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 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.

Step potential problem and harmonic oscillator problem in the minimum length quantum mechanics

2015 ◽  
Vol 30 (20) ◽  
pp. 1550096 ◽  
Author(s):  
Soyeon Park ◽  
Byeong Hyo Woo ◽  
Min Jung ◽  
Eun Ji Jang ◽  
Won Sang Chung
Keyword(s):  
Quantum Mechanics ◽  
Harmonic Oscillator ◽  
Potential Problem ◽  
Quantum Mechanical ◽  
Minimum Length ◽  
Quantum Mechanical System ◽  
Step Potential ◽  
First Order ◽  
Probability Flux ◽  
Ordinary Quantum

In this paper, we use the quasi-position representation of the minimum length quantum mechanics (MLQM) to study the effects of minimum length uncertainty principle (MLUP) on the quantum mechanical system up to a first-order in β. We introduce the probability density and the probability flux to discuss two problems such as particle in a box and step potential problem. For the step potential, we compute the transmission coefficient and the reflection coefficient and compare them with those of the ordinary quantum mechanics. We also discuss the harmonic oscillator problem in MLQM.

scholarly journals Generalized uncertainty principle and its implications on geometric phases in quantum mechanics

2021 ◽  
Vol 136 (2) ◽  
Author(s):  
Giuseppe Gaetano Luciano ◽  
Luciano Petruzziello
Keyword(s):  
Quantum Mechanics ◽  
Phase Shift ◽  
Uncertainty Principle ◽  
Generalized Uncertainty Principle ◽  
Upper Bounds ◽  
The Other ◽  
Quantum Mechanical ◽  
Geometric Phases ◽  
Deformation Parameters ◽  
Aharonov Bohm

AbstractWe study the implications of the generalized uncertainty principle (GUP) with a minimal measurable length on some quantum mechanical interferometry phenomena, such as the Aharonov–Bohm, Aharonov–Casher, COW and Sagnac effects. By resorting to a modified Schrödinger equation, we evaluate the lowest-order correction to the phase shift of the interference pattern within two different GUP frameworks: the first one is characterized by the redefinition of the physical momentum only, and the other is a Lorentz covariant GUP which also predicts non-commutativity of spacetime. The obtained results allow us to fix upper bounds on the GUP deformation parameters which may be tested through future high-precision interferometry experiments.

An uncertainty principle for the basic wavelet transform

2021 ◽  
pp. 2150002
Author(s):  
Javid Ahmad Ganie ◽  
Renu Jain
Keyword(s):  
Quantum Mechanics ◽  
Wavelet Transform ◽  
Weight Function ◽  
Uncertainty Principle ◽  
Signal Analysis ◽  
Integral Transform ◽  
Generalized Uncertainty Principle ◽  
Basic Wavelet ◽  
And Performance ◽  
Definition Of

The aim of this paper is to derive the uncertainty principle which has implications in signal analysis and in quantum mechanics. First, we derive the definition of the wavelet transform in [Formula: see text]-calculus by using some weight function. Certain properties like linearity, scaling, translation, etc. were discussed. Later on, to illustrate this integral transform several results were derived for the effectiveness and performance of the proposed method. Also, this paper surveys recent applications to establish generalized uncertainty principle.

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.

scholarly journals Generalized uncertainty principle corrections to the simple harmonic oscillator in phase space

2016 ◽  
Vol 94 (1) ◽  
pp. 139-146 ◽  
Author(s):  
Saurya Das ◽  
Matthew P.G. Robbins ◽  
Mark A. Walton
Keyword(s):  
Phase Space ◽  
Harmonic Oscillator ◽  
Uncertainty Principle ◽  
Generalized Uncertainty Principle ◽  
Uncertainty Principles ◽  
Marginal Probability ◽  
Wigner Functions ◽  
Simple Harmonic Oscillator ◽  
Probability Densities

We compute Wigner functions for the harmonic oscillator including corrections from generalized uncertainty principles (GUPs), and study the corresponding marginal probability densities and other properties. We show that the GUP corrections to the Wigner functions can be significant, and comment on their potential measurability in the laboratory.

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 An analysis on Quantum mechanical Stability of Regular Polygons on a Point Base Using Heisenberg Uncertainty Principle

2021 ◽  
Vol 9 (10) ◽  
pp. 74-79
Author(s):  
Anurag Chapagain
Keyword(s):  
Quantum Mechanics ◽  
Uncertainty Principle ◽  
Stability Problem ◽  
Classical Mechanics ◽  
Quantum Mechanical ◽  
Heisenberg Uncertainty Principle ◽  
Heisenberg Uncertainty ◽  
Degree Of Stability ◽  
The Stability ◽  
Heisenberg's Uncertainty Principle

Abstract: It is a well-known fact in physics that classical mechanics describes the macro-world, and quantum mechanics describes the atomic and sub-atomic world. However, principles of quantum mechanics, such as Heisenberg’s Uncertainty Principle, can create visible real-life effects. One of the most commonly known of those effects is the stability problem, whereby a one-dimensional point base object in a gravity environment cannot remain stable beyond a time frame. This paper expands the stability question from 1- dimensional rod to 2-dimensional highly symmetrical structures, such as an even-sided polygon. Using principles of classical mechanics, and Heisenberg’s uncertainty principle, a stability equation is derived. The stability problem is discussed both quantitatively as well as qualitatively. Using the graphical analysis of the result, the relation between stability time and the number of sides of polygon is determined. In an environment with gravity forces only existing, it is determined that stability increases with the number of sides of a polygon. Using the equation to find results for circles, it was found that a circle has the highest degree of stability. These results and the numerical calculation can be utilized for architectural purposes and high-precision experiments. The result is also helpful for minimizing the perception that quantum mechanical effects have no visible effects other than in the atomic, and subatomic world. Keywords: Quantum mechanics, Heisenberg Uncertainty principle, degree of stability, polygon, the highest degree of stability

scholarly journals Generation of Time-Independent and Time-Dependent Harmonic Oscillator-Like Potentials Using Supersymmetric Quantum Mechanics

2018 ◽  
Vol 4 (1) ◽  
pp. 47-55
Author(s):  
Timothy Brian Huber
Keyword(s):  
Quantum Mechanics ◽  
Schrödinger Equation ◽  
Harmonic Oscillator ◽  
Mechanical System ◽  
Schrodinger Equation ◽  
Supersymmetric Quantum Mechanics ◽  
Time Dependent ◽  
Quantum Mechanical ◽  
Quantum Mechanical System ◽  
Intertwining Operator

The harmonic oscillator is a quantum mechanical system that represents one of the most basic potentials. In order to understand the behavior of a particle within this system, the time-independent Schrödinger equation was solved; in other words, its eigenfunctions and eigenvalues were found. The first goal of this study was to construct a family of single parameter potentials and corresponding eigenfunctions with a spectrum similar to that of the harmonic oscillator. This task was achieved by means of supersymmetric quantum mechanics, which utilizes an intertwining operator that relates a known Hamiltonian with another whose potential is to be built. Secondly, a generalization of the technique was used to work with the time-dependent Schrödinger equation to construct new potentials and corresponding solutions.

scholarly journals Nonlinear Generalisation of Quantum Mechanics

2021 ◽  
Author(s):  
Alireza Jamali
Keyword(s):  
Hilbert Space ◽  
Quantum Mechanics ◽  
Lorentz Invariance ◽  
Current Theory ◽  
Quantum Mechanical ◽  
Dynamical Condition ◽  
Current Definition ◽  
Klein Gordon ◽  
Definition Of ◽  
Mechanical Momentum

It is known since Madelung that the Schrödinger equation can be thought of as governing the evolution of an incompressible fluid, but the current theory fails to mathematically express this incompressibility in terms of the wavefunction without facing problem. In this paper after showing that the current definition of quantum-mechanical momentum as a linear operator is neither the most general nor a necessary result of the de Broglie hypothesis, a new definition is proposed that can yield both a meaningful mathematical condition for the incompressibility of the Madelung fluid, and nonlinear generalisations of Schrödinger and Klein-Gordon equations. The derived equations satisfy all conditions that are expected from a proper generalisation: simplification to their linear counterparts by a well-defined dynamical condition; Galilean and Lorentz invariance (respectively); and signifying only rays in the Hilbert space.

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