Galilean covariance and the Duffin-Kemmer-Petiau equation

We use a five-dimensional approach to Galilean covariance to investigate the non-relativistic Duffin-Kemmer-Petiau first-order wave equations for spinless particles. The corresponding representation is generated by five 6 × 6 matrices. We consider the harmonic oscillator as an example.

On the regular-geometric-figure solution to the N-body problem

The regular-geometric-figure solution to the N-body problem is presented in a very simple way. The Newtonian formalism is used without resorting to a more involved rotating coordinate system. Those configurations occur for other kinds of interactions beyond the gravitational ones for some special values of the parameters of the forces. For the harmonic oscillator, in particular, it is shown that the N-body problem is reduced to N one-body problems.

Galilean covariance and non-relativistic Bhabha equations

We apply a five-dimensional formulation of Galilean covariance to construct non-relativistic Bhabha first-order wave equations which, depending on the representation, correspond either to the well known Dirac equation (for particles with spin 1/2) or the Duffin-Kemmer-Petiau equation (for spinless and spin 1 particles). Here the irreducible representations belong to the Lie algebra of the 'de Sitter group' in 4 + 1 dimensions, SO(5, 1). Using this approach, the non-relativistic limits of the corresponding equations are obtained directly, without taking any low-velocity approximation. As a simple illustration, we discuss the harmonic oscillator.

Generalized ladder operators for shape-invariant potentials

A general form for ladder operators is used to construct a method to solve bound-state Schrödinger equations. The characteristics of supersymmetry and shape invariance of the system are the start point of the approach. To show the elegance and the utility of the method we use it to obtain energy spectra and eigenfunctions for the one-dimensional harmonic oscillator and Morse potentials and for the radial harmonic oscillator and Coulomb potentials.

Solução da equação de Fokker-Planck para o potencial do oscilador harmônico duplo

Pós-graduação em Biofísica Molecular - IBILCE

O formalismo supersimétrico e suas aplicações em mecânica quântica

Pós-graduação em Biofísica Molecular - IBILCE

Análise comparativa de alguns sistemas dinâmicos clássicos e quânticos

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

Um estudo da dinâmica fracamente não-linear de um sistema nanomecânico

Pós-graduação em Matemática - IBILCE

Modelos descritos por equações diferenciais ordinárias

Pós-graduação em Matemática Universitária - IGCE

O oscilador harmônico singular revisitado

The one-dimensional Schrödinger equation with the singular harmonic oscillator is investigated. The Hermiticity of the operators related to observable physical quantities is used as a criterion to show that the attractive or repulsive singular oscillator exhibits an infinite number of acceptable solutions provided the parameter responsible for the singularity is greater than a certain critical value, in disagreement with the literature. The problem for the whole line exhibits a two-fold degeneracy in the case of the singular oscillator, and the intrusion of additional solutions in the case of a nonsingular oscillator. Additionally, it is shown that the solution of the singular oscillator can not be obtained from the nonsingular oscillator via perturbation theory.

Uma abordagem teórica e experimental do oscilador harmônico em duas dimensões utilizando as curvas de Lissajous

The study of oscillatory phenomena is an important topic in several areas of knowledge. The understanding and the solution of many problems that scientists and engineers confront nowadays can be formulated, at least as an analogy, in terms of an oscillatory movement. The description of the motion of this nature has remarkable contributions to the mathematical and conceptual formation of the students and for solving daily problems in several areas. In this paper we propose a theoretical and experimental approach involving the phenomenon of harmonic oscillations in two dimensions. Lissajous curves were used as experimental demonstration of the resulting trajectories. These curves were obtained from relatively simple experimental apparatus, which is affordable in most teaching laboratories of physics.

One-sided and two-sided Green's functions

We discuss the one-sided Green's function, associated with an initial value problem and the two-sided Green's function related to a boundary value problem. We present a specific calculation associated with a differential equation with constant coefficients. For both problems, we also present the Laplace integral transform as another methodology to calculate these Green's functions and conclude which is the most convenient one. An incursion in the so-called fractional Green's function is also presented. As an example, we discuss the isotropic harmonic oscillator.

Aspects of classical and quantum motion on a flux cone

Motion of a nonrelativistic particle on a cone with a magnetic flux running through the cone axis (a flux cone) is studied. It is expressed as the motion of a particle moving on the Euclidean plane under the action of a velocity-dependent force. The probability fluid (quantum flow) associated with a particular stationary state is studied close to the singularity, demonstrating nontrivial Aharonov-Bohm effects. For example, it is shown that, near the singularity, quantum flow departs from classical flow. In the context of the hydrodynamical approach to quantum mechanics, quantum potential due to the conical singularity is determined, and the way it affects quantum flow is analyzed. It is shown that the winding number of classical orbits plays a role in the description of the quantum Bow. The connectivity of the configuration space is also discussed.

Sudden change in quantum and classical correlations and the Unruh effect

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Elementary particles under the lens of the black holes

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)