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Thesis (MSc (Physical and Mathematical Analysis))--University of Stellenbosch, 2009. / In this thesis we aim to show that the Schr odinger equation, which is a boundary eigenvalue problem, can have a discrete and real energy spectrum (eigenvalues) even when the Hamiltonian is non-Hermitian. After a brief introduction into non-Hermiticity, we will focus on solving the Schr odinger equation with a special class of non-Hermitian Hamiltonians, namely PT - symmetric Hamiltonians. PT -symmetric Hamiltonians have been discussed by various authors [1, 2, 3, 4, 5] with some of them focusing speci cally on obtaining the real and discrete energy spectrum. Various methods for solving this problematic Schr odinger equation will be considered. After starting with perturbation theory, we will move on to numerical methods. Three di erent categories of methods will be discussed. First there is the shooting method based on a Runge-Kutta solver. Next, we investigate various implementations of the spectral method. Finally, we will look at the Riccati-Pad e method, which is a numerical implemented analytical method. PT -symmetric potentials need to be solved along a contour in the complex plane. We will propose modi cations to the numerical methods to handle this. After solving the widely documented PT -symmetric Hamiltonian H = p2 􀀀(ix)N with these methods, we give a discussion and comparison of the obtained results. Finally, we solve another PT -symmetric potential, illustrating the use of paths in the complex plane to obtain a real and discrete spectrum and their in uence on the results.

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PT-symmetric Hamiltonians proposed by Bender and Boettcher can have real energy spectra. As an extension of the Hermitian Hamiltonian, PT-symmetric systems have attracted a great interest in recent years. Understanding the underlying mathematical structure of these theories sheds insight on outstanding problems of physics. These problems include the nature of Higgs particles, the properties of dark matter, the matter-antimatter asymmetry in the universe, and neutrino oscillations. Furthermore, PT-phase transition has been observed in lasers, optical waveguides, microwave cavities, superconducting wires and circuits. The objective of this thesis is to extend the iterative method of solving Schrodinger equation used for an harmonic oscillator systems to Hamiltonians with PT-symmetric potentials. An important aspect of this approach is the high accuracy of eigenvalues and the fast convergence. Our method is a combination of Hill determinant method [8] and the power series expansion. eigenvalues and the fast convergence. One can transform the Schrodinger equation into a secular equation by using a trial wave function. A recursion structure can be obtained using the secular equation, which leads to accurate eigenvalues. Energy values approach to exact ones when the number of iterations is increased. We obtained eigenvalues for a set of PT-symmetric Hamiltonians.

We investigate the some of the many subtleties in taking a microscopic approach to modelling the decoherence of an Open Quantum System. We use the RF-SQUID, which will be referred to as a simply a SQUID throughout this paper, as a non-linear example and consider different levels of approximation, with varied coupling, to show the potential consequences that may arise when characterising devices such as superconducting qubits in this manner. We first consider a SQUID inductively coupled to an Ohmic bath and derive a Lindblad master equation, to first and second order in the Baker-Campbell-Hausdorff expansion of the correlation-time-dependent flux operator. We then consider a SQUID both inductively and capacitively coupled to an Ohmic bath and derive a Lindblad master equation to better understand the effect of parasitic capacitance whilst shedding more light on the additions, cancellations and renormalisations that are attributed to a microscopic approach.

Over the past few decades, single-molecule magnets (SMMs) have been an area of significant interest due to their plethora of potential uses, including possible applications to quantum computing and compact data storage devices. Although theoretical chemistry calculations could aid our understanding of the magnetic couplings present in these types of systems, they are often multiconfigurational in nature, making them difficult to model with tradi- tional single-reference approaches. Methods to handle these types of strongly correlated systems have been developed but often have significant drawbacks, and so these molecules remain difficult to model computationally. In this work, we discuss the application of Fock-space CI approaches to large transition metal complexes. First, we introduce a novel formalism which combines the spin-flip (SF), ioniza- tion potential (IP), and electron affinity (EA) approaches. This redox spin-flip approach, the restricted active space spin-flip and ionization potential/electron affinity (RAS-SF-IP/EA) method, is applied to several molecules exhibiting double exchange behavior. Model Hamil- tonian parameters are extracted from energy gaps and found to be in qualitative agreement with experiment. Having shown the efficacy of this approach, we move on to optimization, using a diagrammatic approach to derive equations for several RAS-1SF-IP/EA schemes. These equations allow direct construction of the most expensive intermediates in the David- son algorithm and should provide significant speedup, allowing application of Fock-space CI approaches to larger systems than ever before. The derived equations are implemented in the LibRASSF package in Q-Chem, as well as in an open-source PyFockCI code, avail- able on GitHub. A Bloch effective Hamiltonian formalism is also utilized to extract model Hamiltonian parameters from RAS-1SF calculations, allowing more nuanced studies of the Heisenberg J couplings present in many molecules with magnetically coupled sites. Over- all, our work with Fock-space CI provides a way to study magnetic couplings in very large strongly correlated systems at relatively low computational cost. This work was supported by a grant from the U.S. Department of Energy: DE-SC0018326. / Doctor of Philosophy / Humans have been familiar with magnets for thousands of years, and we have found a variety of useful applications for them. Magnets are used in everything from navigational devices to credit cards to data storage. Most people are familiar with large, solid magnets, but in the 1990s, it was discovered that individual molecules, called single-molecule magnets (SMMs), could also exhibit magnetic behavior. This means that in the presence of some external magnetic field, like the field caused by the presence of another magnet, the electrons in a SMM will align themselves with the field, and the electrons will maintain that alignment for some period of time after the field is removed. These SMMs have been a significant area of interest to scientists because they have a variety of interesting applications, including applications to quantum computing. In cases such as these, theoretical chemistry can offer useful insight. Broadly, the purpose of theoretical chemistry is to describe chemical problems using mathematical equations. We can use computational models to obtain information about the behavior of electrons in a particular system (the so-called electronic structure) and consequently, we can model the magnetic couplings in a given molecule. However, SMMs are difficult to model using tra- ditional theoretical methods because they often contain multiple orbitals which have nearly the same energy. In these cases, it often becomes ambiguous which orbitals ought to be oc- cupied by electrons; the effect this has on the energy is called "strong correlation". Ideally, one ought to consider all possible fillings of the orbitals, but most methods do not account for this and assume only one configuration is important when solving for the shapes of the orbitals. In this work, we combine two previously-introduced approaches, the spin-flip (SF) and ioniza- tion potential/electron affinity (IP/EA) approaches, to handle strongly correlated systems. In the SF-IP/EA approach, one adds or removes electrons and flips their spins in order to remove all of the ambiguity in orbital occupations. Once we determine the shapes of the or- bitals for this unambiguous state, the electrons are added, removed, or spin-flipped in order to obtain the desired strongly correlated state. We then solve for the energy of the system while considering all possible configurations within the set of ambiguously-occupied orbitals, allowing us to treat them on equal footing. We also study the effect of adding additional configurations to account for contributions from other orbitals, which provides more accu- rate results, albeit at a higher computational cost. Our method is less expensive than many other wavefunction-based methods used for these systems, and it yields qualitatively correct results, allowing theoreticians to study magnetic couplings in SMMs in a straightforward and inexpensive way. We also discuss optimization of our code, as well as an extension of our code that allows us to obtain coupling information for systems containing multiple magnetic sites. It is our hope that these developments will provide useful insights into the electronic structure of these SMM systems.

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