56 
QUANTUM THEORY 
The integral is equal to 2-40411. Hence we find* 
number of quanta per cu. cm. = 20-62 T z (40-92). elec 
Average energy of a quantum is nor] 
aT 4 20-62T 3 = 2-70 RT (40-93). so * 
"th .0 
The maximum value of I (v, T) is at a frequency given by ^ 
hv = 2-821J RT (40-94). ^ 
On the other hand, if Planck’s Law is expressed in terms of A instead of G f t] 
v, so that 1' (A, T) dX is the energy-density between A and A + dX, the by e 
maximum value of 1' (A, T) is at a frequency given by enc( 
hv = 4-965.R1 7 (40-95). 
For comparison it may be added that yellow light is just perceptible 
when 500 quanta per second enter the eye. 
scar 
41. The argument by which we reached Einstein’s equation is plausible; ator 
but it is not contended that the truth of the equation can be demonstrated the 
by a priori reasoning. The particular assumption which might be challenged com 
is that I (j/ 12 ) is involved linearly in (36-21) and (36-23); it is conceivable that plac 
the number of transitions might not be simply proportional to the intensity othe 
of the radiation. But evidently if squared or higher powers of I (v 12 ) had 
appeared in the equation we should not have reached Planck’s Law which radi 
is experimentally confirmed. It appears therefore that the assumptions in tl 
are true in nature, and the whole discussion gives an illuminating idea of j. 
how a diversity of processes leads quite simply to a uniform law of dis- mod 
tribution of radiation. g ar d 
It is remarkable that Einstein’s equation is in a certain sense a violation Lon 
of Law I. Consider the transitions represented by a 21 n 2 I (v 12 ), the number (^> 
being jointly proportional to the number of suitable atoms and to the (l)t 
amount of radiation of relevant frequency. The natural interpretation is * s ^h 
that when a quantum of radiation meets an excited atom, there is a certain *1 
definite probability that a transition will occur leaving us with a normal ve l° 
(or a less excited) atom and two quanta of radiation receding from it, viz. in a 
the original quantum and an emitted quantum. Clearly the reverse process fi uai 
consists in two quanta approaching the normal atom simultaneously, the the 
final state being an excited atom with one quantum of radiation leaving noth 
it. The probability of two quanta colliding with the atom simultaneously on e] 
should be proportional to {I (v 12 )} 2 if the quanta represent independent forc< 
elements of radiation. But in Einstein’s equation we do not balance the the 
transitions a 21 n 2 I (v i2 ) against a term in {I (v 12 )} 2 , we balance them against ^y a 
a portion of the term arising from impacts of single quanta. Formally at men 
least we are balancing a cycle of processes instead of a direct and reverse elect 
process. P°y ] 
* Numerical values of all the physical constants are given in Appendix I.