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We study Magnus expansion (ME) approximation scheme for the interaction between an atom and a single quantized cavity mode (Jaynes-Cumming model) in a closed quantum system in resonance or near resonance for a time-dependent coupling coefficient g(t) in both the interaction and rotating picture by implementing a novel numerical method called MG4 and compare out results to the Runge-Kutta 4th (RK4) order solution to demonstrate the conservation of unitary evolution of the ME. A cursory study of open quantum system is given to encourage the study of ME for dissipative systems. Furthermore, we assume that our time-dependent coupling coefficient g(t) can take on two forms, Gaussian and sinusoidal, which are introduced as pulses to study the behavior and response of the cavity. Our results show that ME is a sufficient approximation scheme in our study of closed quantum systems which may have applications in quantum control.

Author's abstract: We extend previous theoretical investigations of the creation of quantized circulation states by stirring Bose-Einstein condensates (BEC) confined in ``racetrack" potentials. The previous study, {\em Producing Smooth Flow in Atom Circuits by Stirring}, used the Gross-Pitaevskii equation (GPE), which is valid at $T=0$ K. Here we use a non-zero temperature model based on the Zaremba, Nikuni, Griffin (ZNG) theory to simulate stirring racetrack BECs. The two main goals of this thesis are 1) to understand the effects of temperature on the production of circulation and 2) to understand the mechanism by which the circulation is excited. We find that it is possible to produce circulation at non-zero $T$ in much the same way as at absolute zero, and the consideration of thermal effects is important for determining the precise amount of circulation obtained by stirring for a fixed total number of atoms. We also present a case study of the mechanism by which stirring produces flow in the GPE model

Author's abstract: Explicitly correlated coupled cluster singles, doubles, and perturbative triples level [CCSD(T)-F12] and second-order closed-shell Møllar-Plesset perturbative theory (MP2-F12) anharmonic vibrational frequencies are compared with gas phase experiment and higher-level computations. These involve CCSD(T) with the complete basis set limit extrapolation along with, an additive factor for the energy difference between core and noncore electrons and a similar additive difference for scalar relativity to give the CcCR approach. One hundred and sixty-nine vibrational frequencies are computed using CCSD(T)-F12 with the aug-cc-pVTZ basis set. For MP2-F12 twenty-five molecules are examined under aug-cc-pVDZ basis set. The CCSD(T) computational analysis of the closed-shell molecules' anharmonic vibrational frequencies resulted in a mean absolute error of 7.5 cm-1 between the CCSD(T)-F12 and CcCR. Comparison of the CCSD(T)-F12 with gas phase experiment gave mean absolute error of 5.8 cm-1 for twenty-nine closed-shell vibrational frequencies. However, the total computational time is reduced by more than 4 orders of magnitude when utilizing CCSD(T)-F12 as opposed to CcCR. From the computation, it is clear CCSD(T)-F12 correlated wave function is superior to MP2-F12, that this approach is a viable means of computing anharmonic vibrational frequencies of molecules when CcCR is not feasible to use.

We theoretically study the quantum confinement effects and transport prop- erties of quantum ring (QR) systems. In particular, we investigate QRs made out of the following materials: single-layer graphene (SLG), single- layer transition-metal dichalcogenides (TMDs) and narrow-gap semiconduc- tor quantum wells (SQWs). Via perturbation theory and assuming that the ring aspect ratio is small, the general subband dispersion relations of these hard-wall ring confined systems are determined. These dispersion results agree with and extend on previous works. We discover the necessity of including both a size-quantisation energy and an angular momentum dependent energy shift to the dispersion equation due to their sizeable impact on the conductance of the system. The topological properties of these QR systems is also investigated. We find that QR confinement of materials may destroy the topologically non-trivial properties of states. The topological phase can be recovered when the band structure is inverted and the confined material parameters satisfy certain critical widths and gap limits. An analytical expression of the conductance for QRs (with symmetrically- arranged leads), in the presence of the perpendicular magnetic field piercing the centre of the ring, is derived. We study the geometric (i.e. Berry) and dynamic phases of the system that arise from the interference of partial waves in the ring branches. We discover that the Berry phase is modified by a correction term that arises purely from the quantum confinement of the materials. This has generally not been taken into account by previous studies. The explicit analytical expressions of the phase correction term are derived and shown to be proportional to the angular momentum dependent energy shift, present in the dispersion relations, for lead injection energies close to the subband energy. Overall, this study finds that the material-dependent phase plays a significant role in both the dispersion relation and the conductance of QRs and thus provides a useful insight for future experimental efforts with regards to transport in QR systems.

In quantum mechanics the replacement of complex vectors with operators is essential to "quantizing" space. Nonetheless, in many physics textbooks there is no justification for this action. Therefore in this thesis I will attempt to understand the mathematical formalism that allows for such a "replacement" to be rigourous. I will approach this topic by first defining a vector spaces and its dual space, a Hilbert space and a conjugate Hilbert space, and an operator space. Next, I will look at the algebraic tensor product of two vector spaces, two Hilbert spaces, and finally two operator spaces. Ultimately we will look at the completion of the tensor product with resect to the minimal norm and show that the minimal norm of a tensor has an analogous inequality to the Cauchy-Schwarz inequality.

The aim of Physics is to provide a mathematical description of the physical universe and its underlying laws. Currently, the greatest problem lies in uniting the four fundamental forces: gravitation, electromagnetism, strong interaction, and weak interaction. Various methods have been utilized in an attempt to provide a solution, however it is still impossible. The largest difficulty in developing a unified theory has been uniting relativity and quantum physics. Separately, both fields have been extremely successful, however no viable quantum theory of gravity is known. Approaches that look at quantum effects in the context of a gravity have revealed several issues that arise. In particular, the question of how to define a quantum vacuum, in the context of a curved background, has received much attention in the recent years. In this thesis, I will look at the approach to Quantum Field Theory through Classical Field Theory, working with scalar, vector, and tensor fields. I will then quantize a scalar field in a flat spacetime and define the vacuum state. Finally, I will look at quantum fields in the background of an expanding universe and show that our previously defined vacuum state is no longer the "true" vacuum.

Long-lived oscillations lasting up to 1800 fs have been observed in the Fenna--Mathews--Olsen (FMO) Pigment-Protein complex. It is unclear if the oscillations are quantum or classical in origin. Quantized intra-pigment normal modes are responsible for these long-lived oscillations. In this thesis we simulate a bacteriachlorophyll dimer in the FMO Complex at physiological conditions. Our model describes two electronically coupled pigments coupled to the vibrational protein environment. Our goal is to find system parameters that allow long-lived quantum coherences to exist in photosynthetic complexes. We calculate the time evolution of our system's density matrix using the numerically exact quantum adiabatic path integral (QUAPI). The coherence of the system was determined by taking the real component of the density matrices complex off-diagonals. The quantumness of the system was determined by taking the Hilbert-Schmidt distance between the density matrix and its corresponding pointer states. By varying the Huang-Rhys factor, the reorganization energy difference, and the site energy difference we found long-lived coherences with an amplitude of 0.15 and a quantumness of 0.23. The quantum coherence was still present 2500 fs after the initial excitation of the system. This suggests that coherent quantum superpositions can theoretically occur in the FMO Complex at physiological conditions. The period of the coherent oscillations were on the same order of magnitude as the time resolution of our simulation causing aliasing to occur. In order to confirm the observed coherent superpositions we need to increase the temporal resolution of the simulation.

This dissertation describes a theoretical study of strongly correlated electron systems. We present a variational quantum Monte Carlo approach based on matrix-product states, which enables us to naturally extend our work into higher-dimensional tensor-network states as well as to determine the ground state and the low-lying excitations of quasi-one dimensional electron systems. Our results show that the ground state of the quarter-filled zigzag electron ladder is expected to exhibit a bond distortion whose pattern is not affected by the electron-electron interaction strength. This dissertation also presents a new method that combines a quantum Monte Carlo technique with a class of tensor-network states. We show that this method can be applied to two-dimensional fermionic or frustrated models that suffer from a sign problem. Monte Carlo sampling over physical states reveals better scaling with the size of matrices under periodic boundary conditions than other types of higher-dimensional tensor-network states, such as projected entangled-pair states, which lead to unfavorable exponential scaling in the matrix size.

In this dissertation, covariant density functional theory has been applied to a variety of nuclear phenomena in the ground and excited states of rotating and non-rotating nuclei. It has been applied for the interpretation of excited superdeformed bands in 154Dy using the effective alignmentmethods. The properties of the predicted hyperdeformed nuclei at high spin in the Z = 40-58 region were investigated and the spins at which such configuration become yrast were defined. The moments of inertia, the role of single-particle energies and necking degree of freedom have also been studied. It also predicted that 107Cd is the best nucleus for its observation. The impact of time-odd mean fields (nuclear magnetism NM) in both non-rotating and rotating frame works, on physical observables has been studied. It is shown that nuclear magnetism always provide additional binding to the binding energies of odd-mass nuclei. Time-odd mean fields affect odd-even mass differences. However, the modifications of the strength of pairing correlations required to compensate for their effects are modest. In contrast, time-odd mean fields have a profound effect on the properties of odd-proton nuclei in the vicinity of the proton drip line. Their presence can modify the half-lives of proton emitters and considerably affect the possibilities of their experimental observation. They also have a profound effect on the dynamic and kinematic moments of inertia, particle number, configuration, and rotational frequency dependencies of their impact on the moments of inertia. The effect of NM on the binding energy and moments of inertia weakly depend on the choice of the RMF parametrization. Fission barriers are studied systematically with the allowence for triaxial deformations, in the actinide and superheavy regions. It is shown that covariant density functional theory is able to describe fission barriers, in actinides, on a level of accuracy comparable with nonrelativistic calculations. Triaxiality in the region of the first saddle plays a crucial role in achieving that. However, in the Z = 112 - 120 superheavy nuclei, the inner fission barriers are not affected by triaxiality. General trends of the evolution of inner fission heights are discussed.

This dissertation is divided in two parts. In the first part, we study the dynamics of a classical particle system. We calculate the efficiency of a rejection-free dynamic Monte Carlo method in the limit of low temperatures and/or high densities for d-dimensional off-lattice homogeneous particles interacting through a repulsive power-law potential r[superscript -p]. Theoretically we find the algorithmic efficiency is asymptotically proportional to [rho][superscript (p+2)/2]T[superscript -(d/2)] with the particle density [rho] and the temperature T. Dynamic Monte Carlo simulations are performed in 1-, 2- and 3-dimensional systems with different powers p, and the results agree with the theoretical predictions. In the second part, we study the dynamics of a quantum spin 1/2 system. We calculate the average of the z component of the spin operator when one spin 1/2 particle is interacting with one, two, and three baths of 1/2-integer spins. For the cases of one and two baths we find the asymptotic behavior of and the quantum purity P(t) using the density of states of the system. In both cases and P(t) decay, exponentially for a single bath and algebraically for two baths. We present simulations in real time for one spin 1/2 particle coupled to one or two baths of 1/2-integer spins. The simulations were performed using the algorithm and code of Prof. De Raedt. We first simulated one spin coupled to one or two spin-baths with no interactions between the bath spins to compare with the theoretical calculations. We extend these simulations by introducing random interactions between the bath spins, in an attempt to reach the asymptotic decay rate at earlier times and for fewer spins in the baths. We also perform preliminary studies of the correlations of two and six spin 1/2 particles coupled to one, or two bath of spins, as a step to investigate quantum synchronization. We introduce a preliminary result for the spectral density of a spin-bath system, a promising quantity to measure the synchronization of spin systems.

The quantum mechanical wavefunction of a freely falling hydrogen atom will remain unchanged to first order in the presence of sufficiently weak tidal gravitational fields, giving rise to a form of incompressibility not seen in classical systems. The fact that the electron is in a bound state and that only discrete energy level transitions can occur causes the atom to behave differently than a classical object in an inhomogeneous gravitational field under certain circumstances. A time- dependent energy shift that is overlooked in a widely-used gauge choice will be explored, and shown to be a real physical effect through gauge invariance. Quantum incompressibility of the wavefunction governing the electron superfluid of a Type-I superconductor will also lead to non-classical behavior in the presence of tidal gravitational fields, via coupling between gravitational and electromagnetic fields. Theoretical and experimental implications will be discussed.

A general expression for quantum transmission of non-interacting spinless electrons through models of a fully connected network of sites that can be regarded as a nanoparticle is obtained using matrix algebra. This matrix algebra method leads to the same results given by the Green's function method without requiring the mathematical sophistication as required by the later. The model of the nanoparticle in this study comprises a single linear array of atoms that profile the input and output leads connected to a fully connected blob of atoms. A simple tight-binding Hamiltonian motivates the quantum transmission in the discrete lattice system. If there are n atoms in the nanoparticle, the methodology requires the inverse of a n x n matrix. The solution is obtained analytically for different cases: a single atom in the nanoparticle, a single dangle atom, n fully connected atoms in a mean-field type cluster with symmetric input and output connections, and the most general case where the n fully connected atoms can be connected arbitrarily to the input and output leads. A numerical solution is also provided for the case where the intra-bonds among the atoms in the nanoparticle are varied (a case with not-fully connected atoms). The expression for the transmission coefficient thus obtained using the matrix method is compared with the transmission coefficients derived using the real space Renormalization Group method and the Green's function method.

This dissertation examines the suitability of the Copenhagen interpretation of quantum physics for use in Christian theological and apologetic endeavors. Chapter 1 introduces basic issues in quantum physics in non-technical language and defines the basic position of the Copenhagen interpretation. Chapter 2 gives a thorough account of the core interpretational components of the Copenhagen interpretation and indicates that some of these set up potential conflict with Biblical-based Christian beliefs about God and the world He made. The chapter also discusses how famed scientists such as Albert Einstein and Erwin Schrödinger attempted but failed to dissuade the physics and philosophy communities from adopting the Copenhagen interpretation by formulating landmark thought-experiments that aimed to expose absurdities entailed by said interpretation. Chapter 3 examines how scholars and popular-science writers have applied the Copenhagen doctrines to sociology, philosophy, and science. Topics include feminism, race relations, finance, business management, philosophy of science, epistemology, logic, and the sciences of consciousness and cosmology. Chapter 4 discusses applications of the Copenhagen interpretation in metaphysics and theology, with particular emphasis on applications in Eastern and holistic worldviews as well as debates about free-will and divine action in Christian theology. Chapter 5 attempts to show that the Copenhagen interpretation is unsuitable for adoption in Christian theological and apologetic endeavors for several reasons. First, the Copenhagen interpretation was forged by a handful of philosopher-physicists who consciously sought to ensconce indeterminism as an ontological rather than merely epistemological element of quantum theory. Second, there are several scientific and science-historical counter-indicators to the Copenhagen view. Third, several entailments of the CI run counter to important elements in the Christian worldview. Chapter 6 gives a summary assessment of the Copenhagen interpretation and suggests avenues for continuing the science-theology dialogue in light of the current state of quantum science.