Einstein and the Quantum

by Stone, A. Douglas

  Planck had introduced the quantization of energy for the “molecules” (he called them resonators) in his blackbody as a counting device, leaving it quite unclear as to whether this was a physical hypothesis or a mathematical convenience. From this hypothesis he derived the entropy of the resonators and then by further manipulations determined the distribution of energy among frequencies of radiation in the body. Now, in the 1906 paper, Einstein starts from exactly the same equation as Planck, relating the energy of radiation at frequency ν to the average energy of each molecular resonator in the black body. In 1905 he had calculated the average oscillator energy by conventional statistical mechanics and got the answer that each one had the same energy, kT, which when transferred to radiation and added up over an infinite number of possible wavelengths gave infinity. Now he decides to tinker with conventional statistical mechanics. He writes a mathematical expression for the entropy of one of Planck’s resonators, one that he (and independently Gibbs) had found several years earlier, which looks different from Boltzmann’s famous S = k log W but which he shows is mathematically equivalent. In his new expression, instead of counting states, one finds the entropy by adding up contributions from all the possible energies of each resonator.4 He finds that when he allows the energies to take continuous values, as they do in Newtonian physics, he gets an expression for the entropy that leads to the ultraviolet catastrophe. But if he simply uses Planck’s restriction, that energy can increase only in steps of hν, the Planck law follows in a few steps of algebra. He then puts his cards on the table:

  Hence we must view the following proposition as the basis underlying Planck’s theory of radiation: The energy of an elementary resonator can only assume values that are integral multiples of hν; by emission and absorption, the energy of a resonator changes by jumps of integer multiples of hν.

  This in itself is not much of a mathematical advance over Planck; Einstein has just rearranged the mathematical route to Planck’s formula in a way he finds more congenial and intuitive. To Einstein this approach makes it clear that the quantization of energy is not just a mathematical convenience but a hypothesis about nature, and one very closely allied to his hypothesis of quanta of light. Anyone who had read Planck’s derivation carefully might have realized the same thing. In fact, however, we know of no other physicist of the time who did realize this, except for the omniscient Lorentz, and as we shall see, Lorentz ultimately drew the wrong conclusion from this realization—that the Planck formula must be wrong. Einstein does not even entertain the possibility that the Planck formula might be wrong—from his earliest work on the subject he seems to have accepted this law as an established experimental fact that must be dealt with (a view held by a knowledgeable few in Germany, but not more generally). Instead he attempts, as he did earlier in 1905, to find the law’s meaning by linking it to the quantum hypothesis, but he is unsparingly honest about the flaws in his and Planck’s reasoning. The starting point for Planck, and for him in the current paper, is a mathematical relationship between the energy of thermal radiation and the average energy of matter (resonators) in contact with that radiation, a relationship that was found on the basis of assuming the validity of Maxwell’s equations. To then deduce the Planck law, one inserts a quantum hypothesis that is foreign to, and apparently contradicts, Maxwell’s theory—hardly a compelling chain of logic.

  Einstein continues: “For if the energy of a resonator can only change in jumps, then the mean energy of a resonator in a radiation space cannot be obtained from the usual theory of electricity, because the latter does not recognize distinguished [quantized] energy values of a resonator. Thus the following assumption underlies Planck’s theory: although Maxwell’s theory is not applicable to elementary resonators, nevertheless the mean energy of an elementary resonator … is equal to the energy calculated by Maxwell’s theory.” In other words, Planck uses Maxwell when it suits his purposes and doesn’t use him when it doesn’t. But Einstein is not pointing a finger at Planck; instead he is pointing out the need for a fundamental revision of physics to encompass quanta. “In my opinion the above considerations do not at all disprove Planck’s theory of radiation; rather, they seem to me to show that with his theory of radiation Mr. Planck introduced into physics a new hypothetical element: the hypothesis of light quanta.”

  Herr Planck, however, declined to take credit for this outlandish new hypothesis, by all indications immediately upon learning of it, and for certain in his letter of July 1907, when he urged Einstein to focus on the very important issues of relativity theory and to put the potentially lethal case of the contradictory quanta in quarantine. Einstein did not take his advice; on the contrary, he was soon to spread the infection from radiation to matter. But while Einstein was plotting his next radical step, the great men of the field were just waking up to the dangerous paradox of thermal radiation within classical physics, the ultraviolet catastrophe. The eminent British physicist Lord Rayleigh had peeked at it six years earlier but had then averted his eyes; now it was becoming impossible to ignore.

  1 Hans Albert was Einstein’s second child; his first, a daugher named Lieserl, was born to Mileva in Novi Sad, Serbia, in 1902, before their marriage. The child was left in Serbia with friends or relatives under unclear circumstances, and her subsequent history is unknown, although it is believed that she did not survive to adulthood.

  2 The mathematical quantity called “action” mentioned here is the same one that led Planck to call his constant, h, the “quantum of action” because it has the same physical “units” of mass times velocity times distance. In classical physics action is not quantized, and Planck did not invoke the constant h in his work on relativity theory.

  3 Einstein did not actually use the symbol h introduced by Planck for his constant, preferring instead to use a ratio of older constants equal to Planck’s constant. Einstein continued this practice for several more years, perhaps indicating a reluctance to accept h as a fundamental constant of nature.

  4 The method, very common in modern treatments, is to relate the entropy to the “free energy,” which itself is obtained from the “partition function,” a sum over functions of the energy of the system.

  CHAPTER 12

  CALAMITY JEANS

  In June of 1900, six months prior to Planck’s historic act of desperation and long before Einstein’s 1906 clarification of its meaning, John William Strutt, Lord and Third Baron Rayleigh, had noticed that something must be wrong with the Wien law. This was the law that was believed to describe blackbody radiation but would soon be found experimentally to fail at long wavelengths.

  Lord Rayleigh, a member of the British nobility, had overcome his family’s disapproval of the plebeian study of nature to ascend to the very pinnacle of British science. He had been a sickly youth, bouncing from school to school, and thus did not exhibit his talents early, even being turned down for a minor scholarship at Cambridge. Unexpectedly, he had blossomed at Cambridge, winning the top mathematics award (“First Wrangler”) in 1865 under the exacting supervision of the formidable mathematics teacher E. J. Routh.1 Upon graduation he ignored his father’s reservations about a peer joining academia and won a fellowship at Trinity College, Cambridge. In 1871 his theory of the scattering of light waves by pointlike particles provided the first rigorous explanation of the blue color of the sky, and in 1877 he published his masterpiece: The Theory of Sound. By 1879 his reputation as a physicist was so great that he was chosen to succeed Maxwell as the head of the Cavendish Laboratory at Cambridge. After great success in this role he resigned in 1884 to concentrate again on his own research, which spanned not only acoustics and electromagnetism but also arcana such as the “soaring of birds” and the “irregular flight of tennis balls.” At that time he expressed his vocation thus:

  the domain of natural sciences is surely broad enough to satisfy the wildest ambition of its devotees…. Increasing knowledge brings with it increasing power, and great as are the triumphs of the present century, we m
ay well believe that they are but a foretaste of what discovery and invention have yet in store for mankind…. The work may be hard, and the discipline severe; but the interest never fails, and great is the privilege of achievement.

  By 1900, when he enters our story, he had in the previous year received the Copley Medal of the Royal Society, the same high honor it would award to J. W. Gibbs in the following year, and was four years from becoming the fourth recipient of the Nobel Prize in Physics.2 The great generation of Maxwell, Stokes, and Thomson (Lord Kelvin) having relinquished the stage, Rayleigh was now the voice of British physics.

  Rayleigh had been following the investigations of blackbody radiation in Germany and was aware that the Wien law was currently favored both on the basis of experiments and from extensions of the theory (Planck’s extensions, in fact, before his famous retraction in October 1900). But he was unimpressed with the theory leading to the Wien law, dismissing it as “little more than a conjecture.” In fact he had noticed that the mathematical formula expressing this law had an odd feature. Both the incorrect Wien law and the correct Planck law have the feature that at a given temperature there is a particular frequency of radiation that carries the largest fraction of the radiated energy; that is, both mathematical functions have a “peak” at a frequency that depends on the temperature of the body.3 This peak frequency increases with increasing temperature, residing in the infrared at normal temperature but moving into the visible when matter is heated to six thousand degrees kelvin, as at the surface of the sun. Rayleigh was interested in how the peak frequency shifts upward when the temperature is increased. This could happen in two ways, either by the radiation energy increasing only at higher frequencies than the previous peak, or by the energy increasing more at the higher than at the lower frequencies, so that while the entire curve of energy output moves upward, its peak also shifts to higher frequency. Rayleigh’s intuition was that a rising tide lifts all boats; that is, increasing the temperature of the body should increase the radiation energy output at all frequencies, so he thought the second option had to be the correct one. But he noticed that this was not the way the Wien law worked. The Wien law predicts that if one heats an object and measures the low-frequency side of the curve (below the peak), the energy at these frequencies will stop increasing once the temperature is high enough. Rayleigh was sure that this had to be wrong (and he was right; the correct, Planck law does not have this property).

  Rayleigh was very familiar with the fact that the thermodynamic energy of a particle in a gas is simply proportional to its temperature. This fact is closely related to the equation Emol = kT (where k is Boltzmann’s constant) for a simple vibrating molecule, emphasized by Einstein in both his 1905 and 1906 papers on quanta of light. As already mentioned, the classical prediction that all vibrating structures at the same temperature should have the same thermal energy, kT, is known as the equipartition theorem. The term “theorem” is a bit of a misnomer, since the soon-to-emerge quantum theory would disprove this statement, as least in the case of molecular vibrations. But at the time Rayleigh was first studying blackbody radiation, there was not yet even a hint of such drastic measures, so he was willing to rely on the standard classical statistical mechanics. He realized that the equipartition result, Emol = kT, for vibrating structures of different frequency was just one example of the equal sharing of thermal energy. For a gas of freely moving particles, again each particle should have the same amount of thermal energy, but in this case the amount is different than for an oscillator at the same temperature:4 Egas = 3kT/2.

  As suggested by its name, classical statistical mechanics leads to the conclusion that all the thermal energy in the environment is equally shared among similar types of atomic motion. Rayleigh referred to this as “Maxwell’s doctrine of partition of energy,” and he had validated it in detail, using Newtonian mechanics, in a long paper written only a few months earlier in 1900. Such perfect energy sharing would be correct if molecules were made up of little Newtonian billiard balls, but it turns out that this isn’t the world we live in, and equipartition doesn’t hold in our quantum world.

  This is a good thing. Like many egalitarian principles, equipartition can have unanticipated consequences. In this case the consequence would have been the ultraviolet catastrophe pointed out and rejected by Einstein in his paper on light quanta five years later. Unknown to Einstein, Rayleigh had been there first, but he didn’t raise the alarm.

  As Rayleigh pondered the Wien law in June of 1900, he drew on his great expertise in acoustics, which told him that the classical equipartition principle didn’t really describe well the actual behavior of gases. The speed of sound in a gas (or mixture of gases like the atmosphere) depends on how its energy content changes with temperature, which is termed the “specific heat” of the gas. The properties of the specific heat of many gases did not agree with the equipartition concept; many gases had smaller specific heats than they should, if one just counted how many types of vibrations they were expected to have and assigned to each its equal share of energy. Rayleigh, noting this, opined, “what would appear to be wanted is some escape from the destructive simplicity [of the equipartition theorem].” The Great Escape, quantization of energy, would be found by Herr Planck in less than a year, but Rayleigh had no way of knowing this. He did know that equipartition was not working in general, and so he was reluctant to trust it in considering the blackbody law, which he charmingly termed, in his new paper of June 1900, the “law of complete radiation.” Nonetheless he realized that this equipartition principle could fix the odd feature of the Wien law he had noticed.

  While admitting that “the question is one to be settled by experiment … in the meantime,” Rayleigh wrote, “I venture to suggest a modification [of the Wien law] which appears to me more probable a priori.” First he assumed that the radiation in the blackbody cavity could be thought of as the vibrations of an elastic medium, much like sound waves, essentially assuming a mechanical ether to support radiation (as did all the other physicists of his time). He termed each of these elastic vibrations, with different frequencies, “radiation modes.” He then used equipartition to find the energy of those vibrations, but only for the low-frequency modes: “although for some reason not yet explained [the ‘Maxwell-Boltzmann doctrine of the partition of energy’] fails in general, it seems possible that it may apply to the graver [lower-frequency] modes.”

  With this assumption he derives essentially the same expression for the classical blackbody law that Einstein finds in section 1 of his 1905 paper, except … he fudges it. Seeing that this would lead to an absurd result, infinite energy when summed over all wavelengths, he doesn’t write Einstein’s (correct) answer but, instead, adds an additional, completely unjustified factor to the equation that “turns off” the high-frequency modes and avoids the ultraviolet catastrophe. Moreover, in 1900 he does not grace his readers with any explanation for this fudge factor, which is simply introduced in the final answer.

  Perhaps he felt that given his earlier statement about only using equipartition for lower frequencies, readers weren’t going to take his answer seriously at high frequencies anyway. Perhaps this was his best guess about how the true law behaved at high frequencies; we will never know.5 We do know that Rayleigh’s law, with the fudge factor, was taken seriously by the German experimenters Rubens and Kurlbaum, because they compared it to their data just a few months later. They found that while Planck’s newly minted radiation law fit the data at all frequencies, Rayleigh’s did so only at low frequencies, where both laws were the essentially the same.6

  Note the different conclusions drawn by the callow patent officer Einstein and the decorated Lord Rayleigh. Einstein, through his own deep meditations on classical statistical physics, had come to the conclusion that the equipartition theorem was the only possible answer this theory could give; thus the failure of equipartition to apply to the higher frequencies of blackbody radiation meant that there needed to be a fundamentally new the
ory of atoms or electromagnetism, or both. Rayleigh knew that equipartition failed but hesitated to ascribe this to a failure of classical physics. In this he was similar to Planck, who was forced to accept at least the new quantum of action, h, but couldn’t accept its radical implications.

  But this was not the end of the story. A much younger Cambridge physicist, James Hopwood Jeans, had just completed his studies (graduating as Second Wrangler) and took up a research position, devoting himself to the theoretical study of gases. Jeans was a more flamboyant personality than Rayleigh, later specializing in astrophysics and cosmology and introducing the steady-state model of the universe, which was invalidated by the big bang theory.7 It was precisely to blackbody radiation that Jeans turned in 1904 when he published his treatise The Dynamical Theory of Gases. In this work he expressed a more definitive view of the situation than Rayleigh, leading to a shocking conclusion: the equipartition theorem is valid for all frequencies, and the ultraviolet catastrophe is happening. “If an interaction between aether and matter exists, no matter how small this interaction may be … we are led to the conclusion that no steady state is possible until all of the energy of the gas has been dissipated by radiation into the aether.”

  So why wasn’t the entire earth cooled down to absolute zero as the infinity of radiation modes sucked all energy from matter? Easy. It’s all happening very, very slowly.