C. Quanta and gas molecules
Instead of a furnace, consider another container, this one containing air. Most likely, the air molecules in it will be distributed almost evenly; it's extremely unlikely that all the air molecules would be found in one small part of the container if they are free to move anywhere in it and their motions are independent. If there were just one air molecule in the whole container, it would be found in the right half of the container about half the time, and in the left half of the container the other half of the time. If we added a second molecule, we'd expect to find both molecules in the left half one fourth of the time since the first molecule would be there half the time, and the second molecule would be there with it only half of that time>-and one half times one half equals one fourth (one half squared). For three molecules, the chances of finding all three in the left half of the container should be one out of eight-one half times one half times one half again, or one half to the third power. And for N molecules, the chances of finding all N of them in the left half at any given moment should be one half to the Nth power.
Einstein found that light waves in a furnace should act much the same if they behave according to Planck's quantum law. As we noted above, the intensity of the lower-frequency light is described pretty accurately by a theory that doesn't assume the light's energy comes in quanta. But the behavior of high-frequency light (Figure 3) is quite different, and it's here that Einstein focused his attention.
Figure 3. The upper curve is the Planck's-law curve of Figure 2; the lower curve is a close approximation for higher frequencies- so close that the curves can't be distinguished in this figure for frequencies much higher than three units. Einstein found that, at least to the extent the light intensity follows the lower curve, quanta of light should behave in some ways like molecules of a gas.
For light waves of any of the highest frequencies f, the probability that all their quanta will be found at a given instant in the left half of the furnace turns out to be approximately one half to the power
where E is the total energy of all those light waves, f again is their frequency, and h is the ratio of the energy of one quantum to the frequency. So for light with frequency f, the chances of finding all E/hf of its quanta in the same half of the furnace are about as small as the chances of finding all of a gas made of N=E/hf molecules in the same half of its container. Einstein's more general version of this analysis shows that any distribution of higher-frequency light quanta is, approximately, as likely to happen as the same distribution for gas molecules-at least if the frequencies of the light waves considered are high enough.
So, while the law Planck discovered was not what one would expect just from Maxwell's theory of light waves, it was at least close to what one might expect if light quanta had an atom-like distribution in space. Furthermore, Einstein described three other phenomena that seemed better understood under the latter assumption. (.....continued)