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

2015 ◽  
Vol 30 (35) ◽  
pp. 1550206 ◽  
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.

scholarly journals The Minimal Length and the Shannon Entropic Uncertainty Relation

2016 ◽  
Vol 2016 ◽  
pp. 1-8 ◽  
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.

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

2015 ◽  
Vol 24 (11) ◽  
pp. 1550087 ◽  
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].

scholarly journals The Damped Harmonic Oscillator at the Classical Limit of the Snyder-de Sitter Space

2021 ◽  
Vol 13 (2) ◽  
pp. 1
Author(s):  
Lat´evi M. Lawson ◽  
Ibrahim Nonkan´e ◽  
Komi Sodoga
Keyword(s):  
Harmonic Oscillator ◽  
Classical Limit ◽  
Minimal Length ◽  
Poisson Brackets ◽  
Canonical Transformations ◽  
Trigonometric Functions ◽  
De Sitter ◽  
One Dimensional ◽  
Damped Harmonic Oscillator ◽  
Math Phys

Valtancoli in his paper entitled (P. Valtancoli, Canonical transformations and minimal length, J. Math. Phys. 56, 122107 2015) has shown how the deformation of the canonical transformations can be made compatible with the deformed Poisson brackets. Based on this work and through an appropriate canonical transformation, we solve the problem of one dimensional (1D) damped harmonic oscillator at the classical limit of the Snyder-de Sitter (SdS) space. We show that the equations of the motion can be described by trigonometric functions with frequency and period depending on the deformed and the damped parameters. We eventually discuss the influences of these parameters on the motion of the system.

scholarly journals Entropic Analysis of the Quantum Oscillator with a Minimal Length

Proceedings ◽  
2019 ◽  
Vol 12 (1) ◽  
pp. 57
Author(s):  
David Puertas-Centeno ◽  
Mariela Portesi
Keyword(s):  
Excited State ◽  
Harmonic Oscillator ◽  
Deformation Parameter ◽  
Uncertainty Relation ◽  
Minimal Length ◽  
Quantum Oscillator ◽  
Position Distribution ◽  
Entropic Uncertainty Relation ◽  
First Excited State ◽  
Generalized Entropies

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.

scholarly journals Short distance versus long distance physics: The classical limit of the minimal length uncertainty relation

2002 ◽  
Vol 66 (2) ◽  
Author(s):  
Sándor Benczik ◽  
Lay Nam Chang ◽  
Djordje Minic ◽  
Naotoshi Okamura ◽  
Saiffudin Rayyan ◽  
...  
Keyword(s):  
Classical Limit ◽  
Short Distance ◽  
Uncertainty Relation ◽  
Minimal Length ◽  
Long Distance

scholarly journals The Harmonic Oscillator in the Classical Limit of a Minimal-Length Scenario

2016 ◽  
Vol 46 (6) ◽  
pp. 777-783 ◽  
Author(s):  
T. S. Quintela ◽  
J. C. Fabris ◽  
J. A. Nogueira
Keyword(s):  
Harmonic Oscillator ◽  
Classical Limit ◽  
Minimal Length

(ANTI)COMMUTATIVITY OF THE HARMONIC CREATION AND ANNIHILATION OPERATORS

2006 ◽  
Vol 20 (11n13) ◽  
pp. 1808-1818
Author(s):  
S. KUWATA ◽  
A. MARUMOTO
Keyword(s):  
Irreducible Representation ◽  
Harmonic Oscillator ◽  
Linear Algebra ◽  
Commutation Relation ◽  
Complex Number ◽  
Single Mode ◽  
Equation Of Motion ◽  
Sufficient Condition ◽  
Special Linear ◽  
Creation And Annihilation Operators

It is known that para-particles, together with fermions and bosons, of a single mode can be described as an irreducible representation of the Lie (super) algebra 𝔰𝔩2(ℂ) (2-dimensional special linear algebra over the complex number ℂ), that is, they satisfy the equation of motion of a harmonic oscillator. Under the equation of motion of a harmonic oscillator, we obtain the set of the commutation relations which is isomorphic to the irreducible representation, to find that the equation of motion, conversely, can be derived from the commutation relation only for the case of either fermion or boson. If Nature admits of the existence of such a sufficient condition for the equation of motion of a harmonic oscillator, no para-particle can be allowed.

NUMERICAL STUDY OF THE INTERACTION BETWEEN POSITIVE AND NEGATIVE MASS OBJECTS IN GENERAL RELATIVITY

2012 ◽  
Vol 21 (07) ◽  
pp. 1250061 ◽  
Author(s):  
ZHOUJIAN CAO
Keyword(s):  
Boundary Condition ◽  
General Relativity ◽  
Numerical Study ◽  
Naked Singularity ◽  
Newtonian Limit ◽  
Computational Domain ◽  
Positive Mass ◽  
Negative Mass ◽  
Mass Particle ◽  
Causal Property

Based on Baumgarte–Shapiro–Shibata–Nakamura formalism and moving puncture method, we demonstrate the first numerical evolutions of the interaction between positive and negative mass objects. Using the causal property of general relativity, we set our computational domain around the positive mass black hole while excluding the region around the naked singularity introduced by the negative mass object. Besides the usual Sommerfeld numerical boundary condition, an approximate boundary condition is proposed for this nonasymptotically-flat computational domain. Careful checks show that either boundary condition introduces smaller error than the numerical truncation errors. This is consistent with the causal property of general relativity. Except for the numerical truncation error and round-off error, our method gives an exact solution to the full Einstein's equation for a portion of spacetime with two objects whose masses have opposite signs. So our method opens the door for numerical explorations with negative mass objects. Based on this method, we investigate the Newtonian limit of spacetime with two objects whose masses have opposite sign. Our result implies that this spacetime does have a Newtonian limit which corresponds to a negative mass particle chasing a positive mass particle. This result sheds some light on an interesting debate about the Newtonian limit of a spacetime with positive and negative point masses.

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