Heisenberg and Entropic Uncertainty Measures for Large-Dimensional Harmonic Systems
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Entropic uncertainty measuresD-dimensional harmonic oscillatorD-dimensional quantum physicsRadial and momentum expectation valuesHarmonic states at large dimensions
Puertas-Centeno, D.; Valero Toranzo, I.; Sánchez-Dehesa Moreno-Cid, J. Heisenberg and Entropic Uncertainty Measures for Large-Dimensional Harmonic Systems. Entropy, 19(4): 164 (2017). [http://hdl.handle.net/10481/49123]
SponsorshipThis work has been partially supported by the projects FQM-7276 and FQM-207 of the Junta de Andalucía and the MINECO (Ministerio de Economia y Competitividad)-FEDER (European Regional Development Fund) Grants FIS2014- 54497P and FIS2014-59311-P. Irene V. Toranzo acknowledges the support of MEunder the program FPU.
The D-dimensional harmonic system (i.e., a particle moving under the action of a quadratic potential) is, together with the hydrogenic system, the main prototype of the physics of multidimensional quantum systems. In this work, we rigorously determine the leading term of the Heisenberg-like and entropy-like uncertainty measures of this system as given by the radial expectation values and the Rényi entropies, respectively, at the limit of large D. The associated multidimensional position-momentum uncertainty relations are discussed, showing that they saturate the corresponding general ones. A conjecture about the Shannon-like uncertainty relation is given, and an interesting phenomenon is observed: the Heisenberg-like and Rényi-entropy-based equality-type uncertainty relations for all of the D-dimensional harmonic oscillator states in the pseudoclassical ( D → ∞ ) limit are the same as the corresponding ones for the hydrogenic systems, despite the so different character of the oscillator and Coulomb potentials.