scholarly journals Position and momentum uncertainties of the normal and inverted harmonic oscillators under the minimal length uncertainty relation

2011 ◽  
Vol 84 (10) ◽  
Author(s):  
Zachary Lewis ◽  
Tatsu Takeuchi
Keyword(s):  
Uncertainty Relation ◽  
Minimal Length ◽  
Harmonic Oscillators

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 On the Minimal Length Uncertainty Relation and the Foundations of String Theory

2011 ◽  
Vol 2011 ◽  
pp. 1-30 ◽  
Author(s):  
Lay Nam Chang ◽  
Zachary Lewis ◽  
Djordje Minic ◽  
Tatsu Takeuchi
Keyword(s):  
General Relativity ◽  
Quantum Theory ◽  
String Theory ◽  
Momentum Space ◽  
Vacuum Energy ◽  
Uncertainty Relation ◽  
Minimal Length

We review our work on the minimal length uncertainty relation as suggested by perturbative string theory. We discuss simple phenomenological implications of the minimal length uncertainty relation and then argue that the combination of the principles of quantum theory and general relativity allow for a dynamical energy-momentum space. We discuss the implication of this for the problem of vacuum energy and the foundations of nonperturbative string theory.

scholarly journals GENERALIZED HEISENBERG RELATION AND QUANTUM HARMONIC OSCILLATORS

2006 ◽  
Vol 21 (30) ◽  
pp. 6115-6123 ◽  
Author(s):  
P. NARAYANA SWAMY
Keyword(s):  
Quantum Gravity ◽  
Quantum Electrodynamics ◽  
Uncertainty Relation ◽  
Generalized Uncertainty Principle ◽  
Harmonic Oscillators ◽  
Standard Manner ◽  
Generalized Momentum ◽  
Heisenberg Uncertainty ◽  
Heisenberg Relation ◽  
Photon States

We study the consequences of the generalized Heisenberg uncertainty relation which admits a minimal uncertainty in length such as the case in a theory of quantum gravity. In particular, the theory of quantum harmonic oscillators arising from such a generalized uncertainty relation is examined. We demonstrate that all the standard properties of the quantum harmonic oscillators prevail when we employ a generalized momentum. We also show that quantum electrodynamics and coherent photon states can be described in the familiar standard manner despite the generalized uncertainty principle.

DEVIATION OF COHERENT STATE CAUSED BY DISSIPATION

2000 ◽  
Vol 14 (07n08) ◽  
pp. 267-275 ◽  
Author(s):  
Y. H. JIN ◽  
S. P. KOU ◽  
J. Q. LIANG ◽  
B. Z. LI
Keyword(s):  
Spectral Density ◽  
Coherent State ◽  
Small Amplitude ◽  
Uncertainty Relation ◽  
Small Deviation ◽  
Dissipative System ◽  
Harmonic Oscillators ◽  
System A ◽  
Squeezed Coherent State ◽  
Weak Dissipation

The time evolution of a coherent state was studied in the dissipative system, a harmonic oscillator coupling with a bath of harmonic oscillators with Ohmic spectral density. We define a deviation of uncertainty relation versus the squeezed coherent state as [Formula: see text]. It is found that for the case of η ≪ ω0, namely, the weak dissipation, Δ oscillates with a small amplitude. The system is in a squeezed coherent state essentially and the dissipation only leads to a small deviation. For both strong (η ≫ ω0) and critical (η ~ ω0) dissipations, Δ is divergent with respect to t and the coherence of state is destroyed.

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 Minimal length uncertainty relation and ultraviolet regularization

1997 ◽  
Vol 55 (12) ◽  
pp. 7909-7920 ◽  
Author(s):  
Achim Kempf ◽  
Gianpiero Mangano
Keyword(s):  
Uncertainty Relation ◽  
Minimal Length ◽  
Ultraviolet Regularization

scholarly journals Minimal length uncertainty relation and the hydrogen spectrum

2003 ◽  
Vol 572 (1-2) ◽  
pp. 37-42 ◽  
Author(s):  
R Akhoury ◽  
Y.-P Yao
Keyword(s):  
Uncertainty Relation ◽  
Minimal Length

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.

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