The Generalized Uncertainty Principle and the Semi-relativistic Harmonic Oscillator

AbstractSeveral models of quantum gravity predict the emergence of a minimal length at Planck scale. This is commonly taken into consideration by modifying the Heisenberg uncertainty principle into the generalized uncertainty principle. In this work, we study the implications of a polynomial generalized uncertainty principle on the harmonic oscillator. We revisit both the analytic and algebraic methods, deriving the exact form of the generalized Heisenberg algebra in terms of the new position and momentum operators. We show that the energy spectrum and eigenfunctions are affected in a non-trivial way. Furthermore, a new set of ladder operators is derived which factorize the Hamiltonian exactly. The above formalism is finally exploited to construct a quantum field theoretic toy model based on the generalized uncertainty principle.

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Some Implications of Two Forms of the Generalized Uncertainty Principle

Advances in High Energy Physics ◽

10.1155/2014/619498 ◽

2014 ◽

Vol 2014 ◽

pp. 1-8 ◽

Cited By ~ 5

Author(s):

Mohammed M. Khalil

Keyword(s):

Hydrogen Atom ◽

Quantum Gravity ◽

Harmonic Oscillator ◽

Uncertainty Principle ◽

Generalized Uncertainty Principle ◽

Standard Uncertainty ◽

Minimum Length ◽

Length Scale ◽

Minimum Length Scale ◽

Lorentz Force Law

Various theories of quantum gravity predict the existence of a minimum length scale, which leads to the modification of the standard uncertainty principle to the Generalized Uncertainty Principle (GUP). In this paper, we study two forms of the GUP and calculate their implications on the energy of the harmonic oscillator and the hydrogen atom more accurately than previous studies. In addition, we show how the GUP modifies the Lorentz force law and the time-energy uncertainty principle.

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Rényi and von Neumann entropies of thermal state in Generalized Uncertainty Principle-corrected harmonic oscillator

Modern Physics Letters A ◽

10.1142/s0217732321502503 ◽

2021 ◽

Vol 36 (35) ◽

Author(s):

MuSeong Kim ◽

Mi-Ra Hwang ◽

Eylee Jung ◽

DaeKil Park

Keyword(s):

Harmonic Oscillator ◽

Uncertainty Principle ◽

Temperature Region ◽

Thermal State ◽

Generalized Uncertainty Principle ◽

Renyi Entropy ◽

Rényi Entropy ◽

Von Neumann Entropy ◽

Large Temperature ◽

Von Neumann

The Rényi and von Neumann entropies of thermal state in generalized uncertainty principle (GUP)-corrected single harmonic oscillator system are explicitly computed within the first order of GUP parameter [Formula: see text]. While the von Neumann entropy with [Formula: see text] exhibits a monotonically increasing behavior in external temperature, the nonzero GUP parameter makes a decreasing behavior at large temperature region. As a result, for the case of [Formula: see text], the von Neumann entropy is maximized at the finite temperature [Formula: see text]. The Rényi entropy [Formula: see text] with nonzero [Formula: see text] also exhibits similar behavior at large temperature region. In this region, the Rényi entropy exhibits a decreasing behavior with increasing temperature. The decreasing rate becomes larger when the order of the Rényi entropy is smaller.

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

Canadian Journal of Physics ◽

10.1139/cjp-2015-0456 ◽

2016 ◽

Vol 94 (1) ◽

pp. 139-146 ◽

Cited By ~ 9

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.

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Factorization in the quantum mechanics with the generalized uncertainty principle

Modern Physics Letters A ◽

10.1142/s0217732315501175 ◽

2015 ◽

Vol 30 (23) ◽

pp. 1550117 ◽

Cited By ~ 1

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.

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

International Journal of Quantum Information ◽

10.1142/s0219749905001468 ◽

2005 ◽

Vol 03 (04) ◽

pp. 623-632 ◽

Cited By ~ 18

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.

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