Geoscience Reference
In-Depth Information
Let us introduce G
a
.r;R/,
dx
e
x
.x
C
x
0
/
r
x
C
ˇ
j
U.r/
j
.x
C
ˇ
j
U.a
/
j
/
a
2
Z
2
p
G
a
.r;R/
D
r
2
;
ˇjU.R/j
where a
is independent of
E
. Next, we present G
a
.r;R/ in the form
G
a
.r;R/
D
ŒG
a
.r;R/
G
a
.r;R/
C
G
a
.r;R/
We d e fi n e a
from the condition that the expansion of the difference in the square
braces would not contain the term
/
a
2
=r
2
.
Now it is easy to find that
Z
2
p
1
r
2
dx
e
x
.x
C
x
0
/
ŒG
a
.r;R/
G
a
.r;R/
D
ˇ
j
U.R/
j
r
2
.x
C j
U.r
/
j
/
a
2
.x
C j
U.a
/
j
/
q
x
C
ˇ
j
U.r/
j
.x
C
ˇ
j
U.r
/
j
/
r
2
q
x
C
ˇ
j
U.r/
j
.x
C
ˇ
j
U.a
/
j
/
a
2
r
2
C
r
2
Now we neglect all terms contributing additional negative powers of
r
or
R
.We
thus neglect the contribution of the bound states (the lower limit of integration
is replaced by 0), and we neglect the terms containing the ratio a
2
=r
2
in the
denominator. We find then
Z
1
p
1
r
2
dx
e
x
.x
C
x
0
/
ŒG
a
.r;R/
G
a
.r;R/.r;R/
D
ˇ
j
U.R/
j
r
2
.x
C j
U.r
/
j
/
a
2
.x
C j
U.a
/
j
/
p
x
C
ˇ
j
U.r/
j
We can also neglect all terms containing U.r/ and U.R/, for they also introduce
additional negative powers of
R
. We finally find
Z
r
2
U.r
/
e
x
p
x
1
p
dx
D
a
2
U.a
/
0
