372 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
Calculation of the Two-body T-matrix in Configuration Space
A spectral integral method (IEM) for solving the two-body Schroedinger
equation in configuration space is generalized to the calculation of the
corresponding T-matrix. It is found that the desirable features of the IEM,
such as the economy of mesh-points for a given required accuracy, are carried
over also to the solution of the T-matrix. However the algorithm is
considerably more complex, because the T-matrix is a function of two variables
r and r', rather than only one variable r, and has a slope discontinuity at
r=r'. For a simple exponential potential an accuracy of 7 significant figures
is achieved, with the number N of Chebyshev support points in each partition
equal to 17. For a potential with a large repulsive core, such as the potential
between two He atoms, the accuracy decreases to 4 significant figures, but is
restored to 7 if N is increased to 65.Comment: 22 pages, 1 table 8 figure
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