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• W1573229114 abstract "Intuitive arguments involving standard quantum mechanical uncertainty relations suggest that at length scales close to Planck length, strong gravity effects limit 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 structure of such noncommutative spacetimes and their properties, because it not only helps us to improve our understanding of 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 corresponding algebra satisfied by canonical variables describing non-commutative spaces. The representations for 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 solutions are computed in these cases by utilising 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, harmonic oscillator, manifestly non-Hermitian Swanson model and an intrinsically non-commutative model with Poschl-Teller type potential. In many cases operators are not Hermitian with regard to standard inner products and that is 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 metric operator, which is rare in literature, and provide Hermitian versions of non-Hermitian Euclidean Lie algebraic type Hamiltonian systems. We also indicate region of broken and unbroken PT -symmetry and provide a theoretical treatment of gain loss behaviour of these types of systems in unbroken PT -regime, which draws more attention to experimental physicists in recent days.Apart from building mathematical models, we focus on physical implications of noncommutative theories too. We construct Klauder coherent states for perturbative and nonperturbative noncommutative harmonic oscillator associated with uncertainty relations implying minimal lengths. In both cases, uncertainty relations for constructed states are shown to be saturated and thus imply to squeezed coherent states. They are also shown to satisfy Ehrenfest theorem dictating classical like nature of coherent wavepacket. The quality of those states are further underpinned by fractional revival structure which compares quality of coherent states with that of classical particle directly. More investigations into comparison are carried out by a qualitative comparison between dynamics of classical particle and that of coherent states based on numerical techniques. We find qualitative behaviour to be governed by Mandel parameter determining regime in which wavefunctions evolve as soliton like structures. We demonstrate these features explicitly for harmonic oscillator, Poschl-Teller potential and a Calogero type potential having singularity at origin, we argue on fact that effects are less visible from mathematical analysis and stress that method is quite useful for precession measurement required for experimental purpose. In context of complex classical mechanics we also find claim that the trajectories of classical particles in complex potential are always closed and periodic when its energy is real, and open when energy is complex, which is demanded in 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." @default.
• W1573229114 created "2016-06-24" @default.
• W1573229114 creator A5084786513 @default.
• W1573229114 date "2014-10-13" @default.
• W1573229114 modified "2023-09-27" @default.
• W1573229114 title "Solvable Models on Noncommutative Spaces with Minimal Length Uncertainty Relations" @default.
• W1573229114 hasPublicationYear "2014" @default.
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