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arXiv:1108.3828 · Journal of Russian Laser Research

Shannon entropy as a measure of uncertainty in positions and momenta

Łukasz Rudnicki

Published Jul 1, 2011• 19 citations
arXivHTMLPDF
DOI
10.1007/s10946-011-9227-x

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This paper is prepared as a contribution to the proceedings after the 12th ICSSUR/Feynfest Conference held in Foz do Iguacu (Brazil) from 2 to 6 May 2011. In the first part I briefy report the topic of entropic uncertainty relations for position and momentum variables. Then I investigate the discrete Shannon entropies related to the case of fnite number of detectors set to measure probability distributions in position and momentum spaces. I derive an uncertainty relation for the sum of the Shannon entropies which generalizes previous approaches [Phys. Lett. 103 A, 253 (1984)] based on an infnite number of detectors (bins).
This paper is prepared as a contribution to the proceedings after the 12th ICSSUR/Feynfest Conference held in Foz do Iguaçu (Brazil) from 2 to 6 May 2011. In the first part I briefly report the topic of entropic uncertainty relations for position and momentum variables. Then I investigate the discrete Shannon entropies related to the case of finite number of detectors set to measure probability distributions in position and momentum spaces. I derive an uncertainty relation for the sum of the Shannon entropies which generalizes previous approaches [Phys. Lett. 103 A, 253 (1984)] based on an infinite number of detectors (bins).
We study the influence of the interparticle and confining potentials on statistical correlation via the correlation coefficient and mutual information in ground and some excited states of the Moshinsky atom in position and momentum space. The magnitude of the correlation between positions and between momenta is equal in the ground state. In excited states, the correlation between the momenta of the particles is greater than between their positions when they interact through an attractive potential whereas for repulsive interparticle potentials the opposite is true. Shannon entropies, and their sums (entropic formulations of the uncertainty principle), are also analyzed, showing that the one-particle entropy sum is dependent on the interparticle potential and thus able to detect the correlation between particles.