428 THE QUANTUM THEORY- Photon number nonconservation.
- Bose's cell partition numbers p\ are defined by asking how many particles are
in a cell. Boltzmann's axiom of distinguishability is gone. - The Ansatz (Eq. 23.13) implies statistical independence of cells. Statistical
independence of particles is gone.
The astounding fact is that Bose was correct on all three counts. (In his paper,
he commented on none of them.) I believe there had been no such successful shot
in the dark since Planck introduced the quantum in 1900. Planck, too, had
counted in strange ways, as was subtly recalled by Einstein in his review, written
in 1924, of a new edition of Planck's Wdrmestrahlung: 'Planck's law [was] derived
... by postulating statistical laws in the treatment of the interaction between pon-
derable matter and radiation which appear to be justified on the one hand because
of their simplicity, on the other hand because of their analogy to the corresponding
relations of the classical theory' [E5].
Einstein continued to be intrigued by Bose's paper. In an address given in
Lucerne on October 4, 1924, before the Schweizerische Naturforschende Gesell-
schaft, he stressed 'the particular significance for our theoretical concepts' of Bose's
new derivation of Eq. 23.4 [E6]. By this time, he had already published his own
first paper on quantum statistics.
23c. Einstein
As long as Einstein lived, he never ceased to struggle with quantum physics. As
far as his constructive contributions to this subject are concerned, they came to an
end with a triple of papers, the first published in September 1924, the last two in
early 1925. In the true Einsteinian style, their conclusions are once again reached
by statistical methods, as was the case for all his important earlier contributions
to the quantum theory. The best-known result is his derivation of the Bose-Ein-
stein condensation phenomenon. I shall discuss this topic next and shall leave for
the subsequent section another result contained in these papers, a result that is
perhaps not as widely remembered even though it is more profound.
First, a postscript to Einstein's light-quantum paper of 1905.
Its logic can be schematically represented in the following way.
Wien's law 1
Einstein 1905: I -* Light-quanta
Gas analogyAn issue raised in Section 19c should be dealt with now. We know that BE is the
correct statistics when radiation is treated as a photon gas. Then how could Ein-
stein have correctly conjectured the existence of light-quanta using Boltzmann sta-
tistics? Answer: according to BE statistics, the most probable value (n,) of ni for
photons is given by {«,) = [exp (hvJkT) —1]~'. This implies that (nt) <K 1 in