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arXiv:1412.6467 · Canadian Journal of Physics

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

Saurya Das, Matthew P. G. Robbins, Mark A. Walton

Published Oct 8, 2015• 29 citations
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Abstract

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.

DOI
10.1139/cjp-2015-0456

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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.
The generalized uncertainty principle (GUP) is a modification of Heisenberg's Principle predicted by several theories of quantum gravity. It consists of a modified commutator between the position and momentum. In this work, we compute potentially observable effects that GUP implies for the harmonic oscillator, coherent, and squeezed states in quantum mechanics. In particular, we rigorously analyze the GUP-perturbed harmonic oscillator Hamiltonian, defining new operators that act as ladder operators on the perturbed states. We use these operators to define the new coherent and squeezed states. We comment on potential applications.
Abstract The generalized uncertainty principle (GUP) is a natural consequence of many proposed quantum gravity theories which further indicates the modification of the quantum mechanics and classical mechanics. A unified classical-quantum framework also leads to the interpretation of a Winger phase-space distribution of a state as a KvN wave function. In this article, we utilize the classical-quantum unified framework to calculate the expression for the GUP modified Wigner function of a classical harmonic oscillator in the configuration space instead of the conventional approach of utilising the momentum space. Further, we plot the modified Wigner functions and show that they are similar to what has been obtained before in the past literature.
Abstract The generalized uncertainty principle (GUP) is a natural consequence of many proposed quantum gravity theories which further indicates the modification of the quantum mechanics and classical mechanics. A unified classical-quantum framework also leads to the interpretation of a Winger phase-space distribution of a state as a KvN wave function. In this article, we utilize the classical-quantum unified framework to calculate the expression for the GUP modified Wigner function of a classical harmonic oscillator in the configuration space instead of the conventional approach of utilising the momentum space. Further, we plot the modified Wigner functions and show that they are similar to what has been obtained before in the past literature.
In this paper, we study two generalized uncertainty principles (GUPs) including [Formula: see text] and [Formula: see text] which imply minimal measurable lengths. Using two momentum representations, for the former GUP, we find eigenvalues and eigenfunctions of the free particle and the harmonic oscillator in terms of generalized trigonometric functions. Also, for the latter GUP, we obtain quantum mechanical solutions of a particle in a box and harmonic oscillator. Finally we investigate the statistical properties of the harmonic oscillator including partition function, internal energy, and heat capacity in the context of the first GUP.
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.
Nonrelativistic quantum mechanics with a generalized uncertainty principle (GUP) is examined in ddd-dimensional free particle and harmonic oscillator systems. The Feynman propagators for these systems are exactly derived within the first order of the GUP parameter.