Image Processing Reference
In-Depth Information
In general:
1
e
rr
iK rr
− ′
Grr
(, )
Grr
( ,)
=
+
(2.57)
− ′
We note that an equation of the form ∇− ∂∂=
ψ
(
1
/
v
)(
ψ
/
t
)
0
(where
2
2
2
2
∇=∂∂
ψψ
(
/ x
))
is solved by setting x vt = p and x + vt = q ; therefore
2
2
2
∂∂∂=
ψ/ pq
0
. Having the solution ψ = ψ 1 ( p ) + ψ 2 ( q ) = ψ 1 ( x vt ) + ψ 2 ( x + vt ) =
2
+ + , we see that we have, as expected, an ingoing and an outgoing wave.
From this it can be written
ψψ
e
±−
iK rr
ψ
()
r
=
ϕ
()
r
rr Ur
() ()
ψ
r
d 3
r
±
±
(2.58)
and this is an inhomogeneous Fredholm type of equation of the second kind.
If U ( r ′)ψ + ( r ′) decreases rapidly as | r ′| → ∞ (e.g., if it vanishes for | r ′| > R ), then
letting | r | → ∞(which is equivalent to | r ′| → 0) leads to
1
4
e
iKr
ψ
+
()
r
→−
ϕ
()
r
Ur
() ()
ψ
+
r
d
3
r
π
r
constant
1
e
iKr
(2.59)
→+
ϕ
()
r
C
r
rC e
r
iKr
→+
ϕ
()
where C is the scattering amplitude.
Similarly, it follows that
C e
iKr
(2.60)
ψ
()
r
→+
ϕ
()
r
r
At | r | → ∞, by looking at the time dependence and locus of the phase, it
can be determined whether we have an ingoing or outgoing wave; for example,
with Schrödinger's equation,
ψ
(,)
rt
=
ψ
( )
re
iE
(/)
t
=
ψ
( )
re
i
ω
t
Therefore
Ce
iKr
(
ω
t
)
(2.61)
ψ
+
(,)
rt
=
e
iKr
(
ω
t
)
+
r
For ψ + one can see that Kr increases as ω t increases, corresponding to an
outgoing wave (i.e., its constant phase front moves out). This would typically
be written as
e
iKr
(2.62)
ψψ
+ =
()
r
=
e
fiKz
+
f
(, )
θϕ
r

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