Introduction to Quantum Ideas
33
Fig. 2.1 Intensity of radiation from a black-body, I (λ, T), in watts per square
centimetre per micron.
λ m T = constant
(2.3)
This result had been obtained by Wien (1893) from a theoretical analysis.
The initial approach to understand the nature of black-body radiation was in
terms of the standing waves set up in the enclosure and the thermodynamic
energy associated with these waves. Consider a plane wave in a cubic box of
length l. It may be written in the form
ψ = A sin 2π (v t ± k x x ± k y y ± k z z + α)
(2.4)
where
1
= λ
k
is the wave number, v is the frequency of the radiation and
the ± signs define the direction in which the wave is travelling. Standing waves
are obtained by taking linear combinations of the different terms (with x = 0) in
Eq. (2.4). Requiring that the waves vanish at the walls passing through the
origin, we get
ψ s = A cos (2π ν t) sin (2π k x x) sin (2π k y y) sin (2π k z z)
(2.5)
where, in order that the wave should have nodes at the other walls,
,
,
2
2
2
=
=
=
y
x
z
x
y
z
n
n
n
k
k
k
l
l
l
(2.6)