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A LOSS OF IDENTITY: THE BIRTH OF QUANTUM STATISTICS 429

the Wien regime hv{ S> kT. Therefore, up to an irrelevant* factor TV!, Equations

23.2 and 23.3 coincide in the Wien limit. This asymptotic relation in the Wien

region fully justifies, ex post facto, Einstein's extraordinary step forward in 1905!

Bose's reasoning in 1924 went as follows:

Photons 1

Bose 1924: I -» Planck's law

Quantum statistics

and in 1924-5 Einstein came full circle:

Bose statistics 1

Einstein 1924-5: I —>• The quantum gas

Photon analogy

It was inevitable, one might say, that he would do so. 'If it is justified to conceive

of radiation as a quantum gas, then the analogy between the quantum gas and a

molecular gas must be a complete one' [E7].

In his 1924 paper [E8], Einstein adopted Bose's counting formula (Eq. 23.13),

but with two modifications. He needed, of course, the Zs appropriate for nonre-

lativistic particles with mass m:

Second (and unlike Bose!), he needed the constraint that N be held fixed. This is

done by adding a term

(23.16)

(23.15)

inside the parentheses of Eq. 23.14.** One of the consequences of the thus mod-

ified Eq. 23.14 is that the Lagrange multiplier (—ln^4) is determined by

(23.17)

(23.18)

Hence, Einstein noted, the 'degeneracy parameter' A must satisfy

In his first paper [E8], Einstein discussed the regime in which A does not reach

*The TV! is irrelevant since it affects only C in Eq. 23.7. The constant C is interesting nevertheless.
For example, its value bears on the possibility of defining 51 in such a way that it becomes an extensive
thermodynamic variable. The interesting history of these normalization questions has been discussed
in detail by M. Klein [Kl].
*The term A~ is defined as exp (—p,/kT), where n is the chemical potential. Einstein, of course,
never introduced the superfluous X* into the parenthetical term. In Eqs. 23.16-23.22,1 deviate from
Einstein's notation.