<p> Bohmian mechanics is an alternative formulation of quantum mechanics that incorporates the familiar and intuitive picture of particles moving along trajectories and yet predicts the same results as the more widely accepted Copenhagen interpretation.</p> <p> In recent years there has been renewed interest in this Bohmian view, in part for the novel approach that it suggests to certain problems, such as decay processes, both from a theoretical and computational stand point. In this thesis we focus on using the concepts introduced by the Bohmian framework as a practical computational tool.</p> <p> I evaluate a number of implementations of the Bohmian method, get a sense of their strengths and weaknesses and attempt to overcome some stability issues that arise. For problems in one-dimension (lD), accurate solutions of the time-dependent Schrodinger equation produce a wave function from which Bohmian trajectories can be computed by integrating along flux lines. For direct integration of the quantum Hamilton-Jacobi equations, the main problems that arise are related to evauating the quantum potential (QP), especially in regions of low probability density. Sufficient accuracy is required to avoid unphysical trajectory crossings. A number of interpolation schemes were investigated, and smoothed splines with special treatment of edge effects gave the best results.</p> <p> For problems in 2D the alternating direction implicit (ADI) method was employed to produce the wave function. Ways of dealing with unphysical reflections from the boundaries of a finite size domain were studied.</p> <p> The use of cellular automata, especially the lattice-Boltzmann method (LBM) were also considered. Here Bohm trajectories would be propagated by following a small set of rules. The main problem identified is that, unless a scheme can be found in which the quantum potential is self-generating from an equation of continuity, the overhead of computing the QP at each time step, is prohibitive.</p> / Thesis / Doctor of Philosophy (PhD)

In this thesis, four different numerical techniques are implemented for the purpose of simulating resonant tunnelling diodes (RTDs). The chosen methods were: piecewise constant transfer matrix (TMM-C), piecewise linear transfer matrix (TMM-L), quantum transmitting boundary method (QTBM), and the Crank-Nicolson method (CN). The numerical methods converged compared with the known analytic simulation for plane waves tunnelling through a single barrier. To better represent the semiconductor-based RTDs, the effective mass approximation was adopted with accompanying modifications to the Hamiltonian operators to ensure the continuity of the wave function and its derivative. Using the Tsu-Esaki formula, the current density was calculated as a function of bias voltage for two different RTD devices. The numerically obtained current density was of the same order of magnitude as referenced experimental values but differed significantly enough to require better models if engineering applications are decided. The models in this thesis were able to display resonant tunnelling and a negative differential resistance (NDR), giving them plausible educational value.

Indiana University-Purdue University Indianapolis (IUPUI) / In this thesis, we focus on two broad categories of problems, exciton condensation and bound states, and two complimentary approaches, real and momentum space, to solve these problems. In chapter 2 we begin by developing the self-consistent mean field equations, in momentum space, used to calculate exciton condensation in semiconductor heterostructures/double quantum wells and graphene. In the double quantum well case, where we have one layer containing electrons and the other layer with holes separated by a distance $d$, we extend the analytical solution to the two dimensional hydrogen atom in order to provide a semi-quantitative measure of when a system of excitons can be considered dilute. Next we focus on the problem of electron-electron screening, using the random phase approximation, in double layer graphene. The literature contains calculations showing that when screening is not taken into account the temperature at which excitons in double layer graphene condense is approximately room temperature. Also in the literature is a calculation showing that under certain assumptions the transition temperature is approximately \unit{mK}. The essential result is that the condensate is exponentially suppressed by the number of electron species in the system. Our mean field calculations show that the condensate, is in fact, not exponentially suppressed. Next, in chapter 3, we show the use of momentum space to solve the Schr\"{o}dinger equation for a class of potentials that are not usually a part of a quantum mechanics courses. Our approach avoids the typical pitfalls that exist when one tries to discretize the real space Schr\"{o}dinger equation. This technique widens the number of problems that can presented in an introductory quantum mechanics course while at the same time, because of the ease of its implementation, provides a simple introduction to numerical techniques and programming in general to students. We have furthered this idea by creating a modular program that allows students to choose the potential they wish to solve for while abstracting away the details of how the solution is found. In chapter 4 we revisit the single exciton and exciton condensation in double layer graphene problems through the use of real space lattice models. In the first section, we once again develop the equations needed to solve the problem of exciton condensation in a double layer graphene system. In addition to this we show that by using this technique, we find that for a non-interacting system with a finite non-zero tunneling between the layers that the on-site exciton density is proportional to the tunneling amplitude. The second section returns to the single exciton problem. In agreement with our momentum space calculations, we find that as the layer separation distance is increased the bound state wave function broadens. Finally, an interesting consequence of the lattice model is explored briefly. We show that for a system containing an electron in a periodic potential, there exists a bound state for both an attractive as well as repulsive potential. The bound state for the repulsive potential has as its energy $-E_0$ where $E_0$ is the ground state energy of the attractive potential with the same strength.

The traditional explanation of the Stern-Gerlach effect carries with it several very subtle assumptions and approximations. We point out the degree to which this fact has affected the way we practice and interpret modern physics. In order to gain a more complete understanding of the Stern-Gerlach effect beyond the standard approximations and assumptions it typically carries, we introduce the inhomogeneous Stern-Gerlach effect in which the strong uniform field component is removed. This change allows us to easily identify precession as a critical concept. It also provides us with a means by which to study precession and analyze it critically. By applying and comparing several mathematical techniques to this problem we gain insight into the applicability of precession arguments and the role of standard approximations and assumptions in both analytic derivation and interpretation. This approach also allows for a more general discussion regarding the use of representations in physics and teaching.

This licentiate thesis in theoretical physics focuses on the existence of quantum features in physical systems such as axion dark matter and gravity. Our focus is mostly on effects which appear at low energies, a regime in which our models can be confronted with current experiments or within the foreseeable future. In our first project, we focus on squeezing of axion dark matter, a quantum mechanical effect which accompanies the standard mean field description of axions. We have showed that within a reasonable set of assumptions, the quantum state of axions is highly squeezed. This theoretical finding suggests that the mean field description of axion dark matter is incomplete, since the latter conceals many interesting and possibly experimentally relevant phenomena, and paves the way for axion dark matter studies beyond the mean field approximation. Moreover, in this thesis, some ongoing work on axion dark matter decoherence is presented. Our goal is to test whether axion dark matter squeezing is robust against decoherence. Preliminary results indicate that squeezing is not diminished in presence of environmental interactions. Our results stem from an interdisciplinary approach at the intersection between cosmology, quantum optics, quantum open systems and cold atoms. Our second work focuses on quantum features of gravity. An almost century old question is how gravity can be reconciled with the laws of quantum mechanics. This question remains still open and part of the reason is the lack of experimental evidence. However, in recent years, the rapid progress of experimental techniques allows for quantum control and manipulation of larger and larger quantum systems. These new experimental routes have sparkled an interest in testing such fundamental questions with tabletop experiments. One particularly interesting proposal aims to test whether gravity can mediate entanglement between two spatially superposed mesoscopic masses. This proposal, in order to deduce the existence of quantized gravitational mediators, relies on a quantum-information-theoretic argument, the so-called LOCC (Local Operations and Classical Communication). In our work, we critically assess this proposal, its underlying assumptions and what teaches about quantum gravity. We conclude that the LOCC argument is not useful and by invoking it, one cannot unambiguously infer the existence of quantum mediators unless the principle oflocality is elevated to a fundamental principle of nature. We support our claim by explicitly showing that well known relativistic field theories, apart from local formulations can also admit non-local ones. Therefore, the entanglement generating quantum channel can be either local or non-local.

In this thesis I present a derivation of statistical mechanics starting from a closed, isolated quantum many body system. I first establish under what conditions we expect static equilibrium to emerge. It is found that the purity of the diagonal ensemble is a sufficient criteria for equilibration to occur and avoid short time recurrences. I next derive the usual ensembles of statistical mechanics using the principle of maximum entropy. These ensembles are then connected to the diagonal ensemble through the strong and the weak eigenstate thermalization hypothesis (ETH). Counter examples to ETH are discussed along with the process of scrambling. The thesis contains three contributed articles relevant to this introductory chapter studying early time relaxation, recurrences and scrambling. / Thesis / Doctor of Philosophy (PhD)

No description available.

Indiana University-Purdue University Indianapolis (IUPUI) / Over the past two decades, open systems that are described by a non-Hermitian Hamiltonian have become a subject of intense research. These systems encompass classical wave systems with balanced gain and loss, semi-classical models with mode selective losses, and lossy quantum systems. The rapidly growing research on these systems has mainly focused on the wide range of novel functionalities they demonstrate. In this thesis, I intend to present some intriguing properties of a class of open systems which possess parity (P) and time-reversal (T) symmetry with a theoretical background, accompanied by the experimental platform these are realized on. These systems show distinct regions of broken and unbroken symmetries separated by a special phase boundary in the parameter space. This separating boundary is called the PT-breaking threshold or the PT transition threshold. We investigate non-Hermitian systems in two settings: tight binding lattice models, and electrical circuits, with the help of theoretical and numerical techniques. With lattice models, we explore the PT-symmetry breaking threshold in discrete realizations of systems with balanced gain and loss which is determined by the effective coupling between the gain and loss sites. In one-dimensional chains, this threshold is maximum when the two sites are closest to each other or the farthest. We investigate the fate of this threshold in the presence of parallel, strongly coupled, Hermitian (neutral) chains, and find that it is increased by a factor proportional to the number of neutral chains. These results provide a surprising way to engineer the PT threshold in experimentally accessible samples. In another example, we investigate the PT-threshold for a one-dimensional, finite Kitaev chain—a prototype for a p-wave superconductor— in the presence of a single pair of gain and loss potentials as a function of the superconducting order parameter, onsite potential, and the distance between the gain and loss sites. In addition to a robust, non-local threshold, we find a rich phase diagram for the threshold that can be qualitatively understood in terms of the band-structure of the Hermitian Kitaev model. Finally, with electrical circuits, we propose a protocol to study the properties of a PT-symmetric system in a single LC oscillator circuit which is contrary to the notion that these systems require a pair of spatially separated balanced gain and loss elements. With a dynamically tunable LC oscillator with synthetically constructed circuit elements, we demonstrate static and Floquet PT breaking transitions by tracking the energy of the circuit. Distinct from traditional mechanisms to implement gain and loss, our protocol enables parity-time symmetry in a minimal classical system.

Yes / The Desargues property is well known in the context of projective geometry. An analogous property is presented in the context of both classical and Quantum Physics. In a classical context, the Desargues property implies that two logical circuits with the same input, show in their outputs selective correlations. In general their outputs are uncorrelated, but if the output of one has a particular value, then the output of the other has another particular value. In a quantum context, the Desargues property implies that two experiments each of which involves two successive projective measurements, have selective correlations. For a particular set of projectors, if in one experiment the second measurement does not change the output of the rst measurement, then the same is true in the other experiment.

This dissertation explores the properties of sulfonated poly(arylene ether sulfone)s for desalination membrane applications. A multi-scale approach is used to understand the relationships between the chemical structure of the polymer, the equilibrium water content, and the bulk properties. The polysulfones investigated here are aromatic polymers with relatively high ion contenremain in the glassy state at room temperature even when fully hydrated. In order to better understand the effects of water on these ionic polysulfones molecular dynamics (MD) simulation is used to investigate ion aggregation and hydration at the atomic scale. MD simulations show that the sulfonate and sodium ions are not simply paired. Instead, they form an ionic network. The molecular nature of melting water within sulfonated polysulfones is also examined by combining differential scanning calorimetry (DSC), Fourier transform infrared spectroscopy (FTIR), and MD simulation. Experimental evidence shows that at high ion contents, the spacing between the ionic groups impacts the amount of melting water present in the polymer. We conclude that the amount of melting water in the polymer is more closely related to geometric clustering effects than electrostatic effects. Finally, molecular-scale insight is used to understand the trends in hydrated tensile modulus and hydrated glass transition (Tg) temperatures in sulfonated polysulfones. Polymers with a more rigid backbone show different trends compared to those with a more flexible backbone. The modulus and Tg trends for the more flexible backbone are qualitatively consistent with the increase in intra-chain ionic associations (loops) predicted by the sticky Rouse model. / Doctor of Philosophy / This dissertation investigates new materials that could be used to make better membranes that can remove ions (salt) from water. Existing materials are too soft or too brittle when they are fully immersed in water. Consequently, they must be combined with more durable materials in order to make useful membranes. We would like to design durable ionic polymers (large chain-like molecules with ions attached) that interact with water and other ions in a very specific way in order to make membranes that can remove salt efficiently. The goal of this research is to create tools that can describe how changes to the chemical structure of the polymer impact how the polymer, water, and ions interact with each other so that we can improve membrane properties. We find that the ions on the polymer chain interact with each other to form threads, which can form a network inside of the polymer under the right conditions. When the ions are located far apart on the polymer chain, the ion threads link one polymer chain to another polymer chain. These ionic links strengthen the polymer network. However, when the ions are located closer together on the polymer chain, the chain starts to form loops between neighboring ions. As the number of loops increases, the polymer quickly becomes softer and more gel-like. We also find that water molecules are distributed within the polymer and are not always located next to the ions. When there is more water inside the polymer, the water molecules begin to group together to form clusters. At low temperatures, water molecules that have fewer than four neighboring molecules cannot freeze. However, water in a cluster of five molecules or more can freeze into an ice crystal. The insights gained from this research will help the community to design better polymers for desalination membrane applications.