The Last Man Who Knew Everything

by David N. Schwartz

  Pauli tried to find physical interpretations of this fourth number, but eventually gave up. One idea, first suggested by a young colleague of his, Ralph de Laer Kronig, was that electrons actually “spin,” and that the spin can either be “right-handed” or “left-handed,” producing intrinsic angular momentum in opposite directions. Pauli ridiculed the idea, because it implied, at least to him, that if you measured the speed of the electron at its “equator” (think of a spinning globe) that speed would have to be faster than light, which was impossible. Suitably chastened, Kronig dropped the idea. It was picked up again at the end of the year by Fermi’s good friends Uhlenbeck and Goudsmit, who did not know of Pauli’s objection. If they had, they might not have published their idea, but as it turned out, the two of them did not know of Kronig’s spin proposal and are thus often given credit for the idea of electron spin. Pauli still resisted the idea. It was Dirac who, in 1926, pointed out that a relativistic interpretation permitted—even required—electron spin.

  Pauli’s exclusion principle was audacious. To understand his achievement and his particular way of doing physics, one must appreciate that he proposed a mathematical solution that had no physical interpretation at all—it was simply the only way he could imagine to solve the problem he was confronting. It was, in a sense, like Planck’s invention of the quantum, a mathematical solution without an underlying physical interpretation. Like Planck’s idea, it took Pauli enormous courage to propose it and even more to resist the efforts of colleagues to propose the most obvious physical interpretation. That he was wrong in resisting the idea of electron spin takes nothing away from the achievement itself, a beautiful one which bespeaks his own particular genius.

  THE EXCLUSION PRINCIPLE ALONE WAS A GREAT STEP FORWARD. IT provided an explanation of the anomalous Zeeman effect and gave a more complete understanding of how electrons “fill up” the orbits surrounding the nucleus. More breakthrough work was done that year. A young German named Werner Heisenberg, working in Göttingen with Max Born and a young theorist named Pascual Jordan, formulated a method to analyze the way electrons move between orbits, using a mathematical technique called matrix multiplication that successfully explained the varying intensity of spectral lines. Late in the year, a Viennese physicist named Erwin Schrödinger, closeted away in a ski chalet with one of his numerous lovers, came up with a differential “wave” equation that did much the same thing, using techniques more familiar to the average physicist. Physicists now had two highly effective ways of delving into previously incomprehensible physical phenomena.

  Why did both approaches, so different in form and function, provide the same answer to the thorny questions raised by quantum theory? It took yet another genius to show that matrix mechanics and wave mechanics were two sides of the same coin.

  HE WAS, IN THE WORDS OF HIS BIOGRAPHER GRAHAM FARMELO, “the strangest man.”

  Paul Adrien Maurice Dirac was the baby of that generation of quantum pioneers, a year younger than Fermi and Heisenberg, two years younger than Pauli. Born in Bristol, England, he was the son of a strict father whose severe discipline had a lasting effect on the young man. Tall, thin, and extremely reserved, he rarely spoke in sentences longer than two or three words. His teachers at the Merchant Venturer’s Technical College considered him brilliant, and he easily obtained a place at St. John’s College, Cambridge, but his family finances prevented him from attending. Instead, he attended the University of Bristol for his undergraduate education and St. John’s for his graduate degree, the first to be given in the field of quantum theory. In his thesis he provided an independent derivation of Heisenberg’s quantum mechanics based on his observation that the underlying mathematical structure of the matrix algebra, in particular its noncommutative property, was analogous with a particular form of the mathematics physicists used to express classical mechanics.

  Dirac was, in some sense, the anti-Pauli. We might think of him today as suffering from Asperger’s syndrome. He was socially awkward in the extreme and literal-minded to an exasperating degree. He once asked Heisenberg why people danced. Heisenberg replied, “When there are nice girls, it is a pleasure.” Dirac considered this for a moment and blurted out, “But how do you know beforehand that the girls are nice?” Once during a lecture a student raised his hand and said that he did not understand an equation that Dirac had written on the blackboard. Dirac remained silent because, as he explained later, the person in the audience had not asked a question. His colleagues at Cambridge reportedly defined a “dirac” as a unit of one word per hour. He was also aggressively irreligious. In one famous exchange at the 1927 Solvay conference, talk among the younger physicists ventured into the area of philosophy and religion. Heisenberg recounts that Dirac made what was for him an impassioned plea to the effect that religion had no place in the world of a physicist. Pauli, who was silent for much of the discussion, reportedly ventured, “Well, our friend Dirac has got a religion and its guiding principle is ‘There is no God and Dirac is His prophet.’” Heisenberg reported that everyone had a good laugh, none more so than Dirac himself.

  Among the brilliant young theorists of his generation, Dirac may well have been the most brilliant. His PhD thesis was impressive and attracted the attention of the physics world at large. Born was stunned that a doctoral student could master the field so completely, especially since the final Born-Heisenberg-Jordan paper had not yet been completed. Others shared Born’s surprise. Almost immediately, Dirac was propelled into the highest ranks of theoretical physicists. His was the first PhD degree ever granted in the new field of quantum theory and heralded the arrival of a superstar. Later in the year, Dirac followed it up during a post-doc in Copenhagen with an even more impressive paper, which demonstrated mathematically the underlying unity between the Heisenberg and Schrödinger approaches, seeing both as special cases of something called transformation theory. It was a significant finding, but Dirac’s best was yet to come. In 1927, he produced the first paper to develop the concept of a quantum field using the electromagnetic field as his focus. The paper had an historic impact on physics and laid the groundwork for Fermi’s second great contribution.

  IN 1925, HOWEVER, ALL OF DIRAC’S MAJOR CONTRIBUTIONS WERE still in the future. As the year came to an end, while Schrödinger was holed up in the Austrian Alps with his girlfriend during Christmas wrestling with his equation and putting it in the proper form and while young Dirac was grinding out his thesis, Fermi was also thinking deeply about quantum problems. But the problems he considered were slightly different. He had been contemplating them for several years and now the work of Pauli, in particular, showed him the way toward a solution.

  CHAPTER FIVE

  OF GECKOS AND MEN

  WHEN HE ARRIVED IN FLORENCE IN 1925, FERMI ASSUMED HIS position as a lecturer at the Institute of Physics, which served as the University of Florence’s physics department. Perched on a low hill in the Arcetri area just south of the old town, the institute was far away from the rest of the university, which was closer to the city center. The setting may have been inconvenient, but it was quite beautiful and Fermi loved it.

  Andrea Garbasso, the head of the institute, sought to staff the institute with the brightest physicists of the younger generation. The previous year he brought Rasetti on board to teach experimental physics. Now he was delighted to take on Fermi to teach theory. Fermi spent the next two years lecturing on physics to engineering students at the university. He taught two courses, one in mathematical physics and the other in classical mechanics (called “rational mechanics” at the time). The notes for his mechanics lectures were compiled into a beautiful handwritten manuscript, which could serve as a good introduction to the subject even today.

  When he was not teaching and when he was not in the library reading about the pathbreaking work being done in Göttingen, he and Rasetti were hiking in the hills around the institute or playing pranks on members of the institute staff or, occasionally, both at the same time. The two pranksters note
d the somewhat timid nature of the local girls who served meals at the institute cafeteria. Together they hatched a plan: they would collect bagfuls of small lizards, catching the tiny but plentiful creatures using six-foot-long rods of Pyrex glass fitted with silk loops at the end. Then they would release them in the cafeteria and watch the horrified reaction of these unsuspecting and unsophisticated women. It sounds like a Pisa prank, with all the attendant childishness and unspoken misogyny.

  Fermi had little interest in geckos or their habits. Lying in wait for the geckos, he thought about Pauli’s work and its implications for a problem he had been grappling with since 1924—how to describe a “perfect gas” in quantum theoretical terms. A paper he wrote in January 1924, while he was still in Göttingen, is an important piece of evidence. He was struggling with the application of quantum theory to statistical mechanics, particularly with respect to the concept of entropy. He needed a hook on which to hang an analysis that incorporated the work of the Göttingen group into one of his favorite fields.

  Pauli’s work provided just such a hook.

  Given his voracious reading, Fermi almost certainly was aware of Pauli’s original paper when it was published in March 1925. He may, as Segrè speculates, have spent more time discussing it with Pauli’s colleague Kronig during the summer of 1925, when Kronig accompanied Fermi and others—including a young Edoardo Amaldi—on a hiking trip in the Dolomite Mountains. However it happened, he developed a working knowledge of Pauli’s exclusion principle and began to consider how the puzzle over statistical mechanics might be resolved in light of it. The leap of insight he had, waiting quietly for a gecko to ensnare itself in one of his lassos, was that Pauli’s exclusion principle could be extended from electrons surrounding an atomic nucleus to single atoms in a gas. With this insight, he could explain the strange behavior of gases under very high pressure or at very low temperatures, when they seem to lose their thermal capacity, or what physicists call specific heat. They become degenerate, filling all the lowest energy states possible within Pauli exclusion constraints.

  As Fermi’s close friend and colleague Hans Bethe later explained:

  The quantum states of an atom in a gas differ by their velocities; there is one and only one state in which the atom is completely at rest, a second state in which it moves extremely slowly, a third with somewhat higher velocity, and so on. Since no two atoms can be in the same state, only one single atom can be completely at rest, all others must be in motion, even at absolute zero temperature.

  This, in a nutshell, is the difference between traditional statistical mechanics and Fermi’s new interpretation that takes into account the exclusion principle. In the traditional version, the absence of heat implies the absence of motion. In Fermi’s version, even in the absence of heat, atoms jostle for position because no two atoms can share the same state (velocity).

  In his famous paper, “On the Quantization of a Perfect Monatomic Gas,” Fermi laid out the statistical approach required to account for the energy level of such gases, incorporating Pauli’s exclusion principle. The idea must have gelled in early September 1925, before the Arcetri geckos went into hibernation. It took him the rest of the year to get it all down on paper in a form that satisfied him. He presented a short version of the paper at the institute on February 7, 1926, and published it in the journal of the Accademia dei Lincei soon thereafter. Six weeks later he expanded it considerably and sent it off to Zeitschrift für Physik, where it was received on March 26, 1926.

  The paper had an immediate impact, particularly because it gave a mathematical explanation of degeneracy. It was known that electrons in metals exhibit degeneracy. Now people like Pauli, Sommerfeld, and a young graduate student of Sommerfeld’s, Hans Bethe, were able to calculate the behavior of electrons in metals using Fermi’s approach and discovered that it predicted the observable results. Word spread as far as England, where Paul Dirac read the paper—presumably the German version—and promptly forgot about it because it solved a problem that held no interest for Dirac at that particular moment, focused as he was on finishing his PhD thesis. Later in the year, Dirac developed an interest in the problem, approached it from scratch, and came up with a slightly different method that led to the same analytic conclusions. Dirac’s approach was broader, extending to particles that do not obey Pauli’s exclusion principle but rather do obey what was then known as Bose-Einstein statistics, now called “bosons.” He presented his paper to London’s Royal Society on August 26, 1926. The editors of the Royal Society’s Proceedings may not have been aware of Fermi’s prior work on the subject.

  When he read Dirac’s paper, Fermi was surprised and perhaps a bit annoyed that Dirac made no mention of Fermi’s previous paper. He sent a letter in somewhat stilted English to Dirac calling the latter’s attention to his prior work:

  In your interesting paper “On the Theory of Quantum Mechanics”… you have put forward a theory of the Ideal Gas based on Pauli’s exclusion Principle. Now a theory on the ideal gas that is practical [sic] identical to yours was published by me at the beginning of 1926.… Since I suppose you have not seen my paper, I beg to attract your attention to it.

  Contrary to Fermi’s presumption, Dirac had indeed read Fermi’s paper. He had simply forgotten it. As he later explained:

  When I looked through Fermi’s paper, I remembered that I had seen it previously, but I had completely forgotten it. I am afraid it is a failing of mine that my memory is not very good and something is likely to slip out of my mind completely, if at the time I do not see its importance. At the time I read Fermi’s paper, I did not see how it could be important for any of the basic problems of quantum theory; it was so much a detached piece of work. It had completely slipped out of my mind, and when I wrote up my work on the antisymmetric wave functions, I had no recollection of it at all.

  He immediately sent an apology to Fermi and from then on always referred to the statistics as Fermi-Dirac, generously giving Fermi primary credit for it. That we now refer to particles that obey the exclusion principle as fermions and those that do not as bosons is a direct result of Dirac’s acceptance of Fermi’s claim of priority.

  As is the case with many important developments, other talented theorists were thinking about these matters at the exact same time, including Born’s young protégé Pascual Jordan. In December 1925, Jordan gave Born a manuscript to read while Born traveled to the United States, where he was to present lectures at MIT. Born put it in his suitcase and quickly forgot it. Six months later, Born was rummaging through his suitcase and discovered the unread manuscript. As he browsed through it, he realized that he had inadvertently cheated Jordan of credit for the discovery because the Jordan paper proposed using the Pauli exclusion principle in exactly the same way as had Fermi and Dirac, but even earlier. For his part, Fermi would never have allowed six months to elapse before hearing back from Born. Indeed, so great was Fermi’s self-confidence, he probably would not have bothered to ask Born to review it in the first place.

  FERMI HAD NO INTENTION OF REMAINING IN FLORENCE AS A LECTURER for the rest of his career. He was ambitious and wanted a chair at a major university. In this he had an ally in Senator Corbino. Fermi’s work on quantum statistics had earned him a global profile among the elite of the physics world and would, under normal circumstances, have led to offers at any number of prestigious Italian institutions. That it had not yet done so reflected the highly complex, political nature of Italian academic life. Fermi needed the influence and support of someone like Corbino to make his ambition a reality.

  There was a formal system in place in Italy for the recruitment of professors at Italian universities. When an opening was available, the Italian Ministry of Education would run a competition, or concorso, in which it would assign a small committee of senior professors to make a recommendation of a candidate to fill the opening. Each of the committee members would nominate potential candidates, and candidates’ respective qualifications, including publications and any prio
r teaching assignments, would be evaluated and compared. During 1925, a position opened at the University of Cagliari in Sardinia. In the concorso Fermi found himself in competition with Giovanni Giorgi, a man old enough to be his father who had been teaching at the University of Rome since 1913. He was a solid physicist of the old school, with no fundamental contributions to his name. Fermi’s candidacy fell victim to the reluctance of powerful professors to embrace the new physics. The five members of the committee split 3–2 in favor of Giorgi. Levi-Civita and Volterra voted for Fermi; three others put their support behind Giorgi. The three who voted for Giorgi were eminent Italian physicists of the old school; they only reluctantly accepted relativity theory and were relatively immune to the new quantum theory. Levi-Civita, of course, was one of the main advocates of relativity theory, and Volterra was also a strong advocate of the new physics. Segrè notes that Fermi viewed the outcome as “unjust” and never forgave those who voted against him.

  The concorso for the University of Rome was a much bigger prize and Fermi was clearly eager to fill the opening once it was established. Corbino worked hard to get approval for a position in theoretical physics, which would be the first chair of this type in Italy. The other physics chairs were in either experimental physics or mathematical physics. There was some thought that Corbino might get official approval for the position as early as 1925; however, the matter was postponed until the fall of 1926. That Fermi emerged as the concorso’s unanimous first choice was hardly surprising, because Corbino, who chaired the committee, had stacked it with Fermi supporters. Persico and another member of the Rome Institute, Aldo Pontremoli, came in second and third, respectively. Persico went north to Florence, and Pontremoli, to Milan.