The Existence of Light Quanta Postulated   Einstein's Heuristic Argument In 1905 Einstein published three papers which today are regarded as foundation stones of modern physics. The first of these - "Über einen die Erzeugung und Verwandlung des lichtes betreffenden heuristisches Gesichtspunkt" ("Concerning the generation and transformation of light from a heuristic point of view", Annalen der Physik, March, 1905) contains his argument for the postulation of light quanta, and, as an aside, an explanation of the photoelectric effect.

In my school days I used to dread the phrase "thus it can easily be shown that" - because it seemed to me always to involve many pages of closely worked stuff which I never could understand. Einstein's argument however seems - with the necessary hindsight - to have such simplicity, elegance, and inevitability that it does "easily follow that"...

 

Define an entropy density for black body radiation similar to the energy density
From thermodynamics one can show that

 . . . . . Eq. 1

Assuming Wien's radiation formula

 . . . . . Eq. 2

and solving for 1/T, one finds

 . . . . . Eq. 3

and integrating

These relationships correspond to Planck's arguments, and were at the time of Einstein's paper well known
 
Consider, however, a change of volume of a black body, leaving the total energy constant.
This requires that u(v,T)V=U(v,T) remain constant.
 
The entropy in the frequency interval dv is then

 . . . . . Eq. 4

For a finite volume change, keeping U constant, we have

 . . . . . Eq. 5

For a perfect monatomic gas of n atoms, thermodynamics gives for the entropy change by an expansion at constant energy

 . . . . . Eq. 6

This relation is easily connected to Boltzmann's expression S=klogW because for one atom the probablity of its being in a partial volume V of V₀ is V/V₀.
For n independent atoms the probability is thus (V/V₀)ⁿ from which Equation 6 follows immediately.
 
Equations 5 and 6 become the same if one assumes Udv=nhv.
This means that the energy in frequency interval dv is subdivided into n quanta of magnitude
 

E=hv

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And just in case you were wondering (as I was...)
according to Collins English Dictionary (1980):

heuristic

(3, Maths, science) "using or obtained by reasoning from past experience since no algorithm exists or is relevant".

 

Sources
Again, E. Segrè's excellent book "From Quarks to X-Rays" (W.H. Freeman, San Francsico, 1980, ISBN 0-7167-1147-8) (Appendix 3)