Atomic Structure Theory

164 6 Radiative Transitions
U(t, t0) =I − i
t
dt1
¯h t0
ˆ HI(t1)+ (−i)2
¯h 2
t
dt1
t0
ˆ t1
HI(t1) dt2
t0
ˆ HI(t2)U(t2,t0)
∞ (−i)
=
n=0
n
¯h n
t
dt1
t0
ˆ t1
HI(t1) dt2
t0
ˆ tn−1
HI(t2) ··· dtn
t0
ˆ HI(tn) . (6.38)
The S operator is the unitary operator that transforms states prepared in the
remote past (t = −∞), when the interaction HI(t) is assumed to vanish, into
states in the remote future (t = ∞), when HI(t) isalsoassumedtovanish.
Thus
S = U(∞, −∞).
Expanding S in powers of HI, wehave
where
S (n) = (−i)n
¯h n
∞
S = I +
dt1
−∞
ˆ HI(t1)
To first order in HI, wehave
t1
∞
S (n) ,
n=1
dt2
−∞
ˆ HI(t2) ···
∞
tn−1
−∞
dtn ˆ HI(tn) .
S ≈ I − i
dt
¯h −∞
ˆ HI(t). (6.39)
The first-order transition amplitude for a state Φi prepared in the remote past
to evolve into a state Φf in the remote future is
S (1)
fi = 〈Φf |S (1) |Φi〉 = − i
¯h
∞
−∞
6.2.4 Transition Matrix Elements
dt 〈Φ †
f |eiHt/¯h HIe −iHt/¯h |Φi〉 . (6.40)
Let us consider an atom in an initial state Ψi interacting with ni photons. The
initial state is
Φi = Ψi |ni〉.
The operator ci in HI(t) will cause transitions to states with ni − 1 photons,
while the operator c †
i will lead to states with ni + 1 photons. Thus, we must
consider two possibilities:
1. photon absorption, leading to a final state
Φf = Ψf |ni − 1〉, and
2. photon emission, leading to a final state
Φf = Ψf |ni +1〉.