Subatomic Physics

296 The Electromagnetic Interaction
With Eqs. (10.59), (10.61), and (10.63), the matrix element, Eq. (10.60), becomes
〈β|Hem|α〉 = −i�2 � �1/2 � � �
e 2π
i(Eβ − Eγ − Eα)t
exp
m EγV
�
� �� �
i(Eβ + Eγ − Eα)t
+exp
ˆɛ·
�
d 3 xΦ ∗ β∇Φα. (10.65)
The two exponential factors that appear in the matrix element behave very differently.
With Eq. (10.64), the first one becomes exp(−2iEγt/�). Perturbation theory
in the form derived in Section 10.1 is valid only if, according to Eq. (10.16), the
time t is large compared to 2π�/Eγ. For such times, the exponential factor is a
very rapidly oscillating function of time. Any observation involves an averaging
over times satisfying Eq. (10.16), and the rapid oscillation wipes out any contribution
to the matrix element from the first term. The second exponential factor is
unity because of energy conservation, Eq. (10.64), and the emission matrix element
becomes
〈β|Hem|α〉 = −i �2 e
m
� �1/2 �
2π
ˆɛ·
EγV
d 3 xΦ ∗ β∇Φα. (10.66)
If a photon is absorbed rather than emitted in the transition |α〉 →|β〉, Eq. (10.64)
reads Eα + Eγ = Eβ. The first exponential in Eq. (10.65) is then unity, and the
second one does not contribute. The transition rate for spontaneous emission is
now obtained with the golden rule, Eq. (10.19), which we write as
dwβα = 2π
� |〈β|Hem|α〉| 2 ρ(Eγ). (10.67)
With pγ = Eγ/c, the density-of-states factor ρ(Eγ) is given by Eq. (10.28) as
ρ(Eγ) = E2 γVdΩ . (10.68)
(2π�c) 3
Here dwβα is the probability per unit time that the photon is emitted with momentum
p γ into the solid angle dΩ. With the matrix element Eq. (10.66), the transition
rate becomes
dwβα = e2Eγ 2πm2 �
|ˆɛ·
c3 d 3 xΦ ∗ β∇Φα| 2 dΩ. (10.69)
If the wave functions Φα and Φβ are known, the transition rate can be computed.
However, the integral containing the wave functions can be changed into a form that
expresses the salient facts more clearly. Assume that the Hamiltonian H0 describing
the decaying system, but not the electromagnetic interaction, is
H0 = p2
+ V (x),
2m