Einstein and the Quantum

by Stone, A. Douglas

  6 The ratios of the distances in frequency between pairs of lines corresponded to the difference of the inverse of the squares of integers.

  CHAPTER 21

  RELYING ON CHANCE

  Fundamental as [the relativity theory] of Einstein has proved to be for the development of the principles of physics, its applications are for the present still at the very limit of the measurable. His tackling of other questions that are at the moment at the center of attention has proved to be much more significant for applied physics. Thus he was the first to demonstrate the significance of the quantum hypothesis also for the energy of atomic and molecular motions…. That he might sometimes have overshot the target in his speculations, for example in his light quantum hypothesis, should not be counted against him too much.

  —PLANCK, NERNST, RUBENS, AND WARBURG, LETTER NOMINATING EINSTEIN FOR THE PRUSSIAN ACADEMY OF SCIENCES, JUNE 12, 1913

  “Why Planck and I engaged him just as you take on a butler, and now look what a mess he’s made of physics; one can’t turn one’s back for a minute.” This was Nernst’s sardonic appraisal of Einstein’s triumph with general relativity. The esteemed professors of the Prussian Academy had not brought Einstein to Berlin to monkey around with the law of gravity and the geometry of space-time; they were expecting him to lead them to victory in the race to understand the atom. Instead he had ceded the inside rail to the Dane Niels Bohr, and his English collaborators, just as the Great War had begun. Moreover they, along with most of the physics community, had never accepted Einstein’s notion of a new theory in which wave-particle duality would naturally emerge and light would really have particulate properties.

  One might have thought that Bohr’s theory would have advanced the belief in photons. Bohr’s second postulate stated that when an electron makes a transition between two of its allowed orbits, the result is “the emission of homogeneous radiation, for which the relation between the frequency and the amount of energy emitted is the one given by Planck’s theory [ε = hυ].” If one squinted just a little at that sentence (as indeed one often had to do with Bohr’s writing), it sure seemed like he was saying that the atom emitted one Einsteinian light quantum every time an electron changed its energy state. But Bohr did not mean to say this and in fact remained opposed to the particulate concept of light for the next twelve years. He believed that quantum mechanics governed the atom but that the light emitted in the transition between allowed orbits, while it contained the fixed quantum of energy, hυ, was nonetheless solely an electromagnetic wave.

  Einstein himself had been chastened by his failure to find quanta in a modified Maxwellian electrodynamics and since 1911 had spoken of photons only sotto voce. In May of 1912 he expressed his position to Wien: “One cannot seriously believe in the existence of countable quanta, since the interference properties of light emitted by a luminous point in different directions are not compatible with it. Nevertheless, I still prefer the “honest” theory of quanta to the hitherto found compromises meant as its replacements.” Einstein here identifies a puzzle that he had been struggling with for several years already in 1912. In classical electromagnetism a point source of radiation (such as an atom) emits a uniform spherical wave of light traveling outward in all directions; nonetheless if one refocuses some of these diverging rays to a screen, they will show standard interference patterns. Einstein and Lorentz had already agreed that each single light quantum was capable of interfering with itself, so such a picture suggested that light quanta from a point source must “break up” in some manner, belying their existence as “countable quanta.” Another issue Einstein had pondered was the phenomenon of radioactivity, in which one or more particles is emitted from the nucleus, apparently at a random time and in a random direction. Already in 1911 he saw a parallel here with electromagnetic radiation, writing to Besso, “the process of absorption [of light] … really does have similarity with a radioactive process.” His close friend and future Nobel laureate Otto Stern recalls how much this problem vexed him during his Prague years: “Einstein always wracked his brain about the law of radioactive decay. He constructed such models.” After Bohr’s eye-opening atomic theory, it struck him that the new picture of emission and absorption could permit these troublesome puzzle pieces to fit together.

  In February of 1916 Einstein had already put general relativity aside and begun to catch up on the quantum theory of atoms. Sommerfeld had written Einstein in December of 1915 to ask him to look at an improvement he had made in Bohr’s formulas that Einstein would find particularly interesting. Sommerfeld had realized that there was no need for Bohr to restrict the electrons to circular orbits; they could also move on elliptical orbits (as do all the planets1 in our solar system). He then used a variant on Bohr’s approach, which he had invented, to determine the quantized energies of the allowed orbits.2 He found a more general formula than Bohr’s that initially gave the same results for the hydrogen spectra. But in his new approach he was able to take into account an additional effect that Bohr could not. Einstein’s special relativity theory predicts that the measured mass of an electron will increase with increasing velocity. Electrons whizzing around the nucleus were calculated to be moving at a significant fraction of the speed of light, and so this increase in mass should have a measurable effect on the electron’s orbital frequency. Including this effect, as Sommerfeld now had done, caused the spectral lines in the hydrogen series to split into closely spaced groups of lines (“fine structures”), their number depending on the final state of the electron after light has been emitted. Sommerfeld wondered in his letter to Einstein whether the newly minted general theory of relativity would affect his calculations, but Einstein assured him that these effects were too small to matter in this context. The “fine-structure” effect had been previously seen and, just at that time, had been measured carefully by the noted German spectroscopist Friedrich Paschen (after whom one of the original hydrogen series was named). In late December of 1915 he wrote to Sommerfeld, “My measurements … agree everywhere most beautifully with your fine structures.” The experimentalist was so delighted by this transformation of an experimental anomaly into an important discovery that he is reported to have loudly exclaimed, “Now I believe in relativity theory!”

  It was this beautiful marriage of Bohr’s atomic quantum theory and relativity principles that so impressed Einstein, once he had digested it; this is the work that in February of 1916 he called “a revelation” in an ecstatic letter to Sommerfeld. Later, in August, in the midst of his own work, inspired by Bohr’s theory, he wrote Sommerfeld again to say, “your spectral analyses are among my finest experiences in physics. It is just through them that Bohr’s idea becomes entirely convincing. If only I knew which little screws the Lord is adjusting there!” By then he had already shown, in a paper submitted on July 17, that Bohr’s single postulate, that electrons make transitions between allowed stationary energy states via absorption and emission of radiation of energy hυ, had remarkable implications.

  In this first paper of 1916, titled “Emission and Absorption of Radiation in Quantum Theory,” he returns to the theme he first expounded in 1909, that Planck’s derivation of the blackbody law is contradictory because it uses classical electrodynamics to relate the mean energy of an oscillator to the energy density of the radiation field but then departs from classical physics to calculate this mean energy according to a quantum prescription. He again praises Planck’s courage to leap into the unknown—” his derivation was of unparalleled boldness—but adds, “however it remains unsatisfactory that the electromagnetic-mechanical analysis [used] is incompatible with quantum theory.” He continues: “Since Bohr’s theory of spectra has achieved its great successes, it seems no longer doubtful that the basic idea of quantum theory must be maintained.” In the interests of consistency, he says, Planck’s classical assumptions must be “replaced by quantum-theoretical contemplations on the interaction between matter and radiation. In this endeavor I feel galvanized by the follo
wing consideration, which is attractive both for its simplicity and generality.”

  The simple, general consideration he mentions flows from the concept of thermal equilibrium, which we have encountered earlier. The blackbody law holds for radiation in contact with matter, so that the entire system (radiation + matter) has settled into the most probable thermodynamic state. In the thermal equilibrium state the entropy of the system is at its maximum value, its temperature is no longer changing, and the average energy of both the radiation and the matter is not changing in time.3 But since the matter and radiation are continually exchanging energy, this state of equilibrium is not to be conceived of as the absence of interaction but rather as continually compensating change. One can imagine the two systems (matter and radiation) as two swimming pools connected by pipes, so that water is flowing from one pool into the other through certain pipes and being pumped back into the first pool through other pipes, and on average the level of water [energy] in each pool does not change.

  Following Bohr, Einstein can now be more precise about the nature of this dynamical equilibrium state. He assumes the matter in contact with radiation is a gas of molecules with discrete, quantized energy levels, Bohr’s stationary states. His argument is, however, so general that he never needs to assign specific values to the allowed energies: it is enough that they exist. Thus his reasoning becomes independent of the details of Bohr’s method for calculating atomic energy levels. Energy is then exchanged between the molecules and the radiation field according to the Bohr prescription. If a molecule has any energy greater than its minimum energy state (called the “ground state”), then it can either emit radiation of the appropriate energy, hυ1, so as to drop down to a lower energy state, or absorb radiation energy of another specific frequency, hυ2, and jump up to a higher energy state. In order for equilibrium to be maintained, these processes of emission and absorption of energy from the radiation must balance out on average; the same amount of energy must flow into the molecule as out. In fact, Einstein points out, not only must the energy flow balance out overall; the exchange of energy must balance out independently for each pair of molecular levels. For each such pair “upward transitions” (absorption) must equal “downward transitions” (emission) or the system could not maintain its equilibrium.

  So far so good; the logic and math are so simple he can hardly have made a mistake, but he also can hardly have gained much deductive power. Then, a crucial insight: “We shall distinguish here two types of transitions.” When the molecule is in any of its higher-energy, “excited” states, even if there is no radiation at all present, he assumes that there is still some chance that it will emit radiation “without external influence. One can hardly imagine it to be other than similar to radioactive reactions.” What a leap! Radioactive decay was a mysterious nuclear phenomenon that appeared to be completely random. A radioactive substance has a certain half-life, that is, the period of time during which on average half of the nuclei emit radioactive particles. But for any specific nucleus all one could say was that the probability was one-half that it would not decay in one half-life and one-fourth that it wouldn’t decay in two, etcetera. The actual time and direction of decay was (and is still) unpredictable. Now Einstein is claiming that at least some of the events in which atoms emit light are just like this, are what we now call “spontaneous emission” events, and he writes down exactly the same mathematical rule for the number of spontaneous emission events per unit time as for radioactive decay.

  He then considers other emission events of a more conventional sort, which we now call “stimulated emission”; the number of these events depends on the amount of radiation that is already present, which means that in equilibrium this number will be proportional to the blackbody radiation density. These events would have been more familiar to his readers because in classical electrodynamics one pictures the radiation field as “driving” or “being driven by” the electron charges in the atom, increasing or decreasing the amount of energy contained in the electron’s orbital motion. When the radiation field subtracts energy from the electrons, we have “stimulated emission”; when it adds energy to the electrons, we have absorption. Whew, at least some of this sounds familiar, but … he completely throws out the classical method of calculating absorption and emission. Instead he treats these processes as random too.

  Now all he has to do is balance out the various processes; the energy lost by the molecules in spontaneous and stimulated emission must on average equal that gained by absorption. By one last sleight of hand he relates the rate of stimulated emission to that of absorption, simplifying the equation of balance. Two more lines of algebra and, Mein Gott, he has derived Planck’s radiation law! Einstein has ingeniously bypassed all the complicated counting and gone straight to the answer. He can’t resist giving himself a little pat on the back: “The simplicity of the hypotheses, the generality with which the analysis can be carried out so effortlessly, and the natural connection to Planck’s linear oscillator … seem to make it highly probable that these are the basic traits of a future theoretical representation.” In his jubilant letter to Besso a few weeks later he is even more effusive: “A brilliant idea has dawned on me about radiation absorption and emission…. An astonishingly simple derivation, I should say the derivation of Planck’s formula. A thoroughly quantized affair.”

  Einstein was correct; his approach has become the derivation of Planck’s formula. It is completely valid within the modern theory of quantum mechanics and electrodynamics and is in fact the reasoning still used in the majority of textbooks. Einstein introduced two unknown constants of proportionality (one for the rate of spontaneous emission, denoted by A, and one for the rate of stimulated emission, denoted by B), and then used additional arguments to replace them with known constants. These new fundamental quantities can be calculated directly in the modern theory, but in homage to the master are still labeled the “Einstein A and B coefficients.”

  Einstein did not, however, rest on his laurels. Rederiving Planck’s law in a purely quantum framework was progress, but it did not in itself clarify the basic question of the existence of light quanta. He was still hunting for the resolution of the “spherical wave paradox,” that a classical point source emits a uniformly expanding spherical wave front, like the ripples a rock makes when dropped into a pond. Such waves seemed to rule out conceptualizing atomic emission as the release of a localized particle of light, which flies out in a particular direction. It was hard to visualize dropping a rock in a pond and having a single bump of water move out in a specific direction. However, it wasn’t obvious that single atoms really behaved like classical point sources; maybe, he thought, the classical viewpoint was just wrong. Perhaps real atomic emission was a directed process, emitting “bumps” of light in specific directions; only when there were many atoms randomly emitting in all directions (or one emitting repeatedly) would it appear spherical.

  A few weeks after his first 1916 paper Einstein realized, to his delight, that his new hypotheses of quantum emission allowed him to prove just this fact, and he eagerly drafted a second paper containing this demonstration. When he wrote to Besso on August 11, crowing about having found “the derivation,” he added, “I am writing the paper4 right now.” In a follow-up letter to Besso two weeks later he added, “it can be demonstrated convincingly that the elementary processes of emission and absorption are directed processes.”

  So what was the new insight that so excited him? Einstein had realized that he could return yet again to the well that had yielded the theory of Brownian motion of particles in suspension and of radiation energy fluctuations. In this famous second quantum paper of 1916, titled “On the Quantum Theory of Radiation,” he reviews his elegant new derivation of Planck’s law, stating, “this derivation deserves attention not only because of its simplicity, but especially because it seems to clarify somewhat the still unclear process of emission and absorption of radiation by matter.” Now, he says, we must go beyond just cons
iderations of energy exchange. “The question arises: does the molecule receive an impulse (i.e. a push in a specific direction) when it absorbs or emits the energy, hυ? … It turns out that we arrive at a theory which is free of contradictions, only if we interpret those elementary processes as completely directed processes [italics in the original]. Herein lies the main result of the following considerations.”

  We have already seen that radiation exerts pressure even in classical electrodynamics and thus can push on matter, that is, transfer momentum to matter. Einstein had used this fact in his Salzburg “sliding mirror” thought experiment. The effect is similar to the recoil that occurs if you fire a gun; a light wave emitted in a specific direction causes the emitting atom to recoil in the opposite direction. Similarly, the analogue of absorption of light waves by an atom is the unfortunate process of “absorbing” an incoming bullet, which among the less problematic of its effects causes the “absorber” to be pushed in the direction of motion of the bullet. However, if the atom emitted a spherical light wave, the recoil pressure would be equal in all directions and no net momentum would be transferred. One can picture this by imagining a platoon of soldiers on a raft. If they all line up and fire their rifles in the same direction, then the recoil pushes the raft strongly in the opposite direction, but if they form a circle and fire outward at the same time, the raft will not move. (They could also form a circle and all fire inward, and again the raft would not move, but that would have other effects.) So the consequences of directed as opposed to undirected (spherical) emission are different, and in the process of energy exchange between molecules and radiation that Einstein is discussing, different motions of the molecules occur, depending on whether one assumes the emission and absorption is directed or undirected (“isotropic”).

  To prove that each molecular interaction with radiation is a directed process, he makes the following argument. Imagine a gas molecule moving around in an enclosure filled with radiation, both of which are at the same temperature, T (this is the condition for thermal equilibrium). For the radiation this means that its energy density will be described by a universal radiation law, which depends on T and on the frequency, υ. Einstein is not here assuming that this law is given by the (now) well-known Planck formula; his goal here is to derive the Planck law in a new way, based on Bohr’s quantum atom and general considerations from statistical physics. In this effort he assumes that the atoms in the gas have a kinetic energy given by our old friend the equipartition theorem, which he knew failed for the vibrational energy of molecules but was well confirmed for the energy associated with the free motion of atoms in a gas.5