Entropic Analysis of the Quantum Oscillator with a Minimal Length

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.

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The Minimal Length and the Shannon Entropic Uncertainty Relation

Advances in High Energy Physics ◽

10.1155/2016/5101389 ◽

2016 ◽

Vol 2016 ◽

pp. 1-8 ◽

Cited By ~ 7

Author(s):

Pouria Pedram

Keyword(s):

Quantum Mechanics ◽

Harmonic Oscillator ◽

Commutation Relation ◽

Deformation Parameter ◽

Uncertainty Relation ◽

Generalized Uncertainty Principle ◽

Adjoint Representation ◽

Minimal Length ◽

Entropic Uncertainty Relation ◽

Ordinary Quantum

In the framework of the generalized uncertainty principle, the position and momentum operators obey the modified commutation relationX,P=iħ1+βP2, whereβis 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=xandP=tan⁡βp/β, where[x,p]=iħ, the BBM inequality is still valid in the formSx+Sp≥1+ln⁡πas well as in ordinary quantum mechanics. We explicitly indicate this result for the harmonic oscillator in the presence of the minimal length.

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Position and momentum uncertainties of a particle in a V-shaped potential under the minimal length uncertainty relation

International Journal of Modern Physics A ◽

10.1142/s0217751x15502061 ◽

2015 ◽

Vol 30 (35) ◽

pp. 1550206 ◽

Cited By ~ 2

Author(s):

Zachary Lewis ◽

Ahmed Roman ◽

Tatsu Takeuchi

Keyword(s):

Harmonic Oscillator ◽

Commutation Relation ◽

Classical Limit ◽

Uncertainty Relation ◽

Minimal Length ◽

Positive Energy ◽

Positive Mass ◽

Negative Mass ◽

Energy Eigenstates ◽

Theory Comparison

We calculate the uncertainties in the position and momentum of a particle in the 1D potential [Formula: see text], [Formula: see text], when the position and momentum operators obey the deformed commutation relation [Formula: see text], [Formula: see text]. As in the harmonic oscillator case, which was investigated in a previous publication, the Hamiltonian [Formula: see text] admits discrete positive energy eigenstates for both positive and negative mass. The uncertainties for the positive mass states behave as [Formula: see text] as in the [Formula: see text] limit. For the negative mass states, however, in contrast to the harmonic oscillator case where we had [Formula: see text], both [Formula: see text] and [Formula: see text] diverge. We argue that the existence of the negative mass states and the divergence of their uncertainties can be understood by taking the classical limit of the theory. Comparison of our results is made with previous work by Benczik.

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Scheme for steady-state preparation of a harmonic oscillator in the first excited state

Physical Review A ◽

10.1103/physreva.90.023806 ◽

2014 ◽

Vol 90 (2) ◽

Cited By ~ 8

Author(s):

Kjetil Børkje

Keyword(s):

Excited State ◽

Steady State ◽

Harmonic Oscillator ◽

State Preparation ◽

First Excited State

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Massive fermions interacting via a harmonic oscillator in the presence of a minimal length uncertainty relation

International Journal of Modern Physics E ◽

10.1142/s0218301315500871 ◽

2015 ◽

Vol 24 (11) ◽

pp. 1550087 ◽

Cited By ~ 6

Author(s):

B. J. Falaye ◽

Shi-Hai Dong ◽

K. J. Oyewumi ◽

K. F. Ilaiwi ◽

S. M. Ikhdair

Keyword(s):

Energy Spectrum ◽

Harmonic Oscillator ◽

Commutation Relation ◽

Uncertainty Relation ◽

Minimal Length ◽

Harmonic Oscillators ◽

Relativistic Energy ◽

Oscillator Potential ◽

Harmonic Oscillator Potential ◽

Modified Dirac Equation

We derive the relativistic energy spectrum for the modified Dirac equation by adding a harmonic oscillator potential where the coordinates and momenta are assumed to obey the commutation relation [Formula: see text]. In the nonrelativistic (NR) limit, our results are in agreement with the ones obtained previously. Furthermore, the extension to the construction of creation and annihilation operators for the harmonic oscillators with minimal length uncertainty relation is presented. Finally, we show that the commutation relation of the [Formula: see text] algebra is satisfied by the operators [Formula: see text] and [Formula: see text].

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Rosen-Chambers Variation Theory of Linearly-Damped Classic and Quantum Oscillator

JOURNAL OF ADVANCES IN PHYSICS ◽

10.24297/jap.v4i1.6966 ◽

2014 ◽

Vol 4 (1) ◽

pp. 404-426

Author(s):

Vincze Gy. Szasz A.

Keyword(s):

Harmonic Oscillator ◽

Classical Mechanics ◽

Universal Time ◽

Variation Theory ◽

Quantum Oscillator ◽

Variation Principle ◽

Poisson Brackets ◽

Mathematical Meaning ◽

Damped Harmonic Oscillator ◽

Damped Oscillator

Phenomena of damped harmonic oscillator is important in the description of the elementary dissipative processes of linear responses in our physical world. Its classical description is clear and understood, however it is not so in the quantum physics, where it also has a basic role. Starting from the Rosen-Chambers restricted variation principle a Hamilton like variation approach to the damped harmonic oscillator will be given. The usual formalisms of classical mechanics, as Lagrangian, Hamiltonian, Poisson brackets, will be covered too. We shall introduce two Poisson brackets. The first one has only mathematical meaning and for the second, the so-called constitutive Poisson brackets, a physical interpretation will be presented. We shall show that only the fundamental constitutive Poisson brackets are not invariant throughout the motion of the damped oscillator, but these show a kind of universal time dependence in the universal time scale of the damped oscillator. The quantum mechanical Poisson brackets and commutation relations belonging to these fundamental time dependent classical brackets will be described. Our objective in this work is giving clearer view to the challenge of the dissipative quantum oscillator.

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The microwave spectrum of the light molecule of vinyl bromide (CH2 = CH Br), in the first excited state of vibration

Bulletin de la Classe des sciences ◽

10.3406/barb.1972.60566 ◽

1972 ◽

Vol 58 (1) ◽

pp. 956-968

Author(s):

Joseph Maroor ◽

Marc de Hemptinne

Keyword(s):

Excited State ◽

Microwave Spectrum ◽

Vinyl Bromide ◽

First Excited State

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Tripartite entropic uncertainty relation under phase decoherence

Scientific Reports ◽

10.1038/s41598-021-90689-3 ◽

2021 ◽

Vol 11 (1) ◽

Author(s):

R. A. Abdelghany ◽

A.-B. A. Mohamed ◽

M. Tammam ◽

Watson Kuo ◽

H. Eleuch

Keyword(s):

Lower Bound ◽

Uncertainty Relation ◽

Nearest Neighbors ◽

The Other ◽

Uncertainty In Measurement ◽

Entropic Uncertainty Relation ◽

Specific Choice ◽

Phase Decoherence ◽

Time Invariant ◽

Incompatible Observables

AbstractWe formulate the tripartite entropic uncertainty relation and predict its lower bound in a three-qubit Heisenberg XXZ spin chain when measuring an arbitrary pair of incompatible observables on one qubit while the other two are served as quantum memories. Our study reveals that the entanglement between the nearest neighbors plays an important role in reducing the uncertainty in measurement outcomes. In addition we have shown that the Dolatkhah’s lower bound (Phys Rev A 102(5):052227, 2020) is tighter than that of Ming (Phys Rev A 102(01):012206, 2020) and their dynamics under phase decoherence depends on the choice of the observable pair. In the absence of phase decoherence, Ming’s lower bound is time-invariant regardless the chosen observable pair, while Dolatkhah’s lower bound is perfectly identical with the tripartite uncertainty with a specific choice of pair.

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