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Journal of Physics A Mathematical and Theoretical

Two multi-dimensional uncertainty relations

Lubomír Skála, V. Kapsa

Published Jun 11, 2008• 9 citations
Publisher

Abstract

Two multi-dimensional uncertainty relations, one related to the probability density and the other one related to the probability density current, are derived and discussed. Both relations are stronger than the usual uncertainty relations for the coordinates and momentum.
DOI
10.1088/1751-8113/41/26/265302

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In this paper, a three parameter (α, λ, μ) ‘Hartree and Ingman (1933)’ type correlated wave function has been used to derive an analytical model to quantify the two electron systems. The model is used to construct the expressions for the single particle wave functions [ 𝜓(𝑟) , 𝜙(𝑝)] in coordinate space and momentum space respectively with the subscripts marked as correlated (c) and uncorrelated (uc) ones. Further, the correlated and uncorrelated wave functions are used to form the single particle charge densities [𝜌(𝑟) , 𝛾(𝑝)] in coordinate space and momentum space. These wave functions and the single particle charge densities are used to compute the numerical values of the Shannon (S) and Fisher (I) information entropies for both the correlated and uncorrelated systems in coordinate space and momentum spaces and furthermore to examine their correlation effects in coordinate and momentum spaces. The uncorrelated and correlated values of Shannon information entropies indicate the reciprocity between the coordinate and momentum spaces such that low values of the Shannon information entropy in coordinate space ( 𝑆𝜌), are associated with the high values of Shannon information entropy in momentum space (𝑆𝛾) . The uncorrelated and correlated values of Shannon information entropies (S) also validate the entropic uncertainty relations Bialynicki-Birula and Mycielski (BBM) inequality which reads as (𝑆𝜌+𝑆𝛾)≥3(1+𝑙𝑛𝜋), as well as another stronger version of the uncertainty relation that is (𝑆𝜌𝑐+𝑆𝛾𝑐)>(𝑆𝜌𝑢𝑐+𝑆𝛾𝑢𝑐). Further, both the products of uncorrelated and correlated values of Fisher information entropies 𝐼𝜌𝑢𝑐𝐼𝛾𝑢𝑐 and 𝐼𝜌𝑐𝐼𝛾𝑐 satisfy the inequality condition 𝐼𝜌𝑐𝐼𝛾𝑐>𝐼𝜌𝑢𝑐𝐼𝛾𝑢𝑐 in coordinate and momentum spaces. At the same time these products also satisfy the Fisher based uncertainty relation, 𝐼𝜌𝐼𝛾≥36. The consistency of the results satisfying the uncertainty relations can be checked from the data demonstrated in the Table 1 and Table 2.