Position and momentum uncertainties of the normal and inverted harmonic oscillators under the minimal length uncertainty relation
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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) over cap, (p) over cap] = i (h) over bar (1 + beta(p) over cap (2)). This deformed commutation relation leads to the minimal length uncertainty relation Delta x >= ((h) over bar /2)(1/Delta p + beta Delta p), which implies that Delta x similar to 1/Delta p at small Delta p while Delta x similar to Delta p at large Delta p. We find that the uncertainties of the energy eigenstates of the normal harmonic oscillator (m > 0), derived in L. N. Chang, D. Minic, N. Okamura, and T. Takeuchi, Phys. Rev. D 65, 125027 ( 2002), only populate the Delta x similar to 1/Delta p branch. The other branch, Delta x similar to 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) = <(h)over bar>root beta > root 2[(h) over bar (2)/k vertical bar m vertical bar](1/4). Correspondence with the classical limit is also discussed.