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arXiv:1301.6913 · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Some consequences of the generalized uncertainty principle induced ultraviolet wave-vector cutoff in one-dimensional quantum mechanics

Kornél Sailer, Zoltán Péli, S. Nagy

Published Apr 22, 2013• 5 citations
arXivHTMLPDF

Abstract

A projection method is proposed to treat the one-dimensional Schr\"odinger equation for a single particle when the generalized uncertainty principle (GUP) generates an ultraviolet (UV) wave-vector cutoff. The existence of a unique coordinate representation called the naive one is derived from the one-parameter family of discrete coordinate representations. In this bandlimited quantum mechanics a continuous potential is reconstructed from discrete sampled values observed by means of a particle in maximally localized states. It is shown that bandlimitation modifies the speed of the center and the spreading time of a Gaussian wave packet moving in free space. Indication is found that GUP accompanied by bandlimitation may cause departures of the low-lying energy levels of a particle in a box from those in ordinary quantum mechanics to be much less suppressed than commonly thought when GUP without bandlimitation is at work.

DOI
10.1103/physrevd.87.084056

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A projection method is proposed to treat the one-dimensional Schr"odinger equation for a single particle when the generalized uncertainty principle (GUP) generates an ultraviolet (UV) wave-vector cutoff. The existence of a unique coordinate representation called the naive one is derived from the one-parameter family of discrete coordinate representations. In this bandlimited quantum mechanics a continuous potential is reconstructed from discrete sampled values observed by means of a particle in maximally localized states. It is shown that bandlimitation modifies the speed of the center and the spreading time of a Gaussian wave packet moving in free space. Indication is found that GUP accompanied by bandlimitation may cause departures of the low-lying energy levels of a particle in a box from those in ordinary quantum mechanics to be much less suppressed than commonly thought when GUP without bandlimitation is at work.
in ordinary quantum mechanics for the position uncertainty since the minimum value for the uncertainty can be tend to zero. Different theories of quantum gravity, such as string theory, loop quantum gravity, and black hole physics, confirm a minimal observable length. Kempf et al have formulated The minimal length is a fundamental length for position measurement, although there is no lower bound a generalized uncertainty principle (GUP) by modifying the Heisenberg uncertainty principle to include an appearance of the length in quantum mechanics. In this study, we have found out the influence of the GUP on the particle for its confinement in a 1D potential box and calculated energy eigenvalues. The modified Schrödinger equation for the particle is factorized to be of second order to obtain the eigenvalues in a handy way. The obtained energy levels are modified from the usual result with GUP parameter dependency. The modification in the energy spectrum due to the GUP characterized by the presence of the minimal length has been also explored graphically.
Abstract This note aims to elucidate certain aspects of the quasi-position representation frequently used in the investigation of one-dimensional models based on the generalized uncertainty principle (GUP). We specifically focus on two key points: (i) contrary to recent claims, the quasi-position operator can possess physical significance even though it is non-Hermitian, and (ii) in the quasi-position representation, operators associated with the position, such as the potential energy, also behave as a derivative operator on the quasi-position coordinate, unless the method of computing expectation values is modified. The development of both points revolves around the observation that the position and quasi-position operators share the same set of eigenvalues and are connected through a non-unitary canonical transformation. This outcome may have implications for widely referenced constraints on GUP parameters.
This paper provides a concise overview of the comprehensive version of the Generalized Uncertainty Principle (GUP) derived from nonlocal quantum mechanics and extensively addressed by S. Masood, et al. We utilize the specific constraint of this GUP formulation to compute the energy correction and the modified wave function for the linear potential in one dimension. In addition, we examine the impact of deformation on a one-dimensional delta potential well and a one-dimensional delta potential barrier. Furthermore, we specifically focus on the impact of deformation on hydrogen atoms existing in one dimension, and we conduct a distinct investigation into the Stark effect on the atom.
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.
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.
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.