Formulation of the uncertainty relations in terms of the Renyi entropies

Quantum mechanical uncertainty relations for position and momentum are expressed in the form of inequalities involving the Renyi entropies. The proof of these inequalities requires the use of the exact expression for the (p,q)-norm of the Fourier transformation derived by Babenko and Beckner. Analogous uncertainty relations are derived for angle and angular momentum and also for a pair of complementary observables in N-level systems. All these uncertainty relations become more attractive when expressed in terms of the symmetrized Renyi entropies

Pinning and transport of cyclotron/Landau orbits by electromagnetic vortices

Electromagnetic waves with phase defects in the form of vortex lines combined with a constant magnetic field are shown to pin down cyclotron orbits (Landau orbits in the quantum mechanical setting) of charged particles at the location of the vortex. This effect manifests itself in classical theory as a trapping of trajectories and in quantum theory as a Gaussian shape of the localized wave functions. Analytic solutions of the Lorentz equation in the classical case and of the Schr\"odinger or Dirac equations in the quantum case are exhibited that give precise criteria for the localization of the orbits. There is a range of parameters where the localization is destroyed by the parametric resonance. Pinning of orbits allows for their controlled positioning -- they can be transported by the motion of the vortex lines.Comment: This version differs from the printed paper in having the full titles of all referenced pape

Electromagnetic vortex lines riding atop null solutions of the Maxwell equations

New method of introducing vortex lines of the electromagnetic field is outlined. The vortex lines arise when a complex Riemann-Silberstein vector $({\bm E} + i{\bm B})/\sqrt{2}$ is multiplied by a complex scalar function $\phi$. Such a multiplication may lead to new solutions of the Maxwell equations only when the electromagnetic field is null, i.e. when both relativistic invariants vanish. In general, zeroes of the $\phi$ function give rise to electromagnetic vortices. The description of these vortices benefits from the ideas of Penrose, Robinson and Trautman developed in general relativity.Comment: NATO Workshop on Singular Optics 2003 To appear in Journal of Optics

Growth and yield responses of pepper (Capsicum frutescens L.) to varied poultry manure rates in Uyo, southeastern Nigeria

Field trial was carried out during the early cropping seasons of 2007 and 2008 at University of Uyo Teaching and Research Farm, Uyo, Akwa Ibom State, Nigeria to evaluate the effects of poultry manure rates (0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, and 22 t/ha) on growth and yield of pepper (Capsicum frutescens L.). A randomized complete block design with three replicates was used. Results showed that increase in poultry manure rate resulted in increase in growth and yield of pepper up to 8 t/ha rate. It also showed decline in yield with application rate above 8t/ha. Application of 8 t/ha produced 22.75 and 23.56 t/ha of fresh pepper in 2007 and 2008, respectively while 10 t/ha produced 22.70 and 23.91 t/ha of fresh pepper in 2007 and 2008, respectively. This implies that application above 8t/ha rate will not be economical and beneficial to the farmer. Keywords: pepper, poultry manure, growth, yield

Dirac and Weyl Equations on a Lattice as Quantum Cellular Automata

A discretized time evolution of the wave function for a Dirac particle on a cubic lattice is represented by a very simple quantum cellular automaton. In each evolution step the updated value of the wave function at a given site depends only on the values at the nearest sites, the evolution is unitary and preserves chiral symmetry. Moreover, it is shown that the relationship between Dirac particles and cellular automata operating on two component objects on a lattice is indeed very close. Every local and unitary automaton on a cubic lattice, under some natural assumptions, leads in the continuum limit to the Weyl equation. The sum over histories is evaluated and its connection with path integrals and theories of fermions on a lattice is outlined.Comment: 6, RevTe

The Fermi accelerator in atom optics

We study the classical and quantum dynamics of a Fermi accelerator realized by an atom bouncing off a modulated atomic mirror. We find that in a window of the modulation amplitude dynamical localization occurs in both position and momentum. A recent experiment [A. Steane, P. Szriftgiser, P. Desbiolles, and J. Dalibard, Phys. Rev. Lett. {\bf 74}, 4972 (1995)] shows that this system can be implemented experimentally.Comment: 5 pages, 5 figure

ARPN Journal of Agricultural and Biological Science GROWTH AND YIELD RESPONSES OF PEPPER (Capsicum frutescens L.) TO VARIED POULTRY MANURE RATES IN UYO, SOUTHEASTERN NIGERIA

ABSTRACT Field trial was carried out during the early cropping seasons of 2007 and 2008 at University of Uyo Teaching and Research Farm, Uyo, Akwa Ibom State, Nigeria to evaluate the effects of poultry manure rates (0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, and 22 t/ha) on growth and yield of pepper (Capsicum frutescens L.). A randomized complete block design with three replicates was used. Results showed that increase in poultry manure rate resulted in increase in growth and yield of pepper up to 8 t/ha rate. It also showed decline in yield with application rate above 8t/ha. Application of 8 t/ha produced 22.75 and 23.56 t/ha of fresh pepper in 2007 and 2008, respectively while 10 t/ha produced 22.70 and 23.91 t/ha of fresh pepper in 2007 and 2008, respectively. This implies that application above 8t/ha rate will not be economical and beneficial to the farmer