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