where ρ(H i ) is the density of states with respect to energy. Assuming all final energies H i are close to the initial energy H j (weak
perturbation), we can approximate
H i ≈ H j ≈ H.
(4.135)
Additionally, we approximate the matrix element by a single constant value
i|H ˆ 1 |j ≈ (H) .
(4.136)
As a result, we can bring (H) and ρ(H) outside the integral. Substituting,
P (t) = 4 (H)
2 ρ(H)
∞
−∞
dH ij
H
2
ij
sin
2 H ij t
2¯ h
.
(4.137)
Making the substitution
ξ ≡
H ij t
2¯ h
,
(4.138)
we find
P (t) =
2π
¯
h
ρ(H) (H)
2 t,
(4.139)
where we have made use of the integral
∞
−∞
dξ
ξ 2 sin
2 ξ = π.
(4.140)
The transition rate from a single initial state to all final states is
then
dP
2π
=
ρ(H) (H)
2 ,
(4.141)
dt
h ¯
where the transition rate is the probability per unit time for the
transition from a single initial state to all available final states.
This result is quite general, in that it applies to many diverse
phenomena. It is called the golden rule of perturbation theory. In
the following sections we will proceed to apply it to the scattering
problem.
269
4.5. Perturbation theory