Einstein and the Quantum

by Stone, A. Douglas

  In 1921 Bose accepted a faculty position at the new University of Dacca, in East Bengal, recently established by an ambitious vice-chancellor, P. J. Hartog. At Dacca he “spent many sleepless nights” trying to understand the Planck law, while at the same time teaching it to his students. He felt an obligation to present something to them that he himself found clear and consistent: “As a teacher who had to make these things clear to his students I was aware of the conflicts involved…. I wanted to know how to grapple with the difficulty in my own way…. I wanted to know.” In late 1923 he hit upon his new approach to deriving the law and sent off a manuscript to the Philosophical Magazine, where he had previously published papers on quantum theory, only to receive the reply, in the spring of 1924, that the referee’s decision had been negative. It was at this point that Bose took the bold step of sending the paper to Einstein, a strategy so speculative that its success appeared to violate the very principle of maximum entropy employed in the paper itself!

  It was remarkable, but nonetheless true, that Planck’s blackbody formula remained somewhat mysterious a full twenty-four years after Planck’s initial derivation. It was not that anyone doubted any longer the validity of the formula, but the tortured reasoning Planck had used to derive it left physicists unsatisfied for decades. That is why Bose’s paper, titled “Planck’s Law and the Quantum Hypothesis.” was of interest to Einstein and others. As we saw earlier, Planck had been reluctant to treat radiation directly with statistical mechanics, and instead, using classical reasoning, he related the mean energy of radiation at a given frequency to the mean energy of idealized vibrating molecules (“resonators”). He then calculated the entropy of these resonators by introducing into the counting of states his quantized energy “trick.” A key factor in totaling up the number of allowed states in Planck’s method, which was not appreciated for quite some time, was that one could treat the units of energy belonging to each resonator as indistinguishable quantities (i.e., if resonator one had seven units of energy hυ and resonator two had nine units, one didn’t have to ask which units they were). In 1912 Peter Debye, an outstanding young theorist and future Nobel laureate, rederived the Planck law, not by counting resonator states, but by counting states of classical electromagnetic waves that could fit in the blackbody cavity and then ascribing to them the same average energy that Planck had assigned to each resonator. The counting of the number of allowable waves in the cavity led to the factor in the Planck radiation formula, 8πυ2/c3, to which Bose alludes in his letter to Einstein. This factor was very easy to find from classical wave physics but very hard to find from quantum principles, hence Bose’s emphasis on having found a quantum route to it.

  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 8πυ2/c3 “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 [8πυ2/c3] 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 assu
mption that the spherical shell is only one cell thick, he is then able to find the number of states,8 8πυ2/c3.

  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.”

  Despite this missed opportunity, Bose tarried in Paris nearly a full year learning x-ray techniques before working up the courage to move on to Berlin in October of 1925 and finally meeting Einstein a few weeks later. In the intervening year, Einstein had taken up Bose’s novel counting method and extended it to treat the quantum ideal gas, leading to truly remarkable discoveries, about which Bose was unaware. For Bose, “the meeting was most interesting…. he challenged me. He wanted to find out whether my hypothesis, this particular kind of statistics, did really mean something novel about the interaction of quanta, and whether I could work out the details of this business.” During Bose’s visit, Werner Heisenberg’s first paper came out on the new approach to quantum theory known as matrix mechanics (of which we will hear more later). Einstein specifically suggested that Bose try to understand “what the statistics of light quanta and the transition probabilities for radiation would look like in the new theory.”

  However, Bose was not able to make progress. He seems to have had a difficult time assimilating these rapid new developments and wrote somewhat despairingly to a friend: “I have made an honest resolution of working hard during these months, but it is so hard to begin, when once you have given up the habit.” Bose received extensive access to the scientific elite of Berlin through Einstein’s patronage and experienced the whirlwind of excitement around the revolution in atomic theory. But no publication resulted from his stay in Europe, and in late summer of 1926 he returned to Dacca. By then the new quantum mechanics had passed him by.