to other states of the same energy.
9.3.2 Radiative transition rates
We now use equation (9.47) to calculate the rate at which an electromagnetic
wave induces an atom to make radiative transitions between its stationary
states. Our treatment is valid when the quantum uncertainty in the electro
magnetic field may be neglected, and the field treated as a classical object.
This condition is satisfied, for example, in a laser, or at the focus of the
antenna of a radio telescope.
Whereas in our derivation of Fermi’s golden rule, the system had some
density of states at the energy of the initial state and we summed the transi
tion probabilities by integrating over the energy of the final state, now we ar
gue that the electromagnetic field has nonnegligible power over a continuum
of frequencies, and we integrate over frequency. The quantity
(
E
k

V
0

E
N
)
2
that occurs in equation (9.47) is now a function of frequency
ω
. We argue
that each singlefrequency contribution to the electromagnetic field induces
transitions independently contributions at other frequencies. Hence equation
(9.47) is valid if we make the substitution
(
E
k

V
0

E
N
)
2
→
d
ω
d
d
ω
(
E
k

V
0

E
N
)
2
,
(9
.
53)
5
The golden rule was actually first given by Dirac, P.A.M., 1927, Proc. Roy. Soc. A,
114
, 243
9.3
Timedependent perturbation theory
193
which isolates the contribution to
(
E
k

V
0

E
N
)
2
from a frequency interval of
infinitesimal width. Once we have evaluated the transition probability
P
k
(
t
)
that is generated by this frequency interval, we set
t
to a large value and
sum over all intervals by integrating with respect to
ω
. That is we evaluate
integraldisplay
d
ωP
k
=
1
¯
h
2
integraldisplay
d
ω
sin
2
(
xt
)
x
2
d
d
ω
(
E
k

V
0

E
N
)
2
,
(9
.
54)
where
x
≡
(¯
hω
+
E
k
−
E
N
)
/
2¯
h
.
We change the integration variable to
x
using d
x
=
1
2
d
ω
and exploit equation (9.51) to evaluate the integral. The
result is
integraldisplay
d
ωP
k
=
2
πt
¯
h
2
d
d
ω
(
E
k

V
0

E
N
)
2
,
with
ω
= (
E
N
−
E
k
)
/
¯
h.
(9
.
55)
The coefficient of
t
on the right is the rate at which the sinusoidally oscil
lating perturbation causes transitions from

E
N
)
to

E
k
)
. This rate vanishes
unless there is a component of the perturbation that oscillates at the angular
frequency (
E
N
−
E
k
)
/
¯
h
that we associate with the energy difference between
the initial and final states. This makes perfect sense physically because we
know from
§
3.2.1 that when there is quantum uncertainty as to whether the
system is in two states that differ in energy by Δ
E
, the system oscillates
physically with angular frequency Δ
E/
¯
h
.
If the system bears electromag
netic charge, these oscillations are liable to transfer energy either into or out
of the oscillations of the ambient electromagnetic field at this frequency.
To proceed further we need an expression for the derivative of the matrix
element in equation (9.55). In vacuo the electric field of an electromagnetic
wave is divergence free, being entirely generated by Faraday’s law,
∇ ×
E
=
−
∂
B
/∂t
. It follows that the whole electromagnetic field of the wave can be
described by the vector potential
A
through the equations
B
=
You've reached the end of your free preview.
Want to read all 277 pages?
 Spring '15
 Unknow
 Physics, mechanics, The Land, probability amplitudes