arXiv subjects
Publications and source records attributed to Stefano De Leo.
We propose an experiment involving a gaussian laser tunneling through a twin barrier dielectric structure. Of particular interest are the conditions upon the incident angle for resonance to occur. We provide some numerical calculations for a particular choice of laser wave length and dielectric refractive index which confirm our expectations.
The particle approach to one-dimensional potential scattering is applied to non relativistic tunnelling between two, three and four identical barriers. We demonstrate as expected that the infinite sum of particle contributions yield the plane wave results. In particular, the existence of resonance/transparency for twin tunnelling in the wave limit is immediately obvious. The known resonances for three and four barriers are also derived. The transition from the wave limit to the particle limit is exhibit numerically.
In this paper, we analyze, by using a matrix approach, the dynamics of a non-relativistic particle in presence of a quaternionic potential barrier. The matrix method used to solve the quaternionic Schrodinger equation allows to obtain a closed formula for the transmission coefficient. Up to now, in quaternionic quantum mechanics, almost every discussion on the dynamics of non-relativistic particle was motived 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.
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.
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.
Some optical experiments provide the easiest way to test quantum mechanical predictions. Such a situation applies to a laser beam traversing a dielectric structure. The dielectric structure mir- roring quantum mechanical potentials. The simplest of these, studied in this paper, involves a single dielectric block. We exhibit both by analytic and numerical calculations explicit examples of total coherence and total incoherence phenomena. Which case appears depends upon the angle of incidence of the laser beam and/or the dimensions of beam vs block width. Unlike our previ- ous studies our calculations here are three-dimensional, although, with suitable approximations, somewhat simpler planar expressions can be derived.
We present the results of the planar diffusion of a Dirac particle by step and barrier potentials, when the incoming wave impinges at an arbitrary angle with the potential. Except for right-angle incidence this process is characterized by the appearance of spin flip terms. For the step potential, spin flip occurs for both transmitted and reflected waves. However, we find no spin flip in the transmitted barrier result. This is surprising because the barrier result may be derived directly from a two-step calculation. We demonstrate that the spin flip cancellation indeed occurs for each particle (wave packet) contribution.
Recalling the similarities between the Maxwell equations for a transverse electric wave in a stratified medium and the quantum mechanical Schroedinger equation in a piece-wise potential, we investigate the analog of the so called particle limit in quantum mechanics. It is shown that in this limit the resonance phenomena are lost since individual reflection and transmission terms no longer overlap. The result is a stationary zebra-like response with the intensity in each stripe calculable.
The condition for the existence of a bound state between two fermions exchanging massive scalars is derived. For low scalar mass, we reproduce the scalar field model result. The high scalar mass result exhibits a somewhat different inequality condition.
Ladder diagrams are relevant for the study of bound states. The condition upon the coupling strength for the existence of a bound state has been deduced in a scalar field theory for the case of low mass exchanges. We apply this approach to the case of very high mass exchanges.
We compare the behavior of a wave packet in the presence of a complex and a pure quaternionic potential step. This analysis, done for a gaussian convolution function, sheds new light on the possibility to recognize quaternionic deviations from standard quantum mechanics.
We investigate the tunnelling zone V0 < E < V0+m for a one-dimensional potential within the Dirac equation. We find the appearance of superluminal transit times akin to the Hartman effect.
In looking for qualitative differences between quaternionic and complex formulations of quantum physical theories, we provide a detailed discussion of the behavior of a wave packet in presence of a quaternionic time-independent potential step. In this paper, we restrict our attention to diffusion phenomena. For the group velocity of the wave packet moving in the potential region and for the reflection and transmission times, the study shows a striking difference between the complex and quaternionic formulations which could be matter of further theoretical discussions and could represent the starting point for a possible experimental investigation.
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 eigen-vectors 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.
By using the recent mathematical tools developed in quaternionic differential operator theory, we solve the Schroedinger equation in presence of a quaternionic step potential. The analytic solution for the stationary states allows to explicitly show the qualitative and quantitative differences between this quaternionic quantum dynamical system and its complex counterpart. A brief discussion on reflected and transmitted times, performed by using the stationary phase method, and its implication on the experimental evidence for deviations of standard quantum mechanics is also presented. The analytic solution given in this paper represents a fundamental mathematical tool to find an analytic approximation to the quaternionic barrier problem (up to now solved by numerical method).
We study the solutions for a one-dimensional electrostatic potential in the Dirac equation when the incoming wave packet exhibits the Klein paradox (pair production). With a barrier potential we demonstrate the existence of multiple reflections (and transmissions). The antiparticle solutions which are necessarily localized within the barrier region create new pairs with each reflection at the potential walls. Consequently we encounter a new paradox for the barrier because successive outgoing wave amplitudes grow geometrically.
We extend an above barrier analysis made with the Schrodinger equation to the Dirac equation. We demonstrate the perfect agreement between the barrier results and back to back steps. This implies the existence of multiple (indeed infinite) reflected and transmitted wave packets. These packets may be well separated in space or partially overlap. In the latter case interference effects can occur. For the extreme case of total overlap we encounter resonances. The conditions under which resonance phenomena can be observed is discussed and illustrated by numerical calculations.
We report about recent results on Dirac wave packets in the treatment of neutrino flavor oscillation where the initial localization of a spinor state implies an interference between positive and negative energy components of mass-eigenstate wave packets. A satisfactory description of fermionic particles requires the use of the Dirac equation as evolution equation for the mass-eigenstates. In this context, a new flavor conversion formula can be obtained when the effects of chiral oscillation are taken into account. Our study leads to the conclusion that the fermionic nature of the particles, where chiral oscillations and the interference between positive and negative frequency components of mass-eigenstate wave packets are implicitly assumed, modifies the standard oscillation probability. Nevertheless, for ultra-relativistic particles and sharply peaked momentum distributions, we can analytically demonstrate that these modifications introduce correction factors proportional to (m12/p0) square which are practically un-detectable by any experimental analysis