Einstein and the Quantum

by Stone, A. Douglas

  What were the “serious difficulties” of which Lorentz spoke? They arose because of the conflict between two basic categories in the physics of the time: waves and particles. Newton’s laws had introduced the idea of mass, the quality of a material that resists change of motion and that responds to and generates gravitational attraction. Although atomic theory was in its infancy, the idea that everyday massive objects were made of smaller, more fundamental building blocks (i.e., atoms and molecules) had been widely used by physicists since the time of Maxwell and was winning the day by 1905. Atoms were the fundamental particles of physics, although soon to be understood as made up of protons, neutrons, and electrons. (Modern physics has added many other particles to this classification and further subdivided the atomic nucleus into quarks). So the idea that macroscopic “stuff” (solids, liquids, and gases) is an aggregate of atomic-scale particles was commonplace at the time of Einstein’s early work.

  Moreover, it was clear that when these particles aggregate in large quantities, such as in an ocean or in the atmosphere, they create media in which disturbances can propagate, disturbances that in these cases are called water waves or sound waves. It is important to realize that the particles of water or air are the substrate in which the waves propagate; it not possible to generate such “classical” waves without their substrate (in space no one can hear you scream). In the classical view, the fundamental objects are the particles that make up the medium, and the waves are derivative objects. Moreover, it is critical to understand that these waves are collective phenomena; waves move through the medium, but the particles don’t. Waves are not a whole bunch of particles moving in the same direction.

  This is illustrated nicely by a new kind of wave, discovered sometime in the 1980s, which we will refer to as “fan waves.” In the classical fan wave the particles are the sports fans in a stadium, who, because of boredom or some other stimulus, spontaneously generate collective motion. In an ideal, fully developed, clockwise fan wave all the fans from the first to last row in the upper deck stand up and then sit down in a brief period (about two seconds), causing, by poorly understood interactions, the fans immediately next to them on their left to do the same thing immediately thereafter. This disturbance in the crowd then propagates around the stadium, creating a nice visual effect, until it is damped out either by loss of synchronization or loss of interest. Anyone who has contributed to such a wave realizes that the particles (the fans) do not move in the direction of the wave; they just bob up and down. It is the disturbance, the wave, that propagates, not the “particles” of the medium.

  To this extent fan waves are typical classical waves in a medium. To complete the analogy, however, we will have to embellish a bit on the conventional fan wave to allow for a further critical feature, interference of waves. Imagine that all the fans are standing already (it is a particularly exciting moment in the game) and can make two different kinds of waves, by either raising their arms above their heads or lowering them down to their knees, as in a revival meeting. Also allow for the possibility that waves can go either clockwise or counterclockwise. You look to your left or right, and if the fan next to you raises his hands, you do the same; if he lowers them you do the same as well. Now some wise guy starts a clockwise wave by raising his hands, and his friend behind him starts simultaneously a counterclockwise wave by lowering his hands. These two waves propagate around the stadium in opposite directions at the same speed, and so halfway around they meet. At the column where they meet, the fans on the right raise their hands just as the fans on the left lower theirs: the fans in the middle don’t know what to do. So they do nothing. The two waves have met up, “out of phase” as the physicists say, and they cancel each other out.

  This is a somewhat fanciful illustration of the interference of waves, which are extended disturbances in a medium, having both an amplitude (how big the wave is at any given point) and a phase (how close the wave is to a peak or a trough at any given point). Depending on their phases, waves interfere and are larger where the crests coincide and smaller (or zero) where a crest and peak coincide. This is the sine qua non of a wave. But note that when we have destructive interference and two waves cancel out, the particles of the medium are still there (i.e., the fans in our example); they are just undisturbed. Waves are disturbances, so they can be positive or negative and can cancel each other out; you can add one and one and get zero. Particles cannot. (Two fans claiming a single seat will create destructive interference, but of a different kind.)

  This was how all waves were conceived of until 1905. However, a major challenge to this understanding was implicit in Maxwell’s discovery of electromagnetic waves. Here the propagating disturbance was an electric and magnetic field, but there was no obvious medium in which it could propagate. Physicists since the time of Newton had hypothesized that heat and possibly even light propagated through a transparent medium known as the “ether.” Maxwell’s discovery now confirmed its existence and that it was the substrate through which all EM waves propagated.

  The absolute necessity for such a medium was so evident that Heinrich Hertz, the first to demonstrate reception and transmission of radio waves, expressed it thus: “Take electricity out of our world and light vanishes; take the luminiferous ether out of our world and electric and magnetic fields can no longer travel through space.”

  This medium was, however, highly problematic. Despite its stubborn invisibility, it had to be all-pervasive, since apparently EM waves could propagate everywhere. It couldn’t have much (if any) mass, because it would then have gravitational effects, which were not in evidence. And since the earth moves in different directions at different times of year, the velocity of light on earth should vary in some manner, just as a water wave appears to move more slowly to a boat moving in the same direction. However, experiments testing the speed of light showed no hint of this effect.

  But what choice did one have other than postulating an ether? Try creating a fan wave in an empty stadium. You don’t have to be an Einstein to see that you can’t have the wave without the medium. It turns out, however, that you do have to be Einstein to suggest that you can have the wave without the medium.

  As noted above, the familiar waves of classical physics, which propagate as a disturbance in a medium, look different to an observer moving with respect to that medium. The surfer on the crest of a water wave sees an almost stationary wall of water roiling around him. By the same logic the young Einstein, in his school days at Aarau, had imagined moving along next to a light wave at the speed c and seeing a stationary electric field that no longer oscillated. This apparently conceivable physical situation made no sense to him: “But such a thing does not seem to exist, either on the grounds of experience or according to the Maxwellian equations.” The leading theorists of the time, Lorentz and the French mathematical physicist Henri Poincaré, grappled with this conundrum and, while making major mathematical advances, kept the physical picture of electromagnetic waves tied to the ether. Einstein also pondered this puzzle on and off during his student years and afterward, and it was finally in May of 1905, two months after he had submitted his paper on light quanta, that the answer came to him: if time itself were not absolute but “flowed” differently for observers in uniform relative motion, then all the apparent contradictions could be resolved!

  This was the key idea of Einstein’s “rough draft” on the “electrodynamics of moving bodies which employs a modification of the theory of space and time” that he spoke of in his vivacious letter to Habicht in May 1905. Within two months this idea has been developed into his famous paper on what became known as the special theory of relativity. This work has received much attention in the literature, and we will not review it here except to quote part of one critical sentence: “The introduction of a ‘light ether’ will prove superfluous, inasmuch as … no ‘space at absolute rest’ endowed with special properties will be introduced.”

  It is important to understand that the theory of specia
l relativity is completely independent of quantum theory and can be seen (along with the later general theory of relativity) to be the culmination of classical physics and its deterministic worldview. Special relativity makes sense of classical Maxwellian electromagnetic waves, and it does so without the somewhat embarrassing, unobservable ether. Two months after undermining Maxwell’s equations with his heuristic theory of light quanta, he vindicates them from a host of experimental challenges by banishing the ether. Talk about creative tension.

  On the other hand, by getting rid of the ether, it was clear that Einstein was now prepared to accept waves that do not travel in a medium, “fundamental waves.” A few years later he made this thinking explicit: “one can obtain a satisfactory theory only if one drops the ether hypothesis. In that case the electromagnetic fields which constitute the light will no longer appear to be states of a hypothetical medium, but rather independent entities emitted by the source of light.” Since EM waves were, from this point of view, a completely new kind of entity in physics, perhaps they could be something in between a classical particle and a classical wave, as suggested by Einstein’s notion of light quanta. En masse they exhibited interference like classical waves and hence could cancel one another out, but when exchanging energy with matter they acted like localized particles. Einstein was willing to entertain this contradiction. He, of all the physicists of his time, was the only one to really imagine that these two apparently conflicting concepts could be married. Over the next six years Einstein would devote the main part of his energies to consummating this difficult marriage.

  CHAPTER 11

  STALKING THE PLANCK

  “The three of us are fine, as always. The little sprout has grown into quite an imposing impertinent fellow. As for my science, I am not all that successful at present. Soon I will reach the age of stagnation and sterility when one laments the revolutionary spirit of the young. My papers are much appreciated and are giving rise to further investigations. Professor Planck (Berlin) has recently written to me about that.”

  Thus the twenty-seven-year-old Einstein wrote to his former Olympia Academy colleague Maurice Solovine in April 1906, in the after-math of his miracle year. The “little sprout” he spoke of was his first son,1 Hans Albert, who had been born on May 14, 1904, and was now coming up to his second birthday. At this point Einstein remained virtually unknown to the wider physics community, having personally encountered only a handful of physicists during his studies at the Poly and in subsequent research, none of whom were eminent theorists. He was working steadily eight hours a day, six days a week at the patent office, describing himself (with characteristic saltiness) to a friend as “a respectable Federal ink pisser with a decent salary.” His salary had recently grown a bit more respectable. In April of 1905 (in the midst of his creative epiphanies) he had again submitted a dissertation for a PhD to the University of Zurich, choosing his safest work of the moment as the topic, that on irregular molecular movement (“Brownian motion”) and the determination of Avogadro’s number. This time Kleiner and the committee accepted the thesis, and as a consequence he was promoted to technical expert second class at the patent office, with a 15 percent increase in salary.

  FIGURE 11.1. Albert Einstein in 1904 with his wife, Mileva Maric, and his young son, Hans Albert. ETH-Bibliothek Zurich, Image Archive.

  While Einstein personally was unknown to the great men of theoretical physics, his work had already, barely a year later, made a significant impact. As the above quotation makes clear, Planck had already written to tell him his work was very much appreciated (although Planck’s letter has not survived). One might have supposed that Einstein’s work on light quanta, relating as it did to the central achievement of Planck’s career, the blackbody radiation law, would have been the main object of Planck’s attention and appreciation. However this was not the case. Nothing is known of Planck’s reaction to Einstein’s quantum hypothesis until his letter of July 1907, quoted above, definitively rejecting the idea of light quanta in vacuum. In contrast, Planck immediately embraced the special theory of relativity; he, not Einstein, gave the first public lecture on the subject (crediting Einstein of course) in the fall of 1905 shortly after the theory was published in September of that year. (As the theory editor of Annalen der Physik, Planck naturally would have seen the paper when it arrived at the end of June.)

  Not only did Planck quickly publicize relativity theory, but he also paid it the highest compliment possible from a working scientist: he redirected his research to the study and extension of the theory. In 1906 he published the first major contribution to relativity theory not due to Einstein, a proof that relativistic mechanics was compatible with the “principle of least action,”2 an alternative mathematical formulation of classical mechanics that was flexible enough to encompass the alterations of Newton’s laws required by relativity theory. From 1906 to 1908 all of Planck’s new research related to relativity theory. Because of Planck’s stature in the field, his immediate “lively attention” to relativity theory endowed it with a credibility and importance that it otherwise might not have achieved for some time. Einstein acknowledged this in a tribute to Planck in 1913 when he stated, “It is largely due to the determined and cordial manner in which [Planck] supported this theory that it attracted notice so quickly among my colleagues in the field.”

  That Planck reacted in this manner was completely consistent with his personality and philosophy of science; he recognized that relativity theory, as strange as it appeared to laypeople and to some physicists, in fact completed classical mechanics and made it compatible with Maxwell’s electromagnetic theory. Einstein himself described it as “simply a systematic development of the electrodynamics of Maxwell and Lorentz.” He frequently emphasized the continuity of relativity theory with earlier physical principles: “There is a false opinion widely spread among the general public that the theory of relativity is to be taken as differing radically from the previous developments in physics…. The four men who laid the foundations of physics on which I was able to construct my theory are Galileo, Newton, Maxwell, and Lorentz.” Special relativity was a theory a purist like Planck could love. Wayward quanta of light, propagating in vacuum but still interfering like waves, impugning the integrity of Maxwell’s equations; now that was a completely different matter. The best he could do was to forgive this impetuous genius a youthful indiscretion.

  Planck finally did address the light-quantum hypothesis in the letter of July 1907 quoted above (“[I] assume that processes in vacuum are described exactly by Maxwell’s equations”), but apparently only in response to repeated prodding by Einstein, whose own preceding letters to Planck have been lost. After stating his belief in the validity of Maxwell’s equations, and that the “quantum of action” (h) pertained only to the exchange of electromagnetic energy with matter, he continued, “but more urgent than this surely rather old question is at the moment the question of the admissibility of your relativity principle … as long as the proponents of the principle of relativity constitute such a modest little band as is now the case, it is doubly important that they agree among themselves” (italics added). Seven years after slipping discontinuity into physics through the back door, Planck still did not see this for the epoch-making event it was. In contrast, by the end of December 1906, Einstein had already realized that Planck’s quantum of action was not going to remain trapped in Pandora’s radiation cavity, and that the challenge it presented to the worldview of physicists was more fundamental than that of relativity theory.

  As mentioned above, in his 1905 paper on light quanta Einstein had sidestepped a direct confrontation with Planck by crediting the Planck radiation law with being “sufficient to account for all observations made so far” but basing all his conclusions on the Wien approximation to it, which is valid for short-wavelength radiation. His resolute insistence that the statistical mechanics of the time could only give an impossible answer—kT of energy for each allowed wavelength in the cavity, leading
to the ultraviolet catastrophe—indicates that at that time Einstein regarded Planck’s “derivation” of his radiation law, employing the artifice of the energy element, hν, as highly suspect if not downright incorrect. By March of 1906, almost exactly a year after publishing his revolutionary light-quantum hypothesis, he had apparently reconsidered this view, and submitted a paper arguing that the Planck formula requires the concept of light quanta.

  In a study published last year I showed that the Maxwell Theory of electricity in conjunction with the theory of electrons leads to results that contradict the evidence on black-body radiation. By a route described in that study I was led to the view that light of frequency ν can only be absorbed or emitted in quanta of energy3 hν…. This relationship was developed for a range that corresponds to the range of validity of Wien’s radiation formula.

  At the time it seemed to me that in a certain respect Planck’s theory of radiation constituted a counterpart [alternative] to my work. New considerations, which are being reported [here], showed me, however, that the theoretical foundation on which Mr. Planck’s radiation theory is based differs from the one that would emerge from Maxwell’s theory and the theory of electrons, precisely because Planck’s theory makes implicit use of the aforementioned hypothesis of light quanta.

  These are the opening words of the 1906 paper. Note here the consistency with his 1905 paper, beginning first with the statement that conventional theory leads to a blackbody radiation law in contradiction to Planck’s law (and for good measure he restates this incorrect law, which led to the ultraviolet catastrophe in the second section of the 1905 paper). Then he reiterates that his quantum hypothesis was based only on the Wien limit and not on the full Planck law. Finally he explicitly states that when he wrote his 1905 paper he believed that there was a tension, if not an outright contradiction, between the Planck law and the heuristic theory of light quanta. What had changed his mind on this?