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In his famous autobiography The Education of Henry Adams, published only a few years into the twentieth century, the historian Henry Adams shouted alarm that the sacred atom had been split. Since the ancient Greeks, the atom had been the smallest particle of matter, the irreducible and indestructible element, the metaphor for unity and permanence in all things. Then, in 1897, the British physicist J. J. Thomson found electrons, particles far lighter and presumably smaller than atoms. The next year, Marie Sklodovska (Madame Curie) and her husband Pierre Curie discovered that the atoms of a new element, called radium, continuously hurled out tiny pieces of themselves, losing weight in the process. Now, nothing was permanent — nature no more than human civilizations. The solid had become fragile. Unity had given way to complexity. The indivisible had been divided.
As Adams was summing up the nineteenth century, he was evidently unaware of another scientific bombshell that had just exploded, ultimately as earthshaking and profound as the fracturing of the atom. On December 14, 1900, in a lecture to the stodgy German Physical Society in Berlin, Max Planck proposed the astounding idea of the quantum: energy does not exist as a continuous stream, which can be subdivided indefinitely into smaller and smaller amounts. Rather, he suggested, there is a smallest amount of energy that can be divided no further, an elemental drop of energy, called a quantum. Light is an example of energy. The seemingly smooth flood of light pouring through a window is, in reality, a pitter-patter of individual quanta, each far too tiny and weak to discern with the eye. Thus began quantum physics.
At the time of his lecture, Planck was bald from the middle of his head forward, with a sharp aquiline nose, a mustache, a pair of spectacles fastened to his face, and the overall look of a dull office clerk. He was forty-two years old, almost elderly for a theoretical physicist. Newton had been a youth in his early twenties when he worked out his law of gravity. Maxwell had polished off electromagnetic theory and retired to the country by age thirty-five. Einstein and Heisenberg would be in their mid-twenties when they erected their great monuments.
In 1900, Planck was already established as one of the leading theoretical physicists in Europe. Planck himself had helped legitimize the discipline. Fifteen years earlier, when he secured the rare position of professor of theoretical physics at the University of Kiel, theoretical science was considered an impotent profession, inferior to laboratory experiments. Few students clamored to hear Planck’s mathematical lectures. Then, in 1888, after his studies of heat — in which he clarified the Second Law of Thermodynamics and the concept of irreversibility — Planck was appointed professor at the University of Berlin. At the same time, he was made director of the new Institute for Theoretical Physics, founded mainly for him.
At the end of the nineteenth century, physics basked in the glow of extraordinary achievement. Newton’s precise laws of mechanics, which described how particles respond to forces, together with Newton’s law for gravity had been successfully applied to a large range of terrestrial and cosmic phenomena, from the bouncing of balls to the orbits of planets. The theory of heat, called thermodynamics, had reached its climax with the melancholy but deep Second Law of Thermodynamics: an isolated system moves inexorably and irreversibly to a state of greater disorder. Or, equivalently, every machine inevitably runs down. All electrical and magnetic phenomena had been unified by a single set of equations, called Maxwell’s equations after the Scottish physicist James Clerk Maxwell, who completed them. Among other things, these laws demonstrated that light, that most primary of natural phenomena, is an oscillating wave of electromagnetic energy, traveling through space at a speed of 186,282 miles per second. The new areas of physics known as statistical physics and kinetic theory had shown that the behavior of gases and fluids could be understood on the basis of collisions between large numbers of tiny objects, assumed to be the long-hypothesized but invisible atoms and molecules. In short, as Planck scribbled his equations at the dawn of the new century, physics might survey its vast kingdom and be pleased.
Some cracks, however, were starting to show in the marble facade. Aside from the philosophical dismay expressed by Mr. Adams, Thomson’s electron was clearly a new type of matter that demanded explanation and raised other questions about the innards of atoms. The “radioactive” disintegrations observed by the Curies involved the unleashing of huge quantities of energy. What was the nature of this energy and where did it come from? Other emissions of electromagnetic radiation from atoms, the so-called atomic spectra, exhibited surprising patterns and regularities but with no theoretical understanding. Equally perplexing were the repeating patterns in the properties of the chemical elements, a phenomenon that scientists suspected was caused by the structure of atoms.
Finally, physicists had observed that a unique kind of light, called black-body light or black-body radiation, emerged from all hot, blackened boxes held at constant temperature. (Set a kitchen oven at some temperature, leave the oven door closed for a long time, and black-body radiation will develop inside — although at any practical cooking temperature this light will be below the frequencies visible to the human eye.) It was already well known to scientists that all hot objects emit light — that is, electromagnetic radiation. In general, the nature of such light varies with the properties of the hot object. But if the radiating object is additionally enclosed within a box and held at constant temperature, its light assumes a special and unvarying form, the so-called black-body radiation.
A particularly mysterious aspect of black-body light was that its intensity and colors were completely independent of the size, shape, or composition of the container — as surprising as if human beings all over the world, upon being asked a question, uttered the same sentence in reply. A heated black box made of charcoal and shaped like a cigar produces precisely the same light as a black box made of dark tin and shaped like a beach ball, provided that the two boxes have the same temperature. The known laws of physics could not explain black-body light. Even worse, the standard working theories of light and of heat actually predicted that a blackened box held at constant temperature should create an infinite amount of luminous energy! It was the puzzle of black-body radiation that Max Karl Ernst Ludwig Planck had solved for his lecture of December 14, 1900.
A great deal was already known of the subject. With the use of colored filters and other devices, scientists had measured how much energy there was in each frequency range of black-body light. A colored filter allows light of only a narrow range of frequencies to pass through it. (The frequency of light is the number of oscillations per second. Each frequency of light corresponds to a particular color, just as each frequency of sound corresponds to a particular tone.) The amount of energy in a given frequency range of light is measured by a device called a photometer. Photometers gauge the intensity of light falling on a surface — a glass plate, for example — by comparing that light to another beam of light of known intensity. The comparison can be accomplished, for example, by the relative penetrating power of light through a liquid. More intense light beams have greater penetrating power. (Several decades into the twentieth century, light intensities could be measured more accurately by their electrical effects, with photoelectric detectors.)
The breakdown of a light source into the amount of energy in each range of frequency is called a light spectrum. When the light is black-body light, its spectrum is called a black-body spectrum. Figure 1.1 illustrates two black-body spectra, one for a temperature of 50 K and another for a temperature of 65 K. Here the K stands for Kelvin, the unit of temperature on the absolute temperature scale, which is a form of the Celsius scale with the zero point shifted. The coldest possible temperature lies at 0 K and -273 C.
A more familiar example of a spectrum is the graph that shows how many adults there are in each range of heights. Such a spectrum is usually a bell-shaped curve, with few people at very small heights and few people at very tall heights. As one would expect, the height spectrum varies from one country to the next, since human heights are determined by a large number of variables such as genetics and diet. So, it was remarkable when Planck’s predecessor to the Berlin chair, Gustav Kirchhoff, and others, discovered that the black-body spectrum does not vary at all with the details of the container. The black-body spectrum depends only on a single parameter, the temperature.
Planck was much impressed by the uniqueness and universality of the black-body spectrum, reasoning that such a universality must be the result of some fundamental new law of nature. A few weeks prior to his December lecture, the German physicist had in fact guessed a formula for the spectrum of black-body light. Planck’s formula was a mathematical expression for the amount of energy in each range of frequency of black-body light, and it agreed with all experimental measurements. Embracing the aesthetic criteria common to most physicists, Planck found pleasure in the simplicity of his formula, using the word “simple” (einfach in German) twice in the first paragraph of his paper.
But a mathematical formula, in itself, is only a tidy summary of quantitative results, like a sun calendar, which tells the number of daylight hours on each day of the year. Such a calendar is useful for making plans, but it does not explain why the numbers come out as they do. To know why, we need to know what causes day and night, we need to know that the earth spins on its axis at a certain rate, that the earth also orbits the sun at a certain rate, that the earth’s axis is tilted at a particular angle. When we know all of these things, we understand why. With such understanding, we could then predict the sun calendar for any planet anywhere in the universe, given the corresponding astronomical facts.
Planck was not satisfied with merely guessing the right formula for black-body light. What compelled him and haunted him was to answer the deeper question: Why? What fundamental, inviolable principles led to that formula, made it a logical necessity, required it and it alone out of all the possible einfachen formulas that one could imagine? Why was that same formula observed to be true over and over again, from one experiment to the next, even for experiments that had never been done?
To understand the why of his formula, Planck discovered that he had to reject centuries of physical thought that you could chop energy into smaller and smaller pieces indefinitely. Surprisingly, the world did not work in that way. Planck could explain his formula for black-body light only by the radical proposal that there was a smallest piece of energy, called the quantum, which could not be chopped any further. Evidently, energy, like matter, came in granular form. The quantum was the grain of sand on the beach, the penny of currency in the subatomic world. The quantum was indivisible.
Planck was a theoretician, someone who works with pencil and paper and imagines experiments in his mind. To arrive at his conclusions, the German physicist imagined lots of atoms enclosed in a black box, all emitting and absorbing light. In such a situation, the atoms are affected by the surrounding light, and the surrounding light is affected by the atoms. Planck then discovered that if the atoms could absorb or emit energy only in whole chunks, quanta, then the resulting light would necessarily become black-body light.
For much of his life thereafter, Planck was amazed by the success of his quantum proposal. Like other theoretical physicists, he had an almost religious faith in the absolute validity of the laws of nature, which would, as he wrote in 1899, “retain their significance for all times and for all cultures including extraterrestrial and nonhuman ones.” For Planck, “the search for the absolute” was “the loftiest goal of all scientific activity.”
Yet in spite of Planck’s lofty views, he himself did not aspire to make great discoveries. As he told Philipp von Jolly, his professor at the University of Munich, he desired only to understand and perhaps deepen the existing foundations of physics. (In 1878, Jolly actually advised the twenty-year-old Planck not to continue with physics, on the grounds that all the fundamental laws had been discovered.) Planck’s cautious manner of “understanding” was to study a subject slowly and carefully, until he had mastered it. Such a conservative and modest approach seemed to grow naturally out of his background as the descendant of a long line of pastors, scholars, and jurists — Planck’s father, Wilhelm, was a professor of jurisprudence at Kiel and then Munich — and further to resonate with his loyal support of imperial Germany. Planck’s natural restraint carried over to his personal relationships. Marga von Hoesslin Planck, his second wife, wrote to another physicist that her husband was quite proper and reserved with anyone other than his family and could enjoy himself only with people of his own rank, with whom he might take a glass of wine and a cigar and even make a quiet joke.
There were two situations in which Planck abandoned his reserve: with his family and in music. As a young man he wrote to a friend, “How wonderful it is to set everything else aside and live entirely within the family.” Many years later, Marga confirmed this feeling in a letter to Einstein upon the death of her husband: “He only showed himself fully in all his human qualities in the family.” Planck’s other liberation was music. While a student at the University of Munich, he composed songs and a whole operetta; he served as second choirmaster in a school singing group; he played the organ at services in the student’s church; and he conducted. For the rest of his life, he played the piano superbly at small musical gatherings in his home. Music, according to Planck’s nephew-in-law Hans Hartmann, was the “only domain in life in which [Planck] gave his spirit free rein.”
Following Planck’s line of argument will help us understand how theoretical scientists think, how they use models, imagination, and logical consistency through mathematics. As it turns out, Planck’s paper on the quantum is one of the most conceptually difficult and abstract of any in this book, and the reader will need to exercise some patience and good humor. Planck begins his landmark paper by considering the material atoms that make up the inner walls of the blackened box. After all, these atoms are responsible for creating the observed black-body light, by emitting and absorbing electromagnetic radiation. He idealizes each of these atoms as a “monochromatic vibrating resonator,” that is, a system that emits and absorbs light at only a single frequency, say pure red or pure green. A concrete example of one of Planck’s monochromatic vibrating resonators would be an electron bouncing up and down, or “vibrating,” on a spring. As the electron bounces, it emits light of a particular frequency, the precise number of up-and-down bounces each second. Different frequencies correspond to different rates of bouncing, which in turn are determined by different stiffnesses of the springs. Black-body light is then hypothetically produced by a large number of these bouncing electrons at many different frequencies. All of these ideas are in accord with Maxwell’s equations of electromagnetism.