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PHYS 434: Assignment 7 solutions
Problem 7.10:
With no stimulated emission, the term
B
21
in Equation 7.13 is zero
giving
u
(
f
)
=
A
21
/
B
21
e
hf
/
k
B
T
.
This occurs when the photon energy is much larger than
k
B
T
. In this case, the photon
density is very small, and emission is unlikely to be stimulated before spontaneous
emission occurs. This can be seen from Equation 7.15; the ratio of spontaneous to
stimulated emission becomes large when the frequency is large.
Problem 7.11: Assume that the gas particles are described by a Maxwell Boltzmann
distribution of
energies
:
f
(
E
)
=
Ce
−
E
/
k
B
T
where
C
is a normalization constant.
(a) Normalize the distribution:
f
(
E
)
dE
=
C
0
∞
∫
e
−
E
/
k
B
T
dE
=
C
−
k
B
Te
−
E
/
k
B
T
( )
0
∞
∫
0
∞
=
Ck
B
T
=
1
⇒
C
=
1
k
B
T
(b) Calculate the average energy:
E
=
E f
(
E
)
( )
dE
=
0
∞
∫
E
1
k
B
T
e
−
E
/
k
B
T
dE
0
∞
∫
Integrate by parts:
U
=
E
dV
=
1
k
B
T
e
−
E
/
k
B
T
dU
=
dE
V
=
−
e
−
E
/
k
B
T
giving
E
=
Ee
−
E
/
k
B
T
( )
0
∞
−
−
e
−
E
/
k
B
T
( )
dE
0
∞
∫
=
0
+
e
−
E
/
k
B
T
dE
0
∞
∫
=
−
k
B
Te
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 Winter '09
 Kilfoil
 Energy, Photon

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