An Introductory Guide to Computational Methods for the Solution of Physics Problems by George Rawitscher & Victo dos Santos Filho & Thiago Carvalho Peixoto

Author:George Rawitscher & Victo dos Santos Filho & Thiago Carvalho Peixoto
Language: eng
Format: epub
ISBN: 9783319427034
Publisher: Springer International Publishing

8.3.1 The Iterative Method of Seaton and Peach

The iterative method of Seaton and Peach [1] consists in rewriting Eq. (8.5) in the form

(8.7)

where

(8.8)

and calculating the solution of Eq. (8.7) by means of the iteration [1]

(8.9)

Here n denotes the order of the iteration. The initial value of y is given by the WKB approximation [4, 5]

(8.10)

The advantage of formulating the iteration according to Eq. (8.9) is that y varies slowly with r, automatically and adiabatically approaching unity at large distances, and hence is small compared to w. Near the origin of r this term may become large, and the iterations may not converge. In that case the solution of Eq. (8.9) should be started at a point sufficiently far from the origin, in a region where [ is sufficiently small compared to w, depending on the accuracy desired, as will be described further below. A feature of the present Ph-A method is that no initial conditions are required to be imposed. Since the potential in Eq. (8.8) approaches zero asymptotically, the amplitude y automatically approaches unity, asymptotically. This last property is very important, since the wave function derived from the present Ph-A method automatically also has unit amplitude asymptotically, even though the calculation is not required to be performed to asymptotic distances. This property does not hold for finite element or finite difference calculations. Equation (8.6) combined with the first order result is equivalent to the WKB approximation.

In summary, the iteration scheme (8.9) provides a method to iteratively improve the WKB approximation, since after convergence the resulting wave function is in much better agreement with benchmark wave functions than the WKB result for the numerical cases studied below.

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