Supersymmetry and quantum mechanics

In the past ten years, the ideas of supersymmetry have been profitably applied to many nonrelativistic quantum mechanical problems. In particular, there is now a much deeper understanding of why certain potentials are analytically solvable and an array of powerful new approximation methods for handling potentials which are not exactly solvable. In this report, we review the theoretical formulation of supersymmetric quantum mechanics and discuss many applications. Exactly solvable potentials can be understood in terms of a few basic ideas which include supersymmetric partner potentials, shape invariance and operator transformations. Familiar solvable potentials all have the property of shape invariance. We describe new exactly solvable shape invariant potentials which include the recently discovered self-similar potentials as a special case. The connection between inverse scattering, isospectral potentials and supersymmetric quantum mechanics is discussed and multisoliton solutions of the KdV equation are constructed. Approximation methods are also discussed within the framework of supersymmetric quantum mechanics and in particular it is shown that a supersymmetry inspired WKB approximation is exact for a class of shape invariant potentials. Supersymmetry ideas give particularly nice results for the tunneling rate in a double well potential and for improving large N expansions. We also discuss the problem of a charged Dirac particle in an external magnetic field and other potentials in terms of supersymmetric quantum mechanics. Finally, we discuss structures more general than supersymmetric quantum mechanics such as parasupersymmetric quantum mechanics in which there is a symmetry between a boson and a para-fermion of order p

Highly Excited States of Small Molecules and Negative Atomic Ions

Excited states of atoms and molecules exhibit a rich array of diverse phenomena. This dissertation examines two exotic states of atoms at such excited levels: Rydberg molecules and atomic negative ions. Rydberg molecules are formed by a Rydberg atom and one or more ground state atoms, and can be highly polar due to their unusual electronic wave functions and enormous bond lengths. This dissertation expands the theory of these molecules by studying the formation and structure of polyatomic molecules, multichannel Rydberg molecules formed from divalent atoms, and spin effects and relativistic interactions. It also details intermolecular forces between Rydberg molecules, their manipulation via external fields, and their dependence on the intricacies of electron-atom scattering. This electron-atom interaction is also the main component of the latter portion of this thesis, which studies doubly excited states of alkali negative ions in very polarizable and nearly degenerate atomic states. Photodetachment of these states reveals electron correlation and long-range forces stemming from their high excitation

Path Integrals in the Sky: Classical and Quantum Problems with Minimal Assumptions

Cosmology has, after the formulation of general relativity, been transformed from a branch of philosophy into an active field in physics. Notwithstanding the significant improvements in our understanding of our Universe, there are still many open questions on both its early and late time evolution. In this thesis, we investigate a range of problems in classical and quantum cosmology, using advanced mathematical tools, and making only minimal assumptions. In particular, we apply Picard-Lefschetz theory, catastrophe theory, infinite dimensional measure theory, and weak-value theory. To study the beginning of the Universe in quantum cosmology, we apply Picard-Lefschetz theory to the Lorentzian path integral for gravity. We analyze both the Hartle-Hawking no-boundary proposal and Vilenkin's tunneling proposal, and demonstrate that the Lorentzian path integral corresponding to the mini-superspace formulation of the two proposals is well-defined. However, when including fluctuations, we show that the path integral predicts the existence of large fluctuations. This indicates that the Universe cannot have had a smooth beginning in Euclidean de Sitter space. In response to these conclusions, the scientific community has made a series of adapted formulations of the no-boundary and tunneling proposals. We show that these new proposals suffer from similar issues. Second, we generalize the weak-value interpretation of quantum mechanics to relativistic systems. We apply this formalism to a relativistic quantum particle in a constant electric field. We analyze the evolution of the relativistic particle in both the classical and the quantum regime and evaluate the back-reaction of the Schwinger effect on the electric field in $1+1$-dimensional spacetime, using analytical methods. In addition, we develop a numerical method to evaluate both the wavefunction and the corresponding weak-values in more general electric and magnetic fields. We conclude the quantum part of this thesis with a chapter on Lorentzian path integrals. We propose a new definition of the real-time path integral in terms of Brownian motion on the Lefschetz thimble of the theory. We prove the existence of a $\sigma$-measure for the path integral of the non-relativistic free particle, the (inverted) harmonic oscillator, and the relativistic particle in a range of potentials. We also describe how this proposal extends to more general path integrals. In the classical part of this thesis, we analyze two problems in late-time cosmology. Multi-dimensional oscillatory integrals are prevalent in physics, but notoriously difficult to evaluate. We develop a new numerical method, based on multi-dimensional Picard-Lefschetz theory, for the evaluation of these integrals. The virtue of this method is that its efficiency increases when integrals become more oscillatory. The method is applied to interference patterns of lensed images near caustics described by catastrophe theory. This analysis can help us understand the lensing of astrophysical sources by plasma lenses, which is especially relevant in light of the proposed lensing mechanism for fast radio bursts. Finally, we analyze large-scale structure formation in terms of catastrophe theory. We show that the geometric structure of the three-dimensional cosmic-web is determined by both the eigenvalue and the eigenvector fields of the deformation tensor. We formulate caustic conditions, classifying caustics using properties of these fields. When applied to the Zel'dovich approximation of structure formation, the caustic conditions enable us to construct a caustic skeleton of the three-dimensional cosmic-web in terms of the initial conditions

Noise and dissipation in superconducting granular aluminum circuits

In recent years, granular aluminum has emerged to a promising candidate to fill the gap of a high performance, high kinetic inductance material in superconducting detectors, amplifiers, and qubits. In this work, the noise properties of superconducting resonators made from granular aluminum films are studied in detail. The fluctuations of their fundamental frequency are analyzed, and electric fields are employed to investigate dielectric loss, revealing strongly coupled, microscopic two-level defects. The findings lead to a better understanding of disordered materials in the context of superconducting qubits and other quantum circuits

Gauge Fields and Strings

Based on his own work, the author synthesizes the most promising approaches and ideals in field theory today. He presents such subjects as statistical mechanics, quantum field theory and their interrelation, continuous global symmetry, non-Abelian gauge fields, instantons and the quantam theory of loops, and quantum strings and random surfaces. This book is aimed at postgraduate students studying field theory and statistical mechanics, and for research workers in continuous global theory

Fingerprints in the Optical and Transport Properties of Quantum Dots

The book "Fingerprints in the optical and transport properties of quantum dots" provides novel and efficient methods for the calculation and investigating of the optical and transport properties of quantum dot systems. This book is divided into two sections. In section 1 includes ten chapters where novel optical properties are discussed. In section 2 involve eight chapters that investigate and model the most important effects of transport and electronics properties of quantum dot systems This is a collaborative book sharing and providing fundamental research such as the one conducted in Physics, Chemistry, Material Science, with a base text that could serve as a reference in research by presenting up-to-date research work on the field of quantum dot systems

Foundations of Quantum Theory: From Classical Concepts to Operator Algebras

Quantum physics; Mathematical physics; Matrix theory; Algebr

The Fifteenth Marcel Grossmann Meeting

The three volumes of the proceedings of MG15 give a broad view of all aspects of gravitational physics and astrophysics, from mathematical issues to recent observations and experiments. The scientific program of the meeting included 40 morning plenary talks over 6 days, 5 evening popular talks and nearly 100 parallel sessions on 71 topics spread over 4 afternoons. These proceedings are a representative sample of the very many oral and poster presentations made at the meeting.Part A contains plenary and review articles and the contributions from some parallel sessions, while Parts B and C consist of those from the remaining parallel sessions. The contents range from the mathematical foundations of classical and quantum gravitational theories including recent developments in string theory, to precision tests of general relativity including progress towards the detection of gravitational waves, and from supernova cosmology to relativistic astrophysics, including topics such as gamma ray bursts, black hole physics both in our galaxy and in active galactic nuclei in other galaxies, and neutron star, pulsar and white dwarf astrophysics. Parallel sessions touch on dark matter, neutrinos, X-ray sources, astrophysical black holes, neutron stars, white dwarfs, binary systems, radiative transfer, accretion disks, quasars, gamma ray bursts, supernovas, alternative gravitational theories, perturbations of collapsed objects, analog models, black hole thermodynamics, numerical relativity, gravitational lensing, large scale structure, observational cosmology, early universe models and cosmic microwave background anisotropies, inhomogeneous cosmology, inflation, global structure, singularities, chaos, Einstein-Maxwell systems, wormholes, exact solutions of Einstein's equations, gravitational waves, gravitational wave detectors and data analysis, precision gravitational measurements, quantum gravity and loop quantum gravity, quantum cosmology, strings and branes, self-gravitating systems, gamma ray astronomy, cosmic rays and the history of general relativity