Uncertainty and certainty relations for the q-oscillator

Abstract

The uncertainty and certainty relations for the momentum and position operators for the q-oscillator and Fibonacci oscillator are investigated in this thesis. The onedimensional q-oscillator, the two-dimensional q-oscillator which is invariant under the action of the unitary q-deformed quantum group and Fibonacci oscillator are studied. We study the one-dimensional q-oscillator. Firstly, by using the commutation relation for the momentum and position operators, the uncertainty relations for the energy eigenstates and any state which is a superposition of energy eigenstates are calculated. By calculating the upper limit of the expectation value of the hamiltonian, the upper limits of P and X and the certainty relations are obtained in the case in which 0 < q < 1. Then further uncertainty relations for the momentum and position are obtained. Secondly, by calculating P and X directly and by finding their lower and upper limits, the uncertainty and certainty relations for the energy eigenstates are again obtained. As a result, the two ways of finding the uncertainty and certainty relations for the energy eigenstates are true but the most informative results are selected from the two different sets of results obtained by these two methods. Thus the further uncertainty relations and the certainty relations are obtained for the energy eigenstates and an arbitrary state. The classical limits of ("n+1 − "n)/"n where "n are the energy eigenvalues are calculated for the different intervals of q. It is shown that the classical limit of this quantity in the case in which q p 2 is unreasonable. A similar procedure is repeated for the two-dimensional q-oscillator and Fibonacci oscillator.