A Short Course on Quantum Mechanics and Methods of Quantization

Valter Moretti

Narciso N. Garcia, Arthur C. Damask

Elisa Ercolessi

These notes collect the lectures given by the author to the "XXIII International Workshop on Geometry and Physics" held in Granada (Spain) in September 2014. The first part of this paper aims at introducing a mathematical oriented reader to the realm of Quantum Mechanics (QM) and then to present the geometric structures that underline the mathematical formalism of QM which, contrary to what is usually done in Classical Mechanics (CM), are usually not taught in introductory courses. The mathematics related to Hilbert spaces and Differential Geometry are assumed to be known by the reader. In the second part, we concentrate on some quantization procedures, that are founded on the geometric structures of QM — as we have described them in the first part — and represent the ones that are more operatively used in modern theoretical physics. We will discuss first the so-called Coherent State Approach which, mainly complemented by "Feynman Path Integral Technique", is the method which is most widely used in quantum field theory. Finally, we will describe the "Weyl Quantization Approach" which is at the origin of modern tomographic techniques, originally used in optics and now in quantum information theory.

Narciso N. Garcia, Arthur C. Damask, Steven A. Schwarz

Lorenzo J. Curtis, David G. Ellis

There are significant differences in the structures of course presentations of classical mechanics and quantum mechanics, but also essential links that connect the two approaches. A large part of the differences in approach do not involve quantization, but rather the traditional use of instantaneous values to describe macroscopic objects and position probability densities to describe microscopic objects. A pedagogic reformulation of the classical problem to bridge this gap is presented here. The method is numerical in nature, but for illustrative purposes it is herein applied to two specific cases for which analytic solutions also exist, the Kepler–Coulomb potential and the isotropic harmonic oscillator. Using this approach, classical systems can be examined in the presence of perturbations in the same way as is used in quantum mechanics courses. Semiclassical quantization can also be introduced to extend the connection to a quantum-mechanical treatment.

Francisco Castilho Alcaraz, Silvio R. A. Salinas

Mecânica Estatística, EBME)The 2002 version of the Brazilian School on Statistical Mechanics (Escola Brasileira de Mecânica Estatística, EBME), which is hoped to happen every two years, took place from February 18 to March 1st, at the Instituto de Física de São Carlos, University of São Paulo, in the relatively small town of São Carlos, about 250 km from São Paulo.It was conceived as an answer to the need of a thematic school in statistical mechanics, at a national level, in order to promote short graduate courses on a variety of subjects and research topics that are relatively recent and not so well represented in the country.The school consisted of four short courses of approximately 10 hours, attended by about 70 students from several regions of Brazil, and whose lecture notes are finally being published in this special issue of the Braz.J. Phys.In the first week of the school, there were two courses: "Introduction to Bosonization" by Eduardo Miranda, from the University of Campinas, and "Introduction to Quantum Monte Carlo Simulations for Fermionic Systems", by Raimundo R. dos Santos, from the Federal University of Rio de Janeiro.In the second week, there were two other courses: "An Introduction to Numerical Methods in Low-Dimensional Quantum Systems", by André L. Malvezzi, from the Universidade Estadual Paulista, in Bauru, São Paulo, and "Non-Equilibrium Phase Transitions", by Ronald Dickman, from the Federal University of Minas Gerais, in Belo Horizonte.The highly pedagogical notes of Eduardo Miranda, which are selfcontained and assume just some basic knowledge of second quantization, are of particular relevance as an introduction to the use of the modern bosonization techniques.Raimundo Santos presented a tutorial introduction to Monte Carlo techniques for simulations of classical and quantum systems, with particular emphasis on the determinantal quantum Monte Carlo method.This short course was designed to provide the necessary tools for carrying out a numerical analysis of fermionic systems such as the well-known Hubbard model.André Malvezzi presented a comprehensive and systematic overview, including details of implementation, of the Lanczos and the Density Matrix Renormalization methods, which are by far the most common numerical techniques to obtain the eigenspectra of quantum chains.In the last short course, Ronald Dickman presented some applications of several techniques (mean-field, series expansions, renormalization-group) to the analysis of non-equilibrium phase transitions in stochastic systems.Since most of these topics had already been covered in a recent book, Dickman decided to send us a paper on path integrals and perturbation theory for stochastic processes, with a number of applications for nonequilibrium systems.