Anomalies in the Heisenberg Uncertainty Relation for Special Deformed Boson Algebra

The Heisenberg uncertainty relation, concerning the product of the standard deviations of the position and momentum observables, is studied for a quantum oscillator which is characterized by deformed boson algebra. Special forms of the deformed number operator are found to induce the following anomalies: the ground state or any deformed coherent state of the oscillator under study exhibits minimum uncertainty, as the canonical boson algebra provides, but the product of the corresponding standard deviations is less than the canonical lower bound, $\hbar/2$. Above all, each standard deviation and, consequently, the uncertainty become arbitrarily small for the ground state, other states of the Fock basis, and deformed coherent states with arbitrarily small magnitude of the label, if the parameters defining the special deformed number operator are properly chosen.