Geoscience Reference
In-Depth Information
For further needs we also introduce the functions G
o
.r;R/ and G
a
.r;R/
defined as
Z
dx
e
x
p
x
C
ˇ
j
U.r/
j
:
2
p
G
o
.r;R/
D
ˇjU.R/j
Z
2
p
dx
e
x
.r
r
/.x
C
x
0
/
G
a
.r;R/
D
ˇjU.R/j
r
x
C
ˇ
j
U.r/
j
.x
C
ˇ
j
U.r
/
j
/
r
2
r
2
:
Using the identities F.R/
D
G
a
.R;R/ and G
o
.R;R/
D
e
ˇjU.R/jj
yields
n.R/
1
C
e
ˇjU.R/j
F.R/
e
ˇjU.R/j
ŒG
o
.r;R/
C
G
a
.r;R/:
n.r/
D
(3.167)
In the case of repulsion, the conditions L
2
.r/ > 0 and L
2
.a/ > 0 determine
the lower limits of integration in the expression for n.r/. Following the route of
derivation of Eq.
3.164
we obtain
2
2
mˇ
3=2
Z
dx
e
x
p
x
ˇU.r/
1
p
4
C.R/e
ˇU.R/
n.r/
D
ˇU.r/
3
dx
e
x
r
x
ˇU.r/
.x
ˇU.a//
a
2
r
2
Z
5
C
;
ˇU.a/
where, again, C.R/ is given by Eq.
3.165
. The definitions of F.R/, G
o
,andG
a
change. Instead of Eq.
3.166
we have
dx
e
x
r
x
ˇU.R/
.x
ˇU.a//
a
2
Z
2
p
F
rep
.R/
D
R
2
:
ˇU.a/
Next,
Z
dx
e
x
p
x
ˇU.r/
D
e
ˇU.r/
;
2
p
G
o
.r;R/
D
ˇU.r/
