The classical limit of minimal length uncertainty relation: revisit with the Hamilton-Jacobi method

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

Physical Review D ◽

10.1103/physrevd.66.026003 ◽

2002 ◽

Vol 66 (2) ◽

Cited By ~ 145

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

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Homogeneous Field and WKB Approximation in Deformed Quantum Mechanics with Minimal Length

Advances in High Energy Physics ◽

10.1155/2015/718359 ◽

2015 ◽

Vol 2015 ◽

pp. 1-15 ◽

Cited By ~ 7

Author(s):

Jun Tao ◽

Peng Wang ◽

Haitang Yang

Keyword(s):

Wave Function ◽

Quantum Mechanics ◽

Statistical Physics ◽

Turning Point ◽

Minimal Length ◽

Jacobi Method ◽

Homogeneous Field ◽

Quantization Rule ◽

Connection Formula ◽

Forbidden Region

In the framework of the deformed quantum mechanics with a minimal length, we consider the motion of a nonrelativistic particle in a homogeneous external field. We find the integral representation for the physically acceptable wave function in the position representation. Using the method of steepest descent, we obtain the asymptotic expansions of the wave function at large positive and negative arguments. We then employ the leading asymptotic expressions to derive the WKB connection formula, which proceeds from classically forbidden region to classically allowed one through a turning point. By the WKB connection formula, we prove the Bohr-Sommerfeld quantization rule up toOβ2. We also show that if the slope of the potential at a turning point is too steep, the WKB connection formula is no longer valid around the turning point. The effects of the minimal length on the classical motions are investigated using the Hamilton-Jacobi method. We also use the Bohr-Sommerfeld quantization to study statistical physics in deformed spaces with the minimal length.

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Minimal Length Effects on Tunnelling from Spherically Symmetric Black Holes

Advances in High Energy Physics ◽

10.1155/2015/898916 ◽

2015 ◽

Vol 2015 ◽

pp. 1-8 ◽

Cited By ~ 20

Author(s):

Benrong Mu ◽

Peng Wang ◽

Haitang Yang

Keyword(s):

Black Holes ◽

Black Hole ◽

Angular Momentum ◽

Minimal Length ◽

Jacobi Method ◽

Spherically Symmetric ◽

Infinite Time ◽

Quantum Tunnelling ◽

Zero Mass ◽

Jacobi Equations

We investigate effects of the minimal length on quantum tunnelling from spherically symmetric black holes using the Hamilton-Jacobi method incorporating the minimal length. We first derive the deformed Hamilton-Jacobi equations for scalars and fermions, both of which have the same expressions. The minimal length correction to the Hawking temperature is found to depend on the black hole’s mass and the mass and angular momentum of emitted particles. Finally, we calculate a Schwarzschild black hole's luminosity and find the black hole evaporates to zero mass in infinite time.

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

Advances in High Energy Physics ◽

10.1155/2011/493514 ◽

2011 ◽

Vol 2011 ◽

pp. 1-30 ◽

Cited By ~ 23

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.

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

Journal of Mathematics Research ◽

10.5539/jmr.v13n2p1 ◽

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.

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Minimal length uncertainty relation and gravitational quantum well

Physical Review D ◽

10.1103/physrevd.74.036002 ◽

2006 ◽

Vol 74 (3) ◽

Cited By ~ 95

Author(s):

Fabian Brau ◽

Fabien Buisseret

Keyword(s):

Quantum Well ◽

Uncertainty Relation ◽

Minimal Length

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The Minimal Length and the Shannon Entropic Uncertainty Relation

Advances in High Energy Physics ◽

10.1155/2016/5101389 ◽

2016 ◽

Vol 2016 ◽

pp. 1-8 ◽

Cited By ~ 7

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.

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