(Extended) Numerov method for computing eigenvalues of specific Schrodinger equations

Fack, Veerle; Berghe, Guido Vanden

doi:10.1088/0305-4470/20/13/022

Cited by 61 publications

(33 citation statements)

References 15 publications

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“…Phase shifts were calculated using standard Numerov methods (Fack & Berghe 1999) to numerically solve the radial Shrödinger equation up to the asymptotic zone (kr 1) where it was matched to the required asymptotic wave function (Zhang et al 2009;Landau et al 2007). Convergence of Equation (1) was determined as the lth PW for which the phase shift δ l became numerically insignificant in the amplitude summation.…”

Section: Methodsmentioning

confidence: 99%

Energy Relaxation of Helium Atoms in Astrophysical Gases

Lewkow

Kharchenko

Zhang

2012

ApJ

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We report accurate parameters describing energy relaxation of He atoms in atomic gases, important for astrophysics and atmospheric science. Collisional energy exchange between helium atoms and atomic constituents of the interstellar gas, heliosphere, and upper planetary atmosphere has been investigated. Energy transfer rates, number of collisions required for thermalization, energy distributions of recoil atoms, and other major parameters of energy relaxation for fast He atoms in thermal H, He, and O gases have been computed in a broad interval of energies from 10 meV to 10 keV. This energy interval is important for astrophysical applications involving the energy deposition of energetic atoms and ions into atmospheres of planets and exoplanets, atmospheric evolution, and analysis of non-equilibrium processes in the interstellar gas and heliosphere. Angular-and energy-dependent cross sections, required for an accurate description of the momentum-energy transfer, are obtained using ab initio interaction potentials and quantum mechanical calculations for scattering processes. Calculation methods used include partial wave analysis for collisional energies below 2 keV and the eikonal approximation at energies higher than 100 eV, keeping a significant energy region of overlap, 0.1-2 keV, between these two methods for their mutual verification. The partial wave method and the eikonal approximation excellently match results obtained with each other as well as experimental data, providing reliable cross sections in the astrophysically important interval of energies from 10 meV to 10 keV. Analytical formulae, interpolating obtained energy-and angular-dependent cross sections, are presented to simplify potential applications of the reported database. Thermalization of fast He atoms in the interstellar gas and energy relaxation of hot He and O atoms in the upper atmosphere of Mars are considered as illustrative examples of potential applications of the new database.

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Section: Methodsmentioning

confidence: 99%

Energy Relaxation of Helium Atoms in Astrophysical Gases

Lewkow

Kharchenko

Zhang

2012

ApJ

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“…The appropriate identical particle treatments were utilized for collisions in astrophysical environments including H+H, termining the partial wave phase shifts δ l for a given interaction potential to obtain differential and total cross sections. Phase shifts were found numerically using Numerov's method [14]. Numerov's method is suited for numerically solving differential equations of the form…”

Section: Partial Wave Scatteringmentioning

confidence: 99%

“…Numerov's method is a 4 th order method so that the neglected terms are of order h 6 [14]. For a given collision energy, E, Eq.…”

Section: Partial Wave Scatteringmentioning

confidence: 99%

Scattering of Particles and Radiation in Astrophysical Environments

Lewkow¹

2016

Springer Theses

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The scattering of atoms, molecules, and radiation are the most fundamental mechanisms of energy-momentum transfer in nature. Fluxes of particles and radiation may deposit energy into a system as well as be emitted, giving deep insight into fundamental processes occurring within the system. Astrophysics, in particular, may benefit from collisional modeling and transport as remote viewing of distant objects is the only option for the vast majority of astrophysical objects. Practically all interaction processes between atoms, molecules, and radiation require a quantum mechanical description. Using well known quantum analysis as well as newly developed methods, scattering parameters have been obtained over important astrophysical energy ranges not previously investigated. The newly calculated parameters have been used to model transport, collisional heating, and thermalization in a variety of astrophysical systems including the atmosphere of Mars, cometary atmospheres, and the interstellar gas and plasmas. Results of the collisional simulations allow for better understanding of current observational data as well as predictions for next generation in situ measurements.

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“…Numerov's method [29][30][31] is an implicit method for approximating the solution of the second order differential equation of the form Eq. (1).…”

Section: Energy Eigenvalues and Radial Wave Functionsmentioning

confidence: 99%

New analytical independent-particle model potential for atoms

Yu¹,

Sun

2010

Open Physics

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Abstract:A new analytical, independent-particle model potential with four shell-independent parameters is proposed, which is suitable for high, medium, and low Z atoms. The four parameters are determined for 101 atoms from Li to Lr by fitting the results of the X method found in the literature. The average fitting error 0.675% of the new potential for the 101 atoms is far better than 3.92% of the widely used Green's potential. The radial Schrödinger equation with the new potential is solved by using Numerov's numerical method for 7 typical atoms: Ne, Ca, Zn, Zr, Sn, Yb, and Th. The energy eigenvalues, radial wave functions, and atomic ground-state energy are in good agreement with the results of the X method. The new potential here shows greater flexibility and better accuracy compared with the Green's potential. PACS (2008):31.15.B-. Keywords:atomic structure • independent-particle model • energy eigenvalues © Versita Sp. z o.o.

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(Extended) Numerov method for computing eigenvalues of specific Schrodinger equations

Cited by 61 publications

References 15 publications

Energy Relaxation of Helium Atoms in Astrophysical Gases

Energy Relaxation of Helium Atoms in Astrophysical Gases

Scattering of Particles and Radiation in Astrophysical Environments

New analytical independent-particle model potential for atoms

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