332
9 Semiclassical Theory: Path Integrals
σ cs =
(energy/time absorbed by atom in the transition |E ν o →|E ν )
(energy flux of incident radiation field)
.
(9.158)
Let ρ(E ν ) denote the density of final states. The energy flux in the incident radiation
field (energy/area·time) is ω 2
0 |A 0 | 2 /(2μ 0 c), where c is the speed of light and μ 0
is the permeability of free space. Thus, the absorption cross section for incident
radiation with frequency ω 0 can be written
σ cs =
¯
hω 0
dE ν ρ(E ν )w ν o →ν
(ω 2
0 |A 0 | 2 /(2μ 0 c))
=
4μ 0 cπe 2
¯
h
dE ν ρ(E ν ) ω ν,ν o ||E ν |ˆ e·r|E ν o |
2 δ(ω ν,ν o − ω 0 ).
(9.159)
We can express the absorption cross section in terms of the energy Green’s function
if we note that
ρ(E ν )||E ν |ˆ e·r|E ν o |
2
=
ν
δ(E ν − E ν )E ν o |ˆ e·ˆ r|E ν E ν |ˆ e·ˆ r|E ν o
=
1
π
E ν o |ˆ e·ˆ r Im[ ˆ
G + (E ν )] ˆ
e·ˆ r|E ν o
=
1
π
dr
dr
E ν o |r ˆ
e·r (Im[[r| ˆ
G + (E ν )|r
]) ˆ
e·r
r
|E ν o .
(9.160)
Thus, we obtain for the absorption cross section the expression
σ cs =
4μ 0 ce 2
¯
h
dE ν
dr
dr
F ν,ν (r, r
) δ(ω ν,ν o − ω 0 )Im[[r| ˆ
G + (E ν )|r
],
(9.161)
where
F ν,ν (r, r
) = ω ν,ν o E ν o |r ˆ
e·r ˆ
e·r
r
|E ν o .
(9.162)
In Eq. (9.162), we have expressed the absorption cross section directly in terms of
the imaginary part of the energy Green’s function.