A family of high-order multistep methods with vanished phase-lag and its derivatives for the numerical solution of the Schrödinger equation

AbstractMany simulation algorithms (chemical reaction systems, differential systems arising from the modelling of transient behaviour in the process industries etc.) contain the numerical solution of systems of differential equations. For the efficient solution of the above mentioned problems, linear multistep methods or Runge–Kutta single-step methods are used. For the simulation of chemical procedures the radial Schrödinger equation is used frequently. In the present paper we will study a class of linear multistep methods. More specifically, the purpose of this paper is to develop an efficient algorithm for the approximate solution of the radial Schrödinger equation and related problems. This algorithm belongs in the category of the multistep methods. In order to produce an efficient multistep method the phase-lag property and its derivatives are used. Hence the main result of this paper is the development of an efficient multistep method for the numerical solution of systems of ordinary differential equations with oscillating or periodical solutions. The reason of their efficiency, as the analysis proved, is that the phase-lag and its derivatives are eliminated. Another reason of the efficiency of the new obtained methods is that they have high algebraic orde

Probing Quantum Optical Excitations with Fast Electrons

Probing optical excitations with nanometer resolution is important for understanding their dynamics and interactions down to the atomic scale. Electron microscopes currently offer the unparalleled ability of rendering spatially-resolved electron spectra with combined meV and sub-nm resolution, while the use of ultrafast optical pulses enables fs temporal resolution and exposure of the electrons to ultraintense confined optical fields. Here, we theoretically investigate fundamental aspects of the interaction of fast electrons with localized optical modes that are made possible by these advances. We use a quantum-optics description of the optical field to predict that the resulting electron spectra strongly depend on the statistics of the sample excitations (bosonic or fermionic) and their population (Fock, coherent, or thermal), whose autocorrelation functions are directly retrieved from the ratios of electron gain intensities. We further explore feasible experimental scenarios to probe the quantum characteristics of the sampled excitations and their populations.Comment: 13 pages, 6 figures, 56 reference

Photon-induced near-field electron microscopy (PINEM): theoretical and experimental

Electron imaging in space and time is achieved in microscopy with timed (near relativistic) electron packets of picometer wavelength coincident with light pulses of femtosecond duration. The photons (with an energy of a few electronvolts) are used to impulsively heat or excite the specimen so that the evolution of structures from their nonequilibrium state can be followed in real time. As such, and at relatively low fluences, there is no interaction between the electrons and the photons; certainly that is the case in vacuum because energy–momentum conservation is not possible. In the presence of nanostructures and at higher fluences, energy–momentum conservation is possible and the electron packet can either gain or lose light quanta. Recently, it was reported that, when only electrons with gained energy are filtered, near-field imaging enables the visualization of nanoscale particles and interfaces with enhanced contrast (Barwick et al 2009 Nature 462 902). To explore a variety of applications, it is important to express, through analytical formulation, the key parameters involved in this photon-induced near-field electron microscopy (PINEM) and to predict the associated phenomena of, e.g., forty-photon absorption by the electron packet. In this paper, we give an account of the theoretical and experimental results of PINEM. In particular, the time-dependent quantum solution for ultrafast electron packets in the nanostructure scattered electromagnetic (near) field is solved in the high kinetic energy limit to obtain the evolution of the incident electron packet into a superposition of discrete momentum wavelets. The characteristic length and time scales of the halo of electron–photon coupling are discussed in the framework of Rayleigh and Mie scatterings, providing the dependence of the PINEM effect on size, polarization, material and spatiotemporal localization. We also provide a simple classical description that is based on features of plasmonics. A major part of this paper is devoted to the comparisons between the theoretical results and the recently obtained experimental findings about the imaging of materials and biological systems

Studies of Meson Mass Spectra in the Context of Quark-Antiquark Bound States

This dissertation deals with the computation of meson mass spectra in the context of quarkantiquark (q ¯ q) bound-state. Traditionally the q ¯ q bound-state problem is treated by solving the non-relativistic Schrödinger equation in position representation with a linear confining potential and a Coulomb-like attractive potential. For high energy, relativistic kinematics is necessary. It is well known that relativistic kinematics cannot be treated properly in position representation, but it can easily be handled in momentum representation. On the other hand, the linear potential and Coulomb-like potential have singularities in momentum-space and complicated subtraction procedure is necessary to treat the singularities properly. In order to deal with the double conflict, we have developed a method to solve any Schrödinger-like wave equation with/without relativistic kinematics in the mixed-space representation. In this representation, the kinematic term is treated in momentum-space and the potential term is treated in positionspace. The results obtained from the mixed representation are in excellent agreement with the results obtained from the position-space and momentum space representations of the nonrelativistic Schrödinger equation without the spin-dependent terms in potential. The success of our computational scheme encouraged us to extend the investigation towards relativistic treatment of the mesonic systems along with the spin-dependent interactions in potential. We have included relativistic kinematics and spin-dependent potentials along with the regular linear and Coulomb-type potentials in our equation. Our predicted results of meson masses are in excellent agreement with the experimental data

Asymptotic analysis of noisy fitness maximization, applied to metabolism and growth

We consider a population dynamics model coupling cell growth to a diffusion in the space of metabolic phenotypes as it can be obtained from realistic constraints-based modelling. In the asymptotic regime of slow diffusion, that coincides with the relevant experimental range, the resulting non-linear Fokker-Planck equation is solved for the steady state in the WKB approximation that maps it into the ground state of a quantum particle in an Airy potential plus a centrifugal term. We retrieve scaling laws for growth rate fluctuations and time response with respect to the distance from the maximum growth rate suggesting that suboptimal populations can have a faster response to perturbations.Comment: 24 pages, 6 figure