Uhlenbeck, George Eugene

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UHLENBECK, GEORGE EUGENE

(b. Batavia, Java, Dutch East Indies [now Jakarta, Indonesia], 6 December 1900;

d. Boulder, Colorado, 31 October 1988), theoretical physics, statistical mechanics, nuclear and atomic physics.

Uhlenbeck is perhaps most widely known for the discovery of electron spin, a discovery made jointly with Samuel Goudsmit when they were both graduate students at Leiden. In later years he did much important work in atomic structure, nuclear physics, and especially kinetic theory and statistical mechanics. Throughout his career he had a reputation as a superb lecturer and teacher.

Early Life and Education. Uhlenbeck was born into a military family in the Dutch East Indies. His father, Eugenius Marius Uhlenbeck, was a lieutenant colonel, while his mother, Anne Marie Beeger was the daughter of a major general. Of the four surviving children, George was the oldest son; he had an older sister Annie and two younger brothers Willem Jan and Eugenius Marius. In 1907 his father retired and, with the children’s education in mind, the family moved permanently to the Netherlands. There they resided in The Hague, where Uhlenbeck attended elementary and high school. In high school he was influenced by his physics teacher, A. H. Borgesius, who encouraged him to read further in mathematics and physics. He was especially impressed by Hendrik A. Lorentz’s textbook, Beginselen der Natuurkunde(Elements of physics; 2 vols., 1908–1909). In July 1918 Uhlenbeck obtained high marks in his final school examination. However, because his school studies had not included Latin and Greek, he could not enter a university; by law entrance then required proficiency in those languages. Instead he entered the Technical University at Delft to study chemical engineering. Shortly afterward, the law was changed and he entered the University at Leiden in January 1919. There Uhlenbeck studied physics and mathematics under the professors Paul Ehrenfest, Heike Kamerlingh Onnes, and Johannes P. Kuenen, while the retired Lorentz came to lecture once per week. While still an undergraduate he studied Ludwig Boltzmann’s Vorlesungen über Gastheorie (Lectures on gas theory; 2 vols, 1896–1898), which he found hard to understand until he discovered the article Begriffliche Grundlagen (Conceptual foundations; 1912) of Paul and Tatania Ehrenfest, which “was a revelation.” These studies, together with his careful notes on Maxwell theory, so impressed his teachers that in his third and final year he was granted a state fellowship, sparing his parents the expense of tuition. In December 1920 he took his final examinations and became a graduate of Leiden. He then became a graduate student there.

Graduate Studies. As a graduate student Uhlenbeck at first supported himself by teaching part time at a high school in Leiden. He apparently disliked this work, but it gave him enough money to rent a room in Leiden while he continued his studies. He followed lectures by Paul Ehrenfest and was invited to attend the Wednesday evening colloquia, a Leiden tradition. Attendance was by invitation, and once invited, one was expected to continue attending.

Late in his second year of graduate studies, he heard from Ehrenfest of a teaching position in Rome. So it was that in September 1922 he took the post of tutor to a son of the Dutch ambassador to Italy, although he did not entirely neglect his studies. In September 1923 he received from Leiden the degree of doctorandus, roughly equivalent to a master’s degree, or admission to a doctoral program in an American university. While in Rome he studied the Italian language, becoming sufficiently proficient to enable him to attend lectures in mathematics at the University of Rome. Upon his return to Leiden in the summer of 1923, Ehrenfest had asked him to contact Enrico Fermi, so it was that in the fall of 1923 that the two became acquainted. Together with other young Italian physicists, they organized a small Leiden-style colloquium. In this way Uhlenbeck and Fermi became close and longtime friends. It is said that it was Uhlenbeck who introduced Fermi, the native Roman, to the wonders of Roman art. Be that as it may, at about the same time Uhlenbeck became fascinated with art and art history, neglecting completely his studies of physics during his second year in Rome. His first published paper, which appeared in 1924, was on the topic of art. Only his contacts with Fermi and the other young physicists kept him from completely losing touch with physics.

In June 1925 Uhlenbeck returned to Leiden with the intention of studying art history. However, here his old nemesis, the classical languages, came back to haunt him. He began a serious study of Latin but was counseled by an uncle to continue his study of physics as “more practical.” Accordingly, he contacted Ehrenfest, who agreed to take him on as a student. Thus began a relationship that had a profound and lasting influence. At first Uhlenbeck found the daily meetings to be exhausting and the emphasis on clarity as opposed to rigor frustrating, but he soon grew more comfortable. He wrote a paper on the wave equation in odd and even number of spatial dimensions, which was followed by a joint paper with Ehrenfest on the same topic.

Ehrenfest had suggested that he work with a fellow student, Goudsmit, who already was an expert on atomic spectra. The two of them wrote a paper on the spectrum of hydrogen, giving an improved interpretation involving half-integer quantum numbers. Then came their great discovery: electron spin. The quantum numbers they had assigned implied that the electron must have another degree of freedom; it must be rotating. With this idea everything fell into place. They published first a short note and then, at the encouragement of Niels Bohr, who visited Leiden shortly after their discovery, a longer paper that appeared in Nature. The origin of the spin-orbit interaction was apparently suggested by Albert Einstein, who also visited Leiden at the time and pointed out that in its rest frame, the electron sees an orbiting nucleus and hence a magnetic field. There remained an awkward factor of two in the spin precession rate. This was soon explained as a relativistic effect in an elegant paper by Llewellen H. Thomas. In their third and final paper on electron spin, Goudsmit and Uhlenbeck summarized

these results and gave what has become the accepted interpretation of electron spin and atomic spectra.

The discovery of electron spin was in the spirit of the old Bohr quantum mechanics. The new quantum mechanics had been born only a few months earlier with the appearance of Werner Heisenberg’s first paper in July 1925. In common with most physicists of the time, Uhlenbeck found the new theory, with its mathematics of matrices, daunting. When in January 1926 Erwin Schrödinger’s first paper on wave mechanics appeared, it came as a great relief: the mathematics of wave equations was much more familiar. In the spring of 1926 he studied assiduously Schrödinger’s wave mechanics, becoming adept in the new quantum theory. Then Oskar Klein came to Leiden as a visitor, and he and Uhlenbeck had long discussions about the Klein-Gordon wave equation as well as Klein’s ideas of unifying the Maxwell and Einstein equations through a five-dimensional wave equation. Uhlenbeck was much taken with the latter, writing a paper with Ehrenfest.

Since Uhlenbeck’s study of Ludwig Boltzmann in his early student days, he had been fascinated with statistical physics, so when it came time to choose a thesis topic, it was in that field. The topic was the Boltzmann description of an ideal gas and its extension to the new Bose-Einstein and Fermi-Dirac statistics. Through his later writings and lectures, this material became familiar to a generation of physicists. In a brief passage in the thesis, he criticized Einstein’s prediction of what came to be called the Bose-Einstein condensation. This presaged a lifelong interest in the problem of condensation: how does it come about that the equation of state shows a sharp discontinuity when a gas condenses to a liquid? It was another decade before it came to be recognized that the discontinuity occurs only in the thermodynamic limit in which the number of gas particles and the volume containing the gas each become infinite such that the ratio remains finite.

First Period at Michigan. Among the distractions at Leiden, writing the thesis was difficult. Therefore, Ehrenfest arranged a fellowship that allowed Uhlenbeck to spend two intensive months in the spring of 1927 at the Bohr Institute in Copenhagen, Denmark. There he did little but write his thesis. After finishing his dissertation he spent a month in Göttingen, Germany, where he met, among others, J. Robert Oppenheimer. He returned to Leiden with Oppenheimer, who was for a short time an assistant to Ehrenfest. After that things happened quickly. He and Goudsmit defended their theses on the same day, 7 July 1927. It had already been decided that they would go to Ann Arbor as instructors at the University of Michigan. That spring Walter Colby, a professor of theoretical physics at Michigan, had visited Europe seeking a candidate to replace Klein, who had returned to Europe after three years in Ann Arbor. Ehrenfest strongly urged him to consider making an offer to a pair of candidates, so they would have someone to talk to in the “wilderness.” This he was able to do and Uhlenbeck and Goudsmit were both happy to accept, because it meant that they would be in a university and not have to teach in a high school. Then, on 23 August, Uhlenbeck married Else Ophorst, who had been a chemistry student at Leiden. In 1942 their son, Olke Cornelius, was born. He would go on to become a distinguished biochemist. On their arrival in New York, Oppenheimer met the newlyweds; the elegant apartment of Oppenheimer’s parents especially impressed them. A short time later they arrived at Ann Arbor in time for the fall term.

Although at that time the United States had no rival of the European centers of theoretical physics, they found Ann Arbor to be not quite the wilderness they expected. Aside from Colby, there were two young theoretical physicists: Otto Laporte, a student of Arnold Sommerfeld who had arrived the year before, and David Dennison, who had received his PhD at Michigan under the direction of Colby and Klein and was returning after three years in Europe. Through these four young men, theoretical physics at Michigan rapidly grew in importance. There already existed a summer program of visiting lectures, instituted in 1923 by Harrison M. Randall, chairman of the Physics Department, but it had remained modest and mostly of local interest. Under the influence of these younger men, and especially Uhlenbeck, this program changed quickly in character, becoming the Ann Arbor Summer Symposia in Theoretical Physics. These were internationally important events, unique in America, perhaps in the world, with distinguished rosters of lecturers. These summer schools continued until 1941, after which wartime conditions made them impracticable. They resumed after the war but were no longer unique and finally, after 1973, were discontinued.

The first notable work out of Michigan after the arrival of the young physicists was a study of rotational Brownian motion, published in 1929 jointly by Uhlen-beck and Goudsmit. In this work they explained observations by Walter Gerlach of the rotational motion of a tiny mirror fixed on a fine wire, showing how it was that for different pressures of the surrounding gas, the mean square displacement was the same while the displacement in time looked very different. This work presaged the classic paper by Uhlenbeck and Leonard Ornstein on the theory of Brownian motion, which appeared the following year. In that work Uhlenbeck and Ornstein initiated the modern description of Brownian motion, obtaining expressions at all times for the probability distributions for the positions and for the velocity. In doing this they made implicit use of what came to be called the Gaussian property of the Ornstein-Uhlenbeck process. Interestingly, although they obtained expressions for the moments of the joint distribution in position and velocity, they did not obtain either the joint probability distribution or the corresponding Fokker-Planck equation. This was remedied and more by a review article written by Uhlenbeck and Ming Chen Wang and published in 1945. This review article, which is still a standard reference in the field, summarized the theory of Brownian motion within the framework of the general theory of stochastic processes. It is a masterpiece of simplicity and clarity of exposition. These two articles—that with Ornstein and the review article with Wang—established Uhlenbeck as preeminent in the field.

Worth noting from those early days is a paper with Dennison, published in 1932, on the inversion spectrum of ammonia, in which the nitrogen atom makes a transition between equilibrium positions on either side of the plane of the three hydrogen atoms. What is interesting about this paper, aside from the fact that it was the first real application of the quantum mechanical double well problem, is that it sparked the birth of microwave spectroscopy, which resulted in the first direct measurement of the inversion transition. It was, however, ahead of its time, and microwave spectroscopy came into its own only after the World War II developments in radar.

The 1930s saw the rise of interest in nuclear physics, which was the topic of a Summer Symposium lecture in 1931 by Wolfgang Pauli and another in 1933 by Fermi. Inspired by these talks and related discussions, Uhlenbeck wrote, with his student Emil Konopinski, a pair of papers in 1935 on the theory of beta decay, proposing an alternative to the recently appeared Fermi theory involving gradients of the wave function. While at the time this Konopinski-Uhlenbeck theory seemed to give a better fit to the observed spectrum of electron energies, eventually improved techniques gave results supporting the subsequently-accepted Fermi theory. In 1941 Uhlenbeck and Konopinsky returned to the subject, offering a masterful discussion of the Fermi theory in its most general form, describing selection rules and energy distributions for allowed and forbidden transitions and arbitrarily charged nuclei.

Another contribution to nuclear physics appeared in a pair of papers written in 1950 with his student, David Falkoff. The subject was the angular correlation in successive nuclear radiations. Their work was important because it systematized the phenomena in great generality, forming a solid basis for the use of the method in determining the angular momentum and parity of excited states of nuclei.

Return to the Netherlands. In the fall of 1935 Uhlenbeck left Michigan to return to the Netherlands and take up the position of professor at the University of Utrecht. The position had become vacant after Hendrik Kramers, who had been a professor there, moved to Leiden upon the death of Ehrenfest. Soon after arriving, Uhlenbeck wrote a paper with Julian K. Knipp, a visiting fellow from Harvard University, in which they calculated the inner bremsstrahlung accompanying the beta decay of nuclei. At the time this was an important result, because it explained the faint radiation accompanying the decay of radium E.

While Uhlenbeck wrote a number of papers on topics in nuclear physics during his time at Utrecht, his most important work from that time is contained in a pair of papers, in 1937 and 1938, on the theory of condensation based on the thesis of his student, Boris Kahn. In his own thesis, Uhlenbeck had criticized Einstein’s discussion of the Bose condensation, a criticism that at the time even Einstein considered serious. This changed rather dramatically at a 1937 conference in Amsterdam celebrating the one hundredth anniversary of the birth of the physicist Johannes Diderik van der Waals. There the subject of phase transitions was much discussed and Kramers made the important point that the associated sharp discontinuity could occur only in the thermodynamic limit. This Uhlenbeck realized at once, and in the papers with Kahn he not only accepted the Einstein theory but used it as a model for a general theory of condensation. Kahn and Uhlenbeck’s idea was that the phase transition is related to a singularity appearing in the equation of state when expressed in terms of the fugacity, this only in the thermodynamic limit. This Kahn-Uhlenbeck theory remained for a decade and a half the accepted description of the condensation phase transition, despite the fact that it was unable to account for the liquid state. This was remedied by Chen Ning Yang and Tsung-Dao Lee. They showed that if one studied the problem in the entire complex fugacity plane, one could arrive at a simple and correct description of the transition. In the meantime, Uhlenbeck continued to work on the condensation problem. Together with his students Robert J. Riddel and George W. Ford, as well as the mathematician Frank Harary, he embarked on a study of the mathematical theory of linear graphs, which can be used to classify the terms in the virial expansion of the equation of state. This work, which is summarized in a review article with Ford, has had an interest and application well beyond the problem that inspired it.

Uhlenbeck’s interest in the condensation problem persisted. Some years later, in a beautiful series of three papers jointly with Mark Kac and Per C. Hemmer in 1963 and 1964, he discussed a one-dimensional fluid model in which the particle interaction has a hard core and a long-range exponential attraction. It had earlier been shown by Kac that in the thermodynamic limit this is an exactly soluble model. In the first of these papers, the three authors showed that in the further limit in which the range of the exponential becomes infinite while the strength goes to zero, one obtains an equation of state exactly of the well-known van der Waals form. Moreover, because theirs is an exact calculation involving the thermodynamic limit, they obtain the Maxwell equal area rule, which in the usual discussion of the van der Waals equation is obtained by an ad hoc argument. Thus, for this model the problem of condensation could be exactly solved, with an explicit description of the gas-liquid phase transition. In the following two papers they went on to obtain expressions for the two- and three-particle distribution functions and to discuss the critical behavior.

In the fall of 1938, while still a professor at Utrecht, Uhlenbeck was invited to spend a semester at Columbia University in New York. There he shared an office with Fermi, who had recently fled from Italy, and learned about the newly discovered nuclear fission. At the end of the year he returned to Utrecht, but only for a short time. In August 1939 he returned permanently to the United States, taking up again the position of professor at the University of Michigan. Shortly afterward, Adolf Hitler invaded Poland and World War II began.

Second Period at Michigan. Uhlenbeck was soon involved in war work and in 1943 moved to the Radiation Laboratory in Cambridge, Massachusetts, as the leader of a group doing theoretical work on wave guides and other radar-related topics. It was there that he formed his friendship with the mathematician Kac, which remained close and was a source of mutual inspiration until Kac’s death in 1984. Some of the results of this war work are described in a book, Threshold Signals(1950), edited jointly with James Lawson.

In 1945 Uhlenbeck returned to Ann Arbor, taking up again his professorship at the University of Michigan. Not long after he began his long collaboration with C. S. Wang Chang, writing in all some eleven papers on topics in kinetic theory, ranging from sound propagation in rarefied gases to the thickness of shock waves. Although these appeared only in the form of reports of the Engineering Research Institute of the University of Michigan, they were widely circulated and achieved a kind of cult status. Later, about half of these reports were reprinted in Studies in Statistical Mechanics(Vol. 3, 1965).

In the mid-1950s Uhlenbeck embarked on a study of the kinetic theory of dense gases, with the goal of obtaining a density expansion of the transport coefficients (viscosity, heat conduction, etc.) analogous to the well-known virial expansion of the equation of state for a nonideal gas. Together with his student, Soon Tahk Choh, Uhlenbeck used a method first introduced by Nikolai Bogoliubov to obtain the Choh-Uhlenbeck equation, a generalized Boltzmann equation that includes the effects of three-particle collisions. They went on to construct the Chapman-Enskog solution of this equation, obtaining formal expressions for the corrections to the transport coefficients of first order in the density. This work was carried on by others, but the dream of a virial expansion of the kinetic coefficients came to an end with the discovery that the contributions to the generalized Boltzmann equation from four or more particle collisions are divergent. This was a great disappointment, but Uhlenbeck’s efforts remain as an important contribution and catalyst to work on the deep question of the approach to equilibrium.

In the summer of 1960 Uhlenbeck delivered a series of lectures in Boulder, Colorado, at a summer school under the auspices of American Mathematical Society. Out of these came a book, Lectures in Statistical Mechanics(1963), jointly authored with George Ford, in which the general theme is the approach to equilibrium.

Rockefeller Years and Retirement. In 1961 Uhlenbeck left Michigan for the Rockefeller Institute for Medical Research (now Rockefeller University) in New York City, where he was named professor and member of the institute. His reasons for making the move were in part the attraction of joining his old friends Ted Berlin and Kac, who went there at the same time. He and Berlin had long planned to write together a book on statistical mechanics. Another reason might have been the infectious enthusiasm of Detlev W. Bronk, the president of the institute, about plans for building a university. In any event, Uhlen-beck made the move and, in addition to the work with Hemmer and Kac as described above, he directed a number of students on problems of statistical physics and kinetic theory. Unfortunately, with the unexpected death of Berlin in 1962, the book was never written.

In 1971 Uhlenbeck retired from Rockefeller, but he remained active until 1985, when failing health led to a move first to Champagne-Urbana, where his son Olke was professor of microbiology at the University of Illinois, then in 1986 to Boulder when Olke was appointed professor at the University of Colorado. There he died of a stroke at age eighty-seven.

George Uhlenbeck was renowned as a teacher and lecturer. His talks, orderly and systematic, were models of clarity. In his later years he received many honors: Higgins Lecturer, Princeton University, 1954; Member of the U.S. National Academy of Sciences, 1955; Lorentz Professor, Leiden University, 1955; Henry Russel Lecturer, University of Michigan, 1956; Oersted Medal of the American Association of Physics Teachers, 1956; president of the American Physical Society, 1959; Van der Waals Professor, Amsterdam University, 1964; Planck Medal of the German Physical Society (with Samuel Goudsmit), 1965; Lorentz Medal of the Royal Dutch Academy of Sciences, 1970; National Medal of Science of the United States, 1977; commander, Order of Orange-Nassau, the Netherlands, 1977; Wolf Foundation of Israel Prize in physics, 1979.

BIBLIOGRAPHY

A collection of unpublished papers and letters is in the Bentley Historical Library of the University of Michigan, Ann Arbor, where a list of published works will also be found. Other collections are in the Rockefeller University Archives, Tarrytown, New York, and the Niels Bohr Library of the American Institute of Physics, College Park, Maryland.

WORKS BY UHLENBECK

With Samuel Goudsmit. “Spinning Electrons and the Structure of Spectra.” Nature 117 (1926): 264–265. The key paper announcing the discovery of spin.

Over Statistische Methoden in de Theorie de Quanta. PhD diss., Leiden. The Hague: Martinus Nijhoff, 1927.

With Leonard S. Ornstein. “On the Theory of the Brownian Motion.” Physical Review 36 (1930): 823–841.

With Boris Kahn. “On the Theory of Condensation.” Physica 4 (1937) 1155–1156; 5 (1938): 399–416. Kahn’s dissertation has been reprinted in Studies in Statistical Mechanics, edited by Jan de Boer and George E. Uhlenbeck, Vol. 3, Part C. Amsterdam: North-Holland, 1965.

With Ming Chen Wang. “On the Theory of Brownian Motion II.” Reviews of Modern Physics 17 (1945): 323–342. This and the above composed the famous papers on Brownian motion. They are both reprinted in the “noise book”: Selected Papers on Noise and Stochastic Processes, edited by Nelson Wax (New York: Dover, 1954).

With James L. Lawson, eds. Threshold Signals. New York: McGraw-Hill, 1950.

With George W. Ford. “The Theory of Linear Graphs with Applications to the Theory of the Virial Development of the Properties of Gases.” In Studies in Statistical Mechanics, edited by Jan de Boer and George E. Uhlenbeck, Vol. 1, Part B. Amsterdam: North-Holland, 1962.

———. Lectures in Statistical Mechanics. Providence, RI: American Mathematical Society, 1963.

With Mark Kac and Per C. Hemmer. “On the van der Waals Theory of the Vapor-Liquid Equilibrium. I. Discussion of the Distribution Functions.” Journal of Mathematical Physics 4 (1963): 216–228. The work on condensation in a one-dimensional gas model.

With Per C. Hemmer and Mark Kac. “On the van der Waals Theory of the Vapor-Liquid Equilibrium. II. Discussion of the Distribution Functions.” Journal of Mathematical Physics 4 (1963): 229–247.

———. “On the van der Waals Theory of the Vapor-Liquid Equilibrium. III. Discussion of the Cricital Region.” Journal of Mathematical Physics 5 (1964): 60–74.