Atomic Structure Theory

166 6 Radiative Transitions
d 2 wfi =
V
(2π¯h) 2 c 3 ω2 dΩk|Tfi| 2
ni
,
ni +1
where ni is now the number of photons with energy ¯hω = Ef − Ei for absorption
and ¯hω = Ei − Ef for emission. Factoring −ec ¯h/2ɛ0ωV from the
interaction Hamiltonian, we obtain
d 2 wfi = α
2
ωdΩk|Tfi|
2π
ni
, (6.46)
ni +1
where the single-particle interaction Hamiltonian is now replaced by
h I(r,ω) → α · ˆɛλ e ik·r . (6.47)
Let us assume that we have a collection of atoms in equilibrium with a
radiation field. Further, let us assume that the photons of frequency ω in the
radiation field are distributed isotropically and that the number of photons in
each of the two polarization states is equal. In this case, the photon number
n can be related to the spectral density function ρ(ω), which is defined as the
photon energy per unit volume in the frequency interval dω. One finds from
(6.45) that
ρ(ω) =2× n¯hω × 4πω2 ¯hω3
=
(2πc) 3 π2 n. (6.48)
c3 For isotropic, unpolarized radiation, we can integrate (6.22) over angles Ωk
and sum over polarization states ˆɛλ, treating n as a multiplicative factor. The
resulting absorption probability per second, wa→b, leading from an initial
(lower energy) state a to final (higher energy) state b in presence of n photons
of energy ¯hω, is given in terms of the spectral density function ρ(ω) as
w ab
a→b = π2c3 α
ρ(ω) ω dΩk|Tba|
¯hω3 2π 2 . (6.49)
Similarly, the emission probability per second leading from state b to state a
in the presence of n photons of energy ¯hω, is given in terms of ρ(ω) by
w em
b→a = 1+ π2c3
α
ρ(ω) ω dΩk|Tab|
¯hω3 2π 2 . (6.50)
The emission probability consists of two parts, a spontaneous emission contribution
w sp
b→a that is independent of ρ(ω), and an induced or stimulated
emission contribution, wie that is proportional to ρ(ω).
b→a
Let the state a be a member of a g-fold degenerate group of levels γ, and
b be a member of a g ′ -fold degenerate group of levels γ ′ . If we assume that
the atom can be in any of the degenerate initial levels with equal probability,
then the average transition probability from γ → γ ′ is found by summing over
sublevels a and b and dividing by g; whereas, the average transition probability
λ
λ