Interaction with External Fields
183
a m (t) =
0
0
0
exp [
) ]
ω − ω
∫
t
m
m
iqE z
t
cos (ωt) dt, m ≠ 0,
(6.46)
where we have used the boundary conditions in Eq. (6.42), and
z m0 =
0
φ ∗ φ τ
∫ m z d
(6.47)
Carrying out the integration over t leads to
a m (t) =
0
0
0
exp [ (
) ] 1
2
ω − ω + ω −
ω − ω − ω
m
m
m
i
t
qE z
0
0
exp [ (
) ] 1
ω − ω − ω −
+
ω − ω + ω
m
m
i
t
(6.48)
For sufficiently large t [i.e., t >> 1/ω m – ω 0 )], the magnitude of the coefficient
is appreciable only for ω ≈ ± (ω 0 – ω m ). The first case, namely
ω ≈ E 0 – E m
(6.49)
corresponds to emission of radiation of energy ω, while the second case,
ω ≈ E m – E 0
(6.50)
corresponds to absorption of radiation of energy ω. In both the cases, the
probability of finding the particle in state φ m at t, is
| a m (t)|
2
≈
2
2 2
2
0
0
2
2
0
sin [(
) / 2]
|
|
(
)
m
m
m
t
q E z
ω − ω
ω − ω
(6.51)
where ω m0 = | ω m – ω 0 | . Since | z m0 | = | z 0m |, it also follows that if the particle is
originally in state m, the probability of finding the particle in state φ 0 at t, | a 0 (t) |
2
,
is given by the same expression as in Eq. (6.51). Thus, the external field E induces
or stimulates transitions 0 → m and transitions m → 0 with the same probability.
This important conclusion may be stated as an equality
P n → m (E) = P m → n (E)
(6.52)
for induced transition probabilities. Another important fact to be noted is that
transitions are allowed between states which do not conserve energy, i.e.,
| E m – E 0 | ≠ ω. However, for t >> 1/ω m0 , the probability |a m (t) |
2
is significant
only for ω m0 – ω ~ 1/t, (for t → ∞ it is proportional to t
2
), and rapidly goes to
small values as | ω m0 – ω | becomes larger than 1/t. Thus energy conservation is
applicable to the extent of the uncertainty
(∆E) (t) ~
(6.53)
which is a manifestation of the uncertainty relation discussed in Eq. (3.64). For
t → ∞, the uncertainty (ω m0 – ω) tends to zero which essentially restores the
energy conservation relation | E m – E 0 | = ω.
In the analysis so far, it has been assumed that the incident radiation has
only one frequency. In reality, it has a frequency distribution. The changes
183
a m (t) =
0
0
0
exp [
) ]
ω − ω
∫
t
m
m
iqE z
t
cos (ωt) dt, m ≠ 0,
(6.46)
where we have used the boundary conditions in Eq. (6.42), and
z m0 =
0
φ ∗ φ τ
∫ m z d
(6.47)
Carrying out the integration over t leads to
a m (t) =
0
0
0
exp [ (
) ] 1
2
ω − ω + ω −
ω − ω − ω
m
m
m
i
t
qE z
0
0
exp [ (
) ] 1
ω − ω − ω −
+
ω − ω + ω
m
m
i
t
(6.48)
For sufficiently large t [i.e., t >> 1/ω m – ω 0 )], the magnitude of the coefficient
is appreciable only for ω ≈ ± (ω 0 – ω m ). The first case, namely
ω ≈ E 0 – E m
(6.49)
corresponds to emission of radiation of energy ω, while the second case,
ω ≈ E m – E 0
(6.50)
corresponds to absorption of radiation of energy ω. In both the cases, the
probability of finding the particle in state φ m at t, is
| a m (t)|
2
≈
2
2 2
2
0
0
2
2
0
sin [(
) / 2]
|
|
(
)
m
m
m
t
q E z
ω − ω
ω − ω
(6.51)
where ω m0 = | ω m – ω 0 | . Since | z m0 | = | z 0m |, it also follows that if the particle is
originally in state m, the probability of finding the particle in state φ 0 at t, | a 0 (t) |
2
,
is given by the same expression as in Eq. (6.51). Thus, the external field E induces
or stimulates transitions 0 → m and transitions m → 0 with the same probability.
This important conclusion may be stated as an equality
P n → m (E) = P m → n (E)
(6.52)
for induced transition probabilities. Another important fact to be noted is that
transitions are allowed between states which do not conserve energy, i.e.,
| E m – E 0 | ≠ ω. However, for t >> 1/ω m0 , the probability |a m (t) |
2
is significant
only for ω m0 – ω ~ 1/t, (for t → ∞ it is proportional to t
2
), and rapidly goes to
small values as | ω m0 – ω | becomes larger than 1/t. Thus energy conservation is
applicable to the extent of the uncertainty
(∆E) (t) ~
(6.53)
which is a manifestation of the uncertainty relation discussed in Eq. (3.64). For
t → ∞, the uncertainty (ω m0 – ω) tends to zero which essentially restores the
energy conservation relation | E m – E 0 | = ω.
In the analysis so far, it has been assumed that the incident radiation has
only one frequency. In reality, it has a frequency distribution. The changes