A LOSS OF IDENTITY: THE BIRTH OF QUANTUM STATISTICS 425
We owe to Dirac the observation that in the BE case, Eq. 23.2 must be replaced
by
w = 1 (BE statistics) (23.3)
only the single microstate that is symmetric in the N particles is allowed. Dirac
went on to show that Eq. 23.3 leads to the blackbody radiation law, Eq. 19.6 [Dl].
Thus he brought to an end the search—which had lasted just over a quarter of a
century—for the foundations of Planck's law.
Equation 23.3 was of course not known at the time Bose and Einstein com-
pleted the first papers ever written on quantum statistics. Theirs was guesswork,
but of an inspired kind. Let us turn first to Bose's contribution.23b. Bose
The paper by Bose [B3] is the fourth and last of the revolutionary papers of the
old quantum theory (the other three being by, respectively, Planck [PI], Einstein
[E2], and Bohr [B4]). Bose's arguments divest Planck's law of all supererogatory
elements of electromagnetic theory and base its derivation on the bare essentials.
It is the thermal equilibrium law for particles with the following properties: they
are massless, they have two states of polarization, the number of particles is not
conserved, and the particles obey a new statistics. In Bose's paper, two new ideas
enter physics almost stealthily. One, the concept of a particle with two states of
polarization, mildly puzzled Bose. The other is the nonconservation of photons. I
do not know whether Bose even noticed this fact. It is not explicitly mentioned in
his paper.
Bose's letter to Einstein begins as follows: 'Respected Sir, I have ventured to
send you the accompanying article for your perusal. I am anxious to know what
you think of it. You will see that I have ventured to deduce the coefficient 8Tri>^2 /c^
in Planck's law independent of the classical electrodynamics .. .' [Bl]. Einstein's
letter to Ehrenfest contains the phrase, 'the Indian Bose has given a beautiful
derivation of Planck's law, including the constant [i.e., Sirv^2 /^]' [El]. Nei-
ther letter mentions the other parts of Planck's formula. Why this emphasis on
8wv^2 /c}?
In deriving Planck's law, one needs to know the number of states Zs in the
frequency interval between cs and i>s + dv\ It was customary to compute Zs by
counting the number of standing waves in a cavity with volume V. This yields
Bose was so pleased because he had found a new derivation of this expression for
Zs which enabled him to give a new meaning to this quantity in terms of particle
language. His derivation rests on the replacing of the counting of wave frequencies
by the counting of cells in one-particle phase space. He proceeded as follows.
Integrate the one-particle phase space element dxdp over V and over all