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Upper bounds on Shannon and Rényi entropies for central potentials

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10.1063/1.3549585

### Abstract

The Rényi and Shannon entropies are information-theoretic measures, which have enabled to formulate the position–momentum uncertainty principle in a much more adequate and stringent way than the (variance-based) Heisenberg-like relation. Moreover, they are closely related to various energetic density functionals of quantum systems. Here we derive upper bounds on these quantities in terms of the second-order moment 〈*r* ^{2}〉 for general central potentials. This improves previous results of this type. The proof uses the Rényi maximization procedure with a covariance constraint due to Costa *et al.* [in Proceedings of the Fourth International Workshop on Energy Minimization Methods in Computer Vision and Pattern Recognition (EMMCVPR), edited by A.Rangarajan, M. A. T.Figueiredo, and J.Zerubia (Springer-Verlag, Lisbon, 2003), [Lect. Notes Comput. Sci.52, 211 (2003).]] The contributions to these bounds coming from the radial and angular parts of the physical wave functions are taken into account. Finally, the application to the *d*-dimensional (*d* ⩾ 3) hydrogenic and oscillator-like systems is provided.

© 2011 American Institute of Physics

Received 30 July 2010
Accepted 10 January 2011
Published online 11 February 2011

Acknowledgments: P.S.M. and J.S.D. are very grateful to Junta de Andalucía for the Grant Nos. FQM-2445 and FQM-4643, and the Ministerio de Ciencia e Innovación for the Grant No. FIS2008-02380. P.S.M. and J.S.D. belong to the research group FQM-207.

Article outline:

I. INTRODUCTION

II. THE QUANTUM PROBLEM FOR *d*-DIMENSIONAL CENTRAL POTENTIALS

III. RÉNYI ENTROPY AND COVARIANCE MATRIX

IV. UPPER BOUNDS ON RÉNYI’S ENTROPY

V. APPLICATION TO *d*-DIMENSIONAL HYDROGENIC AND OSCILLATOR-LIKE SYSTEMS

A. Hydrogenic systems

B. Oscillator-like systems

VI. CONCLUSIONS AND OPEN PROBLEMS

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2011-02-11

2014-04-20

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