Non-commutative quantum mechanics

Jacobus Daniel Thom

Published Dec 18, 2014• 0 citations

Abstract

The aim of this thesis is threefold. Firstly, I give an overview of non-commutative quan- tum mechanics and build up a description of non-commutative piecewise constant poten- tial wells in this context. Secondly, I look at some of the stationary properties of a finite non-commutative well using the mathematical tools laid out in the first part. Lastly, I in- vestigate how non-commutativity affects the tunneling rate through a barrier. Throughout this work I give the normal commutative descriptions and results for comparsion.

Related papers

Non-commutative quantum mechanics : properties of piecewise constant potentials in two dimensions

Jacobus Daniel Thom

ENGLISH ABSTRACT: The aim of this thesis is threefold. Firstly, I give an overview of non-commutative quan- tum mechanics and build up a description of non-commutative piecewise constant poten- tial wells in this context. Secondly, I look at some of the stationary properties of a finite non-commutative well using the mathematical tools laid out in the first part. Lastly, I in- vestigate how non-commutativity affects the tunneling rate through a barrier. Throughout this work I give the normal commutative descriptions and results for comparsion.

Dec 1, 20101 citation

Noncommutative Mathematics for Quantum Systems

Uwe Franz, Adam Skalski

Noncommutative mathematics is a significant new trend of mathematics. Initially motivated by the development of quantum physics, the idea of 'making theory noncommutative' has been extended to many areas of pure and applied mathematics. This book is divided into two parts. The first part provides an introduction to quantum probability, focusing on the notion of independence in quantum probability and on the theory of quantum stochastic processes with independent and stationary increments. The second part provides an introduction to quantum dynamical systems, discussing analogies with fundamental problems studied in classical dynamics. The desire to build an extension of the classical theory provides new, original ways to understand well-known 'commutative' results. On the other hand the richness of the quantum mathematical world presents completely novel phenomena, never encountered in the classical setting. This book will be useful to students and researchers in noncommutative probability, mathematical physics and operator algebras.

Cambridge University Press eBooksDec 18, 20156 citations

Quasilocality and equilibration in quantum systems

Marko Medenjak

The primary area, on which we focus in the thesis, are the out-of-equilibrium properties of quantum and classical systems. The playground we use to study those properties, are interacting integrable and solitonic systems, whose structure enables us to perform analytical calculations. In the context of the out-of-equilibrium physics we focus on two questions. The first problem is related to the set of stationary states in quantum systems. The question is especially important in the context of the homogeneous quantum quenches, since it makes possible the calculations of expectation values of local observables. The building blocks of stationary states are quasilocal conserved quantities which constitute the information that is preserved under the time evolution. In the first part of the thesis we deal with the construction of quasilocal conserved quantities in one dimensional anisotropic Heisenberg model and bosonic models. Quasilocal charges are put in the context of the standard theory of integrability, which in turn allows for the calculation of physical quantities. The second question we deal with are the transport properties of one dimensional systems. The connection between the local integrals of motion and ideal transport was established in 90's. Here we extend the connection to the diffusive transport. This enables us to obtain explicit lower bound on diffusion constant in Heisenberg model, which proves that the Heisenberg model is not an insulator at finite temperatures. In the last chapter we deal with the transport properties of classical celular automata. The first example we consider is the gas of charged hard core interacting particles, and the second one the gas of interacting solitons. Explicit solution of the dynamics of local observables allows for the explicit calculation of transport coeffcients, and the solution of the inhomogeneous quench problem. Despite their simplicity, the models exhibit wide range of transport phenomena ranging from insulating, through diffusive to ballistic transport.

Jan 1, 20181 citation

Non-locality and back-reaction in acoustic black holes and non-linearity in quantum fluid dynamics

Giorgio Ciliberto

This thesis is mainly about quan- tum correlations and non-linear fluctuations in quantum fluids. It focuses especially on collec- tive quantum fluctuations, i.e. sound waves, in the stationary flow of a 1D Bose-Einstein quasi- condensate which exhibits an acoustic horizon, i.e. a transition from subsonic to supersonic flow. Quantum phenomena generated by the presence of the horizon are investigated. The thesis also presents a brief excursus on two- dimensional turbulence in a quantum fluid. We study quantum non-separability, non- locality and back-reaction of the acoustic Hawking radiation emitted by an analog black hole. The entanglement and non-local correlations within the tripartite system of quasi-particles emitted from the acoustic horizon are first in- vestigated. Back-reaction equations govern- ing the effect of such acoustic radiation on the inhomogeneous background are derived and stationary solutions are considered. Finally, a phenomenological theory based on topological constraints for vortex creation and annihilation is effectively applied to experimental data, thus accounting for the growth and decay of turbulence.

Jun 26, 20250 citations

The formalism of non-commutative quantum mechanics and its extension to many-particle systems

Andreas Hafver

By submitting this thesis electronically, I declare that the entirety of the work contained therein is my own, original work, that I am the owner of the copyright thereof (unless to the extent explicitly otherwise stated) and that I have not previously in its entirety or in part submitted it for obtaining any qualification.

Dec 1, 20100 citations

EXACTLY SOLVABLE MODELS OF STOCH- ASTIC QUANTUM MECHANICS WITHIN THE FRAMEWORK OF LANGEVIN- SCHROEDINGER TYPE EQUATIONS

Parujr Sevak

A closed, uncountably infinite dimensional system of object + is mathematically stated in terms of a complex probabilistic process for the wave function, i.e. the solution of Langevin-Schrodinger (L-Sch)-type stochastic differential equation (SDE). The L-Sch SDE has been studied for nonstationary 1D random potential with quadratic space form. It was proved that when the random processes in poten- tial determined by δ-shaped correlators, then closed, exactly solvable nonrelativistic quantum mechanics may be constructed. In this case all physically measured parameters of the system are built as multiple integrals of the fundamental solution of some second-order partial dif- ferential equations (etalon equations). In the present work we obtain expressions for transition probabilities in a quantum subsystem. It is shown that depending on the coupling constant of the thermostat with the parametric quantum harmonic oscillator (PQHO), some phase tran- sitions of second kind may occur in the expressions for microscopic quan- tum transitions. A method of stochastic density matrix is developed for calculation of thermodynamic potentials of the quantum harmonic oscil- lator (QHO) immersed into a thermostat. The analytic expressions for the ground state energy level widening and shift of QHO are obtained. In other words the possibility of violating the second law of thermody- namics due to quantum fluctuations, (i.e., spontaneous transitions in QHO from vacuum to the excited states) is shown.

Jan 1, 20040 citations