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Construction of the second trial functionDate: 2015-10-07; view: 468. To construct the second trial function χ (x) introduced in (4.9), we define f± (x) by
and
To retain flexibility it is convenient to impose only the boundary condition (4.11) first, but not (4.10); i.e., at x = 0
but leaving the choice of the overall normalization of χ+ (0) and χ− (0) to be decided later. We rewrite the Schroedinger equations Figs. (4.12) and (4.13) in their equivalent forms
and
where
and
Because at x = 0,
So far, the overall normalization of f+ (x) and f− (x) are still free. We may choose
From Figs. (4.6a), (4.47a), (4.49a) and (4.50a), we see that f+ (x) satisfies the integral equation (for x
Furthermore, from Figs. (4.6a) and (4.49a), we also have
Likewise, f− (x) satisfies (for x
and
The function f+ (x) and f− (x) will be solved through the iterative process described in Section 1. We introduce the sequences
for x
for x
and
Thus, Figs. (4.55a) and (4.55b) can also be written in their equivalent expressions
for x
for x
through induction and by using Figs. (4.55a), (4.55b), (4.56a) and (4.56b), we derive all subsequent
at all finite and positive x, and
at all finite and negative x. Since
and
with both v± (0) finite,
both exist. Furthermore, by using the integral equations Figs. (4.55a) and (4.55b) for
also exist. This leads us from the first trial function
and
with
and the boundary conditions at x = 0,
An additional normalization factor multiplying, say, f− (x) would enable us to construct the second trial function χ (x) that satisfies Figs. (4.9), (4.10), (4.11), (4.12) and (4.13).
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