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arXiv:2003.13856 · Physical review. D/Physical review. D.

Generalized uncertainty principle and d -dimensional quantum mechanics

DaeKil Park

Published May 13, 2020• 10 citations
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Abstract

Nonrelativistic quantum mechanics with a generalized uncertainty principle (GUP) is examined in $d$-dimensional free particle and harmonic oscillator systems. The Feynman propagators for these systems are exactly derived within the first order of the GUP parameter.

DOI
10.1103/physrevd.101.106013

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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.
The non-relativistic quantum mechanics with a generalized uncertainty principle (GUP) is examined in the Arthurs-Kelly system. The Feynman propagator for this system is exactly derived within the first order of the GUP parameter βββ. The application of it in the early universe stage is briefly discussed.
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.
The generalized uncertainty principle (GUP) has brought the idea of the existence of a minimum measurable length in quantum physics. Depending on this GUP, the nonrelativistic Hamiltonian at the Planck scale is modified. In this paper, we construct the kernel for this GUP-corrected Hamiltonian for a free particle by applying the Hamiltonian path integral approach and checking the validity conditions for this kernel thoroughly. Interestingly, the probabilistic interpretation of this kernel induces a momentum upper bound in the theory which is comparable with GUP-induced maximum momentum uncertainty.
We propose a simple analytical model for Harmonic oscillator potential that is associated to the generalized uncertainty principle (GUP) in the presence of energy-dependent mass. A formal condition connected with these systems is based on the mathematical structure of modified quantum mechanics. Consequently, thanks to Feynman’s path integrals formalism, we are able to treat the reorganization of wave functions. Furthermore, in order to convert this path formalism into a local representation, we will use a coordinate transformation method at different usual point interval, which is what these systems assume. In each application, the propagator is calculated, the wave functions and the energy eigenvalues are obtained with the corresponding curves. The special cases are then deduced.