Einstein and the Quantum

by Stone, A. Douglas

  Such was the feisty mood of Einstein as he wrote in July of 1901 to an old friend, Jost Winteler. The object of his ire was Paul Drude, theorist and chief editor of Annalen der Physik, the most prestigious physics journal in the world at that time. Drude himself was the author of a well-respected text on optics and Maxwell’s equations (in fact it was Drude who introduced the universal symbol c for the speed of light in vacuum). The “infallible” colleague mentioned by Einstein was none other than Ludwig Boltzmann. Einstein, characteristically, seemed oblivious to the potential consequences of offending such prominent scientists, one of whom was editor of the journal to which he would submit all his original research papers for the next six years.

  Einstein wrote those lines from the small city of Winterthur, about twenty miles from Zurich, where he had a two-month position teaching physics and mathematics at the Technical College while the regular instructor was performing his military service. The teaching load was quite heavy, thirty hours a week, but, undeterred, he reassured Mileva that “the Valiant Swabian is not afraid.” In fact he found that he enjoyed the teaching much more than he had expected, and despite the busy schedule he managed time to study research questions, such as Drude’s new “electron theory of metals.” It was only four years earlier, in 1897, that the English physicist J. J. Thomson had confirmed the existence of electrons, negatively charged particles much lighter than the hydrogen atom itself, and he hypothesized that electrons were constituents of atoms. By 1899 Thomson had shown that electrons could be pulled off atoms (a process we now call “ionization”) and hence that the atom was in this sense divisible. This represented the first crack to appear in the indestructible atoms of Maxwell.

  Drude’s theory was based on a guess about the atomic properties of metals. He hypothesized (correctly) that, in a metal such as copper, one of the electrons in each atom was free to move. While the atoms themselves remained fixed in a solid regular crystalline array, these free electrons formed a gas of charged particles that could move easily through the solid, allowing it to conduct electricity and heat efficiently. Drude’s hypothesis was that many of the important properties of metals arose from this gas of electrons, and he thus could use the kinetic theory of Maxwell and Boltzmann to calculate those properties. This was an important step forward, and some of Drude’s conclusions were based on such general considerations that they remain true and are used in our modern (quantum) theory of metals. Other conclusions he drew from his theory relied on Newtonian mechanics and are now known to be false. Einstein’s letter to Drude, pointing out his “errors,” and Drude’s reply are lost, so nothing is known about the validity of Einstein’s objections. What is known is that around this time Einstein began his own reworking of the basic principles of statistical mechanics.

  Einstein’s very first published work on atomic theory (the one he sent along with his job inquiries) was based on a naive hypothesis about molecular forces: that they behaved similarly to gravity in that they depended only on the distance between molecules, and on the type of molecules involved. He wrote about this work to his former classmate Grossmann in April of 1901: “As for science, I have a few splendid ideas…. I am now convinced that my theory of atomic attraction forces can also be extended to gases…. That will also bring the problem of the inner kinship between molecular forces and Newtonian action-at-a-distance forces much nearer to its solution.” Einstein’s simple attraction hypothesis was wrong, and despite his initial enthusiasm he abandoned it after using it in two articles for Annalen der Physik, which he later referred to as “my worthless first two papers.” These works did emphasize that at the time there was no real understanding among physicists about the origin of molecular forces. After the modern atomic theory was established, it became clear that all the atomic forces important for chemistry or solid-state physics ultimately arise from electromagnetic forces; there are no special new molecular forces.1 However, how atoms behave under the influence of these forces is quite different from what was expected, because they obey a new mechanics (quantum mechanics) and not the classical mechanics of Newton. (In addition, there are new forces within the atomic nucleus, hinted at by the phenomenon of radioactivity, which had just been discovered, but these forces are generally not important for chemistry or solid-state physics.) Einstein praised Boltzmann’s statistical theory of gases precisely because it didn’t rely much on the unknown molecular forces, and after his first immature efforts he decided to pursue the path of statistical mechanics into the atomic realm.

  At the end of 1901 Einstein received “A letter from Marcelius [Marcel Grossmann] … a very kind letter” telling him that the patent office position would soon be advertised and that he would definitely get it. “In two months time we would then find ourselves in splendid circumstances and our struggle would be over,” he wrote to Maric, but, he hastened to add, their bohemian lifestyle would not change “We shall remain students [horribile dictu] as long as we live, and not give a damn about the world.” One more professional disappointment remained. In November 1901 Einstein submitted a PhD thesis on the kinetic theory of gases to Professor Alfred Kleiner at the University of Zurich (the Poly was not yet able to grant PhD degrees, although with Weber in charge it is not likely that Einstein would have tried that route anyway). Only indirect information survives about what transpired, but Einstein withdrew the thesis early in 1902, apparently at Kleiner’s suggestion, “out of consideration for [Kleiner’s] colleague Ludwig Boltzmann,” who, despite Einstein’s admiration for his work, had been “sharply criticized” on certain points.

  By February of 1902 Einstein had relocated to Bern, a picturesque Swiss city on the river Aare. His job at the patent office would not start for five months, and he remained without visible means of support, so he literally hung out his shingle.

  Private Lessons in

  MATHEMATICS and PHYSICS

  for students and pupils

  given most thoroughly by

  ALBERT EINSTEIN, holder of the fed.

  polyt. teacher’s diploma

  GERECHTIGKEITGASSE 32, 1st floor

  Trial Lessons Free

  “The situation with the private lessons isn’t bad at all. I have already found two gentlemen, an engineer & an architect & more in prospect,” Einstein wrote to Maric shortly after arrival. His letter was apparently a bit too cheerful, as he soon received a reply from his fiancée, which is lost, but the content of which is clear from Einstein’s rapid follow-up: “It is true … that it is very nice here. But I would rather be with you in some backwater than without you in Bern.” Actually, although Einstein painted a rosy picture of his life in Bern, an old family friend who visited him there described his condition as “testifying to great poverty … [living in] a small, poorly furnished room.” Strikingly, Einstein never complains in his letters about material conditions, making only occasional humorous allusions to this “annoying business of starving.”

  In Bern Einstein quickly gathered around him a lively circle of friends with shared intellectual interests, several of whom would become lifelong companions. By the end of June 1902 he had taken up his post at the patent office as an expert third class (the lowest rank), and his immediate financial woes were ended. In October of that year he obtained grudging permission from his parents (at his father’s deathbed) to marry Mileva, and on January 6, 1903, with no family present, only two friends, the couple were married with complete lack of ceremony at the Bern registry office. Typically, Einstein had trouble getting into their new apartment that night, as he had forgotten the key. Mileva had gone through many tribulations to get to this point, and it seems that after failing twice to get her teacher’s degree from the Poly, she had now given up her own scientific ambitions. Einstein reports to his friend Michele Besso that she takes good care of him and he “leads a very pleasant, cozy life” with her; in the same letter he tells Besso that he has just sent off his second paper on statistical mechanics, pronouncing the paper “perfectly clear and simple, so that I
am quite satisfied with it.”

  The main point of Einstein’s first two papers on statistical mechanics was to frame all the statistical relations that ultimately underlie the First and Second Laws of thermodynamics in a very general way, no longer referring specifically to gases and collisions in gases, as was done by Maxwell and Boltzmann. Thermodynamics is supposed to apply to everything that stores energy and can absorb or give off heat, which is essentially everything: liquids, solids, machines, … you name it. The Second Law of thermodynamics implies that no engine can change heat into useful work with perfect efficiency. Stated more picturesquely, it is impossible to make a perpetual-motion machine, a machine that, once it gets started, will go forever in a repeated cycle without needing fuel. Einstein, in his new job at the patent office, was regularly coming across proposed “inventions” that, upon closer inspection, were physically impossible because they violated this principle. The generality of the laws of thermodynamics must have been very much on his mind.

  And so he wrote two papers that assume almost nothing about the nature of molecular forces, or the macroscale nature (e.g., gas, solid, etc.) of the thermodynamic system being considered, and that lead to several equivalent forms of the Second Law. The papers make only one assumption, an assumption so subtle that it had been the cause of debate since the time of Maxwell. Einstein appears to have been unaware of this raging debate and does not emphasize this assumption (to be discussed later) or comment on it in any detail. However, the mathematical results of these papers and the formal framework he introduces are quite important, and would alone have made his name known a century later, except for some bad luck.

  Independently, and earlier, Josiah Willard Gibbs at Yale University had established exactly the same principles (“the resemblance is downright startling,” Max Born later commented) and applied them very powerfully to chemical problems. Gibbs, the son of an eminent theologian and scion of an old New England family, received in 1863 the first PhD in engineering granted in the United States. He briefly studied in Europe and became acquainted with the nascent German school of thermodynamics, begun with Clausius and continuing in Einstein’s day with Planck. Very reminiscent of Maxwell in his breadth of interests, Gibbs would make enormous contributions as a physicist, chemist, and mathematician until his death in 1903; he is arguably the greatest American-born scientist of all time. In fact Maxwell himself was so impressed with a clever geometric method devised by Gibbs to determine chemical stability that he made a plaster model illustrating the idea with his own hands and sent it to Gibbs.

  Gibbs introduced the concept of “free energy,” which dominates modern statistical mechanics and is often denoted by the symbol G in honor of Gibbs; this is just one of twelve important scientific contributions bearing his name to this day. His work was initially slow in becoming known in Europe, but just as Einstein was beginning his own statistical studies, Gibbs’s monumental treatise, Elementary Principles of Statistical Mechanics, was published, and he was awarded the Copley Medal of the Royal Society of London. (Before the Nobel prizes, which were first awarded in 1901, this was the most prestigious international science award of the day.)

  Gibbs’s contributions predated and overwhelmed those of Einstein, and Einstein would later comment in print that had he known of Gibbs’s work earlier, he would “not have published those papers at all, but confined myself to a the treatment of some few points [that were distinct].” Einstein’s admiration for Gibbs remained so great throughout his life that when, a year before his death in 1955, he was asked who were the most powerful thinkers he had known, he replied: “[Hendrik] Lorentz,2 I never met Willard Gibbs; perhaps, had I done so, I might have placed him besides Lorentz.” So already, before his twenty-fifth birthday, Einstein had established himself as a deep thinker, on par with the great leaders of his era; unfortunately no scientist of any influence seems to have noticed this at the time. Moreover he was not advancing his career aspirations by telling the leading physicists in Germany of the errors they had made in their earlier work. He would need to devise new theories, which made specific experimental predictions, to get the world’s attention. These would not be long in coming, and when they did come, the free-spirited bohemian outsider would soar above even the great men—Maxwell, Gibbs, and Lorentz—whom he so admired.

  1 There are forces inside the atomic nucleus that were unknown at that time, now called, unimaginatively, the “strong” and “weak” nuclear forces. At that time the existence of the nucleus was itself unknown.

  2 Hendrik Lorentz, a Dutch physicist, was widely regarded as the most eminent theorist of the generation preceding Einstein’s; he will play an important role in our story below.

  CHAPTER 7

  DIFFICULT COUNTING

  The tomb of Ludwig Boltzmann in Vienna is engraved with a very short and simple-looking equation, which, ironically, he never wrote down during his lifetime:

  S = k log W.

  S is the universal symbol for entropy, k is a fundamental constant of nature known as Boltzmann’s constant, and log W is the logarithm1 of a number, W, relating to the physical system of interest, a number that Boltzmann called the number of “complexions.” (The number W can be so devilishly hard to calculate for many physical systems of interest that the greatest mathematical physicists of the twenty-first century, and the most powerful computers as well, are helpless to determine it.) This equation was the lever for setting the quantum revolution in motion. It would form the basis for Planck’s derivation of his radiation law and for Einstein’s first insights into the quantum nature of light.

  Clausius had introduced the concept of entropy to explain heat flow, but he had no idea how a physicist could calculate this quantity from any fundamental mechanical theory (presumably an atomic theory). It was the statistical mechanics of Boltzmann, Maxwell, Gibbs (and Einstein) that gave the recipe. The recipe is deceptively simple sounding. It has two parts: (1) Whatever can happen, will happen. (2) No atom (or molecule) is special.

  FIGURE 7.1. The epitaph on the tomb of Ludwig Boltzmann, S = k log W, expressing the fundamental equation of entropy, which he had discovered. Image courtesy Daderot.

  Imagine shooting gas molecules one by one through a small hole into a box where they bounce around. Mentally divide the box into two equal parts with an imaginary partition. Any part of the box is equally accessible to the molecules (whatever can happen, will), and there is no reason for any molecule to be at one place at a given time versus another (no molecule is special). Suppose for the moment you can actually see the molecules directly. With one molecule in the box you look in periodically, and you find it roughly half the time on the left side of the box and half the time on the right side. Now add a second molecule, wait a bit, and start looking again. Roughly one-fourth of the time both molecules are on the left, one-fourth of the time both are on the right, and half the time one is on the left and one is on the right. Why is the last case more likely than the first two? Because there are two ways that you can get the last case (molecule 1 on right, molecule 2 on left; molecule 2 on right, molecule 1 on left) but only one way you can get the first two cases. This is just like tossing two coins and finding that one heads and one tails happens roughly twice as often as heads-heads and tails-tails. We can now get fancy and define three “states” for the two-molecule gas: in state one, both are on the left; state two, both on the right; and state three, one on the left and one on the right. For the first two states Boltzmann’s W = 1 (there is only one way to get these states), but for the third state2 W = 2. The entropy of this third state is then larger than that of the other two states, according to Boltzmann’s formula (we need not delve into the mysterious properties of the logarithm function to reach this conclusion). In actuality a physicist would specify the states in more detail than just which half of the box the molecules are in, but the underlying concept and method is exactly the same.

  Now imagine we have a few trillion trillion molecules in the box (as indeed we usually d
o). There is still only one way to have all of them on one side; however, W, the number of ways of having about half on the right side and half on the left side, is unspeakably large. We literally have no words, no analogies, for numbers of this magnitude. As a feeble attempt, imagine the following: take all the atoms in the universe and, in one second, clone each of them, so as to create a second “universe.” Now repeat this every second, creating 4, 8, 16, and so on “universes.” Do this every second for the entire age of our current universe. Add up all of the atoms in all these universes and you will arrive at a number that is incredibly big, all right, but still this number is negligibly small compared with the number of states of high entropy of one liter of gas. These high entropy states, in which the gas molecules are roughly equally distributed in each half, have (not surprisingly) enormously higher entropy than the states in which the molecules are all or mostly on one side of the box.

  Suppose that we go to a lot of trouble and evacuate the gas from the box, put an airtight partition in the middle, and put the gas back in on the left side, so that we set the system up in this very improbable (low entropy) state. From Maxwell we know that the gas molecules are flying around at 1,000 mph, colliding with one another and the walls and thus creating pressure on the walls. If we then remove the partition, the gas will rapidly fill the entire box again, approximately equally in each half. The entropy of the system will have increased. After the molecules spread themselves out roughly equally in the box, the molecules will still be colliding and moving around, but on average there will be roughly equal numbers on each side of the box. Intuitively this situation is the most disordered state (you don’t need to make special efforts to achieve it), and according to Boltzmann’s principle, this is the state of maximum entropy. This is the atomic explanation for the Second Law of thermodynamics, that entropy always increases or stays the same. Whenever we try to generate useful work from heat, we are essentially trying to create order out of this molecular chaos, and we are fighting against the laws of probability.