q-Deformed Quantum Coherent States and Some of Their Applications

The concept of q-deformation has found many important applications in a variety of fields in physics, such as quantum optics, atomic physics, solid state physics,nuclear physics and cosmology. This has motivated its extension to many well-established other concepts such as coherent states well-known in quantum optics. On the other hand, the interpretation of the physical meaning of the q-deformation remains an outstanding problem.The present work is an attempt to apply the concept of q-deformed coherent states to solve this interpretation problem. The q-deformed 1-D quantum harmonic oscillator is used as a model for the application of the methodology of using q-deformed coherent states to solve this problem. This is achieved by first deriving the classical Liouville equation for the q-deformed 1-D classical harmonic oscillator in the undeformed and deformed oscillator phase spaces.Then, this equation is solved by using the method of characteristics which gives the classical probability distribution function for this oscillator in phase space.The behavior of this function is then investigated by using a computer visualization method based on a computer program constructed in Mathematica language. On the quantum level, the Heisenberg equation of motion for the density operator corresponding to this 1-D quantum harmonic oscillator is expressed in the present work in terms of the standard siprobability distribution functions, again in the deformed and undeformed phase spaces. This helps to derive the quantum Liouville equations for this q-deformed oscillator in these phase spaces. The classical limits of these resulting Liouville equations are then approached by extending a standard procedure based on the non-deformed coherent states to the q-deformed case. In addition to the application of the standard q-deformed coherent states, a novel approach based on q-deformed coherent states due to Arik and Coon is also employed in this investigation.. Also, the behavior of the classical limits of the quantum Liouville equations for this oscillator is observed to show whorl shapes that can be contrasted with their classical analogs.This whorl shape behavior can be considered as a phenomenon connected with q-deformation in general; the anharmonic oscillator being a special cas The results of detailed mathematical derivations in this process of approaching the classical limit reveal that this limit is statistical in nature. This is similar to the case of the ordinary undeformed oscillator which has been proved previously.They also reveal, together with the complementary computer visualizations, more information about the physical meaning of the q-deformation. This includes the observations that the q-deformed 1-D oscillator can be interpreted as a nonlinear oscillator where the nonlinearity parameter depends on  Some connection with phase space having a non-commutative geometry,resulting from q-deformation, also finds evidence in some of the results presented in this thesis.