/am.2017.83025.

References

[1] Heisenberg, W. (1927) über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik. Zeitschrift für Physik, 43, 172-198.
https://doi.org/10.1007/BF01397280
[2] Heisenberg, W. (1930) The Physical Principles of the Quantum Theory. University of Chicago Press, Chicago. [Reprinted by Dover, New York (1949, 1967)].
[3] Kennard, E.H. (1927) Zur Quantenmechanik einfacher Bewegungstypen. Zeitschrift für Physik, 44, 326-352.
https://doi.org/10.1007/BF01391200
[4] Schürmann, T. and Hoffmann, I. (2009) A Closer Look at the Uncertainty Relation of Position and Momentum. Foundations of Physics, 39, 958-963.
https://doi.org/10.1007/s10701-009-9310-0
[5] This Inequality States That for Any Bounded Interval Q=[a,b] and Any Continuously Differentiable f(x)∈C1(Q) with f(a)=f(b)=0. It Follows That ∫Q∣f∣2 dx≤[(b-a)22] ∫Q ∣f∣2 dx. The Constant (b-a)22 Cannot Be Further Improved [6].
[6] Osserman, R. (1978) The Isoperimetric Inequality. Bulletin of the American Mathematical Society, 84, 1182-1238.
https://doi.org/10.1090/S0002-9904-1978-14553-4
[7] Pólya, G. and Szego, G. (1951) Isoperimetric Inequalities in Mathematical Physics. Annals of Mathematics Studies, 27, Princeton University Press.
[8] Payne, L.E. (1967) Isoperimetric Inequalities and Their Applications. SIAM Review, 9, 453-488.
https://doi.org/10.1137/1009070
[9] Rayleigh, J.W.W. (1894/1896) The Theory of Sound. 2nd Edition, London, 339-340.
[10] Faber, G. (1923) Bayer. Akad. Wiss. München, Math.-Phys., 169.
[11] Krahn, E. (1927) über eine von Rayleigh formulierte Minimaleigenschaft des Kreises. Mathematische Annalen, 94, 97.
https://doi.org/10.1007/BF01208645
[12] Kuttler, J.R. and Sigillito, V.G. (1984) Eigenvalues of the Laplacian in Two Dimensions. SIAM Review, 26, 163-193.
https://doi.org/10.1137/1026033
[13] Krahn, 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.]

  
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