Interpretation of the q-Deformed 1-D Quantum Harmonic Oscillator

The interpretation of the q-deformed 1-D quantum harmonic oscillator is investigated for two definitions of q-deformation. This investigation is achieved by using Zaslavskii’s method to obtain the Heisenberg equations of motion (quantum Liouville equations) in the undeformed phase space. These quantum Liouville equations exhibit a non-commutative geometry produce from the existence of the dilatation operator which is inherent in the q-deformation process. The classical limits of these equations are obtained by applying a special classical limiting condition to produce the classical Liouville equations of the q-deformed oscillator. These classical Liouville equations are solved by using the method of characteristics in order to obtain the classical probability distribution functions for this system. The 2-D and 3-D behaviors of these functions were then investigated using a computer visualization method. The results of the mathematical derivations together with the computer visualization method show that the classical limit of the quantum Liouville equations for the q-deformed 1-D quantum harmonic oscillator are statistical in nature where the nonlinearity parameter for the q-deformed oscillator is connected with h . This result conforms to that obtained by Ghosh et al. for the undeformed 1-D quantum harmonic oscillator. The obtained classical probability distribution functions exhibit whorl shapes that evolve with time in phase space that are similar to the shapes obtained for the 1-D classical q-deformed oscillator. These whorl shapes in phase space are similar to those introduced by Milburn for the 1-D classical anharmonic oscillator. This similarity results from the fact that the anharmonicity itself represents a kind of deformation with a frequency that is a function of amplitude.