424 THE QUANTUM THEORYis discussed in Section 23b with the exception of one major point, which is reserved
for the next chapter: Einstein's last encounter with fluctuation questions. (3) It
will be of help in explaining Einstein's ambivalence to Bose's work. In a letter to
Ehrenfest, written in July, Einstein did not withdraw, but did qualify his praise
of Bose's paper: Bose's 'derivation is elegant but the essence remains obscure' [El ].
(4) It will help to make clear how novel the photon concept still was at that time
and will throw an interesting sidelight on the question of photon spin.
Bose recalled many years later that he had not been aware of the extent to
which his paper defied classical logic. (Such a lack of awareness is not uncommon
in times of transition, but it is not the general rule. Einstein's light-quantum paper
of 1905 is a brilliant exception.) 'I had no idea that what I had done was really
novel. ... I was not a statistician to the extent of really knowing that I was doing
something which was really different from what Boltzmann would have done,
from Boltzmann statistics. Instead of thinking of the light-quantum just as a par-
ticle, I talked about these states. Somehow this was the same question which Ein-
stein asked when I met him [in October or November 1925]: how had I arrived
at this method of deriving Planck's formula?' [Ml].
In order to answer Einstein's question and to understand what gave Bose the
idea that he was doing what Boltzmann would have done, I need to make a brief
digression.
As was discussed in Section 4b, both logically and historically classical statistics
developed via the sequence
fine-grained counting —* course-grained counting
This is, of course, the logic of quantum statistics as well, but its historical devel-
opment went the reverse way, from coarse-grained to fine-grained. For the oldest
quantum statistics, the Bose-Einstein (BE) statistics, the historical order of events
was as follows.
1924-5. Introduction of a new coarse-grained counting, first by Bose, then by
Einstein. These new procedures are the main subject of this chapter.
1925-6. Discovery of nonrelativistic quantum mechanics. It is not at once
obvious how the new theory should be supplemented with a fine-grained counting
principle that would lead to BE statistics [HI].
- This principle is discovered by Paul Adrien Maurice Dirac. Recall first
Boltzmann's fine-grained counting formula for his discrete model of a classical
ideal gas consisting of N particles with total energy E. Let there be n, particles
with energy e, (see section 4b, especially Eq. 4.4 and Eq. 4.5):
(23.1)
Then the corresponding number w of microstates is given by
(Boltzmann statistics) (23.2)