Position and Momentum Uncertainties of the Normal and Inverted Harmonic Oscillators under the Minimal Length Uncertainty Relation

TL;DR

This paper investigates how minimal length uncertainty relations affect the position and momentum uncertainties of harmonic oscillator energy eigenstates, revealing distinct behaviors for normal and inverted oscillators under deformed quantum mechanics.

Contribution

It analyzes the uncertainties in deformed quantum mechanics for harmonic oscillators, identifying different uncertainty branches and conditions for the inverted oscillator's energy spectrum.

Findings

Normal oscillator states populate the /p branch.

Inverted oscillator states populate the p branch.

Inverted oscillator admits an infinite ladder of positive energy states under certain conditions.

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]=i\hbar(1+\beta p^2). This deformed commutation relation leads to the minimal length uncertainty relation \Delta x > (\hbar/2)(1/\Delta p +\beta\Delta p), which implies that \Delta x ~ 1/\Delta p at small \Delta p while \Delta x ~ \Delta p at large \Delta p. We find that the uncertainties of the energy eigenstates of the normal harmonic oscillator (m>0), derived in Ref. [1], only populate the \Delta x ~ 1/\Delta p branch. The other branch, \Delta x ~ \Delta 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 \Delta x_{min} = \hbar\sqrt{\beta} > \sqrt{2} [\hbar^2/k|m|]^{1/4}. Correspondence with the classical limit is also discussed.