5,965 research outputs found
Solving simple quaternionic differential equations
The renewed interest in investigating quaternionic quantum mechanics, in
particular tunneling effects, and the recent results on quaternionic
differential operators motivate the study of resolution methods for
quaternionic differential equations. In this paper, by using the real matrix
representation of left/right acting quaternionic operators, we prove existence
and uniqueness for quaternionic initial value problems, discuss the reduction
of order for quaternionic homogeneous differential equations and extend to the
non-commutative case the method of variation of parameters. We also show that
the standard Wronskian cannot uniquely be extended to the quaternionic case.
Nevertheless, the absolute value of the complex Wronskian admits a
non-commutative extension for quaternionic functions of one real variable.
Linear dependence and independence of solutions of homogeneous (right) H-linear
differential equations is then related to this new functional. Our discussion
is, for simplicity, presented for quaternionic second order differential
equations. This involves no loss of generality. Definitions and results can be
readily extended to the n-order case.Comment: 9 pages, AMS-Te
Right eigenvalue equation in quaternionic quantum mechanics
We study the right eigenvalue equation for quaternionic and complex linear
matrix operators defined in n-dimensional quaternionic vector spaces. For
quaternionic linear operators the eigenvalue spectrum consists of n complex
values. For these operators we give a necessary and sufficient condition for
the diagonalization of their quaternionic matrix representations. Our
discussion is also extended to complex linear operators, whose spectrum is
characterized by 2n complex eigenvalues. We show that a consistent analysis of
the eigenvalue problem for complex linear operators requires the choice of a
complex geometry in defining inner products. Finally, we introduce some
examples of the left eigenvalue equations and highlight the main difficulties
in their solution.Comment: 24 pages, AMS-Te
Quaternionic potentials in non-relativistic quantum mechanics
We discuss the Schrodinger equation in presence of quaternionic potentials.
The study is performed analytically as long as it proves possible, when not, we
resort to numerical calculations. The results obtained could be useful to
investigate an underlying quaternionic quantum dynamics in particle physics.
Experimental tests and proposals to observe quaternionic quantum effects by
neutron interferometry are briefly reviewed.Comment: 21 pages, 16 figures (ps), AMS-Te
Potential Scattering in Dirac Field Theory
We develop the potential scattering of a spinor within the context of
perturbation field theory. As an application, we reproduce, up to second order
in the potential, the diffusion results for a potential barrier of quantum
mechanics. An immediate consequence is a simple generalization to arbitrary
potential forms, a feature not possible in quantum mechanics.Comment: 7 page
Quaternionic eigenvalue problem
We discuss the (right) eigenvalue equation for , and
linear quaternionic operators. The possibility to introduce an
isomorphism between these operators and real/complex matrices allows to
translate the quaternionic problem into an {\em equivalent} real or complex
counterpart. Interesting applications are found in solving differential
equations within quaternionic formulations of quantum mechanics.Comment: 13 pages, AMS-Te
Quaternionic Electroweak Theory and CKM Matrix
We find in our quaternionic version of the electroweak theory an apparently
hopeless problem: In going from complex to quaternions, the calculation of the
real-valued parameters of the CKM matrix drastically changes. We aim to explain
this quaternionic puzzle.Comment: 8, Revtex, Int. J. Theor. Phys. (to be published
Dirac Equation Studies in the Tunnelling Energy Zone
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.Comment: 12 pages, 4 figure
Quantum models related to fouled Hamiltonians of the harmonic oscillator
We study a pair of canonoid (fouled) Hamiltonians of the harmonic oscillator
which provide, at the classical level, the same equation of motion as the
conventional Hamiltonian. These Hamiltonians, say and , result
to be explicitly time-dependent and can be expressed as a formal rotation of
two cubic polynomial functions, and , of the canonical variables
(q,p).
We investigate the role of these fouled Hamiltonians at the quantum level.
Adopting a canonical quantization procedure, we construct some quantum models
and analyze the related eigenvalue equations. One of these models is described
by a Hamiltonian admitting infinite self-adjoint extensions, each of them has a
discrete spectrum on the real line. A self-adjoint extension is fixed by
choosing the spectral parameter of the associated eigenvalue
equation equal to zero. The spectral problem is discussed in the context of
three different representations. For , the eigenvalue equation is
exactly solved in all these representations, in which square-integrable
solutions are explicity found. A set of constants of motion corresponding to
these quantum models is also obtained. Furthermore, the algebraic structure
underlying the quantum models is explored. This turns out to be a nonlinear
(quadratic) algebra, which could be applied for the determination of
approximate solutions to the eigenvalue equations.Comment: 24 pages, no figures, accepted for publication on JM
Barrier Paradox in the Klein Zone
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.Comment: 10 page
Easy preparation of liposome@pda microspheres for fast and highly efficient removal of methylene blue from water
Mussel-inspired chemistry was usefully exploited here with the aim of developing a high-efficiency, environmentally friendly material for water remediation. A micro-structured material based on polydopamine (PDA) was obtained by using liposomes as templating agents and was used for the first time as an adsorbent material for the removal of methylene blue (MB) dye from aqueous solutions. Phospholipid liposomes were made by extrusion and coated with PDA by self-polymerization of dopamine under simple and mild conditions. The obtained Liposome@PDA microspheres were characterized by DLS and Zeta potential analysis, TEM microscopy, and FTIR spectroscopy. The effects of pH, temperature, MB concentration, amount of Liposome@PDA, and contact time on the adsorption process were investigated. Results showed that the highest adsorption capacity was obtained in weakly alkaline conditions (pH = 8.0) and that it could reach up to 395.4 mg g−1 at 298 K. In addition, adsorption kinetics showed that the adsorption behavior fits a pseudo-second-order kinetic model well. The equilibrium adsorption data, instead, were well described by Langmuir isotherm. Thermodynamic analysis demonstrated that the adsorption process was endothermic and spontaneous (∆G0 = −12.55 kJ mol−1, ∆H0 = 13.37 kJ mol−1 ) in the investigated experimental conditions. Finally, the applicability of Liposome@PDA microspheres to model wastewater and the excellent reusability after regeneration by removing MB were demonstrated
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