Introduction

Planck recognised that, in order to give his expression (12.41) for the spectrum of blackbody radiation physical meaning, the way forward involved adopting a point of view which he had rejected in essentially all his previous work. As is apparent from his words quoted at the end of Chapter 12, Planck was working at white-hot intensity, because he was not a specialist in statistical physics. We will find that his analysis did not, in fact, follow the precepts of classical statistical mechanics. Despite the basic flaws in his argument, he discovered the essential role which quantisation plays in accounting for the expression (12.41) – recall that Planck's derivation of October 1900 was essentially a thermodynamic argument.

We have already discussed Boltzmann's expression for the relation between entropy and probability, S ∝ ln W, in Sections 10.7 and 10.8. The constant of proportionality was not known at the time and so we write S = C ln W, where C is some unknown universal constant. First of all, let us describe how Planck should have proceeded, according to classical statistical mechanics.

Boltzmann's procedure in statistical mechanics

Planck knew that he had to work out the average energy Ē of an oscillator in thermal equilibrium in an enclosure at temperature T.