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Early Experiments in Quantum Physics 217

dimensions of the block are L L L ¼ V. While we are interested mainly in the internal waves in
the block, we do need to have an ‘‘observation hole’’ and we might discuss the light waves in the
cavity only but here we assume the waves are part of what is going on inside the whole block and
just continue across the interior hole. That is a key point because Planck assumed the oscillators in
the wall of the observation cavity set up the waves. We will see that Planck derived an exact fitto
the blackbody radiation spectrum but he did not believe it himself for several years and others
regarded it as a sort of parameterized fit due to the use of h as an adjustable constant.
An important point occurs here in that in 1905, Einstein [2] called attention to the idea of
quantized radiation waves ‘‘in the cavity,’’ which are now called ‘‘photons.’’ This is a very difficult
concept but perhaps we can say that if the walls of the cavity are emitting light then the light
observed in the hole will be that emitted light. Then if waves are involved, assume that only an
integer number of wavelengths can fit into each dimension and that the waves continue throughout
the block and across the observation hole (a thinking person will note that automatically

L
introduces integers into the derivation!) we have ¼ n. Multiplying by 2p we obtain
l

L 2p
2p ¼ n2p ¼ L. We can define another characteristic of a wave called the ‘‘wave
l l

2p
and so we get k l L ¼ n2p and this will apply in each of the (x, y, z)
number’’ as k l ¼
l
directions. Thus we have three values for the three dimensions k lx L x ¼ n x 2p, k ly L y ¼ n y 2p, and
k lz L z ¼ n z 2p. However, we know from polarizing sunglasses that light can be polarized in two
planes orthogonal to the direction of propagation so any arbitrary polarization has to be some
combination of two possible polarizations and we will have to multiply our count of possible waves
by 2 later on. For now we see that our scheme for counting the number of waves in (L x , L y , L z )
depends on a triad of values as (n x , n y , n z ) including degeneracies (for a cube) such as (1, 2, 3), (2, 1, 3),
(3, 1, 2), etc. Thus we need a counting scheme that will be able to treat these degeneracies as well as
high values of the indices (n x , n y , n z ). Since we know the n values will be very high for a large
number of possible waves, we need a ‘‘scoreboard’’ representation and a very clever method is given
in [2] (Figure 10.4). The relationship of this scoreboard is similar to the scoreboard in a stadium.
The scoreboard is not the game but it is related to the game! Here we see that as the n values get
2
2
2
2
larger and larger we can define another variable n ¼ n þ n þ n as the square of a radius vector in
x y x
n z

n x n y

FIGURE 10.4 A ‘‘scoreboard’’ to count the energy modes in a radiating blackbody.