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arXiv:2610.29397

The Lévy wave packet and fractional uncertainty relations

Zi-Xuan Li, Xiao-Dong Li, Towe Wang

math-phquant-phSubmitted Oct 21, 2026• Announced Oct 22, 2026
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Abstract

This work revisits the debate on fractional uncertainty relations that arose a decade ago and tries to settle it. After a careful study of the position-momentum uncertainty product of the Lévy wave packet, a novel lower bound is proposed for uncertainty relations in fractional quantum mechanics. This bound is nearly linear in the Lévy index and is satisfied by exact, calculable solutions available in the literature.

Comments
9 pages, 4 figures

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This work revisits the debate on fractional uncertainty relations that arose a decade ago and tries to settle it. After a careful study of the position-momentum uncertainty product of the Lévy wave packet, a novel lower bound is proposed for uncertainty relations in fractional quantum mechanics. This bound is nearly linear in the Lévy index and is satisfied by exact, calculable solutions available in the literature.
The usual Heisenberg uncertainty relations are generalized to the case of nonsimultaneous measurements of position and momentum. The product of the corresponding uncertainties is bounded from below by a quantity depending on the dynamics of the system (the potential), and which may be lower than ħ/2 and even vanish. The case of two successive measurements of momentum and position is also treated, and one recovers the standard phenomenon of spreading of the wave packet. Whereas the case of an arbitrary potential can only be treated to lowest orders in the lime delay, a complete solution is given for the harmonic oscillator.
The authors of this paper [M. Pandit et al., Phys. Rev. A 100, 012131 (2019)] proposed an alternative formulation of the uncertainty relation in quantum mechanics in terms of the Lipschitz constants. They evaluated these constants in various cases and decided that the product of the square roots of Lipschitz constants for position and momentum probability distributions is bounded below by $0.3/\ensuremath{\hbar}$. In this comment, I invalidate this claim by giving an example of wave functions for which the value of the product of the Lipschitz constants for position and momentum does not obey this bound and tends to zero.
A new application of a fractal concept to quantum physics has been developed. The fractional path integrals over the paths of the Levy flights are defined. It is shown that if fractality of the Brownian trajectories leads to standard quantum mechanics, then the fractality of the Levy paths leads to fractional quantum mechanics. The fractional quantum mechanics has been developed via the new fractional path integrals approach. A fractional generalization of the Schrodinger equation has been discovered. The new relationship between the energy and the momentum of the nonrelativistic fractional quantum-mechanical particle has been established, and the Levy wave packet has been introduced into quantum mechanics. The equation for the fractional plane wave function has been found. We have derived a free particle quantum-mechanical kernel using Fox's H-function. A fractional generalization of the Heisenberg uncertainty relation has been found. As physical applications of the fractional quantum mechanics we have studied a free particle in a square infinite potential well, the fractional "Bohr atom" and have developed a new fractional approach to the QCD problem of quarkonium. We also discuss the relationships between fractional and the well-known Feynman path integral approaches to quantum mechanics. (c) 2000 American Institute of Physics.