Energy Eigenequation Expansion for a Particle on Singly Punctured Two-Torus and Triply Punctured Two-Sphere Systems

Ideas from topology have played a major role in physics especially to describe and explain exotic quantum phenomena. There has been a considerable interest among physicists who are working on string theory and quantum gravity to use ideas and results from topology to explain their work. However, often one is limited to the choice of spaces with relatively simpler topologies e.g. sphere, torus etc because more complex spaces are difficult to be characterized or even distinguished. It is our particular interest to consider singularities (i.e. having one or several punctures on it) as a tool to generate a family of complex two-dimensional configuration surfaces. These surfaces may find applications in to mathematical models of quantum chaos, cosmology, particle physics, condensed matter, quantum gravity and string theory. Extensive mathematical studies have been carried out for punctured surfaces but their literature in physics are scarce. Most have tackled the case of quantum mechanical systems on punctured torus with respect to its scattering and chaotic behavior. Of particular interest in the present work are the quantum mechanical systems of singly punctured two-torus and triply punctured two-sphere. They both have two generators and three possible non-contractible loops. Both surfaces can be generated from the same parent generators of the modular group Γ, which is a discrete subgroup of linear fractional transformations of the upper half complex plane H (the universal cover of the punctured surfaces). In this dissertation, we construct both surfaces of singly punctured two-torus and triply punctured two-sphere stepwise using these generators. The main aim however is to construct the energy eigenequation for particle on surfaces of singly punctured two-torus and triply punctured two-sphere. For that purpose, we first identify the configuration space explicitly by considering the tessellation of the upper half-plane and the required surfaces are determined. Next, by using the Fourier expansions, finite Fourier transform of the energy eigenequation is performed to give rise to a sought standard relation for generating the eigenfunction. It is known that the eigenfunction on a punctured system exhibit both discrete and continuous energy spectra. The discrete energy spectrum will correspond to the computation of a countable number of Maass cusp forms while for the continuous spectrum, it is spanned by the Eisenstein series. In this work, we present the expressions for the Maass cusp forms of the singly punctured two-torus and triply punctured two-sphere and the expression of the Eisenstein series for the singly punctured two-torus. At the end of this thesis a unified treatment of the Maass cusp forms and the Eisensteins series for the singly punctured two-torus and the triply punctured two-sphere are presented. The importance of each technique used on the formation of the energy eigenequation are explained in a more physical approach

Two Dimensional Plane, Modified Symplectic Structure and Quantization

Noncommutative quantum mechanics on the plane has been widely studied in the literature. Here, we consider the problem using Isham's canonical group quantization scheme for which the primary object is the symmetry group that underlies the phase space. The noncommutativity of the configuration space coordinates requires us to introduce the noncommutative term in the symplectic structure of the system. This modified symplectic structure will modify the group acting on the configuration space from abelian $\mathbb{R}^2$ to a nonabelian one. As a result, the canonical group obtained is a deformed Heisenberg group and the canonical commutation relation (CCR) corresponds to what is usually found in noncommutative quantum mechanics.Comment: 5 pages. Submitted to Jurnal Fizik Malaysi

some families of complex biorthogonal polynomials in a model of non-commutative quantum mechanics

the main objective of this thesis is to study some new families of complex biorthogonal polynomials which arise in a model of non-commutative quantum mechanics (NCQM) in two dimensions. The NCQM model in which we are interested is a system with an extended Landau Hamiltonian, such as the one which arises when an electron is placed in a constant magnetic field. The new polynomials are deformed versions of the well-known complex Hermite polynomials, generated by the non-commutative raising and lowering operators. We work out in detail the construction of the polynomials and compute some of their useful properties, e.g., the associated generating functions and three-term recurrence relations. It is shown that relative to a non- commutative scaling factor, these polynomials form a class of biorthogonal complex polynomials and in a certain well-defined limit, they reduce to the standard complex Hermite polynomials. The second objective of this thesis is to study the group theoretical properties of certain bilinear combinations of the non-commutative ladder operators. This is done following the well-known manner in which the angular momentum generators of standard quantum mechanics can be obtained from bilinear combinations of the raising and lowering operators of the harmonic oscillator. We study the Lie group structures that are generated by these operators, which again depend on a parameter characterizing the non-commutativity and which, in an appropriate limit, reduce to the standard quantum mechanical operators

Two dimensional plane, modified symplectic structure and quantization

Noncommutative quantum mechanics on the plane has been widely studied in the literature. Here, we consider the problem using Isham's canonical group quantization scheme for which the primary object is the symmetry group that underlies the phase space. The noncommutativity of the configuration space coordinates requires us to introduce the noncommutative term in the symplectic structure of the system. This modified symplectic structure will modify the group acting on the configuration space from abelian R^2 to a nonabelian one. As a result, the canonical group obtained is a deformed Heisenberg group and the canonical commutation relation (CCR) corresponds to what is usually found in noncommutative quantum mechanics

Zero-energy states in graphene quantum dot with wedge disclination

We investigate the effects of wedge disclination on charge carriers in circular graphene quantum dots subjected to a magnetic flux. Using the asymptotic solutions of the energy spectrum for large arguments, we approximate the scattering matrix elements, and then study the density of states. It is found that the density of states shows several resonance peaks under various conditions. In particular, it is shown that the wedge disclination is able to change the amplitude, width, and positions of resonance peaks.Comment: 10 pages, 10 figure

Thermal stability, structural and optical properties of rice husk sillica borotellurite glasses containing MnO2

The quaternary glass system {[(TeO2)0.7(B2O3)0.3]0.8[SiO2]0.2}1-x{MnO2}x where x = 0.00, 0.01, 0.02, 0.03, 0.04 and 0.05 molar fraction was prepared by melt quenching technique. The amorphous nature of the glass is confirmed by X-ray diffraction patterns and Scanning Electron Microscopy(SEM). The prepared glass samples had also been characterized by Differential Scanning Calorimetry(DSC). The glass transition(Tg), onset glass transition(To), crystallization(Tc) and melting temperature(Tm) values were measured from DSC thermo-gram. Results from DSC indicate good thermal stability and low value of fragility (F) of the prepared glass samples. Thermal stability(Ts), Hurby parameter(Kgl), fragility(F) and activation energy(Ea) were calculated for every glass composition. It is observed that the optical band gap decreases with the concentration of MnO2. On the other hand, the refractive index(n) is observed to increase as the concentration of MnO2 increases. Fourier Transform Infrared (FTIR) spectroscopy has been done to identify the functional group in glass sample

Killing tensor of five dimensional Melvin's spacetime

Killing tensors are generalizations of Killing vectors as objects that reflect the symmetries of spacetime. With recent interest in higher-dimensional spacetimes, construction of Killing tensors from lower dimensional ones may be useful. Our focus lies in the (4+1) dimensional Melvin’s spacetime which describes a magnetic universe with a cylindrical symmetry. We constructed the Killing vectors and Killing tensors in 5-dimensional Melvin’s spacetime. The Killing tensors are a linear combination of scalar times a metric and respective symmetric product of Killing vectors similar to those found by Garfinkle and Glass for the 4-dimensional case. It is relatively easy to write down Killing tensors of a particular spacetime admitting both commuting and hypersurface orthogonal Killing vectors

A note on reformed ladder operators for noncommutative morse oscillator

Morse oscillator is one of the known solvable potentials which attracts many applications in quantum mechanics especially in quantum chemistry. One of the interesting results of this study is the generation of ladder operators for Morse potential. The operators are a representation of the shifting energy levels of the states exhibited by the wave function. From this result, we manipulate and deform the operators in such a way that it gives a noncommutative property to promote noncommutative quantum mechanics (NCQM). The resultant NC feature can be shown in the spatial coordinates and finally the Hamiltonian. In this study, we consider two-dimensional Morse potential where the ladder operators are in the form of the corresponding 2D Morse

Effect of colorlessness condition on phase transition from Hadronic Gas to partonic plasma

One of the most important phase transition in physics is the Deconfinement Phase Transition in thermal Quantum ChromoDynamics. Due to the confinement property, we study the effect of colorlessness condition during the Deconfinement Phase Transition from a Hadronic Gas to a Quark-Gluon Plasma. We investigate the behavior of some thermodynamical quantities of the system such as the energy density and the pressure, the colorlessness condition and without colorlessness

Effects of step-potential on confinement strength of strain-induced type-I core–shell quantum dots

In this paper, the transition energy between lowest unoccupied molecular orbital (LUMO) of conduction band and highest occupied molecular orbital (HOMO) of valence band for band structures of type-I core-shell quantum dots (CSQDs) within a strong and weak confinements of charge carriers are estimated using the effective mass approximation together with single-band model. The effect of potential step at the conduction and valence bands on the confinement strength is then properly discussed. Our numerical results show that for a same size of CSQDs, the one with bigger potential steps will have stronger carriers' confinement with more localized excitons