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arXiv:2210.07732 · Foundations of Physics

The Identification of Mean Quantum Potential with Fisher Information Leads to a Strong Uncertainty Relation

Yakov Bloch, Eliahu Cohen

Published Oct 29, 2022• 4 citations
arXivHTMLPDF
DOI
10.1007/s10701-022-00638-x

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The spreading of the quantum-mechanical probability cloud for the ground state of the Morse and modified Pöschl–Teller potentials, which controls the chemical and physical properties of some molecular systems, is studied in position and momentum space by means of global (Shannon's information entropy, variance) and local (Fisher's information) information-theoretic measures. We establish a general relation between variance and Fisher's information, proving that, in the case of a real-valued and symmetric wavefunction, the well-known Cramer–Rao and Heisenberg uncertainty inequalities are equivalent. Finally, we discuss the asymptotics of all three information measures, showing that the ground state of these potentials saturates all the uncertainty relations in an appropriate limit of the parameter.