T
{
¼ T:
ð9:45Þ
Thus, we get
B 21
Ã
¼ B 12 :
ð9:46Þ
But, as in the cases of Sects. 4.2 and 4.3, ψ 1 and ψ 2 can be represented as real
functions. Then, we have
B 21
Ã
¼ B 21 ¼ B 12 :
That is, we assume that the matrix B is real symmetric. In the case of two-level
atoms, as a matrix form we get
B ¼
0 B 12
B 12 0
:
ð9:47Þ
Compare (9.47) with (4.28).
Now, in the thermal equilibrium, we have
W e ¼ W a :
ð9:48Þ
That is,
N 2 B 21 ρ ω 21
ð ÞþN 2 A 12 ¼ N 1 B 21 ρ ω 21
ð Þ,
ð9:49Þ
where we used (9.41) for LHS. Assuming Boltzmann distribution law, we get
N 2
N 1
¼ exp À E 2 À E 1
ð
Þ =k B T
½
Š :
ð9:50Þ
Here if moreover we assume (9.38), we get
N 2
N 1
¼ exp Àħω 21 =k B T
ð
Þ :
ð9:51Þ
Combing (9.49) and (9.51), we have
exp Àħω 21 =k B T
ð
Þ¼
B 21 ρ ω 21
ð Þ
B 21 ρ ω 21
ð ÞþA 12
:
ð9:52Þ
Solving (9.52) with respect to ρ(ω 12 ), we finally get
348
9 Light Quanta: Radiation and Absorption