Environmental Engineering Reference
In-Depth Information
If
H
is defined to be the Hamiltonian for the electron, then
p
=
p
+
e
c
A
e
=
md
r
/dt
=
m
i
h
[
H
,
r
]
,
and
Q
p
may be written
.
(4
.
1
.
10)
m
h
2
κ
κ
×
,
r
]+
iκ
Q
p
=
−
j
0
(
ρ
)[
H
2
{
j
0
(
ρ
)+
j
2
(
ρ
)
}
[
H
,
(
κ
·
r
)
r
]+
···
Considering an arbitrary operator
A
,wehave
, A
]
A
A
A
<i
|
[
H
|
f>
=
<i
|H
−
H|
f>
=(
E
i
−
E
f
)
<i
|
|
f>,
which implies that
Q
p
does not contribute to the cross-section (4.1.4)
in the limit
κ
→
0
. In this limit,
j
n
(0) =
δ
n
0
and, utilizing the energy
δ
-function in (4.1.4), the contribution to the cross section due to
Q
p
is
seen to be proportional to
f>
2
m
h
2
κ
hω
κ
×
<i
|
r
|
→
0 or
κ
→
0
,
(
hκ
)
2
/
2
M
. Introducing the vector operator
K
(
since
|
hω
|≤
κ
), defined
so that
<i
|
κ
×
K
×
κ
|
f>
=
<i
|
Q
p
+
Q
s
|
f>,
(4
.
1
.
11)
we find, neglecting
Q
p
in the limit
κ
→
0,
2
µ
B
K
(
0
)=
µ
B
l
+
hc
r
×
A
e
+2
s
e
≡−
µ
e
,
(4
.
1
.
12
a
)
or
H
int
(
0
)=
−
4
π
µ
n
·
(
κ
×
µ
e
×
κ
)
,
(4
.
1
.
12
b
)
implying that the magnetic cross-section (4.1.4), in the limit where the
scattering vector approaches zero, is determined by the magnetic dipole
moment
µ
e
of the electron. In the treatment given above, we have
included the diamagnetic contribution to
µ
e
, induced by external fields
∝
A
e
. This term may however normally be neglected, as we shall do
from now on.
At non-zero
κ
, we cannot employ directly the above procedure for
obtaining an upper bound on the
Q
p
matrix-element, because
j
n
(
ρ
)does
not commute with
. However, if we restrict ourselves to scattering pro-
cesses in which the
l
quantum number is conserved, the matrix element
of the first term in (4.1.10) vanishes identically, because
j
0
(
ρ
)and
H
H
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