The well-known Heisenberg--Robertson uncertainty relation for a pair of noncommuting observables, is expressed in terms of the product of variances and the commutator among the operators, computed for the quantum state of a system. Different modified commutation relations have been considered in the last years with the purpose of taking into account the effect of quantum gravity. Indeed it can be seen that letting [X,P]=iℏ(1+βP2) implies the existence of a minimal length proportional to β. The Bialynicki-Birula--Mycielski entropic uncertainty relation in terms of Shannon entropies is also seen to be deformed in the presence of a minimal length, corresponding to a strictly positive deformation parameter β. Generalized entropies can be implemented. Indeed, results for the sum of position and (auxiliary) momentum Rényi entropies with conjugated indices have been provided recently for the ground and first excited state. We present numerical findings for conjugated pairs of entropic indices, for the lowest lying levels of the deformed harmonic oscillator system in 1D, taking into account the position distribution for the wavefunction and the actual momentum.
The well-known Heisenberg–Robertson uncertainty relation for a pair of noncommuting observables, is expressed in terms of the product of variances and the commutator among the operators, computed for the quantum state of a system. Different modified commutation relations have been considered in the last years with the purpose of taking into account the effect of quantum gravity. Indeed it can be seen that letting [ X , P ] = i ℏ ( 1 + β P 2 ) implies the existence of a minimal length proportional to β . The Bialynicki-Birula–Mycielski entropic uncertainty relation in terms of Shannon entropies is also seen to be deformed in the presence of a minimal length, corresponding to a strictly positive deformation parameter β . Generalized entropies can be implemented. Indeed, results for the sum of position and (auxiliary) momentum Rényi entropies with conjugated indices have been provided recently for the ground and first excited state. We present numerical findings for conjugated pairs of entropic indices, for the lowest lying levels of the deformed harmonic oscillator system in 1D, taking into account the position distribution for the wavefunction and the actual momentum.
In the framework of the generalized uncertainty principle, the position and momentum operators obey the modified commutation relation X,P=iħ1+βP2 , where β is the deformation parameter. Since the validity of the uncertainty relation for the Shannon entropies proposed by Beckner, Bialynicki-Birula, and Mycielski (BBM) depends on both the algebra and the used representation, we show that using the formally self-adjoint representation, that is, X=x and P=tanβp/β , where [x,p]=iħ , the BBM inequality is still valid in the form Sx+Sp≥1+lnπ as well as in ordinary quantum mechanics. We explicitly indicate this result for the harmonic oscillator in the presence of the minimal length.