Einstein and the Quantum

by Stone, A. Douglas

  Thus matter is continually emitting and absorbing radiation—all objects are glowing, whether we can see their radiation or not. What determines if we can see it is the temperature of the object; at room temperature objects glow with primarily thermal (infrared) radiation, a wavelength that our eyes can’t see (except with “night-vision” goggles). The red glow of heated metal appears when the metal becomes hot enough to emit just a little of its EM radiation as visible light. The surface of the sun, which is even hotter, emits most of its radiation at visible wavelengths.

  The central problem of the physics of heat, the one that Max Planck had worked on for the past five years, was to understand and predict precisely, with a mathematical formula, the amount of electromagnetic energy coming out of an object of a given temperature at each wavelength. This formula is the law of thermal radiation; physicists had known such a formula should exist for over three decades, but finding the correct law and understanding it theoretically had frustrated the best minds of the era. Einstein himself commented somewhat later, “It would be edifying if the brain matter sacrificed by theoretical physicists on the altar of this universal [law] could be put on the scales; and there is no end in sight to this cruel sacrifice!” In 1899, roughly a year and a half earlier, Planck thought he had found the answer, and had proudly announced his conclusions to the very same audience he was scheduled to address this evening. At that earlier meeting he had derived mathematically the equation that generated a universal curve, or graph, with temperature on the horizontal axis and EM energy on the vertical.

  The current speaker, Kurlbaum, was presenting his and Rubens’s measurements of just this curve, as Planck waited in the audience to respond. The data made a neat straight line, showing that the infrared energy radiated by an object increased proportional to the increasing temperature. On the same graph the prediction of the Planck-Wien law was plotted, giving a rainbow-shaped curve with not even a passing resemblance to the actual measured data points.

  Planck had known that this moment was coming. Rubens was a personal friend, and he and his wife had visited Planck twelve days earlier for Sunday lunch. As physicists are wont, Rubens began talking shop and informed Planck that the law of thermal radiation that Planck had defended ardently for the past two years was badly out of agreement with their new data, which instead showed an intriguing linear variation with temperature. It was on this rather dramatic failure of his theory that Herr Planck would soon be asked to “comment.” Thus the impending discussion showed every sign of being exceedingly awkward.

  Planck was no longer a young man, although he was famously vigorous, and would climb mountains well into his seventies. At forty-two his hair was receding above his piercing eyes, and it sometimes pushed straight upward in an unruly shock. He had the bushy handlebar mustache sported by many of his Prussian colleagues and was dressed neatly in the academic style: white shirt with high collar, black bow tie and jacket, and pince-nez glasses. As a young man he had gone into science for the most idealistic of reasons: “my decision to devote myself to science was a direct result of the discovery that … pure reasoning can enable man to gain an insight into the mechanism of [natural laws]…. In this connection it is of paramount importance that the outside world is something independent from man, something absolute, and the quest for these laws … appeared to me as the most sublime scientific pursuit in life.” Early in his academic career he had been attracted to the science of heat, thermodynamics, since it is based on two absolute laws. The First Law states that heat is a form of energy, and the Second Law governs the flow of heat and the possibility of converting heat energy to do useful work, as in a steam engine. The Second Law employs the mysterious concept of entropy (roughly speaking, the amount of disorder in a physical system), and Planck had based his career on the interpretation and applications of this profound notion. That was why he was now in trouble.

  FIGURE 1.1. Original data showing measurements of blackbody radiation energy (vertical axis) as the temperature (horizontal axis) is varied while the frequency is fixed compared to different theories for the Radiation Law. The data points are represented by various types of symbols, with the different types of dashed lines represent different theories. The curve with the larger dashes outlined in gray represents Wien’s Law which disagrees strongly with the data. The small dashes represent an empirical formula of no historical importance. The dash-dotted line is the Raleigh-Jeans Law which works rather well for the long wavelength (low frequency) radiation measured in this experiment (but which fails in other experiments). The solid line is Planck’s Law, which fits the best and also works at higher frequencies. The graph is from 1901, shortly after Planck proposed his law; in October of 1900 he still believed that the Wien Law was correct. More details on the Radiation Laws are given in appendix 2.

  FIGURE 1.2. Max Planck in 1906, six years after he initiated the quantum revolution. Courtsey Archiv der Max-Planck-Gesellschaft, Berlin-Dahlem.

  Planck had not presented the Planck-Wien law of thermal radiation as a conjecture, based on provisional assumptions that he might revise. Quite the contrary. Little more than a year before, standing in front of the very same group of physicists, he had “proved” to them that this law followed from no other assumption than the Second Law of thermodynamics. With crushing certainty he had stated, “the limits of validity of this law coincide with those of the Second Law of Thermodynamics.” This was the heavy artillery; the Planck-Wien law was supposed to be as solid as the Second Law itself! Einstein, also an admirer of thermodynamics, has said it is “the only physical theory of universal content which, within the framework of the applicability of its basic concepts, I am convinced will never be overthrown” (and, so far, he has been right). So if Planck, the world’s expert, said that he had derived the law of thermal radiation directly from the Second Law, the case should have been closed. Unfortunately for Planck, the data disagreed.

  Thus, when Planck stepped to the podium that night, his aim was not scientific revolution but damage control. Nonetheless, he was a truth seeker; he was not willing to run away from unpleasant facts. Later he scorned the English theorist James Jeans for just such behavior: “He is the model of the theorist as he should not be…, [because he believes] so much the worse for the facts if they don’t fit.” Planck stood up and faced the music: “The interesting results of long wavelength spectral energy measurements … confirm the statement … that Wien’s energy distribution law is not generally valid…. Since I myself even in this Society have expressed the opinion that Wien’s law must be necessarily true, I may perhaps be permitted to explain briefly the relationship between the … theory developed by me and the experimental data.”

  The “relationship” between them of course is that the Planck-Wien theory is wrong; Planck could not quite bring himself to say that in his remarks. But he did identify a weak point in his earlier arguments and admitted that the Second Law of thermodynamics does not have enough power, on its own, to answer the question. There had to be some further new principle involved. Having lost his guideposts for the journey, but being under such intense pressure to come up with an answer, Planck did something highly uncharacteristic. Planck was not a man to leap impulsively into the unknown; by his own description he was “by nature … peacefully inclined, and reject[s] all doubtful adventures.” Nonetheless, on that October night he had decided to wing it. What followed was the most fateful improvisation in the history of science.

  Planck had been fortunate that his friend Rubens had given him warning of the failure of his theory. Moreover Rubens’s data provided a huge clue to what was wrong. Earlier experiments had shown that the Plank-Wien law worked very well for visible EM radiation emitted by very hot bodies, that is, for the shorter wavelengths. The new experiments of Rubens and Kurlbaum showed not only that the law failed for the longer, infrared wavelengths emitted by less hot objects, but also showed exactly how it failed. That nice, straight line in the data told Planck that at long wavelengths, co
ntrary to the prediction of the Planck-Wien law, the radiation energy must be proportional to temperature. To Planck the challenge was similar to filling in a line in a crossword puzzle for which the end of the word was known, and now someone had filled in the first letter for him, telling him his original guess was wrong. With a little inspired mathematical insight on the very Sunday night, twelve days earlier, that Rubens had warned him of the problem, Planck had guessed the correct mathematical formula for the law of thermal radiation. Now, at the meeting, he unveiled his new formula, soon to become immortalized as the Planck radiation law.2 Moreover he took the liberty of sketching how his new law compared to the Rubens-Kurlbaum data; it produced a line perfectly matching the data points. He concluded, “I should therefore be permitted to draw your attention to this new formula, which I consider to be the simplest possible, apart from Wien’s expression.”

  With this great leap of intuition Planck had achieved a draw, but not a victory. Theorists are not supposed to just guess the correct formulas to describe data; they are supposed to derive these formulas from the fundamental laws of physics, which at the time were Newton’s laws of mechanics and of gravity, Maxwell’s electromagnetic theory, and the laws of thermodynamics. For Planck’s new law to be anything more than a “curiosity” (as he himself put it), he would have to connect it to the more general laws of physics. As Planck himself said, “even if the absolutely precise validity of the radiation formula is taken for granted, so long as it had merely the standing of a law disclosed by a lucky intuition, it could not be expected to possess more than a formal significance. For this reason, on the very day when I formulated this law, I began to devote myself to the task of investing it with true physical meaning.”

  The details of Planck’s new reasoning will be fully explained later in our story. For the moment it suffices to say that after “some weeks of the most strenuous work” of his life, “some light came into the darkness,” and Planck again went before the German Physical Society to justify his radiation law. In the course of that presentation, on December 14, 1900, he uttered two sentences of incalculable significance for humankind:

  We consider, however—this is the most essential point of the whole calculation—[the energy] E to be composed of a very definite number of equal parts and use thereto the constant of nature h = 6.55 × 10−27 erg-sec. This constant, multiplied by the frequency ν … gives us the energy element, ε.

  Planck showed that, from this assumption and the then-controversial statistical theory of atoms, his new law of thermal radiation followed. But with this cryptic phrase natural science had crossed a philosophical Rubicon: ultimately the exquisitely sharp Newtonian photograph of the natural world would fall out of focus, becoming blurred and uncertain. The even flow of natural processes would give way to an atomic world of sudden jumps and collapses. Light itself would become grainy, belying its wave properties, so brilliantly wrung from the nineteenth-century triumphs of Maxwell and others. And all who look to science to elucidate the universe would have to get used to a worldview that sanctioned “spooky action at a distance,” the modern quantum view of reality.

  Planck’s insight was beyond brilliant; it was an act of genius. The new law he introduced will bear his name and will be used by scientists as long as there is technologically advanced human civilization. In fact, as far as we know, it may be in use right now in nonhuman civilizations.3 The theory that arose from this insight, the quantum theory, is unquestionably the most important theoretical advance in physical science since Newton.

  So by December 1900 Planck had changed everything in physics and chemistry. The only problem was he didn’t realize it. Planck was still recovering from his near-death experience as a reputable theorist. He later described his arrival at the quantum hypothesis as “an act of desperation.” Now he breathed a deep sigh of relief and put the “energy element” out of his mind: “I considered the [quantum hypothesis] a purely formal assumption, and I did not give it much thought except for this: that I had obtained a positive result under any circumstances and at whatever cost.” And the entire physics community went along with this denial, like a family with an unspoken agreement to never again discuss a traumatic event.

  Although he didn’t realize it, Planck had removed a foundation stone from the edifice of classical physics; it would take another twenty-five years for the entire structure to collapse. However, the immediate reaction was … nothing. For the next five years neither Planck nor any of the great physicists of the era took up the meaning and extension of Planck’s ideas. Not the revered Hendrik Lorentz in Holland, nor the profound but impenetrable Ludwig Boltzmann in Vienna, nor any of Planck’s close colleagues picked up the challenge. That was left to a twenty-five-year-old patent examiner and maverick theorist living in Bern, Switzerland. Like Planck, thermodynamics and statistical mechanics had been his first love as a physicist. In addition he had been fascinated from an early age by Maxwell’s equations and EM radiation. Unlike Planck however, he had been rejected by academe and had no reputation to lose. He was on the verge of taking the leap that Planck and the other great physicists of the time had not even considered. He was about to give Planck’s radiation law the most radical interpretation possible: that it implied the discontinuity of motion on the atomic scale. He would begin this uprising with a paper that he himself termed “revolutionary.” His name was Albert Einstein.

  1 Electromagnetic radiation is not the only way heat is transmitted over distances; quite commonly a hot body (e.g., a heating coil in your stove) directly heats the air, which, as it moves around (“convects”), comes in contact with other bodies and heats them up.

  2 Moreover, the old terminology referring to the incorrect law, the “Planck-Wien law,” was quickly adjusted to simply the “Wien law,” erasing from the physics canon the evidence of Planck’s original error.

  3 Surprisingly, the staid Planck had some things to say about extraterrestrials, as we will see in chapter 14.

  CHAPTER 2

  THE IMPUDENT SWABIAN

  We do not know the exact moment when Heinrich Weber began to despise Albert Einstein. It definitely was not at first sight. Professor Weber was the head of the Physics Department of the Federal Institute of Technology, an up-and-coming engineering school in Zurich, Switzerland, now known worldwide as ETH Zurich. In 1895, when Weber and Einstein first met, the “Poly” (as it was called by the locals) had the immense advantage for the young Einstein that it did not require a high school diploma for admission. This was particularly pertinent for Einstein because he had rather recently and without the consent of his parents “excused himself” from the final two years of his well-regarded German high school (the Luitpold Gymnasium in Munich) on the basis of a nebulous medical condition, “neurasthenic exhaustion.” In fact he had hated the school, and once his parents had left Munich for Italy for financial reasons he saw no reason to stick it out. In late December of 1894 the fifteen-year-old Einstein showed up on their doorstep in Milan and “assured them most resolutely” that he would self-school himself in order to qualify for admission to the Zurich Poly by the next fall.

  Indeed Einstein was already an accomplished autodidact, having taught himself differential and integral calculus well ahead of the school curriculum; to qualify for the Poly he had taken the precaution of obtaining a letter of advanced mathematical achievement from his teacher in Munich. Armed with this certificate, he presented himself to Albin Herzog, principal of the Zurich Poly, in October of 1895, as a “prodigy” who should be allowed to take the entrance exams a full year and a half before he would attain the required minimum age. It was at this time that he encountered Professor Weber, a reserved and dignified scientist, who, while not a physicist of historic stature, was a respected experimental researcher in thermodynamics.

  On the entrance exams Einstein confirmed the judgment of his mathematics teacher, performing brilliantly on the math and physics portions of the test. However, he was neither fond of nor talented in subjects
requiring a great deal of memorization, so he failed the general sections of the exam, which covered subjects such as literature, French, and politics. He thus failed to gain admission to the Poly. Yet his strong showing in math and physics so impressed Weber that he invited Einstein, against regulations, to attend his own lectures for second-year physics students. But there was still the minor matter of qualifying for admission, which could not be met by auditing lectures in the subjects where Einstein already excelled. So, at the suggestion of Herzog, Einstein enrolled in the cantonal high school in nearby Aarau for an additional year of formal schooling. He thrived there, finishing first in the final exams and gaining automatic admission to the Poly in October of 1896.

  It was then that his intense relationship with Weber began. Weber was the primary physics instructor, and Einstein took fifteen courses from him, ten in the classroom and five in the lab. He did well in all of them. His very first physics course was with Weber, who immediately impressed him. “Weber lectured on heat … with great mastery. I am looking forward from one of his lectures to the next,” Einstein wrote in 1898 to his fellow student and future wife, Mileva Maric. Einstein had been fascinated with physics since he was a young boy, beginning with his experience of “wonder” at a compass that he received at age five, which revealed to him the existence of unseen forces. Initially at the Poly his youthful love of physics was nurtured, and he responded with a strong academic performance: at the end of his first two years he passed the intermediate diploma exam with an overall grade of 5.7 out of 6, placing him first in his class.