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 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 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 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 Investigation of Free Particle Propagator with Generalized Uncertainty Problem

2016 ◽  
Vol 2016 ◽  
pp. 1-4 ◽  
Author(s):  
F. Ghobakhloo ◽  
H. Hassanabadi
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Quantum Mechanics ◽  
Uncertainty Principle ◽  
Free Particle ◽  
Generalized Uncertainty Principle ◽  
Minimal Length ◽  
Order Ordinary Differential Equation ◽  
Particle Propagator ◽  
Ordinary Quantum

We consider the Schrödinger equation with a generalized uncertainty principle for a free particle. We then transform the problem into a second-order ordinary differential equation and thereby obtain the corresponding propagator. The result of ordinary quantum mechanics is recovered for vanishing minimal length parameter.

Factorization in the quantum mechanics with the generalized uncertainty principle

2015 ◽  
Vol 30 (23) ◽  
pp. 1550117 ◽  
Author(s):  
Won Sang Chung
Keyword(s):  
Quantum Mechanics ◽  
Harmonic Oscillator ◽  
Perturbation Method ◽  
Commutation Relation ◽  
Uncertainty Principle ◽  
Quantum Particle ◽  
Generalized Uncertainty Principle ◽  
Factorization Method ◽  
Momentum Operator ◽  
Energy Eigenvalues

In this paper, we discuss the quantum mechanics with the generalized uncertainty principle (GUP) where the commutation relation is given by [Formula: see text]. For this algebra, we obtain the eigenfunction of the momentum operator. We also study the GUP corrected quantum particle in a box. Finally, we apply the factorization method to the harmonic oscillator in the presence of a minimal observable length and obtain the energy eigenvalues by applying the perturbation method.

scholarly journals QUANTUM MECHANICAL COHERENT STATES OF THE HARMONIC OSCILLATOR AND THE GENERALIZED UNCERTAINTY PRINCIPLE

2005 ◽  
Vol 03 (04) ◽  
pp. 623-632 ◽  
Author(s):  
KOUROSH NOZARI ◽  
TAHEREH AZIZI
Keyword(s):  
Quantum Mechanics ◽  
Harmonic Oscillator ◽  
Uncertainty Principle ◽  
Coherent States ◽  
Equations Of Motion ◽  
Generalized Uncertainty Principle ◽  
Quantum Mechanical ◽  
Simple Harmonic Oscillator ◽  
Definition Of ◽  
Ordinary Quantum

In this paper, dynamics and quantum mechanical coherent states of a simple harmonic oscillator are considered in the framework of the Generalized Uncertainty Principle (GUP). Equations of motion for the simple harmonic oscillator are derived and some of their new implications are discussed. Then, coherent states of the harmonic oscillator in the case of the GUP are compared with the relative situation in ordinary quantum mechanics. It is shown that in the framework of GUP there is no considerable difference in definition of coherent states relative to ordinary quantum mechanics. But, considering expectation values and variances of some operators, based on quantum gravitational arguments, one concludes that although it is possible to have complete coherency and vanishing broadening in usual quantum mechanics, gravitational induced uncertainty destroys complete coherency in quantum gravity and it is not possible to have a monochromatic ray in principle.

scholarly journals A q-OSCILLATOR WITH "ACCIDENTAL" DEGENERACY OF ENERGY LEVELS

2007 ◽  
Vol 22 (13) ◽  
pp. 949-960 ◽  
Author(s):  
A. M. GAVRILIK ◽  
A. P. REBESH
Keyword(s):  
Lie Algebra ◽  
Commutation Relation ◽  
Deformation Parameter ◽  
Uncertainty Relation ◽  
Energy Levels ◽  
The State ◽  
The Real ◽  
Simple Lie Algebra ◽  
Accidental Degeneracy ◽  
Real Value

We study main features of the exotic case of q-deformed oscillators (so-called Tamm–Dancoff cutoff oscillator) and find some special properties: (i) degeneracy of the energy levels En1 = En1 + 1, n1 ≥ 1, at the real value[Formula: see text] of deformation parameter, as well as the occurrence of other degeneracies En1 = En1 + k, for k ≥ 2, at the corresponding values of q which depend on both n1 and k; (ii) the position and momentum operators X and Pcommute on the state|n1> if q is fixed as [Formula: see text], that implies unusual uncertainty relation; (iii) two commuting copies of the creation, annihilation, and number operators of this q-oscillator generate the corresponding q-deformation of the non-simple Lie algebra su(2) ⊕ u(1)whose nontrivial q-deformed commutation relation is: [J+, J-] = 2J0q2J3-1 where [Formula: see text] and [Formula: see text].

scholarly journals Thermal properties of a two-dimensional Duffin–Kemmer–Petiau oscillator under an external magnetic field in the presence of a minimal length

2020 ◽  
Vol 35 (33) ◽  
pp. 2050278
Author(s):  
H. Aounallah ◽  
B. C. Lütfüoğlu ◽  
J. Kříž
Keyword(s):  
Magnetic Field ◽  
External Magnetic Field ◽  
Energy Eigenvalue ◽  
Generalized Uncertainty Principle ◽  
Planck Scale ◽  
Summation Formula ◽  
Minimal Length ◽  
Two Dimensional ◽  
Relativistic Limit ◽  
Ordinary Quantum

Generalized uncertainty principle puts forward the existence of the shortest distances and/or maximum momentum at the Planck scale for consideration. In this article, we investigate the solutions of a two-dimensional Duffin–Kemmer–Petiau (DKP) oscillator within an external magnetic field in a minimal length (ML) scale. First, we obtain the eigensolutions in ordinary quantum mechanics. Then, we examine the DKP oscillator in the presence of an ML for the spin-zero and spin-one sectors. We determine an energy eigenvalue equation in both cases with the corresponding eigenfunctions in the non-relativistic limit. We show that in the ordinary quantum mechanic limit, where the ML correction vanishes, the energy eigenvalue equations become identical with the habitual quantum mechanical ones. Finally, we employ the Euler–Mclaurin summation formula and obtain the thermodynamic functions of the DKP oscillator in the high-temperature scale.

scholarly journals MAXIMALLY LOCALIZED STATES IN QUANTUM MECHANICS WITH A MODIFIED COMMUTATION RELATION TO ALL ORDERS

2012 ◽  
Vol 27 (21) ◽  
pp. 1250113 ◽  
Author(s):  
GLÁUBER CARVALHO DORSCH ◽  
JOSÉ ALEXANDRE NOGUEIRA
Keyword(s):  
Quantum Mechanics ◽  
Commutation Relation ◽  
Casimir Effect ◽  
Minimal Length ◽  
Localized States ◽  
Minimum Length ◽  
Position Measurement ◽  
First Order ◽  
The One ◽  
Physical Consequences

We construct the states of maximal localization taking into account a modification of the commutation relation between position and momentum operators to all orders of the minimum length parameter. To first-order, the algebra we use reproduces the one proposed by Kempf, Mangano and Mann. It is emphasized that a minimal length acts as a natural regulator for the theory, thus eliminating the otherwise ever appearing infinities. So, we use our results to calculate the first correction to the Casimir effect due to the minimal length. We also discuss some of the physical consequences of the existence of a minimal length, culminating in a proposal to reformulate the very concept of "position measurement."

scholarly journals Quantum cosmology with dynamical vacuum in a minimal-length scenario

2021 ◽  
Vol 81 (4) ◽  
Author(s):  
M. F. Gusson ◽  
A. Oakes O. Gonçalves ◽  
R. G. Furtado ◽  
J. C. Fabris ◽  
J. A. Nogueira
Keyword(s):  
Wave Function ◽  
Commutation Relation ◽  
Uncertainty Principle ◽  
Quantum Cosmology ◽  
Generalized Uncertainty Principle ◽  
Minimal Length ◽  
Minimum Length ◽  
Position Space ◽  
Infinite Norm ◽  
The Universe

AbstractIn this work, we consider effects of the dynamical vacuum in quantum cosmology in presence of a minimum length introduced by the GUP (generalized uncertainty principle) related to the modified commutation relation $$[{\hat{X}},{\hat{P}}] := \frac{i\hbar }{ 1 - \beta {\hat{P}}^2 }$$ [ X ^ , P ^ ] : = i ħ 1 - β P ^ 2 . We determine the wave function of the Universe $$ \psi _{qp}(\xi ,t)$$ ψ qp ( ξ , t ) , which is solution of the modified Wheeler–DeWitt equation in the representation of the quasi-position space, in the limit where the scale factor of the Universe is small. Although $$\psi _{qp}(\xi ,t)$$ ψ qp ( ξ , t ) is a physically acceptable state it is not a realizable state of the Universe because $$ \psi _{qp}(\xi ,t)$$ ψ qp ( ξ , t ) has infinite norm, as in the ordinary case with no minimal length.

Sign in / Sign up

Export Citation Format

Share Document