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 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 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 CLASS OF GUP SOLUTIONS IN DEFORMED QUANTUM MECHANICS

2010 ◽  
Vol 19 (12) ◽  
pp. 2003-2009 ◽  
Author(s):  
POURIA PEDRAM
Keyword(s):  
Black Hole ◽  
Quantum Mechanics ◽  
Quantum Gravity ◽  
Uncertainty Principle ◽  
Loop Quantum Gravity ◽  
Black Hole Physics ◽  
Generalized Uncertainty Principle ◽  
Heisenberg Uncertainty Principle ◽  
Deformed Algebra ◽  
Heisenberg Uncertainty

Various candidates of quantum gravity such as string theory, loop quantum gravity and black hole physics all predict the existence of a minimum observable length which modifies the Heisenberg uncertainty principle to the so-called generalized uncertainty principle (GUP). This approach results from the modification of the commutation relations and changes all Hamiltonians in quantum mechanics. In this paper, we present a class of physically acceptable solutions for a general commutation relation without directly solving the corresponding generalized Schrödinger equations. These solutions satisfy the boundary conditions and exhibit the effect of the deformed algebra on the energy spectrum. We show that this procedure prevents us from doing equivalent but lengthy calculations.

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.

RELATIVISTIC PARTICLE IN ELECTROMAGNETIC FIELDS WITH A GENERALIZED UNCERTAINTY PRINCIPLE

2012 ◽  
Vol 27 (15) ◽  
pp. 1250080 ◽  
Author(s):  
M. MERAD ◽  
F. ZEROUAL ◽  
M. FALEK
Keyword(s):  
Electromagnetic Field ◽  
Small Parameter ◽  
Uncertainty Principle ◽  
Relativistic Particle ◽  
Generalized Uncertainty Principle ◽  
Limit Case ◽  
Energy Eigenvalues ◽  
Dirac Equations ◽  
Klein Gordon ◽  
Uniform Electromagnetic Field

In this paper, we propose to solve the relativistic Klein–Gordon and Dirac equations subjected to the action of a uniform electromagnetic field with a generalized uncertainty principle in the momentum space. In both cases, the energy eigenvalues and their corresponding eigenfunctions are obtained. The limit case is then deduced for a small parameter of deformation.

The generalized uncertainty principle with commuting coordinates

2019 ◽  
Vol 34 (33) ◽  
pp. 1950267
Author(s):  
Won Sang Chung
Keyword(s):  
Wave Function ◽  
Quantum Mechanics ◽  
Uncertainty Principle ◽  
Generalized Uncertainty Principle ◽  
Three Dimensional ◽  
One Dimension ◽  
Momentum Representation ◽  
Position Representation

In this paper, we present a new D-dimensional generalized uncertainty principle (GUP) algebra with commuting coordinates which recovers [Formula: see text] in one dimension. We find two representations for this GUP: momentum representation and position representation. We discuss the GUP-corrected three-dimensional quantum mechanics in position representation for a small [Formula: see text]. Finally, we discuss the momentum wave function and GUP-corrected Fermi metal theory.

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