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arXiv:1207.2551

A Mixed-Entropic Uncertainty Relation

Kamal Bhattacharyya, Karabi Halder

Published Jul 11, 2012• 0 citations
arXivHTMLPDF

Abstract

We highlight the advantages of using simultaneously the Shannon and Fisher information measures in providing a useful form of the uncertainty relation for the position-momentum case. It does not require any Fourier transformation. The sensitivity is also noteworthy.
DOI
10.48550/arxiv.1207.2551

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ABSTRACT The dimensional analysis of the position and momentum variances‐based quantum mechanical Heisenberg uncertainty measures was carried out for generalized Yukawa‐type potential. The entropic information measures given by Shannon entropy sums and Fisher information products were obtained out for generalized Yukawa‐type potential. It is interesting to observe that the Shannon entropy in position space, , increases and then decreases with the quantum number , suggesting a growth in delocalization due to a rapid increase in fluctuations. Likewise, the Shannon entropy in momentum space, , decreases and then increases with the quantum number , and vice versa. This unusual phenomenon may be explained by the parity restrictions inherent in the system. The Bialynicki‐Birula and Mycielski (BBM) entropic uncertainty relation was verified to be saturated; however, the sum of the entropies increases with the quantum number up to a certain point and then decreases as continues to increase. We also found that the Fisher information accurately increases with the quantum number . The numerical results that support the validity of the scaling properties for Shannon entropy sum and Fisher information product were presented. The existence of a squeezed phenomenon in the ground and first excited state for some values of the potential screening parameter in momentum space was found. The Heisenberg uncertainty relation is met when the squeeze in one coordinate is compensated by an increase in the other coordinate. The results obtained exceed the theoretical lower bounds for all the information‐theoretic measures considered, confirming their physical validity and accuracy.