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PAUL SOPHUS EPSTEIN March 20, 1883-February 8, Z966 BY lESSE W. M. DuAlOND PAUL SOPHUS EPSTEIN was one of the group of prominent and very gifted mathematical physicists whose insight, creative originality, and willingness to abandon accepted classical con- cepts brought about that veritable revolution in our under- standing of nature which may be said to have created "modern physics," i.e., the physics which has been widely accepted during the Twentieth Century. Paul Epstein's name is closely associ- ated with those of that 'group, such as H. A. Lorentz, Albert Einstein, H. Minkowski, I. I. Thomson, E. Rutherford, A. Sommerfeld, W. C. Rontgen, Max von Lane, Niels Bohr, L. de Broglie, Paul Ehrenfest, and Karl Schwarzschild. Paul Epstein was born in 1883 in Warsaw, which was then a part of Russia. His parents, Siegmund Simon Epstein, a busi- nessman, and Sarah Sophia (Lurie) Epstein, were of a moder- ately well-to-do Jewish family. He himself has told how, when he was but four years old, his mother recognized his potential mathematical gifts and predicted that he was going to be a mathematician. After receiving his secondary education in the Humanistic Hochschule of Minsk (Russia), he entered the school of physics and mathematics of the Imperial University of Moscow in 1901. In the third year of his undergraduate studies he started research in experimental physics under Professor Peter N. Lebedew, who in 1901 had become famous for his ex- 127

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128 perimental demonstration of the pressure exerted on bodies by light or other electromagnetic radiation, an example of the Einstein principle of the inertia of energy. After graduation, in 1905, Epstein served as laboratory in- structor in physics, first at the Moscow Institute of Agriculture and later at the Imperial University, continuing his research at the same time. In 1909 he obtained his master's degree in physics and was appointed assistant professor (Privatdozent) at the Imperial University. In 1910 he decided to specialize in theoretical physics and obtained a leave of absence to do re- search under the famous Arnold Sommerfeld at the University of Munich (Germany). Epstein's early research was in the theory of electromagnetic waves and particularly the theory of their diffraction. Two of his papers of this period were his doctoral thesis (1914), "Dif- fraction from a Plane Screen," and an article in the German Encyclopedia of Mathematical Sciences (1916), "Special Prob- lems of Diffraction." At the beginning of the First World War, in 1914, Epstein was at Munich. Being a Russian, he was regarded as an "enemy alien" and was automatically declared a civil prisoner. How- ever, he was interned in a prisoner's camp only for a short time, thanks to the kindly intervention of Professor Sommerfeld. For the duration of the war he was allowed to live privately in Munich with access to the facilities for doing theoretical work and for publishing it, but of course was restricted from leaving Germany. By 1916 Epstein had become deeply interested in problems of the quantum theory of atomic structure based on classical mechanics, and he shared the early development of this branch of physics with Niels Bohr and Arnold Sommerfeld. His most important paper in this connection was "Zur Theorie des Starkeffektes" (1916~. In this paper he computed the electron orbits, atomic energy levels, and splitting of the spectral lines BIOGRAPHICAL MEMOIRS

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PAUL SOPHUS EPSTEIN 129 for a hydrogen atom in the presence of a superimposed electric field and compared his theoretical predictions with the experi- mentally-observed results then available. The dramatic story of the writing of the paper was told by Epstein years later. The story, which follows, was taken from a tape recording of an interview between the historian I. L. Heilbron and Paul S. Epstein on May 25, 1962. Paul Epstein had been understandably anxious to escape from his captivity as an "enemy alien" in Munich, and to do this he had hopes of finding a position as a theoretical physicist somewhere outside Germany. Two places he had in mind were Leyden and Zurich. But to obtain such a position as the one in Zurich, he must write a habilitationsschrift, that is to say a thesis for becoming Privatdozent. Sommerfeld had just writ- ten his famous paper in which, by introducing the principle of relativity into Bohr's theory of atomic orbits, he had arrived at an explanation of the fine structure-splitting, till then unex- plained by the simpler Bohr theory. A much more complicated case of line-splitting was known, however, and was as yet com- pletely unaccounted for by any theoretical treatment. This was the effect, observed by Stark in 1913, when an atom is in the presence of an externally-imposed electric field. So Epstein proposed to Sommerfeld that he would tackle this difficult problem as the subject of his habilitationsschrift for Zurich, and Epstein's proposal was accepted at Zurich. The Stark effect had been well known for three years and in fact, as chance would have it, at the very time of which we are speaking, Wagner, one of Rontgen's assistants in Munich, put on a demonstration of the Stark effect using a so-called "canal ray" tube. This was a vacuum electrical discharge tube in which the negative electrode or cathode was provided with holes. In such a tube most of the positive ions bombarded the cathode and "splashed out" the electrons from it so as to main- tain the discharge, but a few of the positive ions would go

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130 BIOGRAPHICAL MEMOIRS on through the holes, and these were called "canal rays." Since vacuum technique was in a very primitive stage, the mean free path of an ion in such tubes was short, and the canal ray ions, excited by collisions with other ions, would emit spectral lines of much greater complexity than normal for that atomic species if the electric field in the near vicinity of the cathode were strong. The splitting up of the normally-to-be-expected spectral lines into these complicated spectra was the Stark effect. Wag- ner's timely demonstration of the effect in Munich was probably done with mercury vapor in the tube, but the theoretical ex- planation of the effect, even for the simplest atomic species, hydrogen, was difficult enough to be as yet an unsolved problem in terms of Bohr-Sommerfeld quantum theory. This demonstration stimulated Epstein to start thinking vigorously how he might construct a theory to explain quan- titatively the splitting of the line spectra. He had studied generalized mechanics from the French text of P. Appell, and he knew from this a certain theorem of the famous mathema- tician, Jacoby, furnishing a convenient method of integrating the differential equations of motion for a case such as this. Now at that time there was a famous mathematician, Karl Schwarzschild, of powerful ability whom P. S. Epstein, as be- hooved a much younger and less widely known man, in fact only a young Privatdozent, held in great respect and consider- able awe. Epstein only saw Sommerfeld infrequently, owing to restrictions imposed on him because of his "enemy alien" status in Munich, but at one of the meetings which he was permitted to attend through Sommerfeld's intervention, the latter told Epstein, "I wrote Schwarzschild that he should work on this article " (meaning the Stark effect). Epstein relates that he "was a little crestfallen, because I regarded this as a stab in the back, since he ESommerfeld] knew that I was writing about it and," Epstein continued, "Schwarzschild was a mathematician of un- believable energy; he could do everything in a twinkling; of

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PAUL SOPHUS EPSTEIN 131 course I couldn't reproach him ESommerfeld], but I decided: 'Now I have no prospects unless Schwarzschild should go to heaven.' " Epstein goes on to tell how the next day when he was going to bed he saw his way through what he needed for the solution. He got up at 5 o'clock the next morning and by 10 o'clock he had the formula! And then the same morning he showed his result to Sommerfeld. "And what do you know, the same afternoon he ESommerfeld] got a letter from Schwarzschild, and Schwarzschild had the wrong formula! It was the same order of magnitude, but didn't agree on the positions of the lines. So Sommerfeld wrote Schwarzschild, 'This morning Epstein brought me the formula of the Stark effect, and this afternoon we got your letter. But Epstein's formula agrees with the observations.'" When Schwarzschild had first obtained his result, he im- mediately announced in the Berlin Academy that he would speak about it. He did so, however, before having written to Sommerfeld and Epstein, so the report he gave to the Academy before he actually lectured contained his erroneous result. By that time Epstein had already submitted his announcement of his result for publication, and it came out dated just one day before Schwarzschild delivered the above-mentioned lecture to the Academy. So Epstein had the priority over Schwarzschild by one day. In his lecture Schwarzschild had apparently cor- rected his error verbally (undoubtedly giving credit to Epstein for the correction), and when he received the galley or page proof of the printed version he corrected the error and removed all of the discrepancies. Thus Schwarzschild's final published version came out correctly. In substantially all textbooks and histories of physics the theory of the Stark effect is attributed jointly to Epstein and Schwarzschild It is clear, however, that they really solved the ~ See, for example, History of Physics by Max van Eagle, translated by Ralph E. Oesper, Academic Press, Inc., New York, N.Y., 1950.

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132 BIOGRAPHICAL MEMOIRS problem independently and that Epstein's solution came first and did indeed correct an error in Schwarzschild's solution. This in- cident, recounted directly from Epstein's own lips, illustrates dramatically the competitive tensions that existed among this group of European physicists in those early days of the develop- ment of the quantum theory of atomic structure. Paul Epstein's intimate knowledge of those exciting times and gifted scientists at the turn of the century was a source of great inspiration to us younger men who attended his classes in theoretical or mathematical physics a little later after he had come to Caltech (in 1921~. I shall never forget his account of von Laue's accidental learning of the hypothesis (first clearly formulated by Ludwig A. Sieber) that crystals are latticework structures of atoms. It seems that van Laue first learned of this when, in hopes of a consultation, he sought out Sommerfeld, who happened to be sitting in a little summer pavilion in one of the gardens of the University of Munich with his student, P. P. Ewald, discussing Ewald's famous thesis in which the idea of the "reciprocal lattice" had emerged as a mathematical device of great power. Von Laue was electrified when he overheard the conversation and grasped the idea of the crystalline atomic lattice. Here, made by Nature herself, was the equivalent of the artificial ruled grating (of Henry Rowland), the ideal tool, perhaps, which might indeed have the appropriate fineness of structure to answer the burning question with which van Laue had been deeply occupiedwhether or not the Rontgen rays, discovered 17 years earlier, in 1895, were undulatory in nature and, like visible light waves, capable of being diffracted by a . . grating or lattice. By 1900 Haage and Wind had tried to determine, by dif- fractions of x rays through fine slits, whether Rontgen's radia- tions were undulatory in nature and, if so, of what order of wavelength. These first results were inconclusive, but later, Walter and Pohl repeated the slit-diffraction experiment with

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PAUL SOPHUS EPSTEIN 133 greater refinement. Not until 1912, however, through the good fortune that the microphotometer of P. P. Koch ~ had just been invented and developed, did it become possible to study quan- titatively the slight broadening of the photographically-recorded lines of Walter and Pohl. From this broadening they concluded that the x rays observed had wavelengths of the order of 4 X 10-9 cm.l Von Laue immediately set two young experimental phys- icists, Friedrich and Knipping, at the University of Munich the task of trying to see if a beam of x rays could be diffracted by scattering from a crystalline solid. Their experiment was fraught with many difficulties and tribulations. At that time the only way of getting the high voltage electrical power to operate a Ront<,en ray tube was with a "spark coil" or "Ruhm- korff coil." Public electrical power (for lighting the university) was only of the constant voltage, direct current variety. The light sources were so-called "arc lamps" in which the light came from a direct current arc maintained between two graphite electrodes. Such a lamp has a nonlinear current-voltage char- acteristic which tends strongly to amplify any small accidental fluctuations in the supplied voltage. In order to operate the Ruhmkorff coil, one needed an intermittent electrical supply to it with an appropriate "interrupter" ~ and capacitor for gener- ating high frequency oscillations. But the transient fluctuations of the general voltage supply induced by the interrupter were strongly amplified by all of the arc lights in the university, ~ P. P. Koch, Ann. Phys., 38, 507 (1912) . t I am indebted for my dates and information on these early slit-diffraction experiments to the famous text of A. Sommerfeld, Atomic Structure and Spectral Lines, translation of H. L. Brose, E. P. Dutton and Co., New York, N.Y., 1923. , A "Wehnelt interrupter," which interrupted the current about a thousand times per second, was used. I owe some of these details to a delightful account of the van Laue, Friedrich, and Knipping experiment at the University of Munich written by Max van Laue himself, which was printed and privately dis- tributed by North American Philips, Inc., on the 50th anniversary of Rontgen's famous discovery.

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134 BIOGRAPHICAL MEMOIRS which emitted deafening rattling noises every time the x-ray tube was in action. Knipping had constructed an automatic device which switched on the current from the university's electrical system for about five seconds and then switched it off again for twice that length of time. Fortunately the experiment started during vacation, but before the two scientists got any diffraction photographs, classes started and the nuisance of the "talking arc lamps" drowning out all the lectures can be readily imagined. Quoting from von Laue's account: "Due to some psychological law this primitive music was contagious to the students. They thought it a great joke to hum along with it. The merriment grew greater and greater until finally the whole lecture was ruined." Von Laue continues: "The rector of the university naturally ordered a strict investigation into the cause of the disturbance. All of the many committees which are part of a university were set in motion. But in vain. We physicists, who could have explained the whole thing, knew nothing about it! "At the end of the first three weeks of the new semester the matter accidentally came to light. A mechanic, who had been ordered to look for the source of the disturbance, came into the cellar where the Wehnelt interrupter stood, listened, and at once reported it to the higher authorities. Then the waves of general indignation broke over all of our heads. All the various committees came and certainly did not show us the most agree- able side of their natures." They demanded an immediate remedy or else suspension of the experiments. "Faced with this need, we turned to Rontgen to ask whether we might draw our current from his institute. We needed only to carry a conducting wire across the university court from the window of one institute to that of the other. And as soon as it was established that the university would thus no longer be disturbed, Rontgen gladly gave his consent. "Just as matters had reached this point, the building com-

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PAUL SOPHUS EPSTEIN 135 mittee walked into Sommerfeld's institute. They were the most powerful of all the university committees and apparently the least popular with the professors. They, too, wanted to let us feel the force of their anger, but we did not give them a chance to speak. Instead, we at once told them of the arrangement that we had made with Rontgen. They were nevertheless suspicious. They went to Rontgen themselves to have this confirmed. They returned a few minutes later in a state of indignation. We had deceived them. Rontgen was absolutely opposed to supplying current from his institute. We must therefore discontinue our experiments at once. "So the four of us sat there, Sommerfeld, Friedrich, Knip- ping, and I [von Laue], and did not know what we should do next. Luckily our quandary did not last long. The solution came a few minutes later in the person of a mechanic, a fat, affable Bavarian, from the Rontgen institute. In his deep bass and local dialect, which considerably increased the humor of the situation, he said, 'The Geheimrat (meaning Rontgen) told me to tell you that you can go ahead and put up the wire. He is keeping to his agreement. It is just that whenever the building commission people come to him, the Geheimrat always says NO to them!' " It was thus that the experiments of von Laue, Friedrich, and Knipping were continued until the end. They had tried at first to study the radiation diffracted by a crystal at very large angles, i.e., in the backward direction to the incident beam. When at last they tried placing the photographic plate on the far side of the crystal (copper sulfate), they obtained on the plate a central spot, produced by the direct beam going through the crystal, and, forming a pattern around the central spot, a group of symmetrically arranged spots of lesser intensity whose arrange- ment and symmetry depended on how the crystal was oriented relative to the beam. Rontgen, who was deeply impressed, did not believe at first that the spots represented an interference phenomenon through x-ray diffraction by the crystal lattice. The

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136 BIOGRAPHICAL MEMOIRS complete explanation became evident only after further work by the British physicists W. W. Bragg, his son Lawrence, and H. G. l. Moseley at Cambridge as well as certain other work at Munich by E. Wagner and J. Brentano. The five scientists worked with two crystals which demonstrated the mor~ochrom- atization of the rays in the first crystal. Professor Epstein, after coming to Caltech, would recount to his students very dramatically the occasion of the first suc- cess of the von Laue, Friedrich, and Knipping experiment indisputably one of the truly great "breakthroughs" of early Twentieth Century physicsmuch as I have given it here. The group of physicists from the University of Munich had the pleasant custom of meeting for luncheon and coffee at the little round marble-topped tables out-of-doors in the garden of the Cafe Lutz just across the way. The custom was so well estab- lished and accepted that the waiters of The cafe would dutifully see to it that the particular table for this group, at which on previous days they may have been discussing mathematical physics while writing the equations in pencil on the marble top, would be saved from day to day without washing it off so the discussions could continue. On a certain beautiful warm spring day in the Easter holidays of 1912 van Laue arrived a few minutes late at the accustomed table. Paul Epstein, P. P. Koch, the mathematician Rosenthal, and the physicists E. Wagner and W. Lenz were already there. But an unusual at- mosphere prevailed at the physicists' table. Instead of con- versing as usual, each one silently read a newspaper. Nlon Laue sat down, ordered coffee, and took up a newspaper waiting for a conversation to begin. But none did. One of the company made a remark, shortly after another did the same, and so on around the table, all of which struck van Laue as incompre- hensible and mystifying. Finally what must have happened, but which he had not yet heard about, dawned on him, and he said, "Well, gentlemen, I assume from your remarks that the inter-

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PAUL SOPHUS EPSTEIN 143 In his research career, after his arrival at Caltech, Epstein at first continued his work on Bohr's form of the quantum theory, culminating it in 1922 with three papers in the Zeitschrift fur Physik and one in the Physical Review. Later Epstein took part in the development of quantum mechanics initiated by Heisen- berg and Schroedinger. An important paper in the Physical Review (1926), "The Stark Effect from the Point of View of Schrodinger's Quantum Theory," ~ should be mentioned in . to US COnneCtlOn. In 1930, Epstein was elected to the National Academy of Sciences. P. S. Epstein also devoted considerable attention to border- line problems related simultaneously to both physics and several cognate sciences. Examples are "Zur Theorie des Radiometers" (1929), "Reflection of Waves in an Inhomogeneous Absorbing, Medium" [the Heaviside Layer] (1930), "On the Air Resistance of Projectiles" (1931~. Other examples of borderline problems which Epstein studied were the settling of gases in the atmo- sphere, the theory of vibrations of shells and plates, and the absorption of sound in fogs and suspensions. Two of his articles in this category outside of physics are especially worthy of men- tion. Both appeared in a monthly literary and scientific maga- zine, Reflex, published in the 1930's in Los Angeles, California, and edited by Dr. S. M. Melamed. The first of these articles, "The Frontiers of Science," is a highly scholarly presentation of certain central problems of both philosophy and religion set forth in their relationship to recent concepts on the frontiers of physics and mathematics. His discussion of the old philo- sophical and religious problem of free will vs. the concept of "scientific determinism" and the "law of causality" is particu- larly noteworthy since, in one form or another, all of human- ~ See also in this connection "The New Quantum Theory and the Zeeman Effect" (1926); "The Magnetic Dipole in Undulatory Mechanics" (1927).

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144 BIOGRAPHICAL MEMOIRS kind has struggled for centuries with these questions. Epstein invokes the "principle of indetermination" of Werner Heisen- berg, enunciated in 1927 and points out that, built into the very structure of Nature herself, there is a basic principle which precludes mankind from making with indefinitely high accuracy the requisite physical measurements to predict the future from a knowledge of the present with the ideal certainty postulated by S. Laplace in the Seventeenth Century. This article is indeed a rewarding one to the reader. Epstein's other article in Reflex is "Uses and Abuses of Na- tionalism." In it he reveals a deep and farsighted understanding of certain patterns in the history of the political development of nations. In this discussion Eppie's complete alignment on the side of liberalism becomes self-evident. He takes the history of France as the vehicle for his argument and perceives the Dreyfus affair in the Nineteenth Century as an important turning point, away from imperialism and militarism at home and toward friendly cooperation abroad. In the opinion of the writer this article of Epstein's revealed his deep prescience in world affairs. It was written long before de Gaulle made the wise decision to withdraw France from its military commitments, first in Southeast Asia and later in Algeria. Other nations could well "profit by this example." It is a pity that these two articles, splendidly exemplifying Paul Epstein's remarkable scholarship, erudition, and pre- science in humanistic matters well outside his own fields of specialization, should be lost from the far wider circulation they deserve. The writer wishes to suggest that they be republished. After Paul Epstein's retirement as Emeritus Professor at Caltech in 1953 he served as a consultant for several large indus- trial companies. Prominent among the many reports submitted by him in such work was his "Theory of Wave Propagation in a Gyromagnetic Medium" (1956~.

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PAUL SOPHUS EPSTEIN 145 Paul Epstein died at his home in Pasadena on February 8, 1966, at the age of 83, after suffering with admirable stoicism a prolonged and painful illness (herpes roster or shingles). He was beloved of many students and colleagues, and his long and useful life stands as a splendid tribute to his brilliant mind and his altruistic sharing of it with others.

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146 KEY TO ABBREVIA TIONS BIOGRAPHICAL MEMOIRS BIBLIOGRAPHY ~\m. J. Phys. American Journal of Physics Ann. Physik. Annalen der Physik Naturwiss. Die Naturwissenschaften Phys. Rev.- Physical Review Physik. Blatt. Physikalische Blatter Physik. Z. = Physikalische Zeitschrift Proc. Nat. Acad. Sci. Proceedings of the National Academy of Sciences Rev. Mod. Phys. Review of Modern Physics Verhandl. Deut. Physik. Gesell. = Verhandlungen der Deutschen Physika- lischen Gesellschaft Z. Physik. _ Zeitschrift fur Physik 1911 .. Uber relativistische Statik. Ann. Physik., 36:779-95. Kraftliniendiagramme fur die Ausbreitung der Wellen in der drahtlosen Telegraphie bei Berucksichtigung der Bodenbeschaf- fenheit. In: Jahrbuch der drahtlosen Telegraphie und Tele- phonie, pp. 176-87. Leipzig, Verlag von Johann Ambrosius Barth. 1912 Discussion of some theories of terrestrial magnetism. ~ournal of the Russian Physical Society, 43: 1-15. 1914 Die ponderomotorischen Drehwirkungen einer Lichtwelle und die Impulssatze der Elektronentheorie. Ann. Physik., 44:593-604. Uber die Beugung an Einem Ebenen Schirm unter Berucksichti- gung des Materialeinflusses. (Inaugural dissertation for doc- torate, University of Munich.) Zur Thermodynamik von Systemen mit nicht additiver Entropie. Physik. Z., 15:673-75. Lichtdruck auf die vollkommen leitende Halbebene. Verhandl. Deut. Physik. Gesell., 16:449-56. 1915 With M. v. Laue. Wellenoptik. Mit einem Beitrag uber spezielle

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PAUL SOPHUS EPSTEIN 147 Beugungsprobleme. In: Enzyklopadie der Mathematischen Wissenschaften, Vol. V, pp. 488-525. Leipzig, Druck und Verlag von B. G. Teubner. 1916 Zur Theorie des Starkeffektes. Physik. Z., 17:148-50. Ann. Physik., 50:489-520; also in Uber den lichtelektrischen Effekt und die p-Strahlung radioaktiver Substanzen. Physik. Z., 17:313-16. Versuch einer Anwendung der Quantenlehre auf die Theorie des lichtelektrischen Effekts und der ,8-Strahlung radioaktiver Sub- stanzen. Ann. Physik., 50:815~0. Zur Quantentheorie. Ann. Physik., 51: 168-88. Uber die spezifische Warme des Wasserstoffs. Verhandl. Deut. Physik. Gesell., 18:398~13. 1917 Zur Theorie der Beugung an metallischen Schirmen. Ann. Physik., 53: 33-42. Bemerkung uber das Nernstsche Warmetheorem. Ann. Physik., 53:76-78. Hamilton-lacobische Funktion und Quantentheorie. Verhandl. Deut. Physik. Gesell., 19:116-29. 1918 Anwendungen der Quantenlehre in der Theorie der Serienspektren. Naturwiss., 6: 230-53. Uber die Struktur des Phasenraumes bedingt periodischer Systeme. Sitzungsberichte der Koniglich Preussischen Akademie der Wis- senschaften, 23:435~46. 1919 Uber die Interferenzfahigkeit von Spektrallinien vom Standpunkt der Quantentheorie. Sitzungsberichte der Bayerischen Akademie der Wissenschaften, Vorgelegt am 11 Januar, pp. 73-90. Sur la coherence des lignes spectrales du point de vue de la theorie des quanta. Archives des Sciences Physiques et Naturelles, Societe Suisse de Physique, 5me Periode, Vol. 1, pp. 238-40. Zur Theorie der Raumladungserscheinungen. Verhandl. Deut. Physik. Gesell., 21:85-99.

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148 BIOGRAPHICAL MEMOIRS Bemerkungen zur Frage der Quantelung des Kreisels. Physik. Z., 20:289-94. Theoretisches uber den Starkeffekt in der Fowlerschen Heliumserie. Ann. Physik., 58: 553-76. Uber das Vorzeichen des Lichtdruckes auf kleine Teilchen. Mittei- lungen der Physikalischen Gesellschaft, Zurich, Nr. 19, pp. 30-35. Erweiterung der Quantansatze fur beliebige Systeme. Vortragen zu den Verhandlungen der Schweizerischen Naturforschenden Gesellschaft, Lugano. 1921 Beschouwingen op het geibied van de theorie der quanta. Gewone Vergadering der Wis- en Natuurkundige, Deel, 29:965-79. On the principles of the theory of quanta. Proceedings of the Academy of Sciences of Amsterdam, 23:1193-1205. Uber die Polflucht der Kontinente. Naturwiss., 9:499-502. 1922 Die Storungsrechnung im Dienste der Quantentheorie. I. Eine Methode der Storungsrechnung. Z. Physik., 8:211-28. Die Storungsrechnung im Dienste der Quantentheorie. II. Die numerische Durchfuhrung der Methode. Z. Physik., 8:305-20. Die Storungsrechnung im Dienste der Quantentheorie. III. Kritische Bemerkungen zur Dispersionstheorie. Z. Physik., 9:92-110. Problems of quantum theory in the light of the theory of perturba- tions. Phys. Rev., 19: 578-608. The evaluation of quantum integrals. Proc. Nat. Acad. Sci., 8:166- 67. 1923 Zur Aberrationstheorie. Bemerkung zu einer Abhandlung von A. Kopff. Physik. Z., 24:64-65. Paramagnetism and the theory of quanta. Science, 57:532-33. The Stark effect for strong magnetic fields. Philosophical Magazine 46:964. Simultaneous action of an electric and a magnetic field on a hydro- gen-like atom. Phys. Rev., 22:204. (A) Ferromagnetism and quantum theory. Phys. Rev., 22:204. (A)

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PAUL SOPHUS EPSTEIN 149 1924 With P. Ehrenfest. The quantum theory of the Fraunhofer dif- fraction. Proc. Nat. Acad. Sci., 10: 133-39. On the resistance experienced by spheres in their motion through gases. Phys. Rev., 23:710-33. On the simultaneous jumping of two electrons in Bohr's model. Proc. Nat. Acad. Sci., 10: 337-42. 1926 (centennial of the undulatory theory of light. Science, 63:387-93. The Stark effect from the point of view of Schrodinger's quantum theory. Phys. Rev., 28:695-710. On the evaluation of certain integrals important in the theory of quanta. Proc. Nat. Acad. Sci., 12:629-33. The new quantum theory and the Zeeman effect. Proc. Nat. Acad. Sci., 12:634-38. Second order Stark effect in hydrogen. Science, 64:621-22. 1927 Two remarks on Schrodinger's quantum theory. Proc. Nat. Acad. Sci., 13:94-96. The magnetic dipole in undulatory mechanics. Proc. Nat. Acad. Sci., 13:232-37. With P. Ehrenfest. Remarks on the quantum theory of diffraction. Proc. Nat. Acad. Sci., 13:400-408. The dielectric constant of atomic hydrogen in undulatory mechanics. Proc. Nat. Acad. Sci., 13: 432-38. 1928 On the theory of the radiometer. Phys. Rev., 31:914. (A) Interference of reflected light. Phys. Rev., 32:328. (A) 1929 Zur Theorie des Radiometers. Z. Physik., 54:537-63. With M. Muskat. On the continuous spectrum of the hydrogen atom. Proc. Nat. Acad. Sci., 15:405-11. Konferenz uber den Michelson-Morleyschen Versuch. Naturwiss., 17:923-28.

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150 BIOGRAPHICAL MEMOIRS 1930 Geometrical optics in absorbing media. Proc. Nat. Acad. Sci., 16:37~5. Reflection of waves in an inhomogeneous absorbing medium. Proc. Nat. Acad. Sci., 16:627-37. Note on the nature of cosmic rays. Proc. Nat. Acad. Sci., 16:658-63. 1931 Answer to Prof. Stormer's remark. Proc. Nat. Acad. Sci., 17:160-61. On the air resistance of projectiles. Proc. Nat. Acad. Sci., 17:532~7; also in Phys. Rev., 37:233. (A) 1932 Uber Gasentmischung in der Atmosphare. Gerlands Beitrage zur Geophysik, 35: 153-65. On ferromagnetism and related problems of the theory of electrons. Phys. Rev., 41:91-109. 1933 La resistance de Fair sur les projectiles. Extrait du Memorial de L'Artillerie Franchise, Paris, pp. 635-50. On the temperature dependence of ferromagnetic saturation. Proc. Nat. Acad. Sci., 19:1044~9. 1934 The expansion of the universe and the intensity of cosmic rays. Proc. Nat. Acad. Sci., 20: 67-78. 1935 On the bending of electromagnetic microwaves below the horizon. Proc. Nat. Acad. Sci., 21:62-68. The frontiers of science. Reflex, 6:19-24. 1936 Uses and abuses of nationalism. Reflex, 7:14-18.

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PAUL SOPHUS EPSTEIN 1937 151 Textbook of Thermodynamics. New York, John Wiley & Sons, Inc. 406 pp. Physics and metaphysics. Scientific Monthly, 45:49-54. On the magnetic energy of supraconductors. Proc. Nat. Acad. Sci., 23:604-10. 1938 Influence of the solar magnetic field upon cosmic rays. Phys. Rev., 53:862-66. 1941 Secondary school mathematics in relation to college physics. Am. J. Phys., 9:34-37. On the absorption of sound waves in suspensions and emulsions. In: Theodore son Barman Anniversary Volume (Applied Mechan- ics), by the friends of Theodore van Karman, pp. 162-88. Pasa- dena, California Institute of Technology. 1942 The time concept in restricted relativity. Am. l.--Phys., 10:1-6. The time concept in restricted relativity" A rejoinder. Am. T. Phys., 10: 205-8. On the theory of elastic vibrations in plates and shells. journal of Mathematics and Physics, 21:198-209. 1945 The reality problem in quantum mechanics. Am. J. Phys., 13:127- 36. 1946 On the elastic properties of lattices. Phys. Rev., 70:915-22. 1947 Radio-wave propagation and electromagnetic surface waves. Proc. Nat. Acad. Sci., 33:195-99. 1948 Robert Andrews Millikan as physicist and teacher. Rev. Mod. Phys., 20: 10-25.

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152 BIOGRAPHICAL MEMOIRS 1950 With M. S. Plesset. On the stability of gas bubbles in liquid-gas solutions. Journal of Chemical Physics, 18: 1505-9. 1953 With R. R. Carhart. The absorption of sound in suspension and emulsions. I. Water fog in air. journal of the Acoustical Society of America, 25:553-65. Dialektischer Materialismus und die modernen physikalischen Theorien. Physik. Blatt., 9: 49-55. Zur Situation der Naturwissenschaftler in der Sowjetunion. Physik. Blatt., 9:221-26. 1954 On Planck's quantum of action. Am. I. Phys., 22:402-5. On the possibility of electromagnetic surface waves. Proc. Nat. Acad. Sci., 40: 1158-65. 1956 Theory of wave propagation in a gyromagnetic medium. Rev. Mod. Phys., 28:3-17.

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