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arXiv:1812.06112 · Physical review. D/Physical review. D.

Minimal length effect on the broadening of free wave packets and its physical implications

Carlos Villalpando, Sujoy K. Modak

Published Jul 26, 2019• 21 citations
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Abstract

We study the generalized uncertainty principle (GUP) modified time evolution for the width of wave packets for a scalar potential. The free particle case is solved exactly where the wave-packet broadening is modified by a coupling between the GUP parameter and higher-order moments in the probability distribution in momentum space. We consider two popular forms of deformations widely used in the literature---one of which modifies the commutator with a quadratic term in momentum, while the other modifies it with terms both linear and quadratic in momentum. Unlike the standard case, satisfying Heisenberg uncertainty, here the GUP modified broadening rates, for both deformations, not only depend on the initial size (in both position and momentum space) of the wave packet but also on the initial probability distribution and momentum of the particle. The new rates of wave-packet broadening, for both situations, are modified by a handful of new terms---such as the skewness and kurtosis coefficients, as well as the (constant) momentum of the particle. Comparisons with the standard Heisenberg uncertainty principle--based results show potentially measurable differences in the rates of free wave-packet broadening for physical systems such as the ${C}_{60}$ and ${C}_{176}$ molecules, and more so for large organic molecular wave packets. In doing so, we open a path to scan the GUP parameter space by several orders of magnitude inside the best existing upper bounds for both forms of GUP.

DOI
10.1103/physrevd.100.024054

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arXiv:1205.3919•Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields•Oct 1, 2012•28 citations
We study the generalized uncertainty principle (GUP) modified time evolution for the width of wave packets for a scalar potential. The free particle case is solved exactly where the wave-packet broadening is modified by a coupling between the GUP parameter and higher-order moments in the probability distribution in momentum space. We consider two popular forms of deformations widely used in the literature---one of which modifies the commutator with a quadratic term in momentum, while the other modifies it with terms both linear and quadratic in momentum. Unlike the standard case, satisfying Heisenberg uncertainty, here the GUP modified broadening rates, for both deformations, not only depend on the initial size (in both position and momentum space) of the wave packet but also on the initial probability distribution and momentum of the particle. The new rates of wave-packet broadening, for both situations, are modified by a handful of new terms---such as the skewness and kurtosis coefficients, as well as the (constant) momentum of the particle. Comparisons with the standard Heisenberg uncertainty principle--based results show potentially measurable differences in the rates of free wave-packet broadening for physical systems such as the C60{C}_{60}C60​ and C176{C}_{176}C176​ molecules, and more so for large organic molecular wave packets. In doing so, we open a path to scan the GUP parameter space by several orders of magnitude inside the best existing upper bounds for both forms of GUP.
The most obvious obstacle behind a direct test of Quantum Gravity (QG) is its energy scale (101910^{19}1019 GeV), which remains well outside of any human made machine. The next best possible approach is to provide indirect tests on effective theories of QG which can be performed in a lower energy scale. This paper is aimed in this direction, and shows a promising path to test the existence of the fundamental minimal length scale of Nature by measuring the dispersion of free, large molecular wave-packets. The existence of the minimal length is believed to be the reason for a modified commutation relationship between the position and momentum operators and, in this paper, we show that such a modification of the commutator has a profound effect on the dispersion rate of free wave-packets, and precise measurement on the broadening times of large molecular wave-packets (such as C60C_{60}C60​, C176C_{176}C176​ and large organic molecules) provide a promising path for an indirect test of quantum gravity, in a laboratory setting.
Abstract We study the implications of the generalized uncertainty principle (GUP) with a minimal measurable length on some quantum mechanical interferometry phenomena, such as the Aharonov–Bohm, Aharonov–Casher, COW and Sagnac effects. By resorting to a modified Schrödinger equation, we evaluate the lowest-order correction to the phase shift of the interference pattern within two different GUP frameworks: the first one is characterized by the redefinition of the physical momentum only, and the other is a Lorentz covariant GUP which also predicts non-commutativity of spacetime. The obtained results allow us to fix upper bounds on the GUP deformation parameters which may be tested through future high-precision interferometry experiments.
This work establishes a link between deformed angular momentum operators and modifications of the quantum uncertainty principle. Starting from an algebra in which the standard angular momentum operators are deformed by two parameters, [Formula: see text] and [Formula: see text], we construct the corresponding momentum operators via [Formula: see text] where the radial operator [Formula: see text] is determined by the requirement that the standard momentum operator is recovered in the limit [Formula: see text], [Formula: see text]. The resulting commutation relations take the form [Formula: see text] where [Formula: see text] is a position-dependent deformation matrix. This leads to an uncertainty relation [Formula: see text] which is state-dependent, anisotropic and position-dependent. We use the term Generalized Uncertainty Principle (GUP) in its broad sense referring to any modification of the Heisenberg relation motivated by quantum-gravity ideas. However, the model does not yield a fixed minimal length of the kind familiar from conventional GUP models (e.g. [Formula: see text]). In the isotropic, limit the trace of the deformation matrix simplifies the bound to [Formula: see text] which can be considered as a rescaling of the uncertainty constant. Our work thus provides a geometric origin for modified uncertainty relations from symmetry breaking in the rotation group, and offers a framework in which quantum-gravity effects manifest themselves through state- and direction-dependent measurement limits, without necessarily introducing a universal minimal length.
A momentum-dependent translation operator, inspired by the position-dependent translation operator (PDTO) framework, is introduced to describe a particle in a minimum-length scenario. We show that a chosen momentum-dependent translation profile G ( p ) induces a deformed commutation relation between position and momentum and the corresponding Generalized Uncertainty Principle (GUP), providing a structural construction rather than postulating the commutator independently. The construction is the momentum-space dual of the position-space operator that yields the Extended Uncertainty Principle, so that EUP and GUP emerge as the two faces of a single translation principle. We then adapt the split-operator method, using the eigenfunctions of the momentum operator modulated by a Gaussian in the representation of Brau and Pedram, to describe a particle under three potentials: free, linear, and quadratic. In all cases the quantum mean values track the classical dynamics obtained from the modified Poisson brackets, with a leading correction proportional to β ⟨ p ⟩(Δ p ) 2 that we identify analytically. Propagating a classical ensemble with the same initial spread shows this correction to be statistical rather than quantum: it is reproduced to machine precision for free and linear forces, and the genuinely quantum residue in the oscillator amounts to 5 × 10 − 5 of the amplitude. We also bound the error of the first-order truncation by comparison with the exact tangent Hamiltonian. This places the deformed quantum–classical correspondence on a quantitative footing, complementing earlier work on coherent states and on packet dispersion in a minimum-length scenario.
Several approaches to quantum gravity suggest the emergence of a fundamental minimal length at the Planck scale. In quantum mechanics, this feature is naturally encoded through deformations of the Heisenberg algebra, leading to the Generalized Uncertainty Principle (GUP). While the phenomenological implications of GUP have been extensively explored, a consistent characterization of coherent states in minimal-length quantum mechanics remains elusive. In this work, we present a systematic analysis of coherent states for the one-dimensional harmonic oscillator. We show that the canonical equivalence among their standard characterizations - as eigenstates of the annihilation operator, displaced vacuum states and minimum-uncertainty wave packets - is generically lost in the presence of a minimal length. We then investigate the dynamical and semiclassical consequences of this inequivalence by comparing the evolution of generalized coherent states with that of states saturating the GUP. In particular, we demonstrate that minimal-length effects induce nontrivial deformations of phase-space trajectories and give rise to an intrinsic squeezing mechanism with no counterpart in ordinary quantum mechanics. These results establish a unified framework for coherence in GUP-based quantum theories and identify distinctive semiclassical signatures of minimal-length physics, opening a new avenue for probing quantum-gravitational effects.
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.