Entropic analysis of the quantum oscillator with a minimal length

PORTESI, M.; PUERTAS CENTENO, D.; DEHESA, J. S.

Cagliari

Congreso; Quantum Cagliari 2018: quantum structures & quantum information theory; 2018

Università degli Studi di Cagliari

The well-known Heisenberg-Robertson uncertainty principle for a pair of noncommuting observables, is expressed in terms of the variances and the commutatoramong the operators, computed for the quantum state of a system. Different modified commutation relations have been considered in the last years with the purpose oftaking into account the effect of quantum gravity, and experimental realizations havebeen discussed. In general one can consider [X, P ] = i f (X, P ). It can be seen thata modified commutation relation as [X,P] = i(1+βP2 +αX2) implies the existenceof a minimal length and a minimal momentum proportional to the square roots of βand α, respectively. The Bialynicki-Birula & Mycielski entropic uncertainty relationin terms of the Shannon entropy is seen to be deformed in the presence of a minimallength, corresponding to a strictly positive deformation parameter β (setting α = 0).Generalized entropies can be implemented. Indeed, results for the sum of positionand momentum Rényi entropies with conjugated indices have also been provided recently. We present analytical and numerical findings for arbitrary pairs of entropicindices, analyzing the quantum oscillator with minimal length. We comment alsoon the behavior of other relevant information and complexity measures applied tothis problem.