Position and momentum uncertainties of the normal and inverted harmonic oscillators under the minimal length uncertainty relation

Abstract

We analyze the position and momentum uncertainties of the energy eigenstates of the harmonic oscillator in the context of a deformed quantum mechanics, namely, that in which the commutator between the position and momentum operators is given by [x,p] 2 ). This deformed commutation relation leads to the minimal length uncertainty relation ∆x ≥ ( /2)(1/∆p + β∆p), which implies that ∆x ∼ 1/∆p at small ∆p while ∆x ∼ ∆p at large ∆p. We find that the uncertainties of the energy eigenstates of the normal harmonic oscillator (m > 0), derived in Ref.[1], only populate the ∆x ∼ 1/∆p branch. The other branch, ∆x ∼ ∆p, is found to be populated by the energy eigenstates of the 'inverted' harmonic oscillator (m < 0). The Hilbert space in the 'inverted' case admits an infinite ladder of positive energy eigenstates provided that ∆xmin = √ β > √ 2 [ 2 /k|m| ] 1/4 . Correspondence with the classical limit is also discussed.

Probing deformed commutators with macroscopic harmonic oscillators

“…Because of its importance as a prototype system, several studies have been focused on harmonic oscillators. Modifications of stationary states are calculated in refs 12 , 13 , 14 . Approaches to construct generalized coherent states are proposed in refs 15 , 16 .…”

Bound states of moving potential wells in discrete wave mechanics

“…Fundamental limits to a minimum measurable length were suggested in the early days of quantum physics, notably by Heisenberg, and appear, for example, in modern theories of loop quantum gravity, where spacetime looks granular [13]. A phenomenological approach to account for a minimum length scale is to consider an extension of the uncertainty principle by deforming the canonical commutation relations of position and momentum operators [14], leading to an extended form of the Schödinger equation [15,16]. Other simple models consider the non-relativistic Schrödinger equation defined on a discrete lattice [7][8][9][10][11][12][17][18][19][20][21][22], leading to so-called discrete wave mechanics [7] or discrete quantum mechanics [20].…”

Probing quantum gravity effects with quantum mechanical oscillators

“…Corrections to Landau levels and Lamb shift associated to modified quantum Hamiltonians have been computed in reference [19]. The modified spectrum of a quantum harmonic oscillator has been calculated in references [22,24,25]. Expressions for generalized coherent states has been obtained in references [26,27] and the modified time evolution and expectation values of position and momentum operators are discussed in references [28][29][30].…”