Einstein and the Quantum

by Stone, A. Douglas

  Maxwell’s achievement particularly captivated Einstein. Maxwell, Faraday, and Newton were the three physicists whose picture he had on the wall in his study later in life. Of Maxwell he wrote, “[the purely mechanical world picture was upset by] the great revolution forever linked with the names of Faraday, Maxwell and Hertz. The lion’s share of this revolution was Maxwell’s … since Maxwell’s time physical reality has been thought of as represented by continuous fields…. this change in the conception of reality is the most profound and fruitful that physics has experienced since the time of Newton.” Elsewhere he said, “Imagine his feelings when the differential equations he had formulated proved to him that electromagnetic fields spread in the form of … waves and with the speed of light”; and, “to few men in the world has such an experience been vouchsafed.”

  Maxwell had completed the second pillar of classical physics, what we now call classical electrodynamics, to go along with the first pillar, classical mechanics. But neither his nor Newton’s equations in themselves answered the fundamental question: what is the universe made of? One knew that there were masses and charges and forces and fields, but what were the building blocks of the everyday world? The enormous challenge was to extend these physical laws down to this conjectured “atomic” scale. Were there new, microscale forces not detectable at everyday dimensions? Did Newton’s and Maxwell’s laws still hold there? Were atoms little billiard balls with mass and electrical charge obeying classical mechanics and electrodynamics? Were there atoms at all, or were they just “theoretical constructs,” as many physicists and chemists maintained until the end of the nineteenth century?

  At the time of Maxwell there was no way to probe the internal structure of atoms or molecules directly. As Maxwell put it, “No one has ever seen or handled a single molecule. Molecular science therefore … cannot be subjected to direct experiment.” However physicists, led by Maxwell and Boltzmann, were beginning to use the atomic concept to explain in great depth the macroscopic behavior of gases. In doing so they were inferring properties of atoms and their interactions. This was the work that Einstein never forgave Herr Weber, his erstwhile mentor, for ignoring. It is here that Einstein first put his shoulder to the wheel.

  1 This wonderful incident may well be apocryphal, as there is no contemporaneous account of it.

  CHAPTER 5

  THE PERFECT INSTRUMENTS OF THE CREATOR

  “The Boltzmann is magnificent,” Einstein wrote to Maric in September of 1900. “I am firmly convinced that the principles of his theory are right, … that in the case of gases we are really dealing with discrete particles of definite finite size which are moving according to certain conditions … the hypothetical forces between molecules are not an essential component of the theory, as the whole energy is kinetic. This is a step forward in the dynamical explanation of physical phenomena.” Einstein was reading Boltzmann’s Lectures on the Theory of Gases. The Viennese physicist Ludwig Boltzmann and Maxwell had developed a theory of gases in the 1860s with much the same content, but with the difference that Boltzmann wrote long, difficult-to-decode treatises, while Maxwell’s work was much more succinct. Maxwell commented on this drily: “By the study of Boltzmann I have been unable to understand him. He was unable to understand me on account of my shortness, and his length was and is an equal stumbling block to me.” Einstein, despite the enthusiasm he expressed to his fiancée in 1900, was later to warn students, “Boltzmann … is not easy reading. There are very many great physicists who do not understand it.” It is likely that Einstein had no access to Maxwell’s work on gases in 1900, and as he did not read English until much later in life, he would not have been able to benefit from it anyway (in contrast, Maxwell’s electrodynamics was available to Einstein in German textbooks).

  Maxwell beautifully described the scientific advance he had made in atomic theory in an address to the Royal Society in 1873 titled, simply, “Molecules.”

  An atom is a body which cannot be cut in two. A molecule is the smallest possible portion of a particular substance. The mind of man has perplexed itself with many hard questions…. [Among them] do atoms exist, or is matter infinitely divisible? …

  According to Democritus and the atomic school, we must answer in the negative. After a certain number of sub-divisions, [a piece of matter] would be divided into a number of parts each of which is incapable of further sub-division. We should thus, in imagination, arrive at the atom, which, as its name literally signifies, cannot be cut in two. This is the atomic doctrine of Democritus, Epicurus, and Lucretius, and, I may add, of your lecturer.

  Maxwell goes on to describe how chemists had already learned that the smallest amount of water is a molecule made up of two “molecules” of hydrogen and one “molecule” of oxygen (here he has decided, somewhat confusingly, to use molecule to refer to both atoms and molecules). Then he arrives at his current research.

  Our business this evening is to describe some researches in molecular science, and in particular to place before you any definite information which has been obtained respecting the molecules themselves. The old atomic theory, as described by Lucretius and revived in modern times, asserts that the molecules of all bodies are in motion, even when the body itself appears to be at rest…. In liquids and gases, … the molecules are not confined within any definite limits, but work their way through the whole mass, even when that mass is not disturbed by any visible motion…. Now the recent progress of molecular science began with the study of the mechanical effect of the impact of these moving molecules when they strike against any solid body. Of course these flying molecules must beat against whatever is placed among them, and the constant succession of these strokes is, according to our theory, the sole cause of what is called the pressure of air and other gases.

  This simple picture, that gas pressure arises from the collisions of enormous numbers of molecules with the walls of the container, along with simple ideas of classical mechanics, allows Maxwell to derive Boyle’s law, that the pressure of the gas is proportional to its density. It also allows him to understand the observation that the ratio of volumes of any two gases depends only on the ratio of temperatures of the gases. The relation of temperature to volume of a gas is critical: in this view absolute temperature (what we now call the kelvin scale) is related to molecular motion and is proportional to the average of the square of the molecular velocity in a gas. Since the energy of motion for any mass, called kinetic energy, is just one-half its mass times the square of the velocity, this also means that for a gas its energy is just proportional to temperature. As Einstein had noted in his letter to Maric, in the Maxwell-Boltzmann theory, the entire energy of a gas is the kinetic energy of moving molecules. This principle of the Maxwell-Boltzmann theory, that the energy of each molecule is proportional to the temperature, applies even in the solid state, in which the molecules vibrate back and forth around fixed positions instead of moving freely throughout the substance. This property of the theory would perplex Einstein later, when he was trying to make sense of Planck’s radiation law.

  “The most important consequence which flows from [our theory],” Maxwell continues, “is that a cubic centimetre of every gas at standard temperature and pressure contains the same number of molecules.” This fact about gases was conjectured by the Italian scientist Amadeo Avogadro in 1811. In 1865 Josef Loschmidt, a professor in Vienna and later a colleague of Boltzmann, had estimated this actual number, which is very large: 2.6 × 1019, or roughly five billion squared. (This “Loschmidt number” is closely related to Avogadro’s number, which is the number of molecules in a mole of any gas—both Einstein and Planck were very interested in accurately determining these numbers). With all this information about gas properties, it was possible for Maxwell to determine the average velocity of a molecule in air. He found it to be roughly one thousand miles per hour. He described the implications most picturesquely:

  If all these molecules were flying in the same direction, they would constitute a wind blo
wing at the rate of seventeen miles a minute, and the only wind which approaches this velocity is that which proceeds from the mouth of a cannon. How, then, are you and I able to stand here? Only because the molecules happen to be flying in different directions, so that those which strike against our backs enable us to support the storm which is beating against our faces. Indeed, if this molecular bombardment were to cease, even for an instant, our veins would swell, our breath would leave us, and we should, literally, expire…. If we wish to form a mental representation of what is going on among the molecules in calm air, we cannot do better than observe a swarm of bees, when every individual bee is flying furiously, first in one direction, and then in another, while the swarm as a whole … remains at rest.

  Maxwell goes on to describe how his own experiments and others have determined that the molecules in a gas are continually colliding with one another, moving only about ten times their diameter before changing direction again through a collision, leading to a kind of random motion called diffusion. Because of this constant changing of direction, the actual distance moved from the starting point during a given time is much less than if the molecule were moving in a straight line. This explained why, when Maxwell took the lid off a vial of ammonia in the lecture, its characteristic odor was not immediately detected in the far reaches of the lecture hall. The same kind of diffusion occurs in liquids such as water, but much more slowly. Maxwell then throws off a poetic but profound comment: “Lucretius … tells us to look at a sunbeam shining through a darkened room … and to observe the motes which chase each other in all directions…. This motion of the visible motes … is but a result of the far more complicated motion of the invisible atoms which knock the motes about.” Exactly this process occurs to small particles suspended in a liquid but visible under a microscope, so-called Brownian motion. In one of his four masterpieces of 1905 Einstein would actually take the suggestion of Lucretius and Maxwell seriously and, by careful analysis, turn this into a precise method for determining Avogadro’s number! Experiments by the French physicist Jean Perrin would confirm Einstein’s predictions and determine that number very precisely; as a result Perrin received the Nobel Prize for Physics in 1926, long after his work had permanently put to rest doubts about the existence of atoms.

  The kind of complex, essentially random motion characteristic of gas molecules gave rise to a new way of doing physics, described by Maxwell in the same lecture. “The modern atomists have therefore adopted a method which I believe new in the department of mathematical physics, though it has long been in use in the section of statistics.” Thus was born the discipline of statistical mechanics. Maxwell could only assume that the invisible molecules obeyed Newtonian mechanics; he had no reason to doubt this. But in describing what would happen in a gas, he realized that one must inevitably encounter the weak point in Laplace’s grandiose dictum. Laplace had imagined an intellect that “at a certain moment would know all forces that set nature in motion, and all positions of all items of which nature is composed.” Maxwell realized that getting all the necessary information and using it to predict the future was an absurd proposition. “The equations of dynamics completely express the laws of the historical [Laplacian] method as applied to matter, but the application of these equations implies a perfect knowledge of all the data … but the smallest portion of matter which we can subject to experiment consists of millions of molecules, not one of which becomes individually sensible to us … so that we are obliged to abandon the historical method and to adopt the statistical method of dealing with a large group of molecules.” Maxwell’s point is that for all practical purposes one doesn’t want to know what each molecule is doing anyway; for example, to find the pressure exerted by a gas one needs only to know the average number of molecules hitting the wall of a container per second, and how much momentum (mass times velocity) they transfer to the wall.

  This was the key insight of Maxwell and Boltzmann: to predict the physical properties of a large aggregation of molecules, one needed only to find their average behavior, assuming they were behaving as randomly as allowed by the laws of physics. Calculating these properties was relatively easy for a gas, where most of the time the molecules are not in close contact; for liquids and solids it was much harder and in certain cases still challenges the physicists of the twenty-first century. Tied up with this insight was a new understanding of the laws of thermodynamics. The First Law says that heat is a form of energy, and that the total energy (heat plus mechanical) always stays the same (is “conserved”) even when one form is being changed into the other. For example, when a car is moving at 60 miles per hour, it has a lot of mechanical energy, specifically kinetic energy, ½mv2, where m is the mass of the car and v is its speed (60 mph in this case). When you slam on the brakes, that kinetic energy doesn’t disappear; it is turned into heat in your brakes and tires, due to friction. From the point of view of statistical mechanics, that heat is just mechanical energy transmitted to the molecules of the road and tires, distributed in some complicated and apparently random manner among them. So heat is just random, microscopic mechanical energy, stored in various forms in the atoms and molecules of gases, liquids, and solids.

  This view sheds light on the Second Law, which states that disorder always increases and is measured by a quantity called entropy. This law now can be interpreted as saying that in any process where something changes (e.g., the car coming to a stop), you can never perfectly “reorganize” all the energy that goes into the random motion of molecules. It is always too hard to retrieve all of it in a useful form. Before the car stopped, all its molecules (in addition to some random motion due to its non-zero temperature) were moving together in the same direction at 60 mph, providing a kinetic energy that could be used to do useful work, such as dragging a heavy object against friction. As the car stops, that energy is transformed into the less usable form of heat. It is not that we can’t turn heat back into usable energy (e.g., use it to get the car moving again); it is just that we can’t do it perfectly. We could run some water over the hot brake discs of our stopped car, which could generate steam, which could turn a turbine, and, presto, we would get back some useful mechanical energy. This of course is not the best-designed heat engine one could imagine. But the Second Law says that no matter how carefully or cleverly you design an engine to turn heat into useful mechanical energy, you will always find that you have to put more heat energy in than you get back.

  To make this all precise and tractable in a mathematical theory, the German physicist Rudolph Clausius, while a professor at our familiar Zurich Poly in 1865, introduced the notion of entropy, which is a measure of how much the microscopic disorder increases in every process involving heat exchange. The word entropy was chosen from the Greek word for “transformation,” and indeed Clausius was guided by just the picture we have been painting: heat is the internal energy of atoms or molecules, which can be partially but never fully transformed to usable energy. Now, with their new statistical mechanics, Maxwell and Boltzmann were trying to make this idea of the internal energy of a trillion trillion rocking and rolling molecules precise, and in so doing come to understand entropy and the laws of thermodynamics on the basis of atomic theory. This program was so controversial that even by the end of the century, thirty years later, Planck, the thermodynamicist par excellence, was reluctant to adopt it. It was only his quantum conundrum that forced him to overcome his scruples, as we will see.

  The key point is that the statistical mechanics of Maxwell and Boltzmann was still Newtonian mechanics, just applied to a system so complicated that one imagines it behaving like a massive game of chance, in which each molecular collision with a wall or with another molecule is like a coin being tossed (heads you go to the right, tails you go to the left). The worldview is the same as that of Newton and Laplace; only the method is different. Maxwell, had he lived another two decades, might have begun to recognize the leaks springing in this optimistic vessel, since the basic inconsistency in this v
iew appeared at the intersection of his two great inventions, the theory of electromagnetic radiation and the statistical theory of matter. However, that was not to be; he would pass away a mere five years after his spectacular lecture on molecular science, having spent those final years occupied by his administrative duties. At the end of that same lecture, having anticipated the next twenty-five years of physical theory, the devout Maxwell makes one of the great historical appeals for intelligent design:

  Natural causes, as we know, are at work, which tend to modify, if they do not at length destroy, all the arrangements and dimensions of the earth and the whole solar system. But … the molecules out of which these systems are built … remain unbroken and unworn.

  They continue this day as they were created, perfect in number and measure and weight, and from the ineffaceable characters impressed on them we may learn that those aspirations after accuracy in measurement, truth in statement, and justice in action, which we reckon among our noblest attributes as men, are ours because they are essential constituents of the image of Him Who in the beginning created, not only the heaven and the earth, but the materials of which heaven and earth consist.

  The next century would demonstrate in many ways, culminating in the awesome demonstration of August 1945, that atoms are not as indestructible as Maxwell had supposed. And Einstein would be the first to understand, through his most famous equation, E = mc2, just how much energy would be released when the perfect instruments of the Creator were disassembled.

  CHAPTER 6

  MORE HEAT THAN LIGHT

  “I have again made the acquaintance of a sorry example of that species—one of the leading physicists of Germany. To two pertinent objections which I raised about one of his theories and which demonstrate a direct defect in his conclusions, he responds by pointing out that another (infallible) colleague of his shares his opinion. I will shortly give that man a kick up the backside with a hefty publication. Authority befuddled is the greatest enemy of truth.”