Time-dependent perturbation theory 543∫T/ 2
−T/ 2dt[
ei(ωfi+ω)t+ ei(ωfi−ω)t]
=
T
[
sin[(ωfi+ω)T/2]
(ωfi+ω)T/ 2+
sin[(ωfi−ω)T/2]
(ωfi−ω)T/ 2]
. (14.2.39)
Assumingω >0, in the limit of largeT, the second term on the right in eqn. (14.2.39)
dominates over the first and peaks sharply atω=ωfi. Thus, we can retain only this
term and write the mean transition rate as
R(1)fi(T) =1
̄h^2T|F(ω)|^2 |〈Ef|Vˆ|Ei〉|^2sin^2 (ωfi−ω)T/ 2
[(ωfi−ω)T/2]^2. (14.2.40)
Regarding eqn. (14.2.40) as a function ofω, at largeT, this function becomes highly
peaked whenω=ωfibut drops to zero rapidly away fromω=ωfi. The condition
ωfi=ωis equivalent to the conditionEf=Ei+ ̄hω, which is a statement of energy
conservation. Since ̄hωis the energy quantum of the electromagnetic field, also known
as aphoton, the transition can only occur if the energy of the field frequencyωis
exactly “tuned” for the the transition fromEitoEf. Hence, a monochromatic field of
frequencyωcan be used as a probe of the allowed transitions and hence the eigenvalue
structure ofHˆ 0.
We now consider theT→ ∞limit more carefully. Denoting the rate in this limit
simply asRfi, the integral in eqn. (14.2.39) in this limit becomes
lim
T→∞∫T/ 2
−T/ 2dt[
ei(ωfi+ω)t+ ei(ωfi−ω)t]
=
∫∞
−∞dt[
ei(ωfi+ω)t+ ei(ωfi−ω)t]
= 2π[δ(ωfi+ω) +δ(ωfi−ω)]. (14.2.41)Again, forω >0 andωfi>0, only the secondδ-function is ever nonzero, so we can
drop the firstδ-function. Note that the secondδ-function in eqn. (14.2.41) can also
be written as 2π ̄hδ(Ef−Ei− ̄hω). Therefore, the expression for the mean rate in this
limit can be written as
Rfi(ω) = lim
T→∞Pfi(1)(T)
T= lim
T→∞1
T ̄h^2∣
∣
∣
∣
∣
∫T/ 2
−T/ 2dtei(ωfi−ω)t∣
∣
∣
∣
∣
2
|F(ω)|^2∣
∣
∣〈Ef|Vˆ|Ei〉∣
∣
∣
2= lim
T→∞1
T ̄h^2[∫
T/ 2−T/ 2dtei(ωfi−ω)t]
×
[∫
T/ 2−T/ 2dte−i(ωfi−ω)t]
|F(ω)|^2∣
∣
∣〈Ef|Vˆ|Ei〉∣
∣
∣
2
, (14.2.42)where we have dropped the “(1)” superscript (it is understood that the result is de-
rived from first-order perturbation theory) and indicated explicitly the dependence