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

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. Correspondence with the classical limit is also discussed.