154 research outputs found
An economical method to calculate eigenvalues of the Schroedinger Equation
The method is an extension to negative energies of a spectral integral
equation method to solve the Schroedinger equation, developed previously for
scattering applications. One important innovation is a re-scaling procedure in
order to compensate for the exponential behaviour of the negative energy
Green's function. Another is the need to find approximate energy eigenvalues,
to serve as starting values for a subsequent iteration procedure. In order to
illustrate the new method, the binding energy of the He-He dimer is calculated,
using the He-He TTY potential. In view of the small value of the binding
energy, the wave function has to be calculated out to a distance of 3000 a.u.
Two hundred mesh points were sufficient to obtain an accuracy of three
significant figures for the binding energy, and with 320 mesh points the
accuracy increased to six significant figures. An application to a potential
with two wells separated by a barrier, is also made.Comment: 19 pages, 3 figures, submitted to Eur. J. Phy
Comparison of numerical methods for the calculation of cold atom collisions
Three different numerical techniques for solving a coupled channel
Schroedinger equation are compared. This benchmark equation, which describes
the collision between two ultracold atoms, consists of two channels, each
containing the same diagonal Lennard-Jones potential, one of positive and the
other of negative energy. The coupling potential is of an exponential form. The
methods are i) a recently developed spectral type integral equation method
based on Chebyshev expansions, ii) a finite element expansion, and iii) a
combination of an improved Numerov finite difference method and a Gordon
method. The computing time and the accuracy of the resulting phase shift is
found to be comparable for methods i) and ii), achieving an accuracy of ten
significant figures with a double precision calculation. Method iii) achieves
seven significant figures. The scattering length and effective range are also
obtained.Comment: 22 pages, 3 figures, submitted to J. Comput. Phys. documentstyle
[thmsa,sw20aip]{article} in .te
Inclusion of virtual nuclear excitations in the formulation of the (e,e'N)
A wave-function framework for the theory of the (e,e'N) reaction is presented
in order to justify the use of coupled channel equations in the usual Feynman
matrix element. The overall wave function containing the electron and nucleon
coordinates is expanded in a basis set of eigenstates of the nuclear
Hamiltonian, which contain both bound states as well as continuum states.. The
latter have an ingoing nucleon with a variable momentum Q incident on the
daughter nucleus as a target, with as many outgoing channels as desirable. The
Dirac Eqs. for the electron part of the wave function acquire inhomogeneous
terms, and require the use of distorted electron Green's functions for their
solutions. The condition that the asymptotic wave function contain only the
appropriate momentum Q_k for the outgoing nucleon, which corresponds to the
electron momentum k through energy conservation, is achieved through the use of
the steepest descent saddle point method, commonly used in three-body
calculations.Comment: 30 page
A Novel Method for the Solution of the Schroedinger Eq. in the Presence of Exchange Terms
In the Hartree-Fock approximation the Pauli exclusion principle leads to a
Schroedinger Eq. of an integro-differential form. We describe a new spectral
noniterative method (S-IEM), previously developed for solving the
Lippman-Schwinger integral equation with local potentials, which has now been
extended so as to include the exchange nonlocality. We apply it to the
restricted case of electron-Hydrogen scattering in which the bound electron
remains in the ground state and the incident electron has zero angular
momentum, and we compare the acuracy and economy of the new method to three
other methods. One is a non-iterative solution (NIEM) of the integral equation
as described by Sams and Kouri in 1969. Another is an iterative method
introduced by Kim and Udagawa in 1990 for nuclear physics applications, which
makes an expansion of the solution into an especially favorable basis obtained
by a method of moments. The third one is based on the Singular Value
Decomposition of the exchange term followed by iterations over the remainder.
The S-IEM method turns out to be more accurate by many orders of magnitude than
any of the other three methods described above for the same number of mesh
points.Comment: 29 pages, 4 figures, submitted to Phys. Rev.
Complex Conjugate Pairs in Stationary Sturmians
Sturmian eigenstates specified by stationary scattering boundary conditions
are particularly useful in contexts such as forming simple separable two
nucleon t matrices, and are determined via solution of generalised eigenvalue
equation using real and symmetric matrices. In general, the spectrum of such an
equation may contain complex eigenvalues. But to each complex eigenvalue there
is a corresponding conjugate partner. In studies using realistic
nucleon--nucleon potentials, and in certain positive energy intervals, these
complex conjugated pairs indeed appear in the Sturmian spectrum. However, as we
demonstrate herein, it is possible to recombine the complex conjugate pairs and
corresponding states into a new, sign--definite pair of real quantities with
which to effect separable expansions of the (real) nucleon--nucleon reactance
matrices.Comment: (REVTEX) 8 Pages, Padova DFPD 93/TH/78 and University of Melbourn
- …