Light Quanta, March 1905
As Einstein noted to Habicht, it was the first of these 1905 papers, not the famous final one expounding a theory of relativity, that deserved the designation “revolutionary.” Indeed, it may contain the most revolutionary development in the history of physics. Its suggestion that light comes not just in waves but in tiny packets—quanta of light that were later dubbed “photons”—spirits us into strange scientific mists that are far murkier, indeed more spooky, than even the weirdest aspects of the theory of relativity.
Einstein recognized this in the slightly odd title he gave to the paper, which he submitted on March 17, 1905, to the Annalen der Physik: “On a Heuristic Point of View Concerning the Production and Transformation of Light.”8 Heuristic? It means a hypothesis that serves as a guide and gives direction in solving a problem but is not considered proven. From this first sentence he ever published about quantum theory until his last such sentence, which came in a paper exactly fifty years later, just before he died, Einstein regarded the concept of the quanta and all of its unsettling implications as heuristic at best: provisional and incomplete and not fully compatible with his own intimations of underlying reality.
At the heart of Einstein’s paper were questions that were bedeviling physics at the turn of the century, and in fact have done so from the time of the ancient Greeks until today: Is the universe made up of particles, such as atoms and electrons? Or is it an unbroken continuum, as a gravitational or electromagnetic field seems to be? And if both methods of describing things are valid at times, what happens when they intersect?
Since the 1860s, scientists had been exploring just such a point of intersection by analyzing what was called “blackbody radiation.” As anyone who has played with a kiln or a gas burner knows, the glow from a material such as iron changes color as it heats up. First it appears to radiate mainly red light; as it gets hotter, it glows more orange, and then white and then blue. To study this radiation, Gustav Kirchhoff and others devised a closed metal container with a tiny hole to let a little light escape. Then they drew a graph of the intensity of each wavelength when the device reached equilibrium at a certain temperature. No matter what the material or shape of the container’s walls, the results were the same; the shape of the graphs depended only on the temperature.
There was, alas, a problem. No one could fully account for the basis of the mathematical formula that would produce the hill-like shape of these graphs.
When Kirchhoff died, his professorship at the University of Berlin was given to Max Planck. Born in 1858 into an ancient German family of great scholars, theologians, and lawyers, Planck was many things that Einstein was not: with his pince-nez glasses and meticulous dress, he was very proudly German, somewhat shy, steely in his resolve, conservative by instinct, and formal in his manner. “It is difficult to imagine two men of more different attitudes,” their mutual friend Max Born later said. “Einstein a citizen of the whole world, little attached to the people around him, independent of the emotional background of the society in which he lived—Planck deeply rooted in the traditions of his family and nation, an ardent patriot, proud of the greatness of German history and consciously Prussian in his attitude to the state.”9
His conservatism made Planck skeptical about the atom, and of particle (rather than wave and continuous field) theories in general. As he wrote in 1882, “Despite the great success that the atomic theory has so far enjoyed, ultimately it will have to be abandoned in favor of the assumption of continuous matter.” In one of our planet’s little ironies, Planck and Einstein would share the fate of laying the groundwork for quantum mechanics, and then both would flinch when it became clear that it undermined the concepts of strict causality and certainty they both worshipped.10
In 1900, Planck came up with an equation, partly using what he called “a fortuitous guess,” that described the curve of radiation wavelengths at each temperature. In doing so he accepted that Boltzmann’s statistical methods, which he had resisted, were correct after all. But the equation had an odd feature: it required the use of a constant, which was an unexplained tiny quantity (approximately 6.62607 x 10–34 joule-seconds), that needed to be included for it to come out right. It was soon dubbed Planck’s constant, h, and is now known as one of the fundamental constants of nature.
At first Planck had no idea what, if any, physical meaning this mathematical constant had. But then he came up with a theory that, he thought, applied not to the nature of light itself but to the action that occurred when the light was absorbed or emitted by a piece of matter. He posited that the surface of anything that was radiating heat and light—such as the walls in a blackbody device—contained “vibrating molecules” or “harmonic oscillators,” like little vibrating springs.11 These harmonic oscillators could absorb or emit energy only in the form of discrete packets or bundles. These packets or bundles of energy came only in fixed amounts, determined by Planck’s constant, rather than being divisible or having a continuous range of values.
Planck considered his constant a mere calculational contrivance that explained the process of emitting or absorbing light but did not apply to the fundamental nature of light itself. Nevertheless, the declaration he made to the Berlin Physical Society in December 1900 was momentous: “We therefore regard—and this is the most essential point of the entire calculation—energy to be composed of a very definite number of equal finite packages.”12
Einstein quickly realized that quantum theory could undermine classical physics. “All of this was quite clear to me shortly after the appearance of Planck’s fundamental work,” he wrote later. “All of my attempts to adapt the theoretical foundation of physics to this knowledge failed completely. It was as if the ground had been pulled out from under us, with no firm foundation to be seen anywhere.”13
In addition to the problem of explaining what Planck’s constant was really all about, there was another curiosity about radiation that needed to be explained. It was called the photoelectric effect, and it occurs when light shining on a metal surface causes electrons to be knocked loose and emitted. In the letter he wrote to Mari right after he learned of her pregnancy in May 1901, Einstein enthused over a “beautiful piece” by Philipp Lenard that explored this topic.
Lenard’s experiments found something unexpected. When he increased the frequency of the light—moving from infrared heat and red light up in frequency to violet and ultraviolet—the emitted electrons sped out with much more energy. Then, he increased the intensity of the light by using a carbon arc light that could be made brighter by a factor of 1,000. The brighter, more intense light had a lot more energy, so it seemed logical that the electrons emitted would have more energy and speed away faster. But that did not occur. More intense light produced more electrons, but the energy of each remained the same. This was something that the wave theory of light did not explain.
Einstein had been pondering the work of Planck and Lenard for four years. In his final paper of 1904, “On the General Molecular Theory of Heat,” he discussed how the average energy of a system of molecules fluctuates. He then applied this to a volume filled with radiation, and found that experimental results were comparable. His concluding phrase was, “I believe that this agreement must not be ascribed to chance.”14 As he wrote to his friend Conrad Habicht just after finishing that 1904 paper, “I have now found in a most simple way the relation between the size of elementary quanta of matter and the wavelengths of radiation.” He was thus primed, so it seems, to form a theory that the radiation field was made up of quanta.15
In his 1905 light quanta paper, published a year later, he did just that. He took the mathematical quirk that Planck had discovered, interpreted it literally, related it to Lenard’s photoelectric results, and analyzed light as if it really was made up of pointlike particles—light quanta, he called them—rather than being a continuous wave.
Einstein began his paper by describing the great distinction between theories based on particles (such
as the kinetic theory of gases) and theories that involve continuous functions (such as the electromagnetic fields of the wave theory of light). “There exists a profound formal difference between the theories that physicists have formed about gases and other ponderable bodies, and Maxwell’s theory of electromagnetic processes in so-called empty space,” he noted. “While we consider the state of a body to be completely determined by the positions and velocities of a very large, yet finite, number of atoms and electrons, we make use of continuous spatial functions to describe the electromagnetic state of a given volume.”16
Before he made his case for a particle theory of light, he emphasized that this would not make it necessary to scrap the wave theory, which would continue to be useful as well. “The wave theory of light, which operates with continuous spatial functions, has worked well in the representation of purely optical phenomena and will probably never be replaced by another theory.”
His way of accommodating both a wave theory and a particle theory was to suggest, in a “heuristic” way, that our observation of waves involve statistical averages of the positions of what could be countless particles. “It should be kept in mind,” he said, “that the optical observations refer to time averages rather than instantaneous values.”
Then came what may be the most revolutionary sentence that Einstein ever wrote. It suggests that light is made up of discrete particles or packets of energy: “According to the assumption to be considered here, when a light ray is propagated from a point, the energy is not continuously distributed over an increasing space but consists of a finite number of energy quanta which are localized at points in space and which can be produced and absorbed only as complete units.”
Einstein explored this hypothesis by determining whether a volume of blackbody radiation, which he was now assuming consisted of discrete quanta, might in fact behave like a volume of gas, which he knew consisted of discrete particles. First, he looked at the formulas that showed how the entropy of a gas changes when its volume changes. Then he compared this to how the entropy of blackbody radiation changes as its volume changes. He found that the entropy of the radiation “varies with volume according to the same law as the entropy of an ideal gas.”
He did a calculation using Boltzmann’s statistical formulas for entropy. The statistical mechanics that described a dilute gas of particles was mathematically the same as that for blackbody radiation. This led Einstein to declare that the radiation “behaves thermodynamically as if it consisted of mutually independent energy quanta.” It also provided a way to calculate the energy of a “particle” of light at a particular frequency, which turned out to be in accord with what Planck had found.17
Einstein went on to show how the existence of these light quanta could explain what he graciously called Lenard’s “pioneering work” on the photoelectric effect. If light came in discrete quanta, then the energy of each one was determined simply by the frequency of the light multiplied by Planck’s constant. If we assume, Einstein suggested, “that a light quantum transfers its entire energy to a single electron,” then it follows that light of a higher frequency would cause the electrons to emit with more energy. On the other hand, increasing the intensity of the light (but not the frequency) would simply mean that more electrons would be emitted, but the energy of each would be the same.
That was precisely what Lenard had found. With a trace of humility or tentativeness, along with a desire to show that his conclusions had been deduced theoretically rather than induced entirely from experimental data, Einstein declared of his paper’s premise that light consists of tiny quanta: “As far as I can see, our conception does not conflict with the properties of the photoelectric effect observed by Mr. Lenard.”
By blowing on Planck’s embers, Einstein had turned them into a flame that would consume classical physics. What precisely did Einstein produce that made his 1905 paper a discontinuous—one is tempted to say quantum—leap beyond the work of Planck?
In effect, as Einstein noted in a paper the following year, his role was that he figured out the physical significance of what Planck had discovered.18 For Planck, a reluctant revolutionary, the quantum was a mathematical contrivance that explained how energy was emitted and absorbed when it interacted with matter. But he did not see that it related to a physical reality that was inherent in the nature of light and the electromagnetic field itself. “One can interpret Planck’s 1900 paper to mean only that the quantum hypothesis is used as a mathematical convenience introduced in order to calculate a statistical distribution, not as a new physical assumption,” write science historians Gerald Holton and Steven Brush.19
Einstein, on the other hand, considered the light quantum to be a feature of reality: a perplexing, pesky, mysterious, and sometimes maddening quirk in the cosmos. For him, these quanta of energy (which in 1926 were named photons)20 existed even when light was moving through a vacuum. “We wish to show that Mr. Planck’s determination of the elementary quanta is to some extent independent of his theory of blackbody radiation,” he wrote. In other words, Einstein argued that the particulate nature of light was a property of the light itself and not just some description of how the light interacts with matter.21
Even after Einstein published his paper, Planck did not accept his leap. Two years later, Planck warned the young patent clerk that he had gone too far, and that quanta described a process that occurred during emission or absorption, rather than some real property of radiation in a vacuum. “I do not seek the meaning of the ‘quantum of action’ (light quantum) in the vacuum but at the site of absorption and emission,” he advised.22
Planck’s resistance to believing that the light quanta had a physical reality persisted. Eight years after Einstein’s paper was published, Planck proposed him for a coveted seat in the Prussian Academy of Sciences. The letter he and other supporters wrote was filled with praise, but Planck added: “That he might sometimes have overshot the target in his speculations, as for example in his light quantum hypothesis, should not be counted against him too much.”23
Just before he died, Planck reflected on the fact that he had long recoiled from the implications of his discovery. “My futile attempts to fit the elementary quantum of action somehow into classical theory continued for a number of years and cost me a great deal of effort,” he wrote. “Many of my colleagues saw in this something bordering on a tragedy.”
Ironically, similar words would later be used to describe Einstein. He became increasingly “aloof and skeptical” about the quantum discoveries he pioneered, Born said of Einstein. “Many of us regard this as a tragedy.”24
Einstein’s theory produced a law of the photoelectric effect that was experimentally testable: the energy of emitted electrons would depend on the frequency of the light according to a simple mathematical formula involving Planck’s constant. The formula was subsequently shown to be correct. The physicist who did the crucial experiment was Robert Millikan, who would later head the California Institute of Technology and try to recruit Einstein.
Yet even after he verified Einstein’s photoelectric formulas, Millikan still rejected the theory. “Despite the apparently complete success of the Einstein equation,” he declared, “the physical theory on which it was designed to be the symbolic expression is found so untenable that Einstein himself, I believe, no longer holds to it.”25
Millikan was wrong to say that Einstein’s formulation of the photo-electric effect had been abandoned. In fact, it was specifically for discovering the law of the photoelectric effect that Einstein would win his only Nobel Prize. With the advent of quantum mechanics in the 1920s, the reality of the photon became a fundamental part of physics.
However, on the larger point Millikan was right. Einstein would increasingly find the eerie implications of the quantum—and of the wave-particle duality of light—to be deeply unsettling. In a letter he wrote near the end of his life to his dear friend Michele Besso, after quantum mechanics had been accepted by almost every living physicis
t, Einstein would lament, “All these fifty years of pondering have not brought me any closer to answering the question, What are light quanta?”26
Doctoral Dissertation on the Size of Molecules, April 1905
Einstein had written a paper that would revolutionize science, but he had not yet been able to earn a doctorate. So he tried one more time to get a dissertation accepted.
He realized that he needed a safe topic, not a radical one like quanta or relativity, so he chose the second paper he was working on, titled “A New Determination of Molecular Dimensions,” which he completed on April 30 and submitted to the University of Zurich in July.27
Perhaps out of caution and deference to the conservative approach of his adviser, Alfred Kleiner, he generally avoided the innovative statistical physics featured in his previous papers (and in his Brownian motion paper completed eleven days later) and relied instead mainly on classical hydrodynamics.28 Yet he was still able to explore how the behavior of countless tiny particles (atoms, molecules) are reflected in observable phenomena, and conversely how observable phenomena can tell us about the nature of those tiny unseen particles.
Almost a century earlier, the Italian scientist Amedeo Avogadro (1776–1856) had developed the hypothesis—correct, as it turned out—that equal volumes of any gas, when measured at the same temperature and pressure, will have the same number of molecules. That led to a difficult quest: figuring out just how many this was.
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