Einstein and the Quantum (34 page)

It was clear that Einstein had been troubled by Planck's derivation from his earliest works, but not until 1916 had he even tried to justify this law, succeeding marvelously with his “perfectly quantic” paper, which introduced the concepts of spontaneous and stimulated emission of photons, as well as providing strong arguments for the reality of photons. One major reason that Einstein was so happy with this work, and even called it “
the
derivation” of Planck's formula, was that it did
not
at any point use the strange counting of the distribution of
energy units that Planck had employed. Instead he managed to get the same answer by a different route, based on Bohr's quantized atomic energy levels and his own plausible hypotheses about the balancing out of all emission and absorption processes. Bose, though, was not completely happy even with this method and claimed to have found an even more ideologically pure derivation.

Bose's paper is concise in the extreme, running to less than two journal pages. He begins the work by laying out his motivation for presenting yet another approach to the Planck law. “
Planck's formula … forms the starting point
for the quantum theory … [which] has yielded rich harvests in all fields of physics … since its publication in 1901 many types of derivations of this law have been suggested. It is acknowledged that the fundamental assumptions of the quantum theory are inconsistent with the laws of classical electrodynamics.” However, Bose continues, the factor 8
πυ
²
/
c
³
“
could be deduced only
from the classical theory. This is the unsatisfactory point in all derivations.” Even the “remarkably elegant derivation … given by Einstein” ultimately relies on some concepts from the classical theory, which he identifies as “Wien's displacement law” and “Bohr's correspondence principle,” so that “in all cases it appears to me that the derivations have insufficient logical foundation.”

Einstein did not agree with this criticism and even took time out during his first, quite friendly letter to Bose to dispute it: “
However I do not find your objection
to my paper correct. Wien's displacement law does not presuppose [classical] wave theory, and Bohr's correspondence principle is not used. But this is unimportant. You have derived the first factor [8
πυ
²
/
c
³
] quantum-mechanically…. It is a beautiful step forward.” On both points Einstein was correct.
³
Bose had come up with a more direct method of getting the result, the first to use only the photon concept itself, a tremendously appealing simplification.

Ever since Einstein's 1905 paper on light quanta, there had been a glaring logical problem with taking quanta seriously as elementary particles. In dealing with the statistical mechanics of a gas of molecules, it is possible to derive all the important thermodynamics relations, such as the “ideal gas law” (
PV
=
RT
),
⁴
without ever specifying any other system with which the gas molecules interact. It is enough to simply say that there exist other large systems (“reservoirs”) with which the gas can exchange energy. Then counting the states of the gas, using the classical method (no
h
!) pioneered by Boltzmann, leads to both the entropy and the energy distribution of the gas molecules, and eventually to all the known relations.
⁵
The very same approach appeared to fail for quanta of light; it led to Wien's incorrect radiation law, and not Planck's. This was a major problem, which, along with the difficulty in explaining the interference properties of light, led to the consensus that light quanta weren't “real” particles but some sort of heuristic construct. This consensus had survived even the awarding of the Nobel Prize to Einstein for the photoelectric effect. Bose's work shows how to escape from the first of these dilemmas.

Bose sets out to count the possible states,
W
, into which many light quanta of energy
hυ
and momentum
hυ
/
c
can be distributed according to quantum principles. From this, by a variant of Planck's method, he obtains the average entropy and energy of the photon gas.
⁶
Step one is to consider a
single
light quantum, with energy
hυ
and momentum
hυ
/
c
. If a photon were to be treated as a real, classical particle, one ought to be able to specify its state at each time by its position and its momentum. Physicists refer to such quantities as
vectors
, since they
carry both a magnitude and a direction (e.g., the photon is 5 blocks northeast, with its momentum [always parallel to its direction of motion] due south). The momentum for a massive particle is just its mass times its velocity vector (when its speed is much less than
c
); but a photon's speed (magnitude of velocity) is always equal to
c
, and Einstein has shown (e.g., in his 1916 work) that the magnitude of its momentum is
hυ
/
c
(not
m
times
c
, since the photon has zero mass).

Counting the position states of the photon is not the hard part of Bose's argument; one assumes that the photon gas is enclosed in a box of volume
V
and it can be anywhere within
V
, with equal probability (this point was used by Einstein in his original arguments for the photon concept when analyzing the blackbody entropy back in 1905). Therefore Bose focuses on counting
momentum
states. Since the possible directions of photon motion are continuous and hence infinite, he has to employ an idea already proposed by Planck as early as 1906. Planck's constant,
h
, defines a quantum limit on the smallest difference in momentum that can be resolved.
⁷
Since all photons of frequency
υ
have the same magnitude of momentum,
hυ
/
c
, which is assumed equally likely to point in any direction, Bose can count their states by tiling the surface of a sphere of radius
hυ
/
c
with these “Planck cells.” From basic geometry and the assumption that the spherical shell is only one cell thick, he is then able to find the number of states,
⁸
8
πυ
²
/
c
³
.

Up to this point, what Bose has done is logically appealing but not historic. It was his next step that caused his paper to become “
the fourth and last
of the revolutionary papers of the old quantum theory.” He still has to obtain the Planck form for the radiation law and not the
Wien form. For this next, crucial step he has to calculate how many physically distinct ways there are to put
many
photons at the same time into these available states. But Bose does not appear to realize that the next step is the big one; instead he seems to think that the previous one was the most significant. He begins the relevant paragraph by saying, “
It is now very simple
to calculate the thermodynamic probability of a macroscopically defined state.” After a few definitions, he unveils his answer, a rather obscure combinatorial formula bristling with factorial symbols. This is the key intermediate step. From here on, finding the Planck formula is inevitable and just involves straightforward manipulations, within the competence of scores of his contemporaries.

This paltry written record leaves an enormous historical question. To what extent did Bose understand the key concept in his “revolutionary” derivation? For buried in Bose's factorial formula is a very deep and bold assertion. This formula implies that interchange of two photons in the photon gas does
not
lead to different physical states, unlike the standard, classical, Boltzmann assumption for the atomic gas. Boltzmann, and everyone else after him, assumed that even though atoms were very small and presumably all “looked” the same, one could imagine labeling them and keeping track of them. And if photons were particles like atoms, one should be able to do the same thing. Photon 1 having momentum toward the north and photon 2 having momentum toward the south was a physically different condition from photon 2 north, photon 1 south. Bose, without saying a word about it in his paper, implicitly denied this was so!

Late in his life Bose was asked about this critical hypothesis concerning the microscopic world, which followed from his work. He replied with remarkable candor:

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 about the light quantum just as a particle, I talked about these states. Somehow this was the same question that Einstein asked when I met him: How had I arrived at this method of deriving
Planck's formula? Well, I recognized the contradictions in the attempts of Planck and Einstein, and applied statistics in my own way,
but I did not think that it was different from Boltzmann statistics
[italics added].

Recall that in his derivation Bose was guided by the knowledge of the end point, the precisely known formula for the Planck radiation law. So he did not have to convince himself in advance that his counting method was justified; it turned out to be the method that gave the “correct” answer, so it would seem to be justified a posteriori. He appears to have missed the fact that in asserting this new counting method, he had made a profound discovery about the atomic world, that elementary particles are indistinguishable in a new and fundamental sense.

Einstein, despite his initial enthusiasm for the prefactor derivation, very quickly grasped that the truly significant but puzzling thing about the Bose work was the postclassical counting method. Soon after receiving Bose's paper, he expressed this in a letter to Ehrenfest: “
[the Bose] derivation is elegant
, but the essence remains obscure.” He would pursue and ultimately elucidate this obscure essence for the next seven months.

Bose, not having fully appreciated the novelty of his first paper, placed a great deal of emphasis on his second paper, which was dated June 14, 1924, and sent to Einstein immediately on the heels of the first one. This paper is not a rederivation of a known result, as is his previous paper, but an attempt to reformulate the quantum theory of radiation, in direct contradiction to Einstein's classic 1916 work. The paper, titled “Thermal Equilibrium of the Radiation Field in the Presence of Matter,” proposes a bold hypothesis. While there still is a balance between quantum emission and absorption leading to equilibrium, the emission process is assumed to be
completely
spontaneous and independent of the presence or absence of external radiation. Bose has eliminated Einstein's hypothesis of stimulated emission, which he refers to as “negative irradiation,” saying it is “not necessary” in his theory. To make things balance out, he then has to assume that the probability of absorption also has a different and more complicated dependence on the energy density of radiation than Einstein assumed.

Einstein, who was a virtuoso at finding absurd implications of flawed theories since his days at the patent office, published a decisive critical note appended to his translation of this paper. First, he notes that Bose's hypothesis “
contradicts the generally and rightly accepted
principle that the classical theory represents a limiting case of the quantum theory…. in the classical theory a radiation field may transfer to a resonator positive or negative energy with equal likelihood.” Second, Bose's strange hypothesis about the nature of absorption implies that a “cold body should absorb [infrared radiation] less readily than the less intense radiation [at higher frequencies]…. It is quite certain that this effect would have been already discovered for the infrared radiation of hot sources if it were really true.” Because of these compelling arguments, Bose's only published attempt to
extend
quantum theory had no influence on the field and is solely of historical interest.

Nonetheless, Einstein's recognition of Bose's first paper transformed his professional situation in an instant. Einstein's supportive postcard to Bose congratulating him on his “beautiful step forward” was shown to the vice-chancellor at Dacca, and it “
solved all problems
.” Bose recalled, “that little thing [the postcard] gave me a sort of passport for [a two-year] study leave [in Europe] … on rather generous terms…. Then I also got a visa from the German consulate just by showing them Einstein's card.”

By mid-October of 1924 Bose had arrived in Paris and was introduced to the noted physicist Paul Langevin, who was a personal friend of Einstein's. Bose immediately wrote to Einstein, asking Einstein's opinion on his second paper (which he was unaware had already been published with Einstein's assistance) and expressing his desire to “
work under you
, for it will mean the realization of a long cherished hope.” Einstein quickly replied, “
I am glad I shall have
the opportunity soon of making your personal acquaintance,” then summarized his reasons for rejecting Bose's conclusions in the second paper and concluded by saying, “we may discuss this together in detail when you come here.”

Despite the warmth of Einstein's reply, Bose was reluctant to move on to Berlin immediately, in part because Einstein's potent critique had made him unsure of his new proposal, which he wanted to refine further. In addition he seems to have found the change of cultures
challenging; he decided to settle in Paris in the company of a local circle of Indian compatriot intellectuals. He justified this as follows: “
because I was a teacher
… and had to teach both theoretical and experimental physics … my motivation then became to learn all about the techniques I could in Paris … radioactivity from Madame Curie and also something of x-ray spectroscopy.” An interviewer of Bose in 1972 noted that “
even more than forty years
later one still has the impression that Bose was terribly intimidated by most Europeans.” This no doubt contributed to his disastrous interview with Madame Curie, concerning joining her lab. She had hosted a previous Indian visitor and had fixed it in her mind that the collaboration had failed because of his poor French. Thus she conducted her first interview with Bose entirely in English, and while welcoming him warmly, firmly insisted he would need four months of language preparation before starting work. Although “
she was very nice
,” Bose, who had studied French already for ten years, found no opportunity to interrupt her monologue. And so “I wasn't able to tell her,” he later explained, “that I knew sufficient French and could manage to work in her laboratory.”