Equation 19.6 follows if one can derive Eq. 19.12.
The Statistical Step. I should rather say, what Planck held to be a statistical
step. Consider a large number N of linear oscillators, all with frequency v. Let
UN = NUandSN = NS be the total energy and entropy of the system, respectively
Put SN = kin WN, where WN is the thermodynamic probability. Now comes the
quantum postulate.
The total energy UN is supposed to be made up of finite energy elements c.UN
= Pe, where P is a large number. Define WN to be the number of ways in which
the P indistinguishable energy elements can be distributed over N distinguishable
oscillators. Example: for N = 2, P = 3, the partitions are (3e,0), (2e,e), (6,2e),
(0,3e). In general,
(19.12;
370 THE QUANTUM THEORYoscillator by integrating TdS = dU, where T is to be taken as a function of U
(for fixed v). This yields(19.13)
Insert this in SN = k\n WN, use P/N = U/e, SN = NS and apply the Stirling
approximation. This gives
(19.14)
It follows from Eqs. 19.4 and 19.11, and from TdS = dU, that S is a function
of U/v only. Therefore(19.15)Thus one recovers Eq. 19.12. And that is how the quantum theory was born. This
derivation was first presented on December 14, 1900 [P4].
From the point of view of physics in 1900 the logic of Planck's electromagnetic
and thermodynamic steps was impeccable, but his statistical step was wild. The
latter was clearly designed to argue backwards from Eqs. 19.13-19.15 to 19.12.
In 1931 Planck referred to it as 'an act of desperation. ... I had to obtain a positive
result, under any circumstances and at whatever cost' [H2]. Actually there were
two desperate acts rather than one. First, there was his unheard-of step of attach-
ing physical significance to finite 'energy elements' [Eq. 19.15]. Second, there was
his equally unheard-of counting procedure given by Eq. 19.13. In Planck's opin-
ion, 'the electromagnetic theory of radiation does not provide us with any starting
point whatever to speak of such a probability [ WN] in a definite sense' [PI]. This
statement is, of course, incorrect. As will be discussed in Section 19b, the classical
equipartition theorem could have given him a quite definite method for determin-