Elements of Modern Physics
34
with n x n y and n z being positive integers (negative integers give the same mode).
The wave number and therefore the frequency v = c | k | is obtained from these
relations as
2
2
2 1 / 2
(
)
2
x
y
z
c
c
n
n
n
l
=
+
+
(2.7)
Every possible set of positive integers (n x , n y , n z ) gives a possible standing
wave, which may be depicted by a point in the 3-dimensional plot of (n x , n y , n z ).
Since there is one such point per unit volume, the number of states is essentially
equal to the volume in this space (provided the volume is large). Therefore, the
number of stationary modes with frequency between 0 and v (which corresponds
to the volume in the first octant with n ≤ 2l v/c) is
3
1 4
2
( ) 2 8
3
lv
N v
c
π
=
3 3
3
8
3
l v
c
π
=
(2.8)
where a factor of 2 has been introduced to take into account the fact that for
each frequency v, there are two transverse modes of electromagnetic oscillations.
In the theory of statistical mechanics, the principle of equipartition states
that a mean energy of
1
2
k T (k is the Boltzmann constant which has a value of
1.38 × 10
–23
J K
–1
) is associated with each degree of freedom. For example, for
an ideal gas with molecules treated as geometric points, the mean energy of
each molecule is
3
2
kT corresponding to its translational motion in three
independent directions. However, for a molecule in oscillatory motion.
corresponding to each mode of translational motion there is a potential energy
term which also contributes a mean energy of
1
2
kT . Now, if a mean energy of
kT is assigned to each mode of electromagnetic oscillation, then according to
the principle of equipartition, the energy per unit volume, between frequencies
v and v + dv, is given by
3
( )
( )
( )
=
dN v
u v dv
kT
l
(2.9)
so that energy density per unit volume, per unit frequency is
2
3
8
( )
v kT
u v
c
π
=
(2.10)
34
with n x n y and n z being positive integers (negative integers give the same mode).
The wave number and therefore the frequency v = c | k | is obtained from these
relations as
2
2
2 1 / 2
(
)
2
x
y
z
c
c
n
n
n
l
=
+
+
(2.7)
Every possible set of positive integers (n x , n y , n z ) gives a possible standing
wave, which may be depicted by a point in the 3-dimensional plot of (n x , n y , n z ).
Since there is one such point per unit volume, the number of states is essentially
equal to the volume in this space (provided the volume is large). Therefore, the
number of stationary modes with frequency between 0 and v (which corresponds
to the volume in the first octant with n ≤ 2l v/c) is
3
1 4
2
( ) 2 8
3
lv
N v
c
π
=
3 3
3
8
3
l v
c
π
=
(2.8)
where a factor of 2 has been introduced to take into account the fact that for
each frequency v, there are two transverse modes of electromagnetic oscillations.
In the theory of statistical mechanics, the principle of equipartition states
that a mean energy of
1
2
k T (k is the Boltzmann constant which has a value of
1.38 × 10
–23
J K
–1
) is associated with each degree of freedom. For example, for
an ideal gas with molecules treated as geometric points, the mean energy of
each molecule is
3
2
kT corresponding to its translational motion in three
independent directions. However, for a molecule in oscillatory motion.
corresponding to each mode of translational motion there is a potential energy
term which also contributes a mean energy of
1
2
kT . Now, if a mean energy of
kT is assigned to each mode of electromagnetic oscillation, then according to
the principle of equipartition, the energy per unit volume, between frequencies
v and v + dv, is given by
3
( )
( )
( )
=
dN v
u v dv
kT
l
(2.9)
so that energy density per unit volume, per unit frequency is
2
3
8
( )
v kT
u v
c
π
=
(2.10)