Entropic Analysis of the Quantum Oscillator with a Minimal Length

• Main Authors: David Puertas-Centeno, Mariela Portesi
• Format: Article
• Language: English
• Published: MDPI AG 2019-11-01
• Series: Proceedings
• Subjects: uncertainty relations; information entropy; quantum gravity
• Online Access: https://www.mdpi.com/2504-3900/12/1/57

The well-known Heisenberg&#8722;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 <inline-formula> <math display="inline"> <semantics> <mrow> <mrow> <mo>[</mo> <mi>X</mi> <mo>,</mo> <mi>P</mi> <mo>]</mo> </mrow> <mo>=</mo> <mi>i</mi> <mi>ℏ</mi> <mrow> <mo>(</mo> <mn>1</mn> <mo>+</mo> <mi>&#946;</mi> <msup> <mi>P</mi> <mn>2</mn> </msup> <mo>)</mo> </mrow> </mrow> </semantics> </math> </inline-formula> implies the existence of a minimal length proportional to <inline-formula> <math display="inline"> <semantics> <msqrt> <mi>&#946;</mi> </msqrt> </semantics> </math> </inline-formula>. The Bialynicki-Birula&#8722;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 <inline-formula> <math display="inline"> <semantics> <mi>&#946;</mi> </semantics> </math> </inline-formula>. Generalized entropies can be implemented. Indeed, results for the sum of position and (auxiliary) momentum R&#233;nyi 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.