Problem Set 09
Note:
The following problem set is due November 14 by midnight. Please return directly
to me in my office Rutherford 321. If I’m not there slip your assignment below my door.
1.
Imagine I fill a box with certain number of quantum particles at a temperature
T
.
There is no restriction on energies that the particles can have, but the number of particles
N
(
E
)
dE
lying between energies
E
and
E
+
dE
is restricted to be:
N
(
E
)
dE
=
N
0
E
2
g
(
E/kT
)
dE
where
g
(
x
) is the so called distribution function that is a function of
x
=
E/kT
and
k, N
0
are other constants. Show that the total energy of the particles at a temperature
T
is always proportional to
T
4
no matter what the function
g
is as long as we have the
following bound:
0
<
Z
∞
0
dx x
3
g
(
x
)
<
∞
Considering now the fact the energy of these quantum particles come in packets as
E
=
hν
where
ν
is the frequency, show that the maximum intensity of these particles will be at a
frequency
ν
max
where
ν
max
is given by the following transcendental equation:
ν
max
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 Fall '05
 KeshavDasgupta
 mechanics, Energy, Total Energy, quantum particles, potential steps

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