On 18th March 1905, Einstein sent the Annalen der Physik the first of a series of papers
that made that year his annus mirabilis. In this first paper (Einstein 1905), “On a Heuristic Point
of View Concerning the Production and Transformation of Light,” he advanced his light
quantum hypothesis, that heat radiation of high frequency ν behaves as if it consists of
independent, spatially localized quanta of energy E = hν, where h is Planck’s constant. While his
use of the photoelectric effect to support this hypothesis is widely known, my concern here is
with a much more ingenious and telling argument that forms the centerpiece of the paper and is
laid out in its Section 6.2
The evidential basis of the argument is an expression for the volume dependence of the
entropy of a system of heat radiation of energy E and high frequency ν. If S is its entropy when
the system occupies volume V and S0 its entropy when the system occupies volume V0, then
S - S0 = k (E/ hν) ln (V/ V0) (1)
where k is Boltzmann’s constant. It is important that this result derives from macroscopic
measurements. The experimentalists had made precise measurements of the distribution of
energy over the different frequencies of heat radiation. Wien had fitted a well-known
distrubution formula to the experimental results that worked well for higher frequencies. Using
such formulae, any competent thermodynamicist could infer the corresponding entropy
distibutions, such as (1).
Einstein then used this macroscopic formula to infer directly to the microscopic
constitution of the radiation in an argument that, in my view, is the boldest of his corpus of 1905.
In an earlier section, Einstein had recapitulated what then seemed to be a superficial truism.

Imagine that one has a thermal system that consists of many, independent moving points—n,
say. Such is the constitution, for example, of an ideal gas. If one has a volume V0 with one point
in it, the probability that the point will be found in a subvolume V is just V/V0. Since the points
move independently, it now follows that the probability that all n are to be found in the
subvolume is just
W = (V/V0)n (2)
This formula gives the probability that the volume of the thermal system will spontaneously
fluctuate to the smaller volume V. For a system of molecules comprising an ideal gas, n will be
of the order of 1024 for macroscopic samples of gas. So the probability of any significant volume
fluctuation is unimaginably small. Whether small or not, the probability of the transition is
related to the entropy of the initial and final states by what Einstein called “Boltzmann’s
Principle”:
S - S0 = k ln W (3)
Applying Boltzmann’s Principle (3) to (2) for an ideal gas of n molecules immediately returns
the expression S - S0 = kn ln (V/V0) for entropy of the gas, from which, as Einstein shows in a
footnote, the ideal gas law follows.
Proceeding now to the case of heat radiation, Einstein combined the expressions (1) and
(3) to conclude that the probability that a volume of radiation V0 will spontaneously fluctuate to
the subvolume V is
W = (V/V0)E/hν (4)
Einstein thought the import of this last formula obvious. He wrote without any intervening text:
From this we further conclude:
Monochromatic radiation of low density (in the region of validity of Wien’s
formula) behaves thermodynamically as if it consisted of energy quanta of size
hν that are independent of one another.
The thought is clear. The similarity of expressions (2) and (4) led Einstein to infer that the
radiation consists of n = E/hν independent points; that is the energy E is divided into n
independent quanta of size hν. The only hesitation in Einstein’s inference is the “behaves … as
if” qualification. That qualification is dispensed with elsewhere, such as in the introductory
section, with mention of the full array of evidence of the paper.

While Einstein’s inference to light quanta above seems irresistible, we should recall that
its conclusion directly contradicted the great achievement of nineteenth century optics and
electrodynamics, the wave theory of light.