We study the non-relativistic charge-monopole system when the charged particle has a position-dependent mass written as M(r) = m0rw. The angular wave functions are the well-known monopole harmonics, and the radial ones are ordinary Bessel functions which depend on the magnetic and electric charge product as well as on the w parameter. We investigate mappings—approximate and exact—between the charge-monopole system with constant mass and the charge with a position-dependent mass solving the position-dependent mass Schrödinger equation for the mass distribution.
The main purpose of this work is to reproduce a quantum system charge-monopole utilizing position-dependent effective mass (PDM) system in the nonrelativistic regime via the Pauli equation. In this case, we substitute the exact charge-monopole wavefunction into the free PDM Pauli equation and then solve it for the mass distribution considering a radial dependence only, i.e., M = M(r). The resulting equations are nonlinear, and in such cases, we were able to numerically solve them, fixing θ0 and considering specific values of μ and m satisfying a given condition. The mapping was studied for eigenvalues starting from the minimal value j = μ − 1/2.
The primary purpose of this work is to reproduce the scenario composed of a charge-dyon system utilizing position-dependent effective mass (PDM) background in the non-relativistic and in the relativistic regimes. In the non-relativistic case we substitute the exact charge-dyon eigenfunction into PDM Schrödinger equation, in the Zhu-Kroemer parametrization, and then solve it for the mass distribution considering M = M(r). Analogously, in the relativistic case we study the Klein-Gordon equation for a position-dependent mass, and in this case, we are able to analytically solve the equation for M = M(r,θ).
We investigate the scattering of a plane wave in the hyperbolic plane. We formulate the problem in terms of the Lippmann-Schwinger equation and solve it exactly for barriers modeled as Dirac delta functions running along: (i) N − horizontal lines in the Poincaré upper half-plane; (ii) N − concentric circles centered at the origin; and, (iii) a hypercircle in the Poincaré disk.
We study the non-relativistic quantum mechanical scattering of a plane wave by a shield barrier and an elliptical aperture modeled as Dirac delta functions running along a coordinate surface of the sphero-conal coordinate system. The scattering problem is formulated via Lippmann-Schwinger (LS) equation in the position representation. In order to solve the LS equation, we first calculate the free Green's function of the problem and obtain its bilinear expansion in terms of the eigenfunctions of the scalar Helmholtz equation—which are products of spherical Bessel (or first kind Hankel) functions and Lamé polynomials. Such bilinear expansion allows us to obtain an integral equation with a separable kernel and solve the scattering problem. Then, we calculate the wavefunctions in the internal and external domains and the scattering amplitudes.
We consider a particle with spin 1/2 with position-dependent mass moving in a plane. Considering separately Rashba and Dresselhaus spin-orbit interactions, we write down the Hamiltonian for this problem and solve it for Dirichlet boundary conditions. Our radial wavefunctions have two contributions: homogeneous ones which are written as Bessel functions of non-integer orders—that depend on angular momentum m—and particular solutions which are obtained after decoupling the non-homogeneous system. In this process, we find non-homogeneous Bessel equation, Laguerre, as well as biconfluent Heun equation. We also present the probability densities for m = 0, 1, 2 in an annular quantum well. Our results indicate that the background as well as the spin-orbit interaction naturally splits the spinor components.
Este trabalho tem como objetivo apresentar uma introdução ao grupo S O ( 4 ) e duas aplicações na Física: uma na Mecânica Clássica e outra na Mecânica Quântica. Os geradores do grupo S O ( 4 ) serão determinados, assim como sua álgebra de Lie. A aplicação na Mecânica Clássica será na obtenção da transformação de Galilei homogênea e na Mecânica Quântica ser á na obtenção do espectro de energia do átomo de hidrogênio, no regime não-relativístico. A versão quântica do vetor de Laplace-Runge-Lenz será fundamental para a construção da fórmula de Bohr para os níveis de energia do respectivo átomo.
Neste trabalho apresentaremos uma introdução à Física do Monopolo Magnético no regime não – relativístico. O tratamento clássico de um sistema composto por uma partícula pontual eletricamente carregada na presence de um monopolo magnético na origem, com suas equações de movimento e o estudo do potencial vetor de Dirac, será desenvolvido ao longo do presente trabalho. A questão das singularidades e as transformações de calibre serão também tratadas. Ao final, listaremos alguns modelos análogos ao monopolo magnético, utilizando outros sistemas físicos.
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