22 Wave–particle duality and its manifestation
This volume is often referred to as a unit or primitive cell. The wavenumber of
such a standing wave, k =|k|, is defined as
k =
k
2
x
+ k
2
y
+ k
2
z
= π
n
2
x
L
2
x
+
n
2
y
L
2
y
+
n
2
z
L
2
z
. (2.7)
Blackbody radiation in a rectangular cavity can be considered as a sum of standing
electromagnetic waves with different wavelengths and frequencies. Their values
are defined by the set of numbers n
α
and L
α
. Let us define the number of standing
waves in the cavity with wavenumbers less than the given value of k. For this
purpose let us select in k-space a sphere of radius k with volume V
k
:
V
k
=
4πk
3
3
. (2.8)
Each point inside this sphere, (k
x
, k
y
, k
z
), or more precisely each primitive cell,
corresponds to two independent standing waves with fixed frequencies and with
orthogonal polarizations. For positive numbers n
α
all three projections of the
wavevector, k
α
, are positive, i.e., they are within the first octant of the space of
wavenumbers. The number of standing waves, Z
k
, that correspond to 1/8ofa
sphere, which is the above-mentioned octant, can be defined as 2/8 of the ratio of
V
k
and V
k0
(Eqs. (2.8) and (2.6)). The factor of 2 is due to the two polarizations
of waves (more details about waves and their polarizations are discussed in
Appendix B):
Z
k
=
2
8
4
3
πk
3
π
3
/L
x
L
y
L
z
=
L
x
L
y
L
z
3π
2
k
3
. (2.9)
Taking into account the relation between the wavenumber and frequency, ω = ck,
we can find the number of standing waves, Z
ω
, corresponding to the entire
frequency interval from 0 to ω:
Z
ω
=
V
3π
2
c
3
ω
3
, (2.10)
where V = L
x
L
y
L
z
is the volume of the cavity. We can find the number of
standing waves corresponding to an infinitesimal interval of frequencies from ω
to ω + dω by differentiating Eq. (2.10):
dZ
ω
= V
ω
2
π
2
c
3
dω. (2.11)
Let us introduce the density of states, N
ω
, i.e., the number of standing waves
corresponding to a unit volume of the cavity and to a frequency interval dω:
N
ω
=
1
V
dZ
ω
dω
=
ω
2
π
2
c
3
. (2.12)
Taking into account Eq. (2.12), we can write the expression for the spectral
density of blackbody radiation, u
ω
, defined as
u
ω
(T ) = N
ω
ε
ω
=
ω
2
π
2
c
3
ε
ω
, (2.13)