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

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

2011-09-13

arxiv.org/abs/1109.2680v1

Phys. Rev. D 84, 105029 (2011)

Physics

High Energy Physics

High Energy Physics - Theory

16 pages, 31 eps figures

Scientific paper

10.1103/PhysRevD.84.105029

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.

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