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| HAL: hal-00542576, version 1 |
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| 30th International Workshop on Bayesian Inference and Maximun Entropy Methods in Science and Engineering (MaxEnt'10), Chamonix : France (2010) |
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| On moments-based Heisenberg inequalities |
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| Steeve Zozor 1Mariela Portesi 2 |
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| (2010) |
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| In this paper we revisit the quantitative formulation of the Heisenberg uncertainty principle. The primary version of this principle establishes the impossibility of refined simultaneous measurement of position x and momentum u for a (1-dimensional) quantum particle in terms of variances: <||x||^2> <||u||^2> ≥ 1/4. Since this inequality applies provided each variance exists, some authors proposed entropic versions of this principle as an alternative (employing Shannon's or Rényi's entropies). As another alternative, we consider moments-based formulations and show that inequalities involving moments of orders other than 2 can be found. Our procedure is based on the Rényi entropic versions of the Heisenberg relation together with the search for the maximal entropy un- der statistical moments' constraints ( <||x||^a> and <||u||^b> ). Our result improves a relation proposed very recently by Dehesa et al. [1] where the same approach was used but starting with the Shannon version of the entropic uncertainty relation. Furthermore, we show that when a = b, the best bound we can find with our approach coincides with that of Ref. [1] and, in addition, for a = b = 2 the variance-based Heisenberg relation is recovered. Finally, we illustrate our results in the cases of d-dimensional hydrogenic systems. |
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| 1: | Grenoble Images Parole Signal Automatique (GIPSA-lab) |
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CNRS : UMR5216 – Université Joseph Fourier - Grenoble I – Université Pierre-Mendès-France - Grenoble II – Université Stendhal - Grenoble III – Institut Polytechnique de Grenoble - Grenoble Institute of Technology |
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| 2: | Instituto de Física La Plata, and Departamento de Física, Facultad de Ciencias Exactas, Universidad Nacional de La Plata (CONICET) |
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Universidad Nacional de la Plata |
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| 3: | Instituto Carlos I de Física Teórica y Computacional |
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Universidad de Granada |
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| 4: | Departamento de Matemática Aplicada |
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Universidad de Granada |
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| 5: | Departamento de Física Atómica, Molecular y Nuclear |
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Universidad de Granada |
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| Subject | : | Mathematics/Information Theory Computer Science/Information Theory and Coding Physics/Quantum Physics |
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| Heisenberg uncertainty principle – maximal entropy – statistical moments constraints |
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| hal-00542576, version 1 | |
| http://hal.archives-ouvertes.fr/hal-00542576 | |
| oai:hal.archives-ouvertes.fr:hal-00542576 | |
| From: Steeve Zozor | |
| Submitted on: Friday, 3 December 2010 04:56:21 | |
| Updated on: Tuesday, 26 July 2011 15:48:10 | |
