Enrico Fermi

Quantum Statistics

Fermi reached Pauli's principle in a roundabout way. He aimed to make an entropy calculation for an ideal gas of atoms using Boltzmann's statistical entropy equation,

                                             S = kln W,

with careful attention to the rules of quantum mechanics. One of those rules is that atoms can exist only in certain discrete states and no others. Another is that like atoms in an enclosure cannot be labeled and distinguished from each other. This is because the wave function representing an atom has a long enough reach that it overlaps wave functions for other atoms in the enclosure. (Wave functions for electrons. With a suitably constructed Schrodinger equation, wave functions for atoms, or any other physical entity, can be defined.) Fermi's model thus departed from Boltzmann's, which was based on the assumption that like atoms (or molecules) of a gas are distinguishable from each other.

To succeed in his entropy calculation, Fermi had to include one more departure from Boltzmann. Taking his cue from Pauli, he added the rule that each quantum state can accommodate one and only one atom. Even at low temperatures, the atoms must all be found in different quantum states. With his rejection of Boltzmann's rule of distinguishability, and his adaptation of Pauli's rule, Fermi got the entropy calculation he wanted. He published his new statistical model in 1926.

Fermi was not the first to find uses for statistical models modified to meet the demands of quantum mechanics. Two years before Fermi's paper was written, the Indian physicist Satyendranath Bose proposed a model based on Einstein's concept that light and other forms of radiation behave like an ideal gas of particles, later called photons. Bose found that he could reconcile Einstein's theory of radiation with Planck's by deriving Planck's radiation law with a statistical model that accepted the indistinguishability of photons, and also that each quantum state could accommodate any number of photons, not just one as in Fermi's model.

Bose was beginning his career as a theorist when he found this connection between the otherwise un reconciled theories of Planck and Einstein. He sent his manuscript to Einstein, who was impressed, translated the paper into German, and had it published in the Zeitschrift  fur Physik. Einstein added the note: “In my opinion Bose's derivation of the Planck formula signifies an important advance.”

As Fermi was pursuing his statistical model, Paul Dirac was independently exploring the same territory from a broader point of view. He emphasized the difference between the Bose-Einstein model for photons and a model he proposed for electrons in atoms based, like Fermi's theory, on the requirements of Pauli's principle. Fermi's paper preceded Dirac's, but Dirac failed to mention it in his own paper, even though, as he later admitted, he had seen the Fermi work but failed to appreciate its importance. This brought an objection from Fermi. “Since I suppose that you have not seen my paper,” Fermi wrote to Dirac, “I beg to attract your attention to it.”

The Fermi-Dirac model was limited to atoms and electrons, and the Bose Einstein model to photons, but the two models have proved to be far more encompassing. Contemporary particle physicists assume that all particles not only electrons and photons, but protons, neutrons, neutrinos, and many other particles fit one model or the other. Dirac atoned for his sin of omission by proposing that all particles following Fermi's (and Dirac's own) scheme be called “fermions.” Similarly, he introduced the term “boson” for particles obeying the Bose Einstein model.

Physics Reawakens in Rome

In the fall of 1926, Fermi went back to Rome. Largely through the efforts of Fermi's patron, Orso Corbino, a chair of theoretical physics had been established at the University of Rome, and Fermi easily won the competition for the new post. At age twenty-five, he had, Segre` writes, “practically attained the zenith of a university career in Italy.”

Corbino expected Fermi to bring modern physics to Italy. As Segre` remarks, “a new generation had to take over, and Fermi was to be its leader.” Fermi's first step to make himself and his subject known was to give popular lectures and write textbooks. The writing was done during summer vacations in his favorite mountain country, the Dolomites of northern Italy. There, according to Segre`, he sometimes worked “lying on his stomach in a mountain meadow, armed with an adequate supply of pencils and bound blank notebooks, [writing] page after page, without a book for consultation, without an erasure (there are no erasers on Italian pencils) or a word crossed out.”

A year after Fermi's arrival, Corbino brought another protege to Rome, the young experimentalist Franco Rasetti. He was “an elongated man with thin hair, a determined chin, and a steady gaze that went through people,” Laura Fermi tells us. Rasetti and Fermi had been classmates at the Scuola Normale in Pisa and confederates in mischief-making, ranging from “fights with pails of water on the roofs of Pisa to protect young damsels’ honor, which had never been in danger,” says Laura Fermi, to “make-believe duels for reasons that were unknown both to challengers and to challenged.” One escapade, a stink-bomb in a classroom, nearly brought permanent expulsion from the university. According to Laura Fermi, Rasetti was the ringleader in these merry pranks: “I do not believe Fermi would have given himself so thoroughly to this kind of life if he had not been dragged into it and held fast by . . . Franco Rasetti.”

Fermi and Rasetti, with two more recruits, Edoardo Amaldi, a former engineering student, and Emilio Segre`, Fermi's first graduate student, formed the core of Corbino's School of Rome. Corbino called them “his boys.” They were young, talented, intensely devoted to their work, and convinced that great discoveries would come their way, as indeed they did. In a casual way, Fermi was their leader. In theoretical matters he was infallible, so they called him the “pope.” Otto Frisch, who knew Fermi later, remarked that he had “never met anyone who in such a relaxed and unpretentious way could be so completely dominant.” Fermi's style as a theorist was always pragmatic and as simple as possible. He aimed for the concrete and avoided the abstract. Hans Bethe, another colleague of Fermi's in later work, contrasts Fermi's style with another, mainly German, tradition:

My greatest impression of Fermi's method in theoretical physics was its simplicity. He was able to analyse into its essentials every problem, however complicated it seemed to be. He stripped it of mathematical complications and of unnecessary formalism. In this way, often in half an hour or less, he could solve the essential physical problem involved. Of course there was not yet a mathematically complete solution, but when you left Fermi after one of these discussions, it was clear how the mathematical solution should proceed.

This method was particularly impressive to me because I had come from the school of Sommerfeld in Munich who proceeded in all his work by complete mathematical solution. Having grown up in Sommerfeld's school, I thought that the method to follow was to set up the differential equation for the problem (usually the Schrodinger equation), to use your mathematical skill in finding a solution as accurate and elegant as possible, and then to discuss this solution. In the discussion, you would find out the qualitative features of the solution, and hence understand the physics of the problem. Sommerfeld's way was a good one where the fundamental physics was already understood, but was extremely laborious. It would take several months before you knew the answer to the question.

It was extremely impressive to see that Fermi did not need all this labor. The physics became clear by an analysis of the essentials, and a few order of magnitude estimates. His approach was pragmatic. . . .

Fermi was a good mathematician. Whenever it was required, he was able to do elaborate mathematics; however, he first wanted to make sure that this was worth doing. He was a master at achieving results with a minimum of effort and mathematical apparatus.

On a hot day in July 1928, Enrico Fermi married Laura Capon. They had met four years earlier on an outing of young people to the countryside south of Rome. Laura was not impressed by a “short-legged young man in a black suit and a black felt hat, with rounded shoulders and neck craned forward,” but he took charge and organized a soccer game, and Laura did as she was told when he assigned her to goalkeeping. Two years later they met again, this time on a mountain-climbing excursion. Fermi, whom Laura remembered as “the queer guy who made me play soccer,” was again in command. He mapped out twelve-mile conditioning hikes, and accepted no excuses. “It was always thus,” Laura tells us. “Fermi would propose, and the others would follow, relinquishing their wills to him.”

By the fall of 1926, the soccer captain and hiking companion had become “Professor Fermi” at the University of Rome, but did not wear the “overwhelming halo of importance and solemnity” expected of a full professor. “[The] young physicist who could inspire respect in his older colleagues showed a remarkable ability to put himself on the level of the young,” writes Laura, “and I found I could still talk to him without restraint. Often on Sundays I joined him and his group for a hike in the country or a stroll in Villa Borghese, the main park of Rome. Our companionship did not break up.” Then, on the hot July day in 1928, Laura became a partner in the Fermi enterprise. The story of the marriage is a happy one, and Laura Fermi has told it with style and candor in Atoms in the Family.