Graduate School of Natural Sciences Master's Thesis Mathematics in Physics Education at Leiden University 1912{1940 Supervisor: Author: dr. F.D.A. Wegener Dionijs van Tuijl Second reader: 4021428 dr. G. Bacciagaluppi August 2, 2018 Abstract The aim of this research is to show how the changing role of mathematics in physics in the first half of the 20th century was reflected on the mathematics in physics education. I have focused on Leiden University, in the period between 1912 and 1940. In this period quantum theory changed the mathematics used by physicists. Hendrik Antonie Kramers is studied in detail, because he was both a student and a professor of physics at Leiden University in this period. As a result, he provides the ideal case study to find out what a physics study was like both in the 1910s and the 1930s at Leiden University. In this study is shown how the mathematics curriculum did not change and the mathematical training for the candidates exams stayed the same as well. It was Kramers, as professor of physics, who lectured on subjects of abstract mathematics. But he lectured only on those parts he deemed crucial and the other aspects of modern mathematics did not find their way to Leiden University in the 1930s. iii Acknowledgements First I would like to thank my supervisor Daan Wegener for the helpful advice and encouragement he has given me throughout my research. Next I would like to thank Guido Bacciagaluppi for offering me the opportunity to do research in Copenhagen. Special thanks for Christian Joas for his hospitality at the Niels Bohr Archive, as well as to Rob Sunderland, Lis Rasmussen, and Hakon Bergset. I also want to show my appreciation to all the rendieren, who have always supported me, encouraged me, and served many times as proofreaders. Finally, I would like to thank my parents, for everything. iv Contents Abstract iii Acknowledgements iv Contents v List of figures vii Introduction 1 1 Physics Education at Leiden University in the 1910s 7 Lorentz and Ehrenfest as professors of theoretical physics . .8 A study in Physics in Leiden in the 1910s . 13 Kramers and Kloosterman, students in Leiden . 19 Conclusion . 23 2 Kramers working in Copenhagen 26 From Leiden to Arnhem . 26 Kramers arrival in Copenhagen . 27 The Institute for Theoretical Physics . 30 Kramers's scientific publications . 35 The return to the Netherlands . 38 Conclusion . 40 3 Physics Education at Leiden University in the 1930s 41 Professors of Physics . 41 Kramers's early lectures in quantum mechanics . 43 The role of mathematics . 48 Professor of physics and mechanics . 49 Writing a Textbook . 51 v Ter Haar's Translation . 53 Critical Reviews . 54 Structure and content of Kramers's textbook . 56 Kramers's textbook compared to others . 62 Mathematics in Kramers's textbook . 64 The use of Kramers's textbook . 69 Mathematics in Leiden in the 1930s . 72 Conclusion . 76 Conclusion 77 Bibliography 80 vi List of Figures 1.1 Table of contents of Lorentz' course. .9 1.2 Excerpt of Lorentz' course . 10 1.3 Notes of Ehrenfest's lecture, by Kloosterman . 12 1.4 The studieplan before the candidates exam . 15 1.5 The studieplan after the candidates exam . 15 1.6 Invitation for a meeting of Christiaan Huygens ............... 21 1.7 Kramers's notes on an ellipse . 22 1.8 An exercise done by Kloosterman . 24 2.1 Kramers's first letter to Bohr . 29 2.2 The main building of the Niels Bohr Institute . 31 2.3 Experimental results from Japan . 33 2.4 Front page of Holst's and Kramers's book . 34 3.1 Kramers's first course of quantum mechanics . 45 3.2 Kramers notes on electron spin . 47 3.3 Kramers notes to introduce spin . 48 3.4 Kramers's notes for the colloquium . 52 3.5 Table of contents of Kramers's textbook, (1). 59 3.6 Table of contents of Kramers's textbook, (2). 60 3.7 Excerpt of Kramers's textbook, page 100 . 65 3.8 Excerpt of Kramers textbook, page 102 . 65 3.9 Kramers's notes for a lecture on quantum mechanics . 70 3.10 A brief notation in Blaauw's notes . 70 3.11 Excerpt of Kramers's textbook. 71 3.12 Blaauw's notes on Kramers's quantum lecture . 71 3.13 Exam results of Adriaan Blaauw . 74 3.14 Exercises in Kramers's course on vector analysis . 75 3.15 Kramers's exercises, written down by Adriaan Blaauw . 75 vii Introduction On the 28th of September 1934, professor Hendrik Antonie Kramers (1894-1952) delivered his inaugural address at Leiden University.1 In his speech, Kramers discussed the influence of the personality of a physicist on his work in physics. A wide range of physicists is mentioned, from Isaac Newton to Kramers's predecessor Paul Ehrenfest (1880-1933). Two imaginary professors are included as well, professor Neophilos and professor Palaeophilos. kramers used them to illustrate the different styles of teaching the modern quantum theory. The first, professor Neophilos, would present the modern quantum theory as a com- plete break with classical physics, stating that the modern theory can never be truly un- derstood if one would stick with the old-fashioned dogmas. Therefore, Kramers claimed that Neophilos would switch in his course from physics to mathematics, \om ons een lesje in de niet commutatieve algebra te geven".2 Profesoor Palaeophilos, on the other hand, would present the modern quantum theory in a manner comparable to classical physics, with respect to the mathematics used. He might have presented a new kind of wave equation, but would show in turn how it resembled classical ones mathematically. After describing these two different professors, with their different styles of introducing the quantum theory, Kramers stated: \En het eigenaardige is nu, dat ze beide toch dezelfde theorie hebben voorge- dragen. Beider leerlingen, zullen, zoo ze de noodige bekwaamheid bezitten, vraagstukken der atoomtheorie tot een gelukkige oplossing kunnen brengen, en hun voorspellingen ten aanzien van experimenten zullen, zoo ze zich niet verrekend hebben, precies dezelfde zijn."3 Apparantly there are at least two different ways of lecturing on the quantum theory and both turn out to provide to the students the necessary skills to use the theory. This gives rise to many questions. How could such different lectures present the very same theory? Why would a professor of physics prefer one above the other? And above all, why would Kramers discuss this during his inaugural address? My aim is to answer these questions, by studying physics education, or, even more specifically, mathematics in physics education at Leiden University in the period between 1912 and 1940. 1This speech, named Natuurkunde en Natuurkundigen, was published in Nederlands Tijdschrift voor Natuurkunde, I-8 (1934), but an offprint of the speech was made for the ceremony. One of those offprints, the one that had belonged to George Uhlenbeck, can be found in the Utrecht University Library 2Kramers, Natuurkunde en Natuurkundigen, 256. 3Ibid., 257. 1 In this period the field of physics changed dramatically, due to the theories of relativity and quantum mechanics.4 One of the main changes involved the mathematics in physical theories, physicists started using more abstract mathematics. Examples are the use of non-Euclidean geometry in the theory of general relativity, or the role of matrices in quantum theory, more precisely in matrix mechanics.5 As such, abstract mathematics gained a prominent role in physics. The term abstract mathematics is still rather vague. What I consider abstract math- ematics is the study of formal proofs and theoretical mathematical entities. As a modern textbook in mathematics claims: \All serious students of mathematics eventually reach the stage where they realize that mathematics is not simply the manipulation of numbers or using the right formula".6 In this thesis, I will consider abstract mathematics in this sense, thus the deliberate use of formal proofs, the broader application of mathematics than just picking a formula, and the analytical introduction of mathematical entities, just by definition and without pictures or analogies. Besides abstract mathematics, the terms mathematical physics and theoretical physics are ambiguous too. However, it is impossible to give a clear definition, because the physi- cists in this period used the terms quite confusingly themselves. This confusion in itself is therefore studied as well. I expect that the changing role of mathematics in physics would imply that physicists needed another mathematical basis, since the use of abstract mathematics demands dif- ferent prior knowledge. Examples could be introductory courses in the theory of vector analysis, linear algebra, and the theory of groups. On the other hand, the existing mathematical education had to be preserved as well, for the introduction of abstract mathematics did by no means end the importance of the mathematics used in classical physics. Overall, it seems that the university education in mathematics for physics stu- dents had to be widened. My aim is to study whether this change in mathematics in physics education can indeed be found in the first half of the 20th century, the period in which the theory of quantum physics was established, and if so, how it was implemented in university education. To study this, I look at the case of Leiden University. This was the main center for physics in the Netherlands, which was for a large part due to the prestige of the professors of physics. One of those professors was Hendrik Antoon Lorentz (1853-1928), who might 4See for instance the articles on relativity, Heilbron, `Relativity', 711-713, and quantum physics, Heilbron, `Quantum Physics', 691-694. 5Cao, `Space and Time', 766, Heilbron, `Quantum Physics', 693.