Einstein and the Quantum (8 page)

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.

“
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 logarithm
¹
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.