THE LIGHT-QUANTUM 377Light-quantum hypothesis: Monochromatic radiation of low density [i.e.,
within the domain of validity of the Wien radiation formula] behaves in ther-
modynamic respect as if it consists of mutually independent energy quanta of
magnitude Rftv/N (ft = h/k, R/N = k, Rftv/N = hv).This result, which reads like a theorem, was nevertheless a hypothesis since it
was based on Wien's guess, which itself still needed proof from first principles.
To repeat, the derivation is based on a blend of purely classical theoretical physics
with a piece of experimental information that defies description in classical terms.
The genius of the light-quantum hypothesis lies in the intuition for choosing the
right piece of experimental input and the right, utterly simple, theoretical ingre-
dients. One may wonder what on earth moved Einstein to think of the volume
dependence of the entropy as a tool for his derivation. That choice is less surprising
if one recalls* that a year earlier the question of volume dependence had seemed
quite important to him for the analysis of the energy fluctuations of radiation.
Einstein's introduction of light-quanta in the Wien regime is the first step
toward the concept of radiation as a Bose gas of photons. Just as was the case for
Planck's derivation of his radiation law, Einstein's derivation of the light-quantum
hypothesis grew out of statistical mechanics. The work of both men has a touch
of madness, though of a far more subtle kind in Einstein's case. To see this, please
note the words mutually independent in the formulation of the hypothesis. Since
1925, we have known (thanks to Bose and especially to Einstein) that the photon
gas obeys Bose statistics for all frequencies, that the statistical independence of
energy quanta is not true in general, and that the gas analogy which makes use
of the Boltzmann statistics relation (Eq. 19.20) is not true in general either. We
also know that it is important not to assume—as Einstein had tacitly done in his
derivation—that the number of energy quanta is in general conserved. However,
call it genius, call it luck, in the Wien regime the counting according to Boltzmann
and the counting according to Bose happen to give the same answer while non-
conservation of photons effectively plays no role. This demands some explanation,
which I shall give in Chapter 23.
So far there is still no revolution. The physicist of 1905 could take or leave the
light-quantum hypothesis as nothing more than a curious property of pure radia-
tion in thermal equilibrium, without any physical consequence. Einstein's
extraordinary boldness lies in the step he took next, a step which, incidentally,
gained him the Nobel prize in 1922.
The heuristic principle: If, in regard to the volume dependence of the entropy,
monochromatic radiation (of sufficiently low density) behaves as a discrete
medium consisting of energy quanta of magnitude Rfiv/' N, then this suggests
an inquiry as to whether the laws of the generation and conversion of light are
also constituted as if light were to consist of energy quanta of this kind.*See the discussion following Eq. 4.14.