QUANTUM THEORY 
53 
I 
f atoms 
of dis- 
capture 
try that 
ies; the 
rer con- 
lstances 
3mblage 
lectrons 
ms with 
atom in 
i energy 
rect and 
indicates 
e up the 
estigated 
at, length 
afs arises 
;red here. 
37-3) the 
a definite 
ms corre- 
inate the 
)}• 
where a and k are constants. Wien’s Law then gives 
1 T ^ = de k ^ T - 1 ^ 40 ' 1 ^ 
but Boltzmann’s Law (38-3) is unchanged. 
It is curiously difficult to justify the choice of 1 for the constant a so 
as to obtain Planck’s Law, thereby making I (v,oo) infinite instead of 
having a finite limit. In § 37 we begged the question by assuming as 
obvious that I (v, cc) = oo. The proof cannot be completed without intro 
ducing some additional assumption as to the laws of interaction between 
matter (or electric charges) and radiation. 
For this assumption we may take the Correspondence Principle, which 
asserts that the classical laws of dynamics and electrodynamics represent 
the limit towards which the quantum laws (the actual laws) tend asympto 
tically when the number of quanta involved is very great—that, in fact, 
the older theories are statistically true provided there is sufficient material 
for statistical treatment to be appropriate. Now according to the classical 
law of equipartition of energy I (u, T) should be proportional to T, just as 
the average energy of a molecule is proportional to T. At sufficiently high 
temperature this classical energy will represent a very large number of 
quanta hv, and therefore by the correspondence principle it should agree 
with the true law (40T). There is no such agreement if a 4= 1; but if a = 1 
we have as T -»■ oo 
I(v,T) = °^fT (40-2) 
so that it is proportional to T as in the classical theory. 
Alternatively we may obtain Planck’s Law by considering the mechanism 
of a particular process of transfer, e.g. the scattering of radiation by free 
electrons, since if I (v, T) is determined from any one process all other 
processes must give the same result. But all recognised quantum theories 
of particular processes have been developed in accordance with the 
Correspondence Principle, and it is this feature of the processes which 
settles the value a = 1 . 
The constant C in Planck’s Law can be determined by the Corre 
spondence Principle. Consider the radiation in a cubical enclosure of side 
l. At a given initial instant the electromagnetic vector throughout the 
enclosure can be expressed by a triple Fourier series of which the typical 
term is 
n 2 , n 3 cog ^ 7TUlX ll‘ qq S ^ lTrn 2 yl ^• cog ^ 7 Tn 3 z/l (403), 
where n 1 , n 2 , n z take all integral values. This gives rise to waves of the 
form 
sin 2 tt % J ± n 2 1 ± n z J + I ,