Solvable Models on Noncommutative Spaces with Minimal Length Uncertainty Relations

Intuitive arguments involving standard quantum mechanical uncertainty relations suggest that at length scales close to the Planck length, strong gravity effects limit the spatial as well as temporal resolution smaller than fundamental length scale, leading to space-space as well as spacetime uncertainties. Space-time cannot be probed with a resolution beyond this scale i.e. space-time becomes "fuzzy" below this scale, resulting into noncommutative spacetime. Hence it becomes important and interesting to study in detail the structure of such noncommutative spacetimes and their properties, because it not only helps us to improve our understanding of the Planck scale physics but also helps in bridging standard particle physics with physics at Planck scale. Our main focus in this thesis is to explore different methods of constructing models in these kind of spaces in higher dimensions. In particular, we provide a systematic procedure to relate a three dimensional q-deformed oscillator algebra to the corresponding algebra satisfied by canonical variables describing non-commutative spaces. The representations for the corresponding operators obey algebras whose uncertainty relations lead to minimal length, areas and volumes in phase space, which are in principle natural candidates of many different approaches of quantum gravity. We study some explicit models on these types of non-commutative spaces, in particular, we provide solutions of three dimensional harmonic oscillator as well as its decomposed versions into lower dimensions. Because the solutions are computed in these cases by utilising the standard Rayleigh-Schrodinger perturbation theory, we investigate a method afterwards to construct models in an exact manner. We demonstrate three characteristically different solvable models on these spaces, the harmonic oscillator, the manifestly non-Hermitian Swanson model and an intrinsically non-commutative model with Poschl-Teller type potential. In many cases the operators are not Hermitian with regard to the standard inner products and that is the reason why we use PT -symmetry and pseudo-Hermiticity property, wherever applicable, to make them self-consistent well designed physical observables. We construct an exact form of the metric operator, which is rare in the literature, and provide Hermitian versions of the non-Hermitian Euclidean Lie algebraic type Hamiltonian systems. We also indicate the region of broken and unbroken PT -symmetry and provide a theoretical treatment of the gain loss behaviour of these types of systems in the unbroken PT -regime, which draws more attention to the experimental physicists in recent days. Apart from building mathematical models, we focus on the physical implications of noncommutative theories too. We construct Klauder coherent states for the perturbative and nonperturbative noncommutative harmonic oscillator associated with uncertainty relations implying minimal lengths. In both cases, the uncertainty relations for the constructed states are shown to be saturated and thus imply to the squeezed coherent states. They are also shown to satisfy the Ehrenfest theorem dictating the classical like nature of the coherent wavepacket. The quality of those states are further underpinned by the fractional revival structure which compares the quality of the coherent states with that of the classical particle directly. More investigations into the comparison are carried out by a qualitative comparison between the dynamics of the classical particle and that of the coherent states based on numerical techniques. We find the qualitative behaviour to be governed by the Mandel parameter determining the regime in which the wavefunctions evolve as soliton like structures. We demonstrate these features explicitly for the harmonic oscillator, the Poschl-Teller potential and a Calogero type potential having singularity at the origin, we argue on the fact that the effects are less visible from the mathematical analysis and stress that the method is quite useful for the precession measurement required for the experimental purpose. In the context of complex classical mechanics we also find the claim that "the trajectories of classical particles in complex potential are always closed and periodic when its energy is real, and open when the energy is complex", which is demanded in the literature, is not in general true and we show that particles with complex energies can possess a closed and periodic orbit and particles with real energies can produce open trajectories.