We propose an analytical study of relativistic tunneling through opaque barriers. We obtain a closed formula for the phase time. This formula is in excellent agreement with the numerical simulations and corrects the standard formula obtained by the stationary phase method. An important result is found when the upper limit of the incoming energy distribution coincides with the upper limit of the tunneling zone. In this case, the phase time is proportional to the barrier width.Comment: 11 pages, 3 figure

In this paper, we analyze, by using a matrix approach, the dynamics of a nonrelativistic particle in presence of a quaternionic potential barrier. The matrix method used to solve the quaternionic Schrödinger equation allows us to obtain a closed formula for the transmission coefficient. Up to now, in quaternionic quantum mechanics, almost every discussion on the dynamics of nonrelativistic particle was motivated by or evolved from numerical studies. A closed formula for the transmission coefficient stimulates an analysis of qualitative differences between complex and quaternionic quantum mechanics and by using the stationary phase method, gives the possibility to discuss transmission times. C 2010 American Institute of Physics.

Abstract. The purpose of this paper is to show how the problem of finding the zeros of unilateral n-order quaternionic polynomials can be solved by determining the eigenvectors of the corresponding companion matrix. This approach, probably superfluous in the case of quadratic equations for which a closed formula can be given, becomes truly useful for (unilateral) n-order polynomials. To understand the strehgth of this method, we compare it with the Niven algorithm and show where this (full) matrix approach improves previous methods based on the use of the Niven algorithm. For the convenience of the readers, we explicitly solve some examples of second and third order unilateral quaternionic polynomials. The leading idea of the practical solution method proposed in this work can be summarized in following three steps: translating the quaternionic polynomial in the eigenvalue problem for its companion matrix, finding its eigenvectors, and, finally, giving the quaternionic solution of the unilateral polynomial in terms of the components of such eigenvectors. A brief discussion on bilateral quaternionic quadratic equations is also presented.

After a brief review of the derivation of the standard phase time formula, based on the use of the stationary phase method, we propose, in the opaque limit, an alternative method to calculate the phase time. The new formula for the phase time is in excellent agreement with the numerical simulations and shows that for wave packets whose upper limit of the momentum distribution is very close to the barrier height, the transit time is proportional to the barrier width.

RESUMOA partir do tratado Sobre o Belo (I, 6), de Plotino, é discutida no presente artigo a afinidade entre as vias estética e intelectiva como modo de acesso à união com o Um. Para tanto, propomos que o Belo seja entendido como um esplendor do Um, ou seja, como o que se manifesta concomitante e dependente do Um, atraindo para este o olhar daquele que empreende a ascensão filosófica. Assim, procuramos demonstrar que o Belo se fundamenta na intelecção da unidade, e que a ascensão ao Um é tanto estética quanto intelectiva. PALAVRAS-CHAVE: Estética. Alma. Experiência mística. Ascensão. Plotino. ABSTRACTFrom the treaty On Beauty (I, 6), by Plotinus, the affinity between the aesthetic and intellectual paths as a way of accessing the union with the One is discussed in this article. For that, we propose that the Beauty be understood as a splendor of the One, that is to say, as that one who manifests concomitant and dependent on the One, attracting to the latter the look of one who undertakes the philosophical ascension. Thus, we seek to demonstrate that the Beauty is based on the intellection of unity, and that the ascension to the One is both aesthetic and intellectual. KEYWORDS: Aesthetic. Soul. Mystical experience. Ascension. Plotinus.No tratado I, 6 de Plotino, Sobre o Belo, o autor retoma importantes noções platônicas tais como encontradas, por exemplo, no Banquete e na República, elaborando, a seu modo, um tratado sobre o Belo 1 que o coloca em estreita relação com a ética, a epistemologia e a metafísica platônica em geral, especialmente com o princípio do Um 2a hipóstase  Doutorado em Filosofia. Universidade Federal do Paraná. a casos particulares que não dizem respeito ao Belo em geral. Um dos motivos para isso é que, em Plotino, os sentidos geral e específico se interseccionam e, além disso, o Belo não é uma hipóstase, tal como o Intelecto ou a Alma, para os quais os sentidos geral e específico podem ser mais bem diferenciados. Quando se tratar de um "belo" notadamente particular, falaremos em "beleza". No entanto, nas citações usadas, encontramos "belo" e "bem", mesmo quando em sentido geral, o que obviamente mantemos por respeito à escolha dos tradutores. 2 Escolhemos aqui nomeá-lo como "Um", e não "Uno", já que "Uno" pode conduzir semanticamente à compreensão de que se trata de algo que possui substância passível de receber a qualificação de "una". Não se trata disso, porque o Um não é um ente. Para tentar evitar esse mal entendido, consideramos que "Um" seja mais apropriado.