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Aleph UCLA Undergraduate Research Journal for the Humanities and Social Sciences

Excitação coulombiana da RGD com representação de um espaço de fase molomórfico com a aproximação eikonal

J.S. da Veiga

Published Jan 1, 1993• 0 citations
OpenAlex

Abstract

We extend the Weyl-Wigner transformation to include the second quantization scheme by introducing the Bargmann mapping kernel in the Weyl transform. Besides, we construct a holomorphic phase space in which complex entire analytic functions of two variables, namely, z and w*, stand for the boson creation and annihilation operators representatives, respectively. The dynamical equations are obtained by the mapped Heisenberg equation of motion. So, a holomorphic expression for an excitation probability is presented and an explicit calculation is performed for the forced harmonic oscilator. Finally, we apply the holomorphic phase-space description to a particular collision process, i.e.,the analytical study of the Coulomb excitation for the isovector giant electric dipole ressonance in the eikonal approximation. Comparisons with experimental results are presented and discussed

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We extend the Weyl-Wigner transformation to include the second quantization scheme by introducing the Bargmann mapping kernel in the Weyl transform. Besides, we construct a holomorphic phase space in which complex entire analytic functions of two variables, namely, z and w*, stand for the boson creation and annihilation operators representatives, respectively. The dynamical equations are obtained by the mapped Heisenberg equation of motion. So, a holomorphic expression for an excitation probability is presented and an explicit calculation is performed for the forced harmonic oscilator. Finally, we apply the holomorphic phase-space description to a particular collision process, i.e.,the analytical study of the Coulomb excitation for the isovector giant electric dipole ressonance in the eikonal approximation. Comparisons with experimental results are presented and discussed
The present work aimed to discuss quantum mechanics in phase space, propose a modification of the Wigner function through the position-dependent translation operator (PDTO) and apply the proposed theory to systems of physical interest. Thus, using a one-dimensional non-Euclidean metric, we construct a modified Wigner function whose form can be expressed in terms of the position-dependent translation operator for the case of finite translations. Through this metric, we define a point transformation that acts as a parameterization that preserves the Euclidean form of the equations. Thus, it was possible to find a modified expression for the Weyl-Wigner quantization map, whose objective is to map functions in phase space into operators in Hilbert space. We show that, when using this modified map to quantize the classical momentum, we obtain the same result as the modified momentum operator found via the position-dependent translation operator. Furthermore, through the inverse operation of the parameterized Weyl-Wigner map, it was possible to define a modified form of the Wigner function, whose mathematical form from the Euclidean case was preserved. We apply the proposed Wigner function to the particle trapped in the potential well and to a quantum oscillator-type system, with the aim of evaluating the influence of the metric on the system dynamics. We plotted the graph of the modified Wigner function for these two systems, where it was possible to verify the appearance of deformations due to the influence of the metric used. Thus, we concluded that the dynamics of the system undergoes a change when the structure of the space is deformed.
Nas últimas décadas constata-se um crescente número de artigos em diversas áreas da Física dedicados a desenvolver e aplicar o formalismo de Weyl-Wigner nos mais diferentes sistemas físicos. A possibilidade de aprofundarmos os nossos conhecimentos sobre o princípio da incerteza e recuperarmos a Mecânica Clássica no limite -> 0, faz com que a descrição da Mecânica Quântica no espaço de fase ganhe cada vez mais adeptos. Nesse sentido, este trabalho tem por finalidade apresentar um conjunto básico de resultados que caracterizam o formalismo em questão, possibilitando ao leitor o acesso às idéias principais decorrentes deste fascinante tema.
The Weyl quantization of classical observables on the torus (as phase space) without regularity assumptions is explicitly computed. The equivalence class of symbols yielding the same Weyl operator is characterized. The Heisenberg equation for the dynamics of general quantum observables is written through the Moyal brackets on the torus and the support of the Wigner transform is characterized. Finally, a dequantization procedure is introduced that applies, for instance, to the Pauli matrices. As a result we obtain the corresponding classical symbols.
Nesta tese revemos alguns aspectos cinemáticos da quantização, através da dualidade Fourier generalizada do grupo canônico associado a um dado espaço de fase.As álgebras de Kac, em termo das quais a dualidade Fourier (linear) de um grupo é obtida, são revistas e decompostas segundo o dual unitário do grupo canônico, o que nos permite tentar uma generalização do formalismo de Weyl-Wigner para tais espaços de fase.Com a introdução de álgebras de Kac projetivas^ as quais demonstramos prover uma dualidade Fourier projetiva do grupo abeliano das translações, é possível mostrar como deduzir o formalismo usual de Weyl-Wigner sobre o espaço euclidiano, a partir de sua intrínseca conexão com uma componente irredutível de tal dualidade.
The Moyal description of quantum mechanics, based on the Wigner-Weyl isomorphism between operators and symbols, provides a comprehensive phase space representation of dynamics. The Weyl symbol image of the Heisenberg picture evolution operator is regular in ħ and so presents a preferred foundation for semiclassical analysis. Its semiclassical expansion "coefficients," acting on symbols that represent observables, are simple, globally defined (phase space) differential operators constructed in terms of the classical flow. The first of two presented methods introduces a cluster-graph expansion for the symbol of an exponentiated operator, which extends Groenewold′s formula for the Weyl product of two symbols and has ħ as its natural small parameter. This Poisson bracket based cluster expansion determines the Jacobi equations for the semiclassical expansion of "quantum trajectories." Their Green function solutions construct the regular ħ ⇓ 0 asymptotic series for the Heisenberg-Weyl evolution map. The second method directly substitutes such a series into the Moyal equation of motion and determines the ħ coefficients recursively. In contrast to the WKB approximation for propagators, the Heisenberg-Weyl description of evolution involves no essential singularity in ħ, no Hamilton-Jacobi equation to solve for the action, and no multiple trajectories, caustics, or Maslov indices.
Neste trabalho estudamos uma das formulações da mecânica quântica no espaço de fases. Para tal utilizamos o formalismo de Moyal, obtido através de uma deformação da álgebra do espaço de fases clássico e a transformação de Wigner, um método que associa funções de posição do espaço de Hilbert à funções densidade de quasiprobabilidade do espaço de fases. Com estas ferramentas, estudamos os casos da partícula livre e do oscilador harmônico. Mostramos por fim que os estados coerentes de Glauber estão intimamente relacionados à equação de Liouville no caso do oscilador harmônico.