Time-dependent q-deformed bi-coherent states for generalized uncertainty relations

Abstract

We consider the time-dependent bi-coherent states that are essentially the Gazeau-Klauder coherent states for the two dimensional noncommutative harmonic oscillator. Starting from some q-deformations of the oscillator algebra for which the entire deformed Fock space can be constructed explicitly, we define the q-deformed bi-coherent states. We verify the generalized Heisenberg's uncertainty relations projected onto these states. For the initial value in time, the states are shown to satisfy a generalized version of Heisenberg's uncertainty relations. For the initial value in time and for the parameter of noncommutativity $\theta =0$, the inequalities are saturated for the simultaneous measurement of the position-momentum observables. When the time evolves the uncertainty products are different from their values at the initial time and do not always respect the generalized uncertainty relations.