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. 6Chartrand et al., Mathematical Proofs, 1.

2 be considered the most renowned scientist in Dutch history.7 But Leiden hosted other outstanding physicists as well, such as Nobel laureate Heike Kamerlingh Onnes (1853- 1926) and Paul Ehrenfest (1880-1933). The period is also called the ‘Second Golden Age’, due to the prominence of Dutch physics, with for instance four Nobel Laureates.8 To be able to find changes in mathematics education, I compare a study of physics in Leiden in the 1910s with a study in the 1930s. A natural starting point is 1912, the year wherein Ehrenfest succeeded Lorentz as professor of physics. And the year 1940 demarcates a clear ending point, with the beginning of the Second World War in the Netherlands. Besides a general description of mathematics and physics education in Leiden in the 1910s and the 1930s, I present a case study, in the person of Hendrik Antonie Kramers. Kramers studied mathematics and physics in Leiden in the period 1912-1916 and followed lectures of both Lorentz and Ehrenfest. Thereafter he worked abroad, in Copenhagen with the physicist Niels Bohr (1885-1962). He obtained his Ph.D. in 1919 under supervi- sion of Paul Ehrenfest and stayed in Copenhagen until 1926, when he became professor of physics in Utrecht. In 1934 he left Utrecht to succeed Ehrenfest as professor in Leiden, where he stayed until his death in 1952. Because of this exceptional chain of events, Kramers is the excellent case to study the role of mathematics in Leiden’s physics edu- cation. After all, he both received his education there and would eventually teach there himself.

All in all, I want to give a pedagogical history of mathematics in physics education in Leiden between 1912 and 1940. The canonical work concerning such a history of the pedagogy of mathematics in physics education is Masters of Theory by Andrew Warwick of 2003.9 He discusses mathematical physics in Cambridge, in the period from the end of the eighteenth to the early twentieth century. What is unique in his approach is that it is written from the perspective of pedagogy and training. In this way scientific knowledge is investigated by studying the educational process. As a result, he is able to focus on craft skill, localism, and collective production of knowledge. Using this, Warwick is for instance able to describe how physicists in Cambridge understood the theory of relativity. War- wick exposed their background knowledge by closely studying their training and could

7His fame is shown in many instances. During his funeral, for example, Dutch state telegraph and telephone services were suspended for three minutes in tribute of Lorentz, as is recalled by Owen Richardson (’Lorentz’, 192). An impression of Lorentz’ funeral, with its many carriages and im- pressive crowd, can also be found on Youtube, ‘Begrafenis van prof. Lorentz in Haarlem (1928)’, https://www.youtube.com/watch?v=H2VtrJD0xJk. 8Willink, De tweede Gouden Eeuw, Kox, ‘Lorentz’, 441. 9Warwick, Masters of Theory.

3 therefore pinpoint exactly the parts of the theory that dazzled the Cambridge physicists. In my research, I will use the same perspective on the educational process, to give a clear description of physics education in Leiden. This includes researching the role of profes- sors and lecturers, the use of textbooks and exercises, the amount and style of exams, the structure of the curriculum, and more. This I will use to grasp the understanding of quantum theory of the physicists in Leiden. Suman Seth has written such a history about Arnold Sommerfeld in Munich, in Craft- ing the Quantum, published in 2010. There, Seth describes Sommerfeld’s approach to, and interpreatation of, physics and the style of education this gave rise to. Seth calls Sommerfeld’s professional style the ’Physics of Problems’. Solving problems is consid- ered more important than finding the underlying principles of physics. In his teaching, Sommerfeld concentrated on these problems as well. Seth concludes that this resulted in a different view on the developments in physics, quantum theory was just another, very successful, theory for problem solving. In this way Seth links training and education to an interpretation of physics. Considering Kramers’s stay of ten years with Niels Bohr in Copenhagen, I assume that he must have had a clear idea of the developments in physics as well. In the same way as Seth, I shall describe Kramers’s ideas set against his training and education. About physics education in Leiden, Martin Klein has written ‘Physics in the Making in Leiden: Paul Ehrenfest as Teacher’, as well as a general biography of Ehrenfest, published in 1989 and 1970 (second edition in 1972) respectively.10 In both works, Klein focuses on the specific characteristics of Ehrenfest as a teacher, but no general context of physics education in Leiden is sketched. It does, on the other hand, give great insight in Ehrenfest himself. Such a description of Ehrenfest and his struggle with modern science is also given by Marijn Hollestelle’s.11 There, Ehrenfest’s considerations and his style of physics are thoroughly displayed. There have been two major works concerning Hendrik Kramers. The first is Max Dresden’s biography of Kramers of 1987, which quite extensively describes his life.12 It especially focuses on the remarkable period he was living in and the scientific develop- ments made in that period. As a result, Kramers scientific research is studied, instead of his education. As a matter of fact, his role as professor of physics in Leiden is barely discussed, as is the period of Kramers as a student.13 The second work on Kramers is

10Klein, ‘Physics in the Making in Leiden’, and Klein, Paul Ehrenfest. 11Hollestelle, Worstelingen. 12Dresden, H.A. Kramers. 13Dresden discusses Kramers Leiden years as student in five pages, where he primarily focussed on the relation between Kramers and Ehrenfest, Dresden, H.A. Kramers, 90-94, and Kramers as a teacher in

4 Dirk ter Haar’s book Master of Modern Physics, where he describes all of Kramers’s scientific contributions.14 This book aims to present the complete scientific oeuvre of Kramers and places his publications in the broader context of developments in physics. In the end, both works do not intensively link Kramers education in Leiden to his later work in physics or to his own teaching. A work that does link the training of a physicist to his later publications is Lewis Pyenson’s The Young Einstein, where he gives a biography of Einstein, but with an extensive focus on Einstein’s education.15 He shows the importance of teaching and training in the forming of scientific knowledge. An example is the extensive treatment of the mathematical courses Einstein took with Minkowski and how the mathematics he learned there formed his later work. This seems to be a good model to link Kramers’s student years to his later work in physics and in education. Until now, only the people related to education are discussed. People are indeed the major factor in education, but other aspects are important as well. One example is the role of textbooks. Actually, Kramers has written such a textbook on quantum theory. Such a textbook can provide clear information about the authors ideas on physics and education. Other textbooks on quantum theory are described in Research and Pedagogy, edited by Massimiliano Badino and Jaume Navarro, and published in 2013. The aim of the book is to show the history of quantum mechanics through its textbooks. Examples of such analysis are Helge Kragh’s chapter on Dirac’s Principles of Quantum Mechanics and Don Howards chapter on Jordan’s Anschauliche Quantentheorie.16 These articles present a clear model of analysis of a quantum textbook, which I will use to analyse the textbook of Kramers.

Overall, I present a pedagogical history of physics education in Leiden in the 1910s, focusing on more than the professor of theoretical physics alone. To do so, I will study lectures, textbooks, and the Leiden curriculum, of which many sources are preserved in archives.17 This results in a clear description of the strengths (and weaknesses) of Lei- den’s education. The case of Kramers shows the result of this education, how it was in turn reflected in his later work and teaching. His case will also demonstrate what a

eight pages, in the chapter on obligation and duty, Dresden, H.A. Kramers, 491-498. 14Ter Haar, Master of Modern Physics. 15Pyenson, The Young Einstein. 16Kragh, ‘Paul Dirac’, 249-264, and Howard, ‘Quantum Mechanics in Context’, 265-284. 17For example, Kramers lecture notes in the Kramers Archief, box 4, in the wetenschapsarchieven of the Noord-Hollands Archief and the Leiden curriculum of 1917 in the Archief van de Faculteit wis- en natuurkunde, AFA FA 1.2.1.2.34, in the bijzondere collecties of the Universiteit Leiden.

5 physicist learned after his students years, by doing research. There a main focus will be on the role of Niels Bohr as Kramers’s mentor in physics. The role of Bohr on Kramers has been studied before, for instance by Dresden and Ter Haar, but not yet compared to his education and teaching. Therefore I will link this role of Bohr to Kramers education, for exampale in showing his influence on Kramers’s textbook, to be able to present where Kramers was influenced by his education and where he was influenced by his stay in Copenhagen. There will once again be a description of the physics education in Leiden in the same way as in the beginning, but considering the 1930s. In the end, this will give an overview of mathematics in physics education in Leiden and the changes therein.

To do all of this, the story is presented in three different parts. In the first part, Physics Education at Leiden University in the 1910s, I shall study the education in Leiden, as well as the academic climate in Leiden in that period. I will closely analyze the lecture notes and sources of the education. I will also study Ehrenfest and Lorentz as professors in Leiden, where I make use of the existing secondary literature. I present the cases of two students in Leiden, firstly Hendrik Kramers, who is followed thereafter as well, and secondly Hendrik Kloosterman, who has made many lecture notes as a student in Leiden and can therefore provide a clear view of the education.18 All in all, I present in this part a clear picture of physics and mathematics education in Leiden in the period of the 1910s. In the second part, Kramers working in Copenhagen, I will focus on the period where Kramers worked abroad. I try to research how he got to work there, who he worked with, and what sort of work he did. There I will also look at what he used from his earlier education, and what he did not. In this way I show what kind of scientist Kramers had become and what the role of his education was in that process. Finally, in the third part, Physics Education at Leiden University in the 1930s, I will return to Leiden, to study Kramers (and, as will be shown, Kloosterman) as teacher there. I look at the lecture notes made by himself and his students to analyse his way of teaching, and at the mathematics taught in Leiden in the period. In this part I analyse the textbook written by Kramers about quantum theory, and link it to the education as well. In this manner I hope to produce a clear picture of the physics and mathematics education between 1930 and 1940, and how it differed from the earlier period.

18Those lecture notes can be found in the Kloosterman Archief, box 1, in the bijzondere collecties of the Universiteit Leiden.

6 Chapter 1

Physics Education at Leiden University in the 1910s

When one wants to have a look at physics education in Leiden in the 1910s, one natu- rally turns to the chair of theoretical physics. The professor of theoretical physics was responsible for the education of physics students, as well as for the surrounding organisa- tion of this education. But those responsibilities are probably not the main reason most researchers have focused on the respective professors of theoretical physics. The main reason would generally be the enormous status in science of these professors in Leiden in the 1910s: Hendrik Antoon Lorentz and Paul Ehrenfest.1 About the first of those Albert Einstein, whose name has become almost synonymous with ‘genius’, said: “At the turn of the century, H.A. Lorentz was regarded by theoretical physicists of all nations as the leading spirit; and this with the fullest justification.”2 And of the second Hendrik Kramers, who would succeed Ehrenfest as professor, stated: “Those who saw and heard Ehrenfest could not escape the feeling that they were subject to a whirlwind which would bring promise and novelty to all the corners of their soul.”3 Apparently, both professors had some characteristic traits that made them extraordinary. In this chapter, my aim is to present physics education in Leiden in the 1910s, looking beyond Lorentz and Ehrenfest. I will present the curriculum in Leiden and show the responsibilities of the the professors of mathematics in the training of physicists. This is in turn used to describe a physics study through the eyes of the students.

1For instance in Anne Kox’s articles, Kox, ‘Hendrik Antoon Lorentz’, and Kox, ‘Lorentz en Ehrenfest’, or in Martin Klein’s works, Klein, Paul Ehrenfest, and (most notably) Klein, ‘Physics in the making in leiden’. 2Einstein, ‘H.A. Lorentz’, 5. 3Kramers, ‘In memoriam P. Ehrenfest’, Physica 13 (1933) 273, as quoted in: Dresden, H.A. Kramers, 92.

7 To do this, I start with the ‘familiar’ part, concerning Lorentz and Ehrenfest. I will describe their style of teaching and education and thier influence in Leiden as a whole. Afterwards, I will shift my focus to a more general description of a physics study in Leiden, where I will explain the role of other professors and lecturers. Finally, I will give two case studies of students in Leiden in the 1910s, Hendrik Kramers and Hendrik Douwe Kloosterman.4

Lorentz and Ehrenfest as professors of theoretical physics

In 1912, Paul Ehrenfest succeeded Hendrik Antoon Lorentz as professor of theoretical physics in Leiden. At that moment, Lorentz had held the chair for thirty-five years, and he was considered one of the most prominent physicists in the world.5 The year before, he had been chairman of the first Solvay conference, where the world’s leading physicists discussed new developments in the field of physics. Lorentz had been a very busy man and when the Teylers museum in Haarlem offered him a position where he could spend more time on this kind of activities, he gratefully accepted. As a result, Leiden had to find a new professor of theoretical physics, something Lorentz organised himself.6 At first, he tried to convince Einstein to succeed him, but Einstein refused the offer and went to Berlin. Afterwards, Lorentz asked Ehrenfest, who at the time lived with his wife in Saint Petersburg. Ehrenfest was highly honored that Lorentz would choose him as a successor, and he accepted the offer. With the arrival of Ehrenfest as professor in Leiden, a new period of physics teaching started off. To understand what was new in this period, I present a brief account of Lorentz’s teaching first.

Lorentz was widely known as an outstanding teacher. His use of the blackboard was renowned, due to its clarity and concise structure. From 1883 onwards, Lorentz had given the introductory course on physics, for future medical students and first year students in mathematics and physics.7 Some of this work is written down in textbooks. The most important one was written by Lorentz in 1893 and called Beginselen der Natuurkunde. It would go through nine editions, the last of which appeared in 1929. Because the course was also aimed at medical students, the textbook did not make use of more mathematical

4It is rather remarkable that three of the main protagonists, Hendrik Lorentz, Hendrik Kramers, and Hendrik Kloosterman, are all named Hendrik. One would expect this to be linked to the marriage between Queen Wilhelmina and Prince consort Hendrik van Mecklenburg-Schwerin, but all three were born before the engagement on October 16th, 1900. 5Kox, ‘Lorentz’, 441. 6Kox, ‘Lorentz en Ehrenfest’, 43. 7De Haas-Lorentz, ‘Reminiscences’, 43.

8 Figure 1.1: Table of contents of Lorentz’ course on the theory of quanta, 1916-1917. From Lorentz, Theorie der Quanta, v. prior knowledge than that obtained in secondary education, ‘not even in the deduction of the velocity of light’.8 In the book, one could also find parts in different font sizes, medical students were advised to skip the parts in the smallest font size completely.9 Teaching this elementary course had become harder and harder for Lorentz. Therefore, in 1906, Johannes Petrus Kuenen (1866-1922) was appointed as a third professor of physics, with the responsibility of teaching the first year students, to ease Lorentz’s burden.10 This did not result in the ending of Lorentz lecturing in Leiden. Once every week, he would travel to leiden to give a lecture. This was the famous ‘Monday morning lecture of professor Lorentz’, which was every Monday at eleven. In these lectures Lorentz would

8De Haas-Lorentz, ‘Reminiscences’, 84. 9Lorentz, Beginselen der Natuurkunde I, ii. 10De Haas-Lorentz, ‘Reminiscences’, 97.

9 Figure 1.2: Excerpt of Lorentz’ course on the theory of quanta. From Lorentz, Theorie der Quanta, 77. sketch the current developments in the field of physics, which was quite exceptional, and he would always, in his own words, ‘turn the subject round and round and over and over’.11 Often, this would include his own opinion about recent developments in different fields of physics. This can for example be seen in Lorentz’s lectures on the theory of quanta in 1916-1917, of which the contents are depicted in figure 1.1. Lorentz introduced modern developments in this theory, such as the Stark-effect, shown in figure 1.2, which would win its discoverer, Johannes Stark (1874-1957), a Nobel prize in 1919. Note the part Slotopmerkingen as well, apparently Lorentz concluded with some of his own re- marks. De Haas-Lorentz, Lorentz’s daughter, remembered ‘how surprised and interested Max Planck was when during a visit to Leiden, he discovered that my father was lectur- ing on Planck’s concentration currents’.12 This shows again how up-to-date his lectures were. In this manner Lorentz stayed connected with the physics education in Leiden. In fact, Lorentz remained professor, but got the special status of extraordinary professor (buitengewoon hoogleraar).

Ehrenfest had a different style of teaching from Lorentz. His style of teaching and his role for Leiden University as a professor have been studied extensively.13 For him, there was no distinction between the scientific and the private life. He was much closer to his students, trying to guide them through their education. De Haas-Lorentz stated:

11De Haas-Lorentz, ‘Reminiscences’, 102. 12Ibid., 83. 13See for instance Martin Klein’s biography of Ehrenfest, Paul Ehrenfest, or his specific contribution considering Ehrenfest’s teaching, ‘Physics in the Making in Leiden’, 29-44.

10 “Whenever Ehrenfest met a young man in whom he saw great possibilities as far as physics was concerned, but who spent too much time and energy on other things, he would spare time nor effort to influence him to take up the study of physics seriously. Lorentz, facing the same situation, would regret the fact, but would come to the conclusion that, for better or for worse, this young man was more interested in other matters than in physics, and that this was his own business.”14

Ehrenfest had already declared this interest in his students during the public lecture of his official inauguration. There he stated, in the end: “I am to devote all the knowledge and ability I have to assisting each of you in finding, with as little damage as possible, the path that corresponds to the very essence of his talent”.15 In the same sense, Ehrenfest was always helping the students by creating optimal conditions to study. He established for instance the Bosscha library, such that the physics students would have a good place to study. And he also convinced his students to become part of discussion groups, such as Christiaan Huygens or De Leidsche Flesch. In this way he would make sure that the students would also become researchers. Creating such conditions was necessary because many of the students did not live in Leiden. In the city of Leiden itself, it was incredibly hard to find affordable places to live for students. Therefore they would often live in surrounding areas, such as the city of Den Haag. By creating discussion groups that would regularly meet, and a place where the students could study, Ehrenfest tried to create some sort of community for his students. Probably his most important contribution was the introduction of a regular collo- quium. Almost immediately after his arrival in Leiden, he set up a colloquium that would meet every Wednesday afternoon. It was not the first time that he had established such a colloquium, in Saint Petersburg he had done the same, together with his wife. This colloquium also fitted Ehrenfest’s his style of personal interaction in science. There had been a colloquium held by Lorentz, but it was not held regularly and did not have a well-defined structure. The audience of Ehrenfest’s colloquium consisted of students who had already passed their first examinations, former students, colleagues and other physicists from nearby villages, and foreigners who came to visit Ehrenfest in Leiden.16 Ehrenfest made sure that the meetings were well prepared and that everybody in the room would be able to follow the presentation. If he considered some parts too difficult for the younger visitors, he would anticipate and already give an explanation in his lecture

14De Haas-Lorentz, ‘Reminiscences’, 107. 15Ehrenfest, Zur Krise der Lichtaether-Hypothese, as quoted in: Klein, ‘Ehrenfest as Teacher’, 29. 16De Haas-Lorentz, ‘Reminiscences’, 105.

11 or in a previous session.17 He also made a list of attendees of the colloquium, something he did not do for his lectures. This shows that Ehrenfest deemed attendance in the collo- quium more important than in his lectures. It is linked to another important aspect of the colloquium too, student members had to be admitted by Ehrenfest himself.18 Ehrenfest had to consider the students ready for attending his colloquium. As a teacher, Ehrenfest always tried to focus on the main points of a theory. He wanted his students to get the main conceptual points of a theorem, such that they would understand its importance in physics. He would not use a common systematic

Figure 1.3: Notes made by Hendrik Kloosterman of Ehrenfest’s lecture on quantum theory in 1919-1920. From the Archief Kloosterman, box 1, in the bijzondere collecties of the Universiteit Leiden.

17De Haas-Lorentz, ‘Reminiscences’, 106. 18Klein, ‘Ehrenfest as Teacher’, 36.

12 development of a subject, but he would rather focus on the salient points. Therefore mathematics was only a tool to come up with the results and often Ehrenfest did not produce these proofs on the blackboard.19 An example can be found in figure 1.3, where Ehrenfest presents new mathematical formulas without giving rigorous proofs. Rather, he focused on the strengths and the weaknesses of a theory, on what the theory tells us. He also tried to keep his education as up-to-date as possible, lecturing in many re- cent results, such as the theory of quanta shown in figure 1.3. In this way he wanted to show his students the fields of physics where they would be able to work themselves. He stressed that it was ‘physics in the making’, that physics had not yet been completed or codified, but was an ongoing discussion.20 As a result he did not give exercises to his pupils, simply because he did not believe that solving the problems of others would teach someone anything. Ehrenfest himself was always carrying a small booklet in his pocket, where he would write questions that arose during the day.21 Those were the questions he himself worked on and he expected something similar from his students.

Lorentz and Ehrenfest were excellent teachers, both in their own characteristic style. They focused in their lectures on current developments, to show their students what it was like to be a physicist. But for Ehrenfest, being a physicist was more than it meant for Lorentz, it was a calling that demanded almost strict devotion. Ehrenfest tried to educate his students to become researchers in a field of physics in the making.

A study in Physics in Leiden in the 1910s

Evidently, the lectures of Ehrenfest were not meant for first year students. They were part of the education for the doctoraalexamens (doctoral exams), already for higher level students. First, you had to study for your candidaatsexamen (candidate exams), which would normally take upon three years. In that period the students were trained to be capable of solving simple problems in mathematics and physics, as a foundation for their further education. This was done in lectures that were not given by the professor of theoretical physics, but by the professors and lectors (lecturers) of ‘mathematics and physics’. These all had their own leeropdracht, which stated what they should lecture in. In the period between 1912 and 1919, six professors were together responsible for the education in mathematics, physics, and astronomy. Of course, Ehrenfest had the leeropdracht the-

19Klein, ‘Ehrenfest as Teacher’, 31. 20Ibid. 21Ibid., 40.

13 oretical physics. As noted earlier, the elementary physics course was given by professor Johannes Kuenen. This included a practical and a large demonstration course, which was also meant for medical students. The lectures in experimental physics were given by professor Heike Kamerlingh Onnes (1853-1926), and in astronomy by professor Willem de Sitter (1872-1934). Pieter Zeeman Gzn. (1850-1915) was responsible for the leerop- dracht mathematics, including geometry and mechanics, which became the leeropdracht of Willem van der Woude (1876-1974) after he succeeded Zeeman in 1916. Finally, Jan Cornelis Kluyver (1860-1932) was responsible for the rest of the mathematics education, which mainly included mathematical analysis and probability theory. Those were the teachers responsible for the education of the young mathematics and physics students. At first, there was no strict table of courses and lectures, and the teachers were free to decide for themselves how many hours they would lecture. This changed in 1917-1918, after long deliberation between the many faculties of science in the Netherlands. Lorentz played a large role in these debates.22 In 1917 the faculty started with a more strict program, the studieplan (see figures 1.4 and 1.5), also induced by a new law in the Netherlands that allowed more students to go to the universities. This law resulted in a larger group of youngsters that were allowed in university education. Before, only students who had gone to a gymnasium, a school with Latin and Greek, were allowed in the universities, but from 1917 onwards also students of the Hogere Burgerschool (HBS), a kind of school comparable to the german Realschule, were allowed to enter. In the new program, a science study was still divided in two parts, the study for the candidates exams and for the doctoral exams. Both parts were divided in a recommended study schedule, which would ideally take up three years. In the first three years, before the candidates exams, students would follow courses in geometry, analysis, elementary physics, and a practical. One could also choose extra courses in astronomy, chemistry, or geology. Most of the time would be spent in the mathematics courses, with six hours each week in geometry for the first two years and three hours in analysis in the first year, which was raised to four hours in the second and third year. The elementary physics course would have three hours each week in the first two years. In this way, students were prepared for their candidates exams. As a candidate, the students could choose more specific paths. Those were divided in mathematics and astronomy, mathematics and mathematical physics, and mathematics

22The new studieplan, as well as the debates about it, can be found in the Archief van de Faculteit der wis- en natuurkunde, AFA FA 1.2.1.2.34, in the bijzondere collecties van de Universiteit Leiden. The role of Lorentz in the realisation of the studieplan has not yet been studied.

14 Figure 1.4: The studieplan before the candidates exam. From the Archief van de Faculteit der wis- en natuurkunde, AFA FA 1.2.1.2.34, in the bijzondere collecties van de Universiteit Leiden.

Figure 1.5: The studieplan after the candidates exam. From the Archief van de Faculteit der wis- en natuurkunde, AFA FA 1.2.1.2.34, in the bijzondere collecties van de Universiteit Leiden. and experimental physics. Notice how mathematics was obligatory in all paths, yet it was not possible to study mathematics alone. In Leiden, mathematics had not become a discipline on its own, but was a necessary part of the physical sciences.

It is remarkable that the studieplan spoke of mathematical physics and that theoreti- cal physics was not mentioned at all. This is even more surprising considering the fact

15 that Leiden had a professor of theoretical physics and that the man who had once held this chair, Hendrik Lorentz, was one of the authors of the studieplan. Apparently, he considered his lectures to deal with mathematical physics. The philosophical debates concerning those concepts might show how they were regarded. The simple part is the division between experimental physics and mathematical physics, which had been much clearer. Henri Poincar´e(1854-1912) described this division a couple of years earlier with an analogy of a library. Deciding which books had to be bought was a decision of experimental physics, the role of mathematical physics was creating a catalog, which would not immediately enrich the library, but aid the librarian (thus the experi- mentalist) in choosing the necessary books.23 Experimental physics could judge whether a theory was true or false, whether it was part of the ‘library of knowledge’, but which theories were possible and how they might be judged was a task of the mathematical physicist. Poincar´emade his point even more specific considering the role of mathemat- ical physics in the theory of electrons. He stated, considering himself a mathematical physicist:

“We can, for example, prepare the way by studying thoroughly the dynamics of the electrons; not, be it well understood, by staring from a single hypothesis, but by multiplying the hypotheses as much as possible. It would then be the part of the physicist to use our work in searching for the crucial experiment which would decide between them.”24

This distinction is simple and precise, and explains why the studieplan considered a study of experimental and mathematical physics. But where can theoretical physics be found? Ludwig Boltzmann (1844-1906) stated that the role of theoretical physics had always been to come up with assumptions for which it “seemed up to a point likely that they correspond exactly with reality; they were therefore called hypotheses”.25 In the quote of Poincar´ethese hypotheses can be found as well. It seems that theoretical physics had to come up with these hypotheses. Experimental and mathematical physics required simple training, how to set up an experiment, how to do the calculations, et cetera. This also implied making exercises and endless repetition of basic experiments. But the training of a theoretical physicist was different, it was becoming one of Eherenfest’s researchers in a field of ‘physics in the making’. The mere courses offered at Leiden University did not create such physicists,

23Poincar´e,‘Relations’, 518-519. 24Poincar´e,‘Present and the future’, 36. 25Boltzmann, Theoretical Physics, 7.

16 but they trained physicists in experimental and mathematical physics. It is likely reason for the studieplan not to mention theoretical physics, although the distinction between mathematical and theoretical physics remained vague.

When one had chosen mathematics and mathematical physics, one would start with Ehrenfest’s theoretical physics course, which also became the main course with four hours in the first two years, and even five in the third year. But one would also continue the mathematics courses, with geometry, analysis and probability theory. The majority of the courses were still in mathematical analysis, with only one hour of probability theory scheduled in the second year Also the colloquium was scheduled, one evening every four- teen days. One was also invited to visit the Monday morning lecture of Lorentz. It is surprising to see that the studieplan spoke of a mathematical colloquium and mathemat- ical physics, while both were the responsibility of the professor of theoretical physics. In this way, one would come more and more in contact with the scientific community. Although this schedule seems rather strict, it presents a picture that hardly corre- sponded to the day-to-day practice. The lectures themselves remained optional, atten- dance was by no means mandatory, and the described hours do not seem to correspond to the actually given lectures. George Uhlenbeck (1900-1988), who studied mathematics and physics in Leiden from 1919 to 1923, said about the situation before his candidates exams:

“Mainly because there were practically no lectures, you had nothing to do. There were only lectures in mathematics, analytic geometry and calculus and analysis and so, and they were only about four hours a week altogether. Two hours geometry, two hours analysis. Then there was physics. The general lec- ture, precisely as in Germany. There was the big lecture with demonstration, which I thought was quite dull and I didn’t go to that. I didn’t have to. I mean there was no attendance, nobody took any kind of responsibility for the student.”26

The content of the given lectures corresponds to the studieplan, but the hours were far less. Besides, Uhlenbeck followed only courses in mathematics, the physics course was too dull. For the candidates exams, one did not need to follow the physics course. There were also no centralized exams. The candidates exams consisted of oral exams, in

26Interview of George Uhlenbeck by Thomas S. Kuhn on 1962 April 5, Niels Bohr Library & Archives, American Institute of Physics, URL: https://www.aip.org/history-programs/niels-bohr-library/oral- histories/4922.

17 the main courses of mathematics, mechanics and physics. To do your exam, you had to make an appointment with the professor responsible for the corresponding field. Thus one would have oral exams with Van der Woude, Kluyver, and Kuenen. The content of the lectures was meant for all students of the science faculty, including students in chemistry, pharmacy, and geology, it started off rather simple and general, and only became more specialized later on. As a result, many students would choose not to follow the lectures in the recommended order, but for instance all in one year, which was possible due to the relatively few lecture hours. Therefore some students did their exams already after their first or second year. Many of the science students would use the newly established Bosscha library to study. The library made sure that it was as up-to-date as possible, and that various textbooks of all fields of physics were available. In that manner, students were able to choose what they would study exactly, and which style they preferred. Uhlenbeck for instance seems to have studied for his physics exams all by himself. As mentioned before, Ehrenfest never gave problems to his students, they had to come up with problems themselves. He seems not to have been unique in that point: other professors did not give the students problems as well. This was part of the responsibility of the students. As Uhlenbeck says about it: “They treated us completely as grown-ups. You could take the responsibility. You made your own problems. . . ”27

Although it was not mandatory, many students did follow the lectures, especially those of Ehrenfest, Kluyver, and Van der Woude. The last two were mathematicians, although Van der Woude taught mechanics too. At the time, mechanics was considered part of the mathematical background. Van der Woude published on relativity theory, differential geometry, dynamics, and kinematics, though his publications did not present his own work. Instead, he tried to systematically summarize the results known in the countries surrounding the Netherlands.28 Kluyver can be considered a pure mathematician, ac- tively promoting mathematics as an independent discipline.29 Mathematics had been prospering in Europe, with for instance Felix Klein (1849-1925) and David Hilbert (1862- 1943) in G¨ottingen,and Henri Poincar´ein Paris. One of the highlights of the period was Hilbert’s talk in Paris at the International Congress of Mathematicians, where he presented 23 unsolved problems in mathematics. Most of the main mathematicians of the period worked in the fields of elliptic functions, number theory, and invariant theory. But in the Netherlands those fields did not get any attention. Kluyver, who was actually

27Interview of George Uhlenbeck by Thomas S. Kuhn, see footnote 26. 28Dijk, Hoogleraren Wiskunde, 55. 29Ibid., 51.

18 a geometer, specialised in analysis and number theory, so as to raise the level of those fields in the Netherlands.30 Thus Kluyver’s course on mathematical analysis was highly mathematical, and there- fore rather strict and rigid. Of this course Samuel Goudsmit (1902-1978) remembers that he had to take “mathematics, which was analytical geometry and calculus, rather stiff, and I did not do well in that”.31 It is clear that the mathematics courses gave the students a solid mathematical foundation and that the courses were an important part of the curriculum, especially in the first years. The importance of this mathematical basis is made clear by Goudsmit, who was not at all embraced by Ehrenfest, but sent away by Ehrenfest to Pieter Zeeman (1865-1943) in Amsterdam to do experimental physics. When he was called a theoretical physicist, he would state that he “never, never was, because I never could do the mathematics sufficiently well for a theorist”.32 Here again, the difficulty with the distinction of theoretical and mathematical physics is apparent. In the image of Goudsmit, a theoretical physicist should have a good understanding of mathematics, much more than the experimental physicist, and he himself did not have such an understanding of mathematics. Uhlenbeck, who would later work together with Goudsmit, had been much better in mathematics. About a special topics course, which was in his opinion “straight forward business — Bolzano-Weierstrass and the Dedekind cut, and all that sort of thing”, Uhlenbeck stated that he “thought all that was wonderful, because it was so rigorous”.33 Goudsmit, on the other hand, did not do well in his oral exam with Ehrenfest, where he remembers Ehrenfest saying: “The trouble with you is I don’t know what I can ask you, all you know is spectral lines. Can I ask you Maxwell equations and things about that?” Goudsmit claims he replied with: “No, please don’t.” But there is even an anecdote claiming he responded with: “That’s the part George [Uhlenbeck] always does.”34 The example shows how mathematics proved to be a necessary basis for a theoretical physicist, and how for Ehrenfest, who did not even use formal proofs in his own lectures, this mathematical basis was vital.

30Dijk, Hoogleraren Wiskunde, 52. 31Interview of Samuel Abraham Goudsmit by Thomas S. Kuhn on 1963 December 5, Niels Bohr Library & Archives, American Institute of Physics, College Park, MD USA, URL: https://www.aip.org/history- programs/niels-bohr-library/oral-histories/4640. 32Ibid. 33Interview of George Uhlenbeck by Thomas S. Kuhn, see footnote 26. 34Interview of Samuel Abraham Goudsmit by Thomas S. Kuhn, see footnote 31.

19 Kramers and Kloosterman, students in Leiden

Hendrik Antonie Kramers was born on the 17th of February 1894. He was raised in the city of Rotterdam, where he went to secondary school. Because he went to the HBS, and not to a gymnasium, he had to study Latin and Greek by himself to be allowed in university education (he graduated just before the new law was introduced). Kramers managed to prepare for his exams in Latin and Greek in merely one year, and learned it so well that he was able to read classics in Latin or Greek for his own pleasure.35 When he entered Leiden University, he also joined the Leidsche Studenten Corps. This was an exclusive fraternity, whose members devoted a great deal of their free time to drinking and partying. Kramers did participate in some of those activities, but after a while he quit, because he himself preferred debates and discussions.36 In that time, he became a member of Christiaan Huygens, the association advocated by Ehrenfest, where physics students would discuss topics in various fields of physics. An invitation for a meeting of Christiaan Huygens is depicted in figure 1.6. From the invitation it is clear that the students gathered in one of their houses, where they held small lectures. In Leiden, Kramers followed many different courses, of which an impressively large number in mathematical analysis. His lecture notes show for instance that he followed Kluyver’s courses on integrals, function theory, elliptic functions, and algebra.37 The last two of them were certainly neither mandatory nor necessary for a physicist at that time. It shows Kramer’s early interest in the field of mathematical analysis. As a student, Kramers was not the most disciplined and organized. For example, his lecture notes are full of little drawings and short poems, of moments where his mind obviously wandered off. He also often noted that he had missed a lecture, and sometimes even many lectures. There are cases where his notes show that he was trying to solve some problems that were not part of the lectures themselves. An example is shown in figure 1.7, where Kramers tried to solve a problem in differential equations, concerning some ellipse. He stated that he had ‘once more different and new ideas’. Afterwards he stated his problem, and he gave this new idea, the use of a differential equation. The problem is found in his lecture notes on differential equations, giving the impression that he was using some new techniques discussed in that course on a problem he had encountered earlier. This was of course completely in line with Ehrenfest’s booklet of problems. Kramers was a gifted student and was recognized as such by Ehrenfest. But, as

35Dresden, H.A. Kramers, 90. 36Ibid., 91. 37Lecture notes of Kramers’s student years can be found in the Kramers Archief, box 4, in the weten- schapsarchieven of the Noord-Hollands Archief.

20 mentioned before, Ehrenfest demanded a complete devotion to physics. Even though Kramers was fascinated both by Ehrenfest and by physics, he was not able to give up everything else.38

Figure 1.6: Invitation for a meeting of Christiaan Huygens, the invitation is for Hendrik Kramers, and the subjects of this meeting are R¨ontgen radiation and tables of logarithms. From the Kramers Archief, box 4, in the wetenschapsarchieven of the Noord-Hollands Archief.

He even often missed Lorentz’s Monday morning lecture, and, even worse, Ehrenfest’s colloquium. This gave rise to a growing estrangement between Ehrenfest and Kramers. After his doctoral exams with Ehrenfest, which Kramers passed but where he did not do exceptionally well, Ehrenfest told him that he did not see a future for Kramers as a physicist.39 Apparently, in Ehrenfest’s eyes, devotion and passion were more important for a physicist than talent and skill.

Hendrik Douwe Kloosterman was born in the small, Frisian village Rottevalle, on the 9th of April 1900. After secondary school, when he was eighteen, he went to Leiden to study Mathematics and Physics. Because it was rather hard to find a space to live in Leiden itself, and it was impossible to travel back and forth to Friesland, he lived in the city of The Hague. There he lived on the Laan van Meerdervoort, close to the sea, and to the just finished Peace Palace. In 1918, five students, including Kloosterman, started their study in mathematics, as-

38Dresden, H.A. Kramers, 92. 39Ibid., 94.

21 Figure 1.7: Lecture notes of Hendrik Kramers, made in Kluyver’s course on differential equa- tions. From the Kramers Archief, box 4, in the wetenschapsarchieven of the Noord-Hollands Archief. tronomy, and physics. But because the lectures in the first year were also aimed at other science students, probably more students than those five would attend them. Klooster- man himself was both a very disciplined and a very gifted student. In his first year in Leiden, he followed the course on integrals and analysis of Kluyver, as well as differential geometry by Van der Woude and thermodynamics of Kuenen. His lecture notes show that he followed the lectures neatly and that he did not miss many lectures. After this

22 first year, in 1919, he already passed his candidates exams.40 In the academic year 1919-1920, he again followed a large amount of courses. These included courses by Van der Woude, mechanics and algebraic geometry, and differen- tial equations by Kluyver. And Kloosterman followed Ehrenfest’s course on theoretical physics, which in 1919-1920 was about statistical mechanics. Both the course on mechan- ics and on theoretical physics would continue in the next academic year, where Ehrenfest altered the subject and would focus on the theory of electromagnetism. Kloosterman also attended lectures by Lorentz, such as one on the constitution of crystals. What is noteworthy is that Kloosterman reentered Kluyver’s course on integrals and analysis. His lecture notes in 1920-1921 of this course correspond almost one-to-one to his lecture notes two years earlier. It is not clear why Kloosterman started this course all over again. It seems unlikely that he did not pass the course the first time, since it would become the subject of his dissertation. A possible explanation is that he wanted to improve himself, to get even more hold on the subject, to prepare himself even better for his doctoral exams and probably even his dissertation. In 1921-1922, the only course he seemed to have followed is Kluyver’s course on elliptic functions, a very detailed mathematical subject. Over and over Kloosterman’s lecture notes show that he followed the lectures carefully and in a disciplined way. This is also visible in the booklet where he did his exercises. These are incredibly neat, especially compared to the ones found in the notes of Kramers. An example can be found in figure 1.8. Here, Kloosterman started with clearly stating the problem at hand. He also stated that W. Kapteyn, at that moment lecturer of mathematics at Utrecht University, came up with the problem. Afterwards, he solved the problem in simple and straightforward steps, naturally leading up to the conclusion.

In the end, both Kramers and Kloosterman were very talented students, who would master different aspects of their scientific training without many difficulties. As stu- dents, they seem to have been influenced much by the courses of Kluyver, considering their main interest in mathematical analysis. But they also had very different personal- ities, which seems to have made them completely different scientists at the end of their study.

40Dijk, Hoogleraren Wiskunde, 58.

23 Figure 1.8: An exercise done by Hendrik Kloosterman in his booklet with exercises. From the Kloosterman Archief, box 1, in the bijzondere collecties of the Universiteit Leiden.

Conclusion

As the examples of Kramers and Kloosterman demonstrates, the mathematics and physics education in Leiden could produce very different scientists. But they also show that both had mastered the same set of basic mathematical skills. This included above all the mastery of different branches of mathematical analysis. In the next chapter I will present how the mastery of mathematical analysis helped Kramers. The prior knowledge necessary to do your candidates exams and to follow higher level courses in physics came from the mathematicians. This was the training necessary to become a physicist. All in all, this reveals the importance of the training the students got before, and besides, the lectures by the professor of theoretical physics. This training made the students physicists, or mathematicians. The professor of theoretical physics had the responsibility to create physicists that could work in the scientific field of physics. They taught the students to ask question and kept them up-to-date with recent developments in physics. They presented a field of

24 ‘physics in the making’. Lorentz and Ehrenfest, in their lectures and colloquia, created critical research scientists. In this sense Lorentz and Ehrenfest are the main protagonists in physics education in Leiden in the 1910s. But they could only do their part if the students had received their background training. And it was this training that gave the students skill in a particular part of mathematics and physics.

25 Chapter 2

Kramers working in Copenhagen

The previous chapter described in detail physics education in Leiden in the 1910s. This education was structured to produce critical scientists. At the end of the chapter, the examples of two of Leiden’s students, Hendrik Kramers and Hendrik Kloosterman, was showed how the university could produce such scientists. In this chapter, we will proceed in following Kramers, from the end of his studies until his eventual return to Leiden.1 The main focus of the chapter will be on his growth as a scientist and the influence of Niels Bohr (1885-1962) on his development.2 The aim of this chapter is to explore Kramers’s continued educational progress after his doctoral exam until he became a professor himself, to be able to relate this to his teaching in the final chapter. In order to do this, I look at the following. First, I describe how Kramers finished his studies and became a physicist, against the expectation of Paul Ehrenfest. Next, I focus on his collaboration with Niels Bohr. Thereafter, I present Kramers’s scientific work in Copenhagen and his scientific development. Finally, I explain how Kramers returned to the Netherlands, and how this change in position also implied a change in importance and scientific style.

From Leiden to Arnhem

In the previous chapter, the growing estrangement between Ehrenfest and Kramers was described. Kramers apparently was unable to live up to Ehrenfest’s demands and to show

1Here, I do not imply that Kramers’s return to Leiden was some sort of necessity, that he was destined to become professor in Leiden. It is the fact that he did return that makes Kramers relevant for this study. 2This is the part of Kramers’s life that has been studied intensively, see for instance Dresden, H.A. Kramers, Ter Haar, Master of Modern Physics, and Pais, ‘Physics in the Making in Bohr’s Copenhagen’.

26 complete devotion to physics. This resulted in a very disappointing doctoral exam with Ehrenfest. The fact that Kramers did this exam was already quite uncommon and it showed the distance between him and professor Ehrenfest. Normally, Ehrenfest would let talented students carry out research until they could achieve a Ph.D. degree, without having done the doctoral exam.3 This was possible due to the extremely loose structure of the university and the influence of Ehrenfest as professor of theoretical physics. In the end, Ehrenfest decided whether students had become the scientists he wanted them to be. This stands in clear contrast to Ehrenfest’s attitude towards the candidates exam, since he considered passing these exams as the minimal requirement to even be allowed in his courses or his seminar. In the end, Ehrenfest told Kramers that he would never become a scientist. Even more, he explicitly told him that there was no future for him as a physicist and that he should become a teacher in secondary education.4 After his exams, Kramers did exactly so and became a teacher in mathematics and physics at the Stedelijk Gymnasium Arnhem. The gymnasium is the highest level of secondary education in the Netherlands com- parable to the British grammar schools, offering preparatory academic education to the most promising pupils. The Stedelijk Gymnasium Arnhem is one of the oldest gym- nasiums of the Netherlands, dating back to a Latin school from at least 1310. At the gymnasium, it again became apparent that Kramers was still not the most disciplined and organised. This he combined with some air of arrogance considering his capabilities as a teacher. When, for instance, the principal harshly criticised him for regularly be- ing late for class, he stated that ‘the principal should realize how much more he taught the students in half an hour than most teachers did in an hour’.5 Clearly, Ehrenfest’s critiques and his performance at his exams had not deteriorated his self-confidence. But Kramers also felt highly dissatisfied with his current position and longed for a return to physics and research. In the summer of 1916 an unique opportunity emerged, when the international student organisation planned a conference in Copenhagen. Be- sides, the principal of the gymnasium had some contacts in Copenhagen. He recognised Kramers’s talents and considered him more of a scientist than a secondary school teacher. Therefore he encouraged Kramers to go to Copenhagen and even gave him some letters of introduction.6

3Dresden, H.A. Kramers, 94. 4Ibid. 5Ibid., 95. 6Ibid., 96.

27 Kramers arrival in Copenhagen

In 1913, the Danish physicist Niels Bohr (1885-1962) deduced from his atom theory, which combined Ernest Rutherford’s (1871-1937) atom model with Max Planck’s (1858-1947) quantum hypothesis, the correct Balmer formula for the hydrogen spectrum.7 Bohr had worked in Manchester with Rutherford, but he returned to Copenhagen in April 1916, to become professor of theoretical physics at the University of Copenhagen. Just a couple of months later, Kramers arrived in Copenhagen. At the time, he had no other contacts apart from the letters of introduction he got from the principal in Arnhem. Thus he had no introductory letter from for instance Ehrenfest or Lorentz. Therefore, on the 25th of August 1916, he wrote a letter to Niels Bohr, which is depicted in figure 2.1. In this letter he introduced himself as a student in physics and mathematics. He stated to have followed lectures by prof. Kuenen, prof. Lorentz, and prof. Ehrenfest in physics, and also that he followed the lectures in mathematics by prof. Kluyver. Afterwards, he explained that he wanted to write his dissertation, and that he always wanted to go to a foreign university. Because of the war, his choice was limited, and therefore he went to Copenhagen. At the end of the letter, he gave his current address, and asked Bohr to write or call ‘when I may come to see you’. It shows how eager Kramers was to meet Bohr and to work with him. Another striking aspect of the letter is the fact that Kramers stated ‘I hope to study now theo mathematical physics. I’ve not yet specialized myself much in that branch’. This is remarkable because what Kramers studied would later be called mathematical physics itself. Taking into account the courses Kramers followed as well, it is curious that he did not consider himself specialized in mathematical physics. This again shows the difficulty in the definitions between theoretical and mathematical physics. Unfortunately, Bohr’s response to this letter is lost, but it is clear that he invited Kramers to meet with him. This would in turn lead to a collaboration that would last for ten years, until 1926 when Kramers would return to the Netherlands. Kramers was Bohr’s first collaborator and student, and their relationship became close very quickly.8 He also swiftly gained the role of promoting Bohr’s ideas abroad. This becomes clear from a couple of letters dated March 1917, in which Kramers describes his experiences in Stockholm and Uppsala.9 There he met Swedish physicists, such as professor Carl Oseen

7Heilbron, ’Quantum Physics’, 691-692, Van der Waerden, Sources, 2, and Duncan and Janssen, ‘Stark effect’, 1. 8Kragh, ‘Niels Bohr’, 143. 9Bohr-Kramers correspondence, Niels Bohr Archive, see for instance the letter from Kramers to Bohr, BSC-KRA, dated March 12 1917.

28 Figure 2.1: Kramers’s first letter to Bohr. Bohr-Kramers correspondence, BSC-KRA, dated August 25, 1916.

(1879-1944), and discussed with them Bohr’s model of the atom. It follows from the letters that Bohr’s theory was not widely accepted at the time and Kramers apparently met with much skepticism.10 Having those discussions and propagating Bohr’s ideas was

10In his letters to Bohr, Kramers wrote about this skepticism. There had been a ‘certain Dr. Juennson’ who did not believe that Bohr’s theory adequately fitted the experiments and professor Benedicks was ‘very skeptical of the theory in general’. Bohr-Kramers correspondence, Niels Bohr Archive, letter from Kramers to Bohr, BSC-KRA, dated March 12 1917.

29 part of Kramers’s collaboration with Niels Bohr. Kramers did also write his dissertation, which he finished in 1919. Although he wrote his whole thesis in Copenhagen, he still promoted in Leiden under Ehrenfest.11 His dissertation was titled Intensities of spectral lines. On the application of the quantum theory to the problem of the relative intensities of the components of the fine structure and of the Stark-effect of the lines of the hydrogen spectrum. The mentioned Stark-effect concerns the splitting of spectral lines by electric fields, and is the electric analogue of the Zeeman effect, the splitting of spectral lines by magnetic fields. The latter was discovered by Pieter Zeeman in 1896, and won Zeeman the Nobel prize in 1902, together with Lorentz, whose electron theory explained Zeeman’s (original) foundings.12 Because of Lorentz’ influence in Leiden and the importance of this discovery, Kramers mastered the theory in every detail. In the first chapter it is also shown that around the time Kramers started with his dissertation, Lorentz lectured in Leiden on the Stark-effect (recall figure 1.2). Besides his knowledge of the Zeeman-effect, Kramers had become during his studies in Leiden a master in Fourier series and, even more specialised, in Hamiltonian mechanics.13 The link between those and the Stark effect was made by Bohr, who made a connection between his own theory and classical electrodynamics, which later became his ‘corre- spondence principle’.14 Therefore, it turned out that those were exactly the techniques necessary to make a description of the Stark-effect. The Stark-effect won its discoverer, Johannes Stark, a Nobel prize in 1919. Stark was no supporter of Bohr’s atom theory, moreover, he publicly denounced it during his Nobel lecture.15 Thus, when Kramers in his dissertation used Bohr’s idea to account for the polarizations and intensities found by Stark, this was a key triumph for Bohr’s theory.16 Arnold Sommerfeld (1868-1951), who had further developed Bohr’s ideas, concluded that ‘the theory of the Zeeman-effect and especially the theory of the Stark-effect belong to the most impressive achievements of our field and form a beautiful capstone on the edifice of atomic physics’.17

11Dresden, H.A. Kramers, 113. 12Duncan and Janssen, ‘Stark-effect’, 1. 13Dresden, H.A. Kramers, 100. 14Kragh, Bohr and the Quantum Atom, 198. In earlier articles, Bohr mentioned the concept of the ‘corre- spondence principle’, but as will become clear, he was rather incoherent in his use of this principle, and Rynasiewicz claims that the way it is used here, would eventually be the basis for ‘the’ correspondence principle. Rynasiewicz, ‘The (?) correspondence principle’, 193. 15Duncan and Janssen, ‘Stark-effect’, 2. 16Ibid., 20. 17Sommerfeld, 1919, 457-458, cited in: Duncan and Janssen, ‘Stark-effect’, 2.

30 Figure 2.2: The main building of the Niels Bohr Institute (formerly Institut for Teoretisk Fysik). Picture made by Dionijs van Tuijl.

The Institute for Theoretical Physics

In the first period of Kramers’s collaboration with Bohr in Copenhagen, the physical facilities were minimal. They shared for instance a single room, which was only fifteen square meters in size.18 There were also no laboratory facilities or some equipment to carry out experiments.19 Already in 1917 Bohr began with planning a new physics institute and he managed to get 200.000 kroner from the university. Above that, Bohr got 80.000 kroner of privately raised funding.20 The building would have been completed in 1919, but due to the shortages in building materials after the First World War and several strikes, it was only finished in 1921. But finally, in March 1921, the Institut for Teoretisk Fysik was inaugurated. This drastically changed the working conditions. On the top two floors there were living areas for the Bohr family, on the ground floor there were several offices, a library, and a lecture theater, and in the basement multiple laboratories.21 Suddenly Bohr, and Kramers with him, would have the means to do experiments themselves and, furthermore,

18Dresden, H.A. Kramers, 98. 19Robertson, ‘Niels Bohr Institute’, 482. 20Ibid. 21Ibid., 485.

31 to invite scientists from all over the world.22 This was further increased by the rise of Bohr’s status after his Nobel prize win in 1922. The number of international visitors rose from five in 1921 to fifteen in 1924, with a total of 63 visitors between 1921 and 1930, only counting those who visited the Institute for at least a month.23 The Institute became one of the most important centres for the international cooperation in science. The collaboration between Kramers and Bohr was a very strong one, they clearly complemented each other. The general ideas about the quantum theory all were Bohr’s, Kramers learned and applied them. On the other hand, because of his training and mathematical talent, Kramers was able to closely analyse Bohr’s ideas and examine their implementability.24 In the Bohr Memorial Volume, Heisenberg recalls a discussion he had with Bohr. In the end he states:

“For the first time I understood that Bohr’s view of his theory was much more sceptical than that of many other physicists - e.g. Sommerfeld - at that time, and that his insight into the structure of the theory was not a result of a mathematical analysis of the basic assumptions, but rather of an intense occupation with the actual phenomena, such that it was possible for him to sense the relationship intuitively rather than derive them formally. Thus I understood: knowledge of nature was primarily obtained in this way, and only as the next step can one succeed in fixing one’s knowledge in mathematical form and subjecting it to the complete rational analysis. Bohr was primarily a philosopher, not a physicist, but he understood that natural philosophy in our day and age carries weight only if its every detail can be subjected to the inexorable test of experiment.”25

The description of Bohr in the citation clarifies the relationship between Bohr and Kramers. Bohr came up with the general, almost philosophical ideas, and Kramers did the ‘next step’. He could arrange them in mathematical form, such as he did in his dissertation with the Stark effect, which in turn opened the opportunity to expose Bohr’s ideas ‘to the inexorable test of experiment’. This is exactly as Poincar´ehad described the

22Bohr and Kramers would not have been very fond to do experimental research themselves. Why an institute of theoretical physics would have laboratories might not be evident at first sight, but as has been discussed earlier experiements were seen as the ultimate judge between different theories. The definition of theoretical physics remained quite vague. One might say that the institute had better be named The Institute of Fundamental Physics, or something alike, to include all branches of physics necessary. 23Robertson, ‘Niels Bohr Institute’, 486, 492. 24Dresden, H.A. Kramers, 102. 25Heisenberg, Bohr Memorial Volume, cited in: Van der Waerden, Sources, 21-22.

32 interplay of different physicists. Bohr came up with the hypotheses, he was the theoretical physicist, Kramers did the calculations, as the mathematical physicist, and his results were eventually tested in the laboratory by experimental physicists. Besides assisting Bohr in these theoretical works, Kramers also played other important roles in the Institute. He gave for instance lectures in Danish on various subjects, such as relativity theory in 1920 and 1923, electron theory in 1922, and hydrodynamics in 1923.26 The amount of lectures he gave also seems to be gradually increasing each year, which points at a growing role for Kramers in the educational part of the Institute. From 1923 onwards, Kramers even took over all of Bohr’s teaching duties.27 As mentioned earlier, Kramers already went to Sweden in his first year, and he made many more travels in the following years. Kramers held many contacts outside the Insti- tute, and corresponded with many scientists concerning the developments made by him and by Bohr. This correspondence included of course close collaborators and visitors of the Institute, but also rather unknown scientists. Prof. Hughes from Queen’s Uni- versity, Ontario, asked for instance for ‘some information as to a difficulty I am having with the application of the methods of Professor Bohr’.28 Kramers kindly responded to this request, explaining the mentioned difficulties Hughes had with Bohr’s paper.29 An-

(a) About the Hα-atom. (b) About the Hβ-atom

Figure 2.3: Experimental results about the Hα-atom and the Hβ-atom send by Takamine to Kramers. From the Kramers Correspondence, Niels Bohr Archive, microfilm 8, section 10, letter from Takamine to Kramers, dated February 29, 1924.

26Notes for those lectures are preserved in the Archive for the History of Quantum Mechanics, Niels Bohr Archive, microfilm 25, sections 8–11. 27Aaserud and Heilbron, Love, 194. 28Kramers correspondence, Niels Bohr Archive, microfilm 8, section 6, letter from Hughes to Kramers, dated June 25 1924. 29Kramers correspondence, Niels Bohr Archive, microfilm 8, section 6, letter from Kramers to Hughes, dated July 8 1924.

33 other example are letters from Takamine Toshio (1885-1959), who sends Kramers some experimental results on the Stark-effect pattern of hydrogen, ‘just as predicted by your theory’.30 Once again, the interaction between mathematical and experimental physics is clear. Kramers had made his predictions and they were tested in different laboratories, apparently even in Japan. And the experimental physicists were eager to present their results to Kramers. Besides keeping other scientists up to date with the developments of the Niels Bohr Institute, Kramers also played a role in propagating Bohr’s theory outside the scien- tific world. He for instance toured through Denmark to give popular lectures to non- specialists.31 Moreover, he wrote, together with the Danish librarian Helge Holst (1871- 1944), a popular science book called Bohrs Atomteori Almenfatteligt Fremstillet, trans-

Figure 2.4: Front page of Holst’s and Kramers’s book. This is a copy of the Utrecht University Library. The stamps show the book used to be in the physics section, then in the science section, and eventually in the section for the history of science.

30Kramers correspondence, Niels Bohr Archive, microfilm 8, section 10, letter from Takamine to Kramers, dated August 24 1924. 31Kragh and Nielsen, ‘Spreading the Gospel’, 269.

34 lated in English to The Atom and the Bohr Theory of its Structure, see figure 1.4 for the front page. It was originally published in Danish in 1922, and translated to English in 1923, to German and Spanish in 1925, and to Dutch in 1927. The book insists on the Copenhagen origins of the Bohr theory, which shows the propagating role of Kramers. It is even called ‘spreading the gospel’.32 The book was also well received in the physics community, where it was considered a good example of a popular science book.33 In the book, Kramers and Holst did not simply describe Bohr’s theory as a complete and undeniable truth, as one might expect in a popular science book that is ‘spreading the gospel’. In the book they also pointed to the key problems of the theory, describing the research that still needed to be done. Helge Kragh and Kristian Nielsen conclude that they ‘used the popular format to present both scientific strengths and weaknesses of the theory, thus stimulating scientifically relevant, public discussion’.34 Here again we see the influence of Bohr’s scientific style, with its sceptical and philosophical side.

Kramers’s scientific publications

After his dissertation, Kramers worked in Copenhagen on different subjects, all related to the quantum theory. Because of the internationalization of the Institute, this was often in collaboration with other scientists. Dirk ter Haar has divided the topics of Kramers’s works into four categories.35 In the first category are the works that concern the quantum theory as it was when Kramers arrived in Copenhagen, such as his thesis, a paper on the helium atom, and two papers on molecular spectra, one of which was written in collaboration with Wolfgang Pauli (1900-1958). The second category consists of papers on x-ray absorption, where Kramers treated continuous spectra. The third category is a paper written by Bohr, Kramers, and John Slater (1900-1976).36 This paper proposed a new program for quantum theory. And in the last category are Kramers papers on dispersion theory, of which one was in close collaboration with Werner Heisenberg (1901-1976). These four categories are also mainly in chronological order. Pauli, Slater, and Heisenberg were just a younger generation, all in their early twenties when they came to Copenhagen, just as Kramers himself had done a couple of years earlier. These collaborations thus also show Kramers’s maturation, from the young assistant to the

32This was first done by: Dresden, H.A. Kramers, 132, and later acknowledged by: Kragh and Nielsen, ‘Spreading the Gospel’, 283. 33Kragh and Nielsen, ‘Spreading the Gospel’, 280. 34Ibid., 283. 35Ter Haar, Master of Modern Physics, 9-10. 36It seems rather strange to make a category of one single paper, but this is justified by the impact of the Bohr-Kramers-Slater theory, both on the ‘Old Quantum Theory’ as a whole, as on the life of Kramers.

35 scientist with young collaborators. A closer look at the publications shows Kramers’s development as a physicist. The papers in the first category, such as Kramers’s thesis, show his mastery of classical mechanics in combination with Bohr’s atom theory, as it was when Kramers arrived in Copenhagen. In his thesis, Kramers had dealt with the hydrogen atom, which was an one- electron system. Bohr had intuitively made a qualitative description for multiple-electron systems, which are also discussed in the popular work of Holst and Kramers. Kramers tried to give a quantitative theory of the helium spectrum, a two-electron system, but the results were contradictory with experimental results. For this, Kramers did not blame the quantum theory, but classical mechanics.37 None of the papers in the first category had lasting value, for the problems they dealt with were solved differently in the theory of quantum mechanics. This is different for Kramers’s work on the absorption of x-rays of 1923, for its results are still relevant.38 In that paper, Kramers used his mastery of classical mechanics and his mathematical talent to combine both Bohr’s correspondence principle with Einstein’s theory on spontaneous and stimulated emission and absorption of radiation, which led to results that were in precise agreement with experimental findings.39 In the mentioned papers, Kramers repeatedly uses Bohr’s correspondence principle. It seems to be a well-defined principle, that constraints the different theories and compu- tations, but it turns out not to be so simple. Bohr himself was never really clear about his correspondence principle, nor was he consistent.40 In some cases it is merely a selection principle, in others it is formulated as a postulate or axiom. It seems that the use of the correspondence principle by Kramers is often a result of long conversations with Bohr, rather than the use of a strictly defined principle.

Just before Christmas 1923, John Slater arrived in Copenhagen. By that time, there was no theory of electromagnetic radiation in the quantum theory. Different theories were given to explain electromagnetic phenomena in radiation. One of those theories was given by Arthur Holly Compton (1892-1962) in 1923. It dealt with the change in the wavelength of x-rays when scattered by electrons, which was later called the Compton- effect.41 The theory used the ‘photon’ concept to explain this effect. Bohr had a strong opposition against the idea of photons and he had convinced Kramers of this opposition

37Ter Haar, Master of Modern Physics, 16. 38Ibid. 39Ibid., 16-17, 20. 40Rynasiewicz, ‘correspondence principle’, 186. 41Ter Haar, Master of Modern Physics, 22.

36 as well.42 Slater had a different solution than Compton, which introduced a ‘virtual field of radiation’.43 He discussed this idea at length with Kramers and Bohr, who were con- vinced that it would imply two fundamental consequences, only statistical conservation of energy and momentum and only statistical independence of the processes of emission and absorption in distant atoms.44 In their joint paper of 1924, they introduced their proposal for a theory, and showed that the effect in the experimental results described by Compton could be interpreted as a statistical one. This divided the world of theo- retical physics. Max Born (1882-1970) and Erwin Schr¨odinger(1887-1961) believed this to be correct, whereas Einstein and Pauli felt it to be wrong.45 Eventually, two exper- imental results from 1925 decided in favor of Compton, leading to an early end for the Bohr-Kramers-Slater proposal. Eventually, the papers of Kramers that turned out to be the most influential for the development of quantum mechanics, were the ones he wrote in his last years in Copenhagen, on his dispersion theory.46 This theory would fit dispersion phenomena into the framework of quantum theory, since the electromagnetic radiation still did not fit into the framework completely. To do so, Kramers claimed that the quantum dispersion formula in the limit completely corresponds to the classical formula. Kramers thus again used the correspondence principle, but explicitly imposed a constraint on further theory construction, which can be seen as the first use of ‘the’ correspondence principle.47 In his paper of 1924, Kramers gave a first outline for the derivation of the dispersion formula, but a full derivation is given in a paper of 1925, in cooperation with Heisenberg. The redaction for this paper was actually entirely due to Kramers.48 About Kramers dispersion theory, Born noted:

“One can say that Kramers, guided by Bohr’s principle of correspondence, guessed the correct expression for the interaction between the electrons in the atom and the electro-magnetic field of the light wave. That is at least the way I regarded his results. It was the first step from the bright realm of classical

42Ter Haar, Master of Modern Physics, 21. Bohr actually convinced Kramers by talking him out of the predictions he had made, exactly the same as found by Compton. Thus Kramers had also found what would become the Compton-effect, which won Compton the Nobel prize in 1927. Dresden argues that this is one of the many ‘near misses’ of Kramers, which would have an impact on his general mood. Dresden, H.A. Kramers, 446. 43Van der waerden, Sources, 11. 44Ibid., 12. 45Ter Haar, Master of Modern Physics, 22-23. 46Ibid., 23. 47Rynasiewicz, ‘correspondence principle’, 193, Jammer, Conceptual Development, 117. 48Van der Waerden, Sources, 15.

37 mechanics into the still dark and unexplored underworld of the new quantum mechanics.”49

All together, these papers show Kramers’s development in his years in Copenhagen. Whereas he started with adding mathematical proofs to parts of Bohr’s theory of the atom, in the end he was able to make one of the first important step towards the theory of quantum mechanics. His physical argumentation was in all his papers based on Bohr’s ideas, what mostly implied some form of the correspondence principle. But also there Kramers made a development, stating in the end a clearer and more coherent version of this correspondence principle than Bohr had ever done.

The return to the Netherlands

In 1925 professor Willem Henri Julius (1860-1925) passed away, leaving open a faculty position at the Utrecht University. A committee was appointed to make recommendations for a successor, with Ehrenfest as its chair. Ehrenfest was very eager to bring Kramers back to the Netherlands, because it would grant the Netherlands a major quantum the- orist.50 He asked Planck, Einstein, Lorentz, and Bohr for letters of recommendations, to convince the Utrecht faculty. Planck and Einstein both wrote excellent letters of recom- mendation for Kramers, whereas Lorentz wrote in his characteristic, reluctant manner, even mentioning Kramers missing some monday morning lectures. But Bohr’s response was even worse, actually not recommending Kramer at all in his letter of ‘recommenda- tion’.51 But Ehrenfest remained sustained, urging Bohr to write a better letter. Bohr’s second letter was finally much clearer, which meant that Ehrenfest had recommendation letters from Planck, Bohr, and Einstein.52 This must have made the decision easy for the Utrecht faculty and, on the other hand, illustrated a new relationship between Ehren- fest and Kramers. It had taken Ehrenfest quite some time to cope with Kramers as a physicist, but in this seems to have changed. For Kramers, the decision to come to Utrecht was probably strengthened by the tragic fate of the BKS theory, and by the growing status of Heisenberg. Heisenberg even took over Kramers’s position after he had left to Utrecht. Taking the position in Utrecht would give Kramers the chance for a new start, moreover, it would give him the opportunity to

49Van der Waerden, Sources, 20. 50Dresden, H.A. Kramers, 311. 51It is not evident why Bohr’s letter was vague and guarded. This might imply a growing distance between Bohr and Kramers, but it might as well point to the fact that Bohr was afraid to loose Kramers. 52Dresden, H.A. Kramers, 312.

38 begin on his own. In a letter to Ralph Fowler (1889-1944), Kramers stated that leaving Copenhagen is ‘not very agreeable, but on the other hand I feel it as a wholesome change to obtain an independent post’.53 This would also imply a new status for Kramers. In Copenhagen, he would always have Bohr for counseling and discussing. In Utrecht, actually, in the Netherlands, he was the main modern quantum expert, which was a huge contrast with Copenhagen. This was made even more apparent by Ehrenfest’s behavior, who would constantly ask Kramers for help. Ehrenfest himself felt at the time unable to follow the developments of physics himself, observing about himself, Planck, Schr¨odinger,and Einstein that “the young can hardly suppress merciful smiles about the old-fashioned backwardness of these men”.54 In this quote, the young referred directly to Kramers. This clearly shows the shifted relationship between Ehrenfest and Kramers. This does not suggest that Utrecht University was some small provincial university without expertise in modern quantum physics. The Utrecht laboratories were renowned and the photometric techniques developed there had led to extraordinary results.55 But whereas Utrecht’s experimental physics department flourished, theoretical physics had received less attention. The faculty committee in Utrecht doubted whether to give the open position to an experimentalist or a theorist, which in turn explains why Ehrenfest needed those excellent letters of recommendation.56 Due to his important status in quantum theory, Kramers did not only lecture at Utrecht University. He also gave lectures in Delft, Wageningen, and Leiden. All of those lectures considered quantum mechanics.57 And he would even give lectures for the Volksuniversiteit (Community College), which offered courses for adults from the lower social classes. In the spring and summer of 1928, Kramers travelled through the United States and Canada, lecturing on quantum mechanics in Toronto and Chicago, and giving a summer course in Ann Arbor. Thus Kramers had become one of the main propagators of quantum mechanics. This is also clear in the Dutch translation of the popular book on the Bohr theory. It is the first version where the name Bohr no longer appears in the Dutch title, which was De Bouw der Atomen, which might suggest that Kramers moved away from Bohr in his

53Kramers correspondence, Niels Bohr Archive, microfilm 8, section 6, letter from Kramers to Fowler, dated December 9 1925. 54Van Lunteren and Hollestelle, ‘Ehrenfest’, 528. 55Blum et al, ’Translation as heuristics’, 7. 56Dresden, H.A. Kramers, 312. 57Notes for those lectures are preserved in the Archive for the History of Quantum Mechanics, Niels Bohr Archive, microfilm 26, sections 3–8.

39 work as well. However, most of the book is a mere translation, translated by dr. H.C. Sluijter.58 Kramers himself has edited some parts of the fifth, sixth, and seventh chapter, and added a new chapter, chapter eight. These adaptations plainly show Kramers’s views on the new quantum mechanics. In the chapter on Bohr’s theory of the hydrogen atom, Kramers described the correspondence principle, after which he stated:

“Een beschouwing over het correspondentie-principe, geschreven n`a1925, kan niet anders dan uitmonden in een beschouwing over de quantummechanica. In den herfst van dat jaar toch verscheen van de hand van den jongen Duitscher Heisenberg een artikel in de “Zeitschrift f¨urPhysik”, dat den weg zou openen naar een theoretische behandeling der atoomproblemen, welke de idealen, die Bohr bij de opstelling van zijn correspondentie-principe voor oogen zweefden, in vervulling zou brengen.”59

By placing this section after his section on the correspondence principle, Kramers presented the reader the important role the correspondence principle had played in the forming of quantum mechanics. And above all, Kramers stated that quantum mechanics completely satisfied Bohr’s ideal about the atom. In the end, Kramers thus still propa- gated these ideals and still presented Bohr as the main authority in the theories of the atom.

Conclusion

Because of his specific training and mathematical talent, Kramers quickly became impor- tant for the scientific theories of Niels Bohr. He was able to derive precise results based on Bohr’s often intuitive ideas, that would correspond to experimental results. His specific training in mathematical analysis was crucial for this work. Through his mathematical skill Kramers was able to complement Bohr. With the rising influence and status of Bohr, Kramers became more and more promi- nent as well. Where in the beginning he followed Bohr’s theories quite meticulously, he would become more and more independent, while still following Bohr’s general line of argumentation. After his move to Utrecht he would gain even more independence, and suddenly became a main authority in his scientific field. But he remained true to Bohr’s approach in physics and kept propagating Bohr’s general ideas.

58Kramers and Holst, Bouw der Atomen, iii. 59Ibid., 121.

40 Chapter 3

Physics Education at Leiden University in the 1930s

In the preceding chapter Hendrik Kramers was followed in his coming of age as a physicist. Especially the importance of Niels Bohr and his style of physics in this process were discussed. Kramers would develop from a secondary school teacher without any prospects of a career in science, to a professor of physics and a main authority in the field of quantum mechanics. This chapter will in turn be about Kramers as professor of physics. The focus will be on his lectures and, furthermore, on the mathematics needed for and present in those lectures. The aim will be to show the kind of mathematics necessary to follow the developments in quantum physics, the prior knowledge expected by Kramers from his students, and the way his students acquired this knowledge. To do all this, the chapter will start with describing generally the professors in physics in the Netherlands and the links between them. Afterwards, Kramers’s early lectures on quantum mechanics are discussed, with attention for their structure, content, and math- ematical level. Thereafter, the switch of Kramers from Utrecht to Leiden is discussed, together with a description of the mathematics and physics studies in Leiden at the mo- ment of that switch. Then a comprehensive analysis of Kramers’s textbook on quantum mechanics is given, focussing again on structure, content, and mathematical level. Fi- nally, an account of the courses on mathematics is given, to link those to the discussed contents and mathematical level of the lectures on quantum physics.

41 Professors of Physics

As mentioned in the previous chapter, Kramers became professor in Utrecht in 1926. There, he became professor of theoretical physics and theoretical mechanics. Especially the second part is interesting. Before, mechanics had always been part of the mathematics education and had been taught by a mathematician. Recall that in Leiden in the 1910s the leeropdracht of mechanics had been that of Pieter Zeeman, who was succeeded by Willem van der Woude. The fact that Kramers became professor of theoretical mechanics thus implied a new status of mechanics, within the field of physics, which was due to the new field of quantum mechanics. It also strengthens the image of Kramers as an authority in the field of quantum mechanics in the Netherlands, as the only professor of theoretical physics and theoretical mechanics. At the time Kramers returned to the Netherlands and became professor in Utrecht, most of the chairs of (theoretical) physics were held by men who had been in Leiden in the 1910s, just like Kramers. In Leiden, Ehrenfest was still professor of theoretical physics, and in that position he was the main advocate for Kramers to attain the chair in Utrecht. Close to Leiden, at the Delft Institute of Technology, which had received full university rights only in 1905, Johannes Burgers (1895-1981) was professor of mechanical engineer- ing, shipbuilding, and electrical engineering.1 Although this does not seem related to Kramers’s field of study at first sight, Burgers had a background quite compatible with Kramers’s. He had studied mathematics and physics in Leiden from 1914 and already obtained his Ph.D. in 1918 with professor Ehrenfest.2 His dissertation was entitled Het atoommodel van Rutherford-Bohr (The atomic model of Rutherford-Bohr), which shows how closely related his research was to that of Kramers. Even though their research was related, they did not correspond much during Kramers’s stay in Copenhagen, besides a few letters about ongoing research.3 Another example is Dirk Coster (1889-1950), who became professor of physics and meteorology in Groningen in 1924. Coster had started to study mathematics and physics in Leiden in 1913 and experimental physics in Deft in 1916. Just like Kramers, he went to Copenhagen to work with Bohr, but a much shorter period, from 1922 to 1923. There, he worked with Bohr on X-ray spectra. He is also famous as the co-discoverer of the element of Hafnium, together with George de Hevesy (1885-1966) in 1923.4 During their study in Leiden, Coster and Kramers had also become very good friends. Kramers kept

1Alkemade, ‘Burgers’. 2Actually, he was the first Ph.D. student of Ehrenfest. 3Alkemade, ‘Burgers’. 4Brinkman, ‘Coster’.

42 Coster constantly up to date with his work in Copenhagen, and Coster did the same. The Kramers-Coster correspondence is therefore one of the most comprehensive to be found in the Niels Bohr Archive.5 All in all, the professors of physics formed a close society of men with the very same background. But it were not only the ordinary professors of physics. In 1928 Adri- aan Fokker (1887-1972) succeeded Hendrik Lorentz as director of research of the Teylers Museum in Haarlem. Lorentz had build up the laboratory there to one of the most out- standing in the Netherlands and Fokker was trained in both experimental and theoretical physics. He had started studying experimental physics in Delft in 1904, but continued his studies in Leiden in 1906 and received his Ph. D. under professor Lorentz in 1913. Afterwards, he went to work with Albert Einstein in Z¨urich and Ernest Rutherford in Manchester.6 Just like Lorentz before him, his position at the teylers Museum also im- plied a position as extraordinary professor at Leiden University. In this way he would thus also stay closely related to the university and to the rest of the physics society. As discussed in the previous chapter, Kramers was the main authority in quantum physics in the Netherlands. His long collaboration with Bohr and his publications had placed him in that position. But as all the examples show, it did not place him in isolation. There were still many colleagues he could discuss with, many physicists close by working in the same field. The main difference was that most of the time they would ask for his judgment, instead of Kramers going to Bohr. This had an influence on Kramers’s teaching duties as well.

Kramers’s early lectures in quantum mechanics

Considering the short links between the professors of physics in the Netherlands, it is not surprising that Kramers, with his great knowledge of quantum mechanics, did not only lecture in Utrecht. From 1929 onwards, he would give elementary courses in quantum mechanics in Leiden and Delft as well. These lectures were by no means mere presen- tations once in a while, but full weekly courses.7 This was possible not only because of these close links between the physicists, but also because of the small distances between the university cities of Utrecht, Leiden, and Delft. Kramers did not lecture in Groningen, where his close friend Dirk Coster was professor. Instead, these lectures were given by a close co worker of Kramers, Ralph Kronig (1904-

5Kramers-Coster correspondence, in the Niels Bohr Archive, microfilm 8, section 4. 6Snelders, ‘Fokker’. 7Kramers’s lecture notes for these courses are preserved in the Archive for the History of Quantum Mechanics, Niels Bohr Archive, microfilm 26, section 3.

43 1995). Kronig was born in Dresden, but did his study in physics in New York, at Columbia University. He had come back to work in Europe in 1927 and stayed a while with Kramers in Utrecht. In 1931 he became lecturer in Groningen, with the leeropdracht mechanics and quantum mechanics. As a theoreticus, he was also important in interpreting Coster’s experimental results in X-rays.8 Thus Kramers gave lectures in elementary quantum mechanics in Utrecht, Leiden and Delft. The name ‘elementary quantum mechanics’ gives the impression that the course only dealt with some elementary problems in quantum mechanics, but this was not at all the case. In the course, Kramers gave a complete overview of the field of quantum mechanics in 1929-1930. In his own lecture notes, he starts with an overview of this course, which is presented in figure 3.1. As is shown in his planning, Kramers started the course with a historical overview of the development of quantum mechanics, with an emphasis on the main problems that gave rise to the theory. Kramers states what this new theory encompasses:

“Mogelijkheid heden een soort van elementaire beschrijving der deeltjes te geven, waarin de ruimte-tijd en dynamische eigenschappen als complementair worden beschouwd; de logische mogelijkheid dit te aanvaarden, komt zodra men met Bohr de mogelijkheid van definitie en waarneming analoog vat.”9

Thus the new atomic theory gives an opportunity to give an elementary description of the particles. This of course is the basis of quantum mechanics, which is the mechanics of ‘the smallest possible’. In this theory, spacetime and dynamical properties are comple- mentary, something which is apparently striking, because it is given a justification. The justification in turn comes from Bohr, in an analogy between observation and definition. This link to Bohr becomes even more clear, when he states that any further approach can only be heuristic and based on the correspondence principle. Further in his lecture notes, the link with Bohr is even stronger. There he states about the build up of the course:

“Wij zullen in deze colleges trachten langs een standpunt, dat enigszins op Bohr’s gelijkt, de quant. mech. opbouwen. Dus meer heuristisch dan ax- iomatisch; zodoende tevens idee van de moeilijkheden die nog voor de boeg staan.”10

8Brinkman, ‘Coster’. 9Kramers’s lecture notes, page 1, in the Archive for the History of Quantum Mechanics, Niels Bohr Archive, microfilm 26, section 3. 10Ibid., page 2.

44 Figure 3.1: Kramers planning for his first course of quantum mechanics in Leiden, autumn 1929. From the Archive for the History of Quantum Mechanics, Niels Bohr Archive, microfilm 26, section 3.

Kramers wants to build up his course heuristically, with an emphasis on experimental results and problems presented by the theory. The method resembles the one he had used for many years working in Copenhagen and is without any doubt an important intellectual heritage of Niels Bohr. For Kramers, it seems to be not only a fruitful method to use in his scientific research, but also the most natural way to educate the theory of quantum

45 mechanics! The first page of Kramers’s lecture notes in turn explain how this would look like in the course on elementary quantum mechanics. This of course is Kramers planning at the start of the course. After the general introduction, Kramers planned to give a general reflection on the wave function, and its contribution to the classical mechanics of point particles. Then Kramers deals with the interpretation of this wave function, leading to the quantum theory of free particles. There, also operators are introduced, together with the importance of eigenvalues in quantum theory. This section ends with Dirac’s theory, assessing the probability distribution. Afterwards, Kramers would take on general correspondence, Ehrenfest’s theorem, and the introduction of matrices. The last section finally includes bound particles, with the introduction of a radiation field, group theory and perturbation theory. Towards the end, the notes become more and more just general names for theories, instead of describing the line of the course. It is therefore not surprising that the course in theory did not completely follow the line Kramers had planned in the beginning. Most of the theories mentioned after point ‘4)’ are not treated in the mentioned fashion, or not treated at all. And Kramers spends much more time on matrices, which he in his course denotes as ‘Heisenberg’s matrices’. But most importantly, he introduces a complete new subject, which was not planned at all. Shortly after his introduction of Heisenberg matrices, Kramers focuses on ‘Dirac’s & Goudsmit-Uhlenbeck’s spin’. His notes show how this part of the course did not follow a structure planned in advance. A part of these notes can be seen in figure 3.2. As it states for lecture eleven, on the eleventh of Februari 1930, ‘Alleen over Dirac’s & Goudsmit- Uhlenbeck’s spin gesproken’. And for the next lecture, a week later, Kramers writes ‘Dit goede manier om spin in te leiden: . . . ’. Together, this gives the impression that Kramers changed his course at that very moment, possibly to adapt to the students present in the lectures and questions raised by them. Kramers course was after all the first course on quantum mechanics in the Netherlands, thus an adaptation of the material discussed is not surprising. The same holds for Kramers quantum mechanics lecture in 1931-1932. The notes for this lecture follow upon the lecture notes for the course in 1929-1930 and the same pattern is shown. The first lectures are written down very precisely, following the plan he had already set out in 1929. But after the fourth lecture already, his notes become much shorter and for some lectures Kramers has made no notes at all. Kramers could at that point of course have followed his notes of the earlier lectures, though if that is the case, it is quite unclear why he would have made new notes for the first couple of lectures. In

46 Figure 3.2: An extract of Kramers notes on electron spin. From the Archive for the History of Quantum Mechanics, Niels Bohr Archive, microfilm 26, section 3. the first chapter, Kramers general sloppiness from time to time and the inconsistency in his notes when he was a student was already mentioned. It is of course possible that the inconsistency in his notes is a part of this personality trait. It is also possible that Kramers, after giving his general introduction to the theory, adapted his course completely on the students present in his lecture. Unfortunately, the number of students that followed his early lectures is unknown, but for later lectures in Delft Kramers kept the number of students.11 Kramers’s course on quantum mechanics

11Kramers Archief, box 2, in the wetenschapsarchieven of the Noord-Hollands Archief. Kramers kept the numbers of students of lectures in Delft.

47 in 1932-1933 in Delft started with an audience of around 30 in October and November. After the Christmas break, this number had dropped to around ten, and the last four lectures had an audience of four. The course of 1933-1934 presents to a certain extent the same tendency. It starts of with an audience of around 25, drops to around fifteen and ends with five students. A steady decline, which does make it possible to adapt to the audience in the later stage of the lectures. On the other hand, it is a decline that also presents information about the education of physics.

The role of mathematics

Kramers’s method to build up quantum theory like Bohr resulted in a course starting from classical physics. As a result, the mathematical tools Kramers used in his course correspond to those of classical physics. But as his course blueprint shows, he planned on discussing subjects of abstract and modern mathematics as well. This included matrix mechanics, something without a classical counterpart, and the theory of groups. Both are mentioned explicitly by Kramers, in points 4 and 6 respectively. In the course itself, however, Kramers did discuss Heisenberg’s matrices, but the theory of groups is used nowhere. Apparently Kramers had changed his mind, just as he did with the subjects he treated in his course.

Figure 3.3: An extract of Kramers notes on electron spin. From the Archive for the History of Quantum Mechanics, Niels Bohr Archive, microfilm 26, section 3.

The matrices were of course necessary in introducing spin. In figure 3.2 Kramers wrote that he had spoken on Dirac’s and Goudsmit-Uhlenbeck’s spin. But before that, he had given an introduction on the specific matrices used in the spin calculations. Kramers’s notes on this preparatory part are depicted in figure 3.3. Kramers introduced spin there- after, starting with Goudsmit’s and Uhlenbeck’s assumption that electrons have intrinsic

48 angular momentum, and deriving the Pauli spin matrices, the ones Kramers had prepared (in figure 3.3), from a commutator. This all implied the use of abstract mathematics and Kramers made many notes for the derivations in this part of his course. The majority of the lectures did not use these abstract mathematics. Kramers in- troduced quantum mechanics from a classical point of view and used the mathematics used in classical physics. The importance of the mathematical tools of classical physics in Kramers lectures is quite logical. Kramers had been an expert of mathematical analysis since he was a student and he had used this field of mathematics to solve most of the prob- lems he had encountered. His expertise in the field of analysis was not only remarkable for a physicist, but for a mathematician as well.12 As a result, Kramers would continue to use above all else methods from analysis to solve problems in physics and, moreover, con- sidered that young students should do the same, especially in the foundations of physical theories. This by no means implies that Kramers did not master modern mathematics. In fact, he showed knowledge of mathematics in all kinds of fields. Besides his teaching duties, Kramers kept doing research as well. In his publications, this broad knowledge of mathematics is displayed. For instance in Die Multiplettaufspaltung bei Koppelung zweier Vektoren, Kramers derived the formulas for multiplets using a method based on group theoretical methods.13 And his paper including some remarks on Heisenberg quantum mechanics showed his complete understanding of matrix techniques.14 His choice whether or not to extensively deal with those mathematical methods thus was by no means made by an argument based on his knowledge of abstract mathematics. It was based on an argument of style, or perhaps even on the knowledge of his students.

Professor of physics and mechanics

On the 25th of September 1933, the physics community was shocked by the tragic death of Paul Ehrenfest. In their search for a worthy successor of Ehrenfest’s chair, Leiden Uni- versity eventually contacted Kramers. As a result, Kramers would eventually succeed the very professor that deemed him unfit for a scientific career. Kramers would, however, not succeed Ehrenfest as professor of theoretical physics. That chair remained empty at Lei- den University until Sybren Ruurds de Groot (1916-1994) became professor of theoretical

12Dresden, H.A. Kramers, 466. 13Ter Haar, Master of Modern Physics, 45. Original paper: Kramers, H.A., ‘Die Multiplettaufspaltung bei Koppelung zweier Vektoren’, Amsterdam Proc. 34 (1931), 965. 14Ter Haar, Master of Modern Physics, 32. Original paper: Kramers, H.A., ‘Enige opmerkingen over de quantummechanica van Heisenberg’, Physica 5 (1925), 369.

49 physics in 1953. Rather than professor of theoretical physics, Kramers became professor of physics and mechanics. Again, this shows both the changed status of mechanics, as well as the role of Kramers in relation to quantum mechanics in the Netherlands. In the introduction, Kramers’s inaugural speech was already discussed. The speech bore the striking name Natuurkunde en Natuurkundigen and Kramers started with hon- oring Paul Ehrenfest. He stated that it was Ehrenfest’s personality and style of physics that once made him wonder about the connection between physics and physicists. It is the special link between style and content that differs for each physicist and that can present someone’s personality even in the strict defined rules of scientific publications. It is exactly this link which Kramers praised about Ehrenfest. After having discussed this link in multiple moments in the history of physics, Kramers returned to his own time and the theory of quantum mechanics. There, Kramers praised Bohr’s scientific style as well and above all, his ability to describe that which is not yet known, where there is not yet a definite conclusion. Kramers stated that it is precisely this characteristic of Bohr that enabled him to constitute modern quantum theory. Afterwards, Kramers discussed the two professors lecturering in quantum theory, pro- fessor Neophilos and professor Palaeophilos. Both would give completely different lec- tures, but in the end their students would be able to predict the results of the same experiments. It is now clear why Kramers would discuss those professors, he had lec- tured on quantum theory himself, and thus had made the decission which of the styles he would use in his lecture. Although he did not mention himself in his speech, his stance is evident, he lectured in the style of Palaeophilos. But as Kramers remarked himself, both styles of lecturing had the same results. It was, in the eyes of Kramers, a matter of personality of the physicist involved. Later on it will become clear that it is in fact not simply the personality of the physicist, but that pedagogical arguments, based on the prior knowledge of the students, play their role as well. At the end of his speech, Kramers addressed his colleagues in Leiden. Firstly, the full professors of physics in Leiden, professor Wilhelmus Keesom (1876-1956), who had become professor of physics in 1923 as successor of professor Kuenen, and professor Wan- der de Haas (1878-1960), who had become professor of physics and meteorology in 1924 as successor of Kamerlingh-Onnes. Secondly, Kramers addressed the extraordinary pro- fessors, professor Adriaan Fokker, the aforementioned director of research of the Teylers Museum, and professor Gilles Holst (1886-1968), who had been the director of the Philips NatLab (Physical Laboratory) and for whom Ehrenfest had created a chair in 1928. Be- sides these professors, there were also three lecturers of physics in Leiden at the time of Kramers inauguration, Geertruida de Haas-Lorentz (1885-1973), Claude August Crom-

50 melin (1878-1965), and Eliza Cornelis Wiersma (1901-1944). Those were Kramers closest colleagues as professors in physics. Besides these colleagues in physics, also the mathematical staff was important in the physics education in Leiden. In 1934, this staff consisted of three men. The first of them, professor Willem van der Wouden, had already become professor of mathematics in 1916, and still held the leeropdracht mathematics and mechanics. This meant that he had to work closely together with Kramers, who became the professor of physics and mechanics. At the moment of Kramers’s inauguration, professor Van der Woude was the rector of Leiden University. Next, there was professor Johannes Droste, who had succeeded professor Kluyver in 1930. And finally, Hendrik Douwe Kloosterman, who became lecturer of mathematics in 1930. To make the picture complete, there were also three colleagues in the astronomy department, professor Jan Hendrik Oort (1900-1992), professor Ejnar Hertzsprung (1873-1967), and lecturer Jan Woltjer (1891-1946). This was the complete staff responsible for the education in mathematics, physics, and astronomy in Leiden. The university had certainly grown, as we have seen earlier with the increased number of students. It is also resembled by the growth of staff, from a total of six between 1912 and 1919, to the fourteen members in 1934. The curriculum, on the other hand, had not changed much since the 1910s. One still had to start with elementary courses in mathematics and physics, to complete the candidates exams. Afterwards, one would proceed to the more advanced courses, of which there was a larger variety due to the growth of lecturers and professors. Kramers also kept the colloquium on the wednesday, which had been set up by Ehrenfest. In his honour, Kramers called it the Colloquium Ehrenfestii.15 In the colloquium, just as it happened under Ehrenfest, students would present recent research. And sometimes visitors would discuss their own research. In figure 3.4(a) the subjects discussed in a couple colloquia are noted down by Kramers. These subjects included for instance a new article of Slater, work of Heisenberg and Dirac, and the infinitesimal rotation spinor. In figure 3.4(b) are in turn the visitors of those colloquia written down. Besides students, one also recognises the names of Fokker, Casimir and Kronig. Apparently, these professors would visit the colloquium as well. All in all, besides the larger scale, the structure of a study in mathematics and physics in the 1930s resembled hugely the structure in the 1910s. The content, on the other hand, did not. But before we can take a closer look at Kramers’s lectures as professor in Leiden, we first have to go a couple of years back.

15Van Baal, Colloquium Ehrenfestii.

51 (a) Kramers’s notes of attendance. (b) Kramers’s notes on the subjects.

Figure 3.4: Kramers’s notes of attendance and of the subjects of some of the colloquia of early 1939. From the Kramers Archief, box 1, in the wetenschapsarchieven of the Noord-Hollands Archief.

Writing a Textbook

On the 31st of January 1930, professor Kramers received a letter from professor Arnold Eucken (1884-1950).16 Eucken was at that moment professor of chemistry and physics at the Technische Hochschule Breslau, although later that year he would receive a chair in G¨ottingen. Eucken had started a project, together with Karl Lothar Wolf (1901- 1969), lecturer at the Technische Hochschule Karlsruhe, to write a series of textbooks on chemical physics. The series would be named Hand- und Jahrbuch der chemisch Physik and would in turn be published by the Akademischen Verlagsgesellschaft in Leipzig. In the letter, Kramers is invited to write the first part of this series, which would constitute “Allgemeine Theorie ¨uber den Aufbau der Materie”. In the impression of the

16Letter Eucken to Kramers, dated Januari 31, 1930, in Kramers Archief, box 2, in the wetenschap- sarchieven of the Noord-Hollands Archief.

52 publishers, this could in turn be divided in two smaller parts, one about the structure of atoms and molecules (where they also expected a basis of quantum theory), and another part on the behavior of atoms and molecules (thus mostly probability theory). Besides, they state clearly that the part should treat the theoretical basis, whereas the “Synthese zwischen Theorie und Erfahrung” would be considered in a third part. The aim was a general introductory chapter where the structure of atoms and molecules is explained, as part of a series of textbooks for chemical physics. At the end of the letter, Eucken expressed his hope that Kramers would agree in writing this part. He was not disappointed, because Kramers did indeed agree to write the part and a contract was signed on the 16th of October 1930.17 In addition, Kramers agreed to divide the part into the two smaller parts, just as the publishers had hoped. The first of those was published already in 1933 and was called Die Grundlagen der Quantentheorie. The second part took a while longer, and was not published until 1938. There was some disagreement about its title as well.18 Kramers had wanted to name the part Erweiterung der Quantentheorie, because in his opinion, the part was a continuation of the first part. But the publishers wanted another title, a title that resembled the possibility of studying the second part without knowledge of the first. Therefore, Kramers changed the title, which would eventually become Quantentheorie des Elektrons und der Strahlung. All in all, in 1938, there was a textbook, as part of the series Hand- und Jahrbuch der chemisch Physik, where Kramers first set forth the foundations of quantum theory and following this foundation, discussed electrons and radiation.

Ter Haar’s Translation

As part on the series Hand- und Jahrbuch der chemisch Physik, Kramers’s textbook was published in German. Kramers himself also never made a translation of his textbook. Only in 1957, five years after the death of Kramers, an English translation of the textbook appeared. The textbook was translated by Dirk ter Haar (1919-2002) and published under the name Quantum Mechanics, although a version including only the first part appeared as well, called Foundations of Quantum Mechanics. Ter Haar had studied mathematics and physics in Leiden from 1937 until 1941. Afterwards he wrote his dissertation, entitled Studies on the origin of the solar system under supervision of Kramers. During the research for his dissertation, he stayed, just like Kramers, a while in Copenhagen to work with Niels Bohr. Ter Haar and Kramers thus knew each other well and have even made

17Contract in the Kramers Archief, box 2, in the wetenschapsarchieven of the Noord-Hollands Archief. 18Letter Eucken to Kramers, dated August 13, 1937, in the Kramers Archief, box 2, in the wetenschap- sarchieven of the Noord-Hollands Archief.

53 two publications together. The choice for Ter Haar as translator of Kramers textbook therefore is no surprising one. In the Translator’s Preface, Ter Haar stated his reasons for translating Kramers work. As he claimed himself:

“The main reason was that I felt that this book still represents the best available exposition of quantum theory and that the English speaking world was the poorer for not having it readily available.”19

Ter Haar had obviously high regard of Kramers and his work. Later in his life, in 1998, he would even write the book on Kramers’s scientific contributions, which was strikingly named Master of Modern Physics.20 The translation of Kramers’s textbook was not the only translation made by Ter Haar. He had a great knowledge of Russian and therefore translated many works from Soviet scientists to make them available in the Western science community.21 As a result, Ter Haar was one of the most prominent scientists in the dissemination of theoretical physics. About the translation itself, Ter Haar stated that it “is a literal one, except where the literature has been brought up to date and the few places where the text was rather cryptic”. The translation does indeed follow Kramers textbook very neatly, and it is hard to find any of the places where Ter Haar has altered the original text. What is surprising though is that in the translation almost no trace of the original series Hand- und Jahrbuch der chemisch Physik can be found. Only in Kramers’s Preface, there still is a reference to the fact that the work is the first volume in this series. The translation itself appears in a series of physics, instead of chemical physics.

Critical Reviews

Kramers’s textbook was received very positive by reviewers in numerous journals. Most praised is the neat structure which Kramers used to build up the theory. Professor Friedrich Hund (1896-1997) opened his review of Die Grundlagen der Quantentheorie in Die Naturwissenschaften with the current status of quantum theory and the hope that one would write a complete overview of the theory. He then stated about Kramers work: “Diese Hoffnung wird hier in sch¨onsterWeise erf¨ullt”.22 Hund asserted that the clarity of the book is primarily due to the fact that Kramers choose the structure of a

19Kramers, Quantum Mechanics, vii. 20Ter Haar, Master of Modern Physics. 21Lamb et al, ‘Dirk ter Haar’, 79. 22Hund, ‘Grundlagen’, 208.

54 textbook, wherein Kramers presented the theory systematically. In the systematic build- up, Hund recognised the fact that Kramers had been a co worker of Bohr, “man sp¨urt jenen ‘Kopenhagener Geist’, den Geist des Korrespondenzprinzip”.23 In the end, Hund recommended the book to all who wanted to learn quantum theory, from experimental physicists and physical chemists to theoretical physicists.24 Ludwig Ebert (1894-1956), from the Kaiser-Wilhelm-Institut in Berlin, wrote a re- view for the more specialised Zeitschrift f¨urElektrochemie und angewandte Physikalische Chemie. He agreed with Hund that Kramers work should be considered a work in the Copenhagen style, “Kramers’s Buch ist ein Dokument dieser Methode, und zwar ein unge- mein lehrreiches, daher der weitesten Beachtung wert.”25 But opposite to Hund, he would not just recommend it to experimental physicists or physical chemists. One needed the help of colleagues in theoretical physics or mathematics to truly understand the work, and even then a lot of effort was required. If one would be able to put this effort in the work, “die M¨uhewird sich jedoch durch einen reichen Gewinn bester physikalischer Begriffsbildung belohnt finden.”26 This sentiment, that the book is quite unfit for chemists, can also be found by Frederick Matsen (1913-2006), who reviewed the first part of the English translation for the Journal of the American Chemical Society. He even went a step further and claimed that the work ‘“does not contain treatment of a number of subjects of interest to chemists”.27 He called the style ‘leisurely’, while claiming on the other hand that the book “is not for a beginner”.28 Some of the subjects mentioned by Matsen are in fact treated in the second part of the book, where electrons and radiation are discussed. The general consensus however still considers the book less interesting for physical chemists. This is for instance shown in the review of Ulrich Dehlinger (1901-1981), from the Technische Hochschule Stuttgart, for the Zeitschrift f¨urElektrochemie und angewandte Physikalische Chemie.29 He made clear that this is not because of the style and structure of the book, which he instead praised, but due to the mathematical level of the book. Even though the method of groups was left out, he considered the mathematical level still too high.30 He did in the end recommend the book to students of theoretical physics, because it offered one of the

23Hund, ‘Grundlagen’, 208. 24Ibid. 25Ebert, ‘Grundlagen’, 194. 26Ibid. 27Matsen, ‘Foundations’, 1774. 28Ibid. 29Dehlinger, ‘Quantentheorie’, 284. 30Ibid.

55 best possible overviews of the foundations of the electron and of radiation, “ein Gebiet, das im Gegensatz zu den Einzelanwendungen der Quantentheorie ja noch ganz unfertig ist und heute wieder vielfach bearbeitet wird”.31 The same sentiment can be found in the review of Carl Friedrich von Weizs¨acker (1912- 2007) in Die Naturwissenschaften. Again the structure is praised, which is systematical rather than historical. As a result, the one who would read the book as careful as the author has written it, “der wird auch scheinbar l¨angstBekanntes in neuem Lichte sehen lernen”.32 Weizs¨acker praised above all else the last chapter, where Kramers discussed electromagnetic radiation. Frederik Belinfante (1913-1991), who had studied mathematics and physics in Lei- den and had written his Ph.D. dissertation, Theory of Heavy Quanta, under supervision of Kramers, wrote a review of the complete translation for Science. He started with complementing Ter Haar with the outstanding job he had done in translating Kramers textbook. Afterwards, he assessed that some students might criticize the book for its lack of exercises, but Belinfante claimed that

“working out in detail some mathematical derivations which the book gives merely in the form of an outline may be an assignment more useful than some of the useless ‘exercises for the sake of an exercise’ found in certain other introductory textbooks. By not burdening the student with such use- less material, this book finds space for a thorough discussion of a number of important aspects of wave mechanics and of matrix mechanics which in many other textbooks are neglecting.”33

Thereafter Belinfante discussed the book in detail, focussing both on the important subjects discussed, as well as those not included. Just as Weizs¨acker, he expressed his praise for the last chapter of the book. In the end, Belinfante concluded with: “This is a fine translation of a remarkable book, which is recommended to every serious student of theoretical physics.”34

Structure and content of Kramers’s textbook

The recurrent theme in all the reviews is the praise for the structure of Kramers’s text- book. Apparently, Kramers has structured his book in a non-trivial way, which could

31Dehlinger, ‘Quantentheorie’, 284 32Weizs¨acker, ‘Quantentheorie’, 646. 33Belinfante, ‘Quantum’, 705. 34Ibid.

56 bring about different views about the theory. In this way Kramers aimed to give an overview of the theory that formed a connected whole. Here Kramers had indeed deliber- ately chosen for the parts of the theory that do form this coherent whole. In the preface, Kramers stated about the structure:

“It seemed to me that the best way to do this would be to present a textbook describing the unified, physical points of view of this theory. The present representation is thus neither historical nor axiomatic. Starting from experi- mental evidence and theoretical considerations about the wave nature of free particles we develop modern quantum mechanics more or less heuristically, using as far as possible the approach suggested by Bohr.”35

It is striking how much this introduction has in common with Kramers introduction in his first course on elementary quantum mechanics. He used the same kind of words and expressed the same general idea. The only important difference is the attention for problems. Whereas in his course Kramers denoted that in this way he was able to pay attention to the problems still ahead, in the textbook he explicitly wanted to create the image of a coherent theory. But still, the book seems to have been based on these early courses. The book is, as previously mentioned, divided into two parts. These in turn are divided into chapters as follows:36

Part One: The Foundations of Quantum Theory

Introduction

I. Quantum theory of free particles (9 sections)

II. Non-relativistic quantum theory of bound particles (16 sections)

III. The non-relativistic treatment of the many-body problem (8 sections)

IV. Transformation theory (13 sections)

V. Perturbation theory (9 sections)

35Kramers, Quantum Mechanics, v. 36The chapters and sections are the same in the original textbook and the translation.

57 Part Two: Quantum Theory of the Electron and of Radiation

VI. The spinning electron (11 section)

VII. The exclusion principle (14 sections)

VIII. Electromagnetic radiation (15 section)

The first chapter considered de Broglie waves and their superposition. Besides, it handled the uncertainty relations and introduced the Schr¨odingerwave equation. In the second chapter, the physical interpretation of the wave function is analyzed, followed by some discussions on eigenvalue and eigenfunction problems. The third chapter treated many particle systems, describing the (semi-classical) motion of wave packets for inter- acting particles. Operators were introduced and the chapter ended with a discussion on causality in quantum mechanics. The fourth chapter dealt with transformation theory, with the introduction of Hermitian operators and Heisenberg’s matrix mechanics. There are some practical examples in the end, such as the Schr¨odingertheory of the hydrogen atom. The last chapter of the first part considered perturbation theory. Once again, the remarkable similarity to the structure of Kramers early courses on quantum mechanics stands out. In his course, he had formulated quantum mechanics in the same fashion, starting with mass points in classical physics. From there, Kramers would introduce the wave equations and its interpretation, first for free particles and then for many-body problems. There he would for the first time consider operators and continued with Heisenberg’s matrices. The only differences were in his treatment of bound particles and perturbation theory, which both received less attention in his course. The link between Kramers’s course and his textbook is unsurprising, it seems only logical that Kramers would use these courses as blueprint for his textbook. After all, these courses presented his ideas on educating quantum mechanics. Since Kramers would deal with spin as the next subject in his course, it is equally unsurprising that the first chapter in the second part of his textbook dealt with spin as well. It started with examining Goudsmit-Uhlenbeck’s spin, from a classical point of view. Afterwards the theory of spinors is introduced, both non-relative and relative. The relativistic spinor calculus is in turn used to derive Dirac’s equations. In the next chapter, Kramers considered Pauli’s exclusion principle. There Kramers also regarded singlet, triplet, and multiplet situations in the N-electron problem. The chapter is concluded with Russell-Saunders coupling and an introduction to homopolar chemical bonds. So

58 Figure 3.5: Table of contents of Part One.

59 Figure 3.6: Table of contents of Part One (continued).

60 these first two chapters of the second part did indeed deal with problems more interesting for chemical physicists, with its focus on spin and the exclusion principle. The final chapter dealt with quantum electrodynamics. Again, Kramers started with a classical description of radiation and gave a semiclassical treatment of quantum electro- dynamics. The justification for this treatment was given by, of course, the correspondence principle, as well as Wentzel-Kramers-Brillouin method. Thereafter Kramers examined the interaction between photons and electrons, and ended with some applications, such as absorption and emission of photons, the Compton effect, semiclassical scattering the- ory, coherent scattering, and disperspersion. What remains remarkably undiscussed in the second part of Kramers’s textbook is the quantum mechanical Stark-effect. In 1926 Erwin Schr¨odinger had applied his wave mechanics to the Stark-effect, solving many of the problems still present.37 Considering the importance of the Stark-effect in Kramers research in Copenhagen, the lack of it in his textbook is rather surprising. What, on the other hand, is unsurprising is the constant mentioning of Bohr and the correspondence principle, something Kramers did time and again. One can indeed clearly find the ‘Kopenhagener Geist’, recognised by the various reviewers as well. Most of the chapters are structured to begin with classical examples or theories, which are in turn extended to cope with the quantum theory. And Kramers choose in almost all cases for a proof based on a heuristic method, instead of a historical or axiomatic one. A fine example is his derivation of the Dirac equations, which Belinfante in his review praised as “elegant, clear, and concise”.38 Kramers had derived those from the theory of relativistic spinors, whereas Dirac had chosen a more axiomatic approach. This Kramers also mentioned in his textbook, where he denoted that his own method “is clearly not according to such clear principles”.39 The great advantage of Kramers’s method is that the Lorentz invariance of the derived equations follows directly, whereas Dirac had to show that these equations did indeed satisfy Lorentz invariance.40 Besides the intellectual legacy of Niels Bohr in Kramers’s textbook, the book likewise contains traces of Lorentz influence on Kramers, albeit less on the surface. During his career, Kramers remained engaged in research concerning the interactions of electrons in an electromagnetic field, the research area where Lorentz has made some of his major contributions.41 In his textbook, Kramers also spend a large section on quantum elec- trodynamics, in fact the longest chapter of the book. And even further, in his work,

37Duncan and Janssen, ‘Stark effect’, 23, 29. 38Belinfante, ‘Quantum’, 705. 39Kramers, Quantum Mechanics, 284. 40Ter Haar, Master of Modern Physics, 44. 41Dresden, H.A. Kramers, 325.

61 Kramers adopted the exact same picture of the electron as Lorentz had done before.42 In his textbook, Kramers described this picture as a picture where “one tries - as Lorentz once did - to discuss the situation using a definite model of the electron which satisfies an ingenious definition of rigidity”.43 It is fair to state that Kramers was in search for a quantum electrodynamics that could lend dignity to Hendrik Lorentz.

Kramers’s textbook compared to others

In 1930, another textbook in quantum mechanics was published, Paul Dirac’s (1902-1984) Principles of Quantum Mechanics. Dirac was doing his Ph.D. under supervision of Ralph Fowler (1889-1944) in Cambridge, when he was asked to lecture about quantum theory in autumn 1926, actually the first lecture on quantum theory at a British university.44 These early lectures would in turn form the basis of Dirac’s textbook, just as they had done for Kramers. But whereas Kramers’s book was praised for its neat structure, Dirac’s textbook was far from reader-friendly.45 The book contained no references or illustrations, and did not even have an index. Dirac’s textbook could be considered divided into two parts, in a manner compara- ble with Kramers, one part discussing the general formalism and a second one giving applications. In the preface of the 1930 edition, Helge Kragh claims, Dirac “stressed the abstract and unvisualizable nature of quantum mechanics and how different it was from classical physics”.46 This is a complete different sentiment than the one found in Kramers’s preface. He did give an impression of a strong link between classical and quantum mechanics, and used this link continuously in building up the quantum theory. It is therefore surprising that in his textbook, Dirac does the exact same thing. Kragh summarizes Dirac’s build up of the quantum theory as follows:

“[...] in Dirac’s presentation the analogy with classical mechanics played an important role. Although quantum mechanics differed radically from the laws and concepts of classical physics, on the formal level there was a great deal of similarity. “practically all the features of the classical theory to which it owes its attractiveness can be taken over unchanged into the new theory,” he wrote.”47

42Dresden, H.A. Kramers, 328. 43Kramers, Quantum Mechanics, 228. 44Kragh, ‘Dirac’s Principles’, 250. 45Ibid., 256. 46Ibid. 47Dirac, Principles of Quantum Mechanics (1930), 1, as cited in: Kragh, ‘Dirac’s Principles’, 258.

62 It seems that Dirac thus did use the same manner in presenting quantum theory as Kramers. However, Dirac’s analogy had a completely different basis. As mentioned be- fore, Kramers’s derivation of Dirac’s equations already differed greatly. There Kramers used a heuristic method, whereas Dirac’s method was based on some principles. This dif- ference is often encountered. Kramers stated another example in his textbook, concerning the probability that some mechanical quantity is equal to its eigenvalue:

“Several authors (Weyl, Dirac) have used the above probability postulate - or a very similar one - as the basic axiom of quantum theory together with the general principle of superposition [...]. This has turned out to be fully justified in the non-relativistic description of many particle systems. We shall show by using the correspondence principle how it is closely connected with the classical description of such systems and also that it contains the previously discussed distribution laws for coordinates and momenta as special cases.”48

This quote is a great summary of the difference between Kramers and Dirac. Firstly, Dirac’s ‘basic axiom’ is the end result of a derivation made by Kramers. Kramers had build up his theory as to finally derive the probability. Secondly, and closely related, Dirac now had to derive the distribution laws out of this axiom, whereas those distribu- tion laws , related to experimental data, are the starting point in the work of Kramers. And thirdly, Kramers needed to justify the results found by linking them to the classical description, using the correspondence principle.

A criticism of Kramers’s work had been its lack of relevance for chemistry students, considering that it was part of the series Hand- und Jahrbuch der Chemisch Physik. As has become clear, Kramers’s textbook was indeed mostly focussed on theoretical physics, without many links to chemistry. In fact, the word ‘chemistry’ did not appear in the index at all. This was rather different in another textbook, Introduction to Chemical Physics, pub- lished in 1939 and written by John Slater. Slater was trained as a physicist and had been in Europe for a couple of years, a period in which he also worked with Kramers and Bohr on the Bohr-Kramers-Slater theory. After returning to the United States, Slater first worked at Harvard University. From 1931 onwards, Slater would head the physics depart- ment at Massachusetts Institute of Technology, with a mission to strengthen theoretical physics and fundamental physics as a whole.49 To succeed in the second part, Slater

48Kramers, Quantum Mechanics, 142. 49Gavroglu and Sim˜oes,‘One Face or Many’, 428.

63 realised that cooperation between chemistry and physics was necessary. His textbook must therefore be seen as pursuing this cooperation. In the words of Kostas Gavroglu and Ana Sim˜oes:“The Introduction to Chemical Physics was an attempt to bridge the gap that had grown up between a largely empirical and non-mathematical chemistry and a physics hitherto unable to deal with atomic forces.”50 As a result, Slater treated more subjects relevant for chemistry. He also build up quantum chemistry in a manner comprehensible for students of chemistry. But Slater stayed a theoretical physicist, and theoretical physics remained the basis. Therefore Gavroglu and Sim˜oesconclude:

“Quantum chemistry was not considered to be a sub-discipline of chemistry but an instance of the application of quantum mechanics to chemical prob- lems. Unification was therefore to be attained through the reduction of chem- istry to physics, [...]”51

It seems that Slater would have been content with Kramers’s textbook as introduction to quantum mechanics for chemists. After all, it was precisely the theoretical fundamen- tals described by Kramers that Slater deemed important for a unification of chemistry and physics. Maybe the criticism in reviews of Foundations of Quantum Mechanics, which appeared in 1957, that the book was of little relevance for chemists, indicate that Slater’s attempt of unification had not yet brought what Slater had hoped for.

Mathematics in Kramers’s textbook

It has become quite clear that Kramers introduced quantum mechanics systematically and in a heuristic manner, repeatedly starting off with classical physics. Since the basis of the textbook lay in his lectures, the book is structured with a pedagogical perspective too. It is therefore to be expected that the mathematics used, especially in the first part, resembles mostly the mathematics of classical physics. This holds to a large extent. Kramers assumed a high level of prior knowledge in the field of mathematical analysis. More specifically, in the branches of real and complex analysis, differential equations and Fourier analysis. Almost immediately the students would encounter triple integrals, Fourier integrals, and partial differential equations. Kramers made this explicit in many places, for instance when he wrote “which is a natural generalization of the well known formal transition from Fourier series to Fourier integrals”.52

50Gavroglu and Sim˜oes,‘One Face or Many’, 429. 51Ibid., 431. 52Kramers, Quantum Mechanics, 74, my emphasis.

64 Figure 3.7: Excerpt of Kramers’s textbook, page 100.

Figure 3.8: Excerpt of Kramers textbook, page 102.

His expectation of prior knowledge of analysis is in complete contrast with his expec- tation of prior knowledge in the use of operators and matrices, or more generally, of the field of linear algebra. Before introducing operators, Kramers first defined an Hermitian operator in terms of a differential operator in an integral equation. Only thereafter, in section 29 on page 100, did Kramers define in operators independently (see figure 3.7). Then he defined a linear operator, with its calculation laws (for instance the distributive law, associative law, and non-commutativity), followed by the definition of the unit op-

65 erator. After that, Kramers stated that “each operator corresponds to a matrix”, where in a footnote the definition of a matrix is given (see figure 3.8). Altogether, the pace of the book has completely changed. Where before Kramers could assume prior knowledge and therefore could proceed rapidly to physical interpretations, all of a sudden Kramers had to explain underlying mathematics. The slower pace remains throughout the chapter on transformation theory. At the end of the chapter, Kramers made a clear statement concerning matrix mechanics:

“In the original matrix mechanics of Born, Jordan, and Heisenberg the main problem of quantum mechanics was formulated as follows. Find solutions of the quantum mechanical equations of motion such that all observable are represented by Heisenberg matrices and that the matrices of the p and q satisfy

the commutation relations. The energy becomes a diagonal matrix Elδll0 . Although in principle many problems of quantum theory can be solved on this basis, it is hardly possible to derive in this way a theory of atomic phenomena which obeys clearly the correspondence principle. For this purpose one uses the Schr¨odingerequation.”53

In other words, Kramers preferred the Schr¨odingerpicture to the Heisenberg picture, because it fitted his general, ‘Copenhagen’ view. And the necessary mathematics for this picture included mathematical analysis, rather than linear algebra. Pascual Jordan (1902-1980) chose for a complete different picture. As Kramers noted, Jordan was one of the main contributors to matrix mechanics. He had stud- ied in G¨ottingen,where he had been the assistant of the mathematician Richard Courant (1888-1972) and where he had written his Ph.D. dissertation under the supervision of Max Born (1882-1970). Jordan wrote a textbook that was published in 1936 and named Anschauliche Quantentheorie. In the textbook, Jordan presented the mathematical fun- damentals of quantum physics and the detailed introduction of matrix and wave mechan- ics is given in a comparatively high level of mathematical abstraction.54 He did show the equivalence of matrix and wave mechanics as well and the picture he chose to present applications such as electron spin is one called statistical transformation theory, in which both matrix and wave mechanics are contained.55 But choosing this picture meant for Jordan by no means a break with Bohr, moreover, he introduced electron spin using arguments based on the correspondence principle.56

53Kramers, Quantum Mechanics, 161-162. 54Howard, ‘Jordan’s Quantentheorie’, 274-275. 55Ibid., 275. 56Ibid, 273.

66 Thus for Jordan, the correspondence principle did not offer an argument for using the Schr¨odingerpicture. He showed that the abstract formulation of matrix mechanics could be reconciled with the correspondence principle. Kramers deliberately omitted abstract mathematics, such as the theory of groups. In his preface he stated: “We have not used the theory of groups explicitly.”57 And the introduction of the more abstract mathematics in Kramers’s textbook is delivered quite slow. Even though Kramers claimed that using wave mechanics was due to his heuristic structure and the use of the correspondence principle, it seems that it mostly was a de- liberate choice based on other arguments!

In Dirac’s Principles, the level of mathematical abstraction is high as well. He had selected a more axiomatic approach, as mentioned earlier, different from Kramers. And where Kramers and Jordan both ‘picked’ a preferred picture of quantum theory, Dirac chose a more general representation, one that he called the symbolic method.58 This method was not reviewed positively, mostly due to its overestimation of the powers of abstraction and its inadequacy to function in a textbook.59 As a result, the second edi- tion of the principles was written in a “less abstract and symbolic form”, but still “not all reviewers were impressed by the pedagogical quality of the new, more menschlich edition”.60 It was exactly this pedagogical quality for which Kramers’s textbook had been praised. Kramers did have this pedagogical point of view in mind while writing his work. He stated in the preface:

“The apparent lack of mathematical morals which is contritely pointed out re- peatedly in the text is not exclusively due to the incompetence of the author. Physical morals, even (or rather especially) in their purest form, that is, un- encumbered by pedagogical afterthoughts, do not live happily together with their mathematical relations in the restricted mansion of the human mind - and neither in the restricted volume of a monograph.”61

57Kramers, Quantum Mechanics, v, in original: Expliziter Gebrauch der Gruppentheorie ist vermieden.”, Kramers, Grundlagen, v. 58Kragh, ‘Dirac’s Principles’, 256. 59Ibid., 257. 60Ibid., emphasis in the original. 61Kramers, Quantum Mechanics, v, in original: “Der offenbare Mangel an mathematischer Moral, auf den im Text wiederholt mit Schuldbewusstsein hingewiesen wird, ist nicht ausschließlich auf das Un- verm¨ogendes Verfassers zur¨uckzuf¨uhren;die physikalische Moral, sogar (oder besonders) in ihrer re- insten, d. h. von p¨adagogischen Hintergedanken ungehemmten, Erscheinungsform vertr¨agtsich in

67 The fact that the lower level of mathematical abstraction is definitely not ‘due to the incompetence of the author’ is clear from Kramers mathematical talent and his pub- lications. These show that Kramers mastered, quite to the contrary, all mathematics necessary. It has therefore been Kramers decision to leave abstract mathematics out, apparently because these abstract mathematics make it harder to understand the phys- ical concepts. The physical morals, the conceptual lessons Kramers wanted to educate his students, would be hindered when they would be introduced by the use of higher mathematics, because there is only a limited space in the human mind, and in the text- book itself! Thus because of a pedagogical argument, and not because of arguments based on the systematic build up or the correspondence principle, Kramers had made the decision to consider the wave picture in order to introduce quantum theory. This points to an important influence, not of interpretation or ‘Kopenhagener Geist’, but of Leiden-Copenhagen education, compared to for instance G¨ottingen-Copenhagen or Cambridge-Copenhagen education.

The story is considerably different for the second part of Kramers textbook. There the level of mathematical abstraction is much higher. Examples can be found in all chapters of the second part, for instance the introduction of spinors in the chapter on electron spin, Kramers’s own symbolic method in the chapter on the exclusion principle, and Kramers’s way of discussing the interactions of photons and electrons. This higher level was also stated by the reviewers, such as Dehlinger, who considered the mathematical level too high for chemists. Kramers himself acknowledged this, claiming that he was “well aware of the fact that the mathematical arguments - especially in the second part [...] - may sometimes put a strain on the reader”.62 For the applications of the quantum theory, one seemingly needed a more abstract mathematical basis than the one necessary to explain the fundamentals and the physi- cal concepts. In the second part, matrix calculus is assumed prior knowledge, and new concepts are introduced much swifter than in the first part, and not always from a phys- ical point of view. For example, spinors are introduced as “mathematical entities”, as is the concept of the “spinor plane”, and only afterwards their physical significance is derived.63 As mentioned earlier, these spinors would also be the key components of Kramers’s derivation of the Dirac equations. The spinors also played a key role in Kramers’s symbolic method, a method used to

der beschrankten Wohnung des menschlichen Geistes (sowie auch im beschr¨anktenBogenumfang einer Publikation) nicht gut mit ihrer mathematischen Schwester.” Kramers, Grundlagen, v. 62Kramers, Quantum Mechanics, v. 63Ibid., 57, 260.

68 derive the formulas for multiplet situations. Kramers designed this method to avoid the use of group theory explicitly.64 Although avoiding the abstract theory of groups, the method itself contained some formal trickery as well, especially the complete mastery of the spinors. Apart from a few of those cases, the second part was mostly based on mathematical analysis, just as the first part. Most of the proofs still consisted of mathematical tricks taken from that field of mathematics. The deviating parts appear to be well prepared, always from a pedagogical point of view. This might in turn explain why the second part arrived years after the first. It seems that Kramers needed time to consider the mathematics needed in the proofs and to arrive at a structure where the students could make use of the second part with prior knowledge of mathematical analysis and the first part of the textbook. It is expected that Kramers took his knowledge of the necessary prior knowledge from the students in his lectures. To better understand how, a look at the use of the textbook in Kramers’s lectures is required.

The use of Kramers’s textbook

Kramers had used his early lectures on quantum mechanics to structure his textbook. As a result, his textbook corresponded to his planning on lecturing about elementary quantum theory. Kramers no longer had to prepare lecture notes, if he would use the textbook as a guideline for his lectures. From his lecture notes, it is clear that he did so. Lectures about the fundamentals of quantum theory would follow Kramers’s textbook from 1932 onwards, as is shown in figure 3.9. There were still differences between the textbook and Kramers’s lectures, most notably, the language. Whereas the textbook was written in German, Kramers would still lecture in Dutch. But he would refer to his textbook and the structure of the course was the structure of the textbook. The fact that Kramers did indeed follow the book neatly can be seen in student notes. Adriaan Blaauw (1914-2010) studied in Leiden between 1932 and 1938. He followed courses in mathematics, physics, and astronomy, and did his candidates exams in 1935. Afterwards, he could attend the lectures of Kramers. Thus, in the year 1935-1936, Blaauw followed Kramers’s course on elementary quantum theory. His notes show how Kramers used his textbook as blueprint and background of his course. There are multiple references made to the book, as seen in figure 3.10, and even the pictures displayed in the book are recognisable in Blaauw’s lecture notes, compare for instance figures 3.11 and 3.12. Kramers’s textbook did not consist of any exercises or other problems-to-be-solved.

64Dresden, H.A. Kramers, 321, Ter Haar, Master of Modern Physics, 45.

69 Figure 3.9: Kramers’s notes for his lectures on quantum mechanics. From the Archive for the History of Quantum Mechanics, Niels Bohr Archive, microfilm 26, section 3.

Figure 3.10: A brief notation in Blaauw’s notes of Kramers lecture on quantum mechanics. Note the explicit reference to kramers’s textbook. From the Blaauw Archief, section A), in the Groninger Archieven.

Recall that Frederik Belinfante as a reviewer already discussed this issue. He claimed that this was an advantage of the book, whereas other books would present useless ‘exercises for the sake of an exercise’. Since Belinfante had studied in Leiden and therefore followed Kramers lecture from this textbook himself, this argument seems to be one made from

70 Figure 3.11: Excerpt of Kramers’s textbook, page 73.

Figure 3.12: Blaauw’s notes on Kramers’s quantum lecture. From the Blaauw Archief, section A), in the Groninger Archieven. experience, rather than a mere comment of a reviewer. Besides Belinfante, Dirk ter Haar, the translator of the textbook, and Max Dresden, author of Kramers’s biography, must have had their first introduction to quantum theory through Kramers textbook. Considering that the second part of the textbook only appeared in 1938, Kramers

71 must have given his lectures about electron spin from his lecture notes. This is also visible in figure 3.9, where he continues his notes for the lecture of 1934-1935. The first part was used as prior knowledge, which Kramers in turn could use when he lectured on applications of quantum theory. Kramers probably adapted this part of his course to the discussion in the colloquium as well. This makes it likely that Kramers wrote the second part aware of the prior knowledge of his students. The necessary mathematical prior knowledge was in turn granted by the mathematical education of the students.

Mathematics in Leiden in the 1930s

The mathematical staff in Leiden consisted of three men, Willem van der Woude, Jo- hannes Droste, and Hendrik Kloosterman. Van der Woude had become professor of mathematics and mechanics in 1916, and lectured on geometry and classical mechanics. As mentioned in the first chapter, he had made publications in the fields of relativity and differential geometry, although those did not include any new theories, but merely a sys- tematic analysis in Dutch of results already known.65 The elementary course on algebra was one of his responsibilities as well, where he treated systems of linear operations, as well as permutations and combinations. Droste had studied mathematics and physics in Leiden and wrote his dissertation under Lorentz, about the gravitational field in general relativity. In 1919 he became lecturer in mathematics and in 1930 professor, succeeding Jan Cornelis Kluyver. He lectured on complex analysis and Fourier analysis.66 As the subject of his dissertation shows, Droste was trained in physics too and was not a pure mathematician. Kloosterman had studied in Leiden as well and his years in Leiden have already been discussed in the first chapter. After his studies in leiden, Kloosterman would study several years abroad. As is shown on multiple occasion, for instance considering the visitors of the institute of Niels Bohr, this was very common for students in theoretical physics. But at the time, it was highly uncommon for students in mathematics.67 Kloosterman would stay in Copenhagen with Harald Bohr (1887-1951), Niels Bohr’s brother, from 1922 until 1924, and in G¨ottingenwith Edmund Landau (1877-1938) from 1925 until 1928. Especially in G¨ottingen,Kloosterman studied many of the developments in mathematics, mostly abstract algebra (including branches such as group theory, vector spaces, and algebraic number theory).68 In 1930 he became lecturer of mathematics in Leiden, when Droste

65Dijk, Hoogleraren Wiskunde, 55. 66Ibid., 7. 67Ibid., 58. 68Ibid.

72 succeeded Kluyver. As lecturer, Kloosterman was mainly responsible for the education of first- and second-year students, lecturing in elementary mathematical analysis. The analysis course was split in two, an A version and a more complex B version. Most students eventually did their candidates exam with Kloosterman in the B version.69 Students studying for their candidates exam would accordingly attend the lectures of Kloosterman, together with some introductory courses on physics. These lectures would be the training necessary to attend the higher level courses. Adriaan Blaauw, for example, before his candidates exam attended the mathematics lectures of Kloosterman, lectures on physics of De Haas-Lorentz, and elementary astronomy courses of De Sitter and Woltjer.70 This does not imply that all students did follow those courses. Another example is Pierre van Laer (1906-1989), who only followed Kloosterman’s mathematics courses before his candidates exam.71 As had been the case in the 1910s, the curriculum still was quite loose, as Blaauw mentioned as well.72 The candidates exam can best be considered as an internal university exam, a test whether one was ready for the higher level courses. The candidates exam in the 1930s consisted of both written and oral exams. Thus before students like Blaauw and Van Laer could continue their studies, they would have to pass those exams first (which both of the students in fact did). The course of elementary mathematical analysis therefore was the prior mathematical knowledge for courses such as Kramers’s elementary quantum mechanics. In figure 3.13, the exam results of Adriaan Blaauw are shown. He made two exams, both of the B version of Kloosterman’s course, and passed them both. The short notations of the question asked, visible in figure 3.13, show once again that the course had discussed real and complex analysis. Kloosterman’s teaching responsibilities were limited to those elementary courses. As a lecturer, he did not have the right to supervise Ph.D. students too. However, he did lecture on other subjects, in his capita selecta presented outside of the curriculum. Here Kloosterman could lecture on any possible subject, which indeed he did. Every year, he would present a modern mathematical subject in such a course, with topics such as Lebesgue integration and the theory of groups.73 The subjects were not limited to

69Kloosterman kept the results of all the exams, Kloosterman Archief, box 10, in the bijzondere collecties of the Universiteit Leiden. 70Notes of Adriaan Blaauw in the Blaauw Archief, section A), in the Groninger Archieven. 71Notes Pierre van Laer in Collegedictaten Van Laer, in the bijzondere collecties of the Universiteit Leiden. 72Interview of Adriaan Blaauw by David DeVorkin on 1979 August 19, Niels Bohr Library & Archives, American Institute of Physics, College Park, MD USA, URL: https://www.aip.org/history- programs/niels-bohr-library/oral-histories/5002. 73Dijk, Hoogleraren Wiskunde, 59.

73 Figure 3.13: Exam results of Adriaan Blaauw, notes made by Hendrik Kloosterman. From the Kloosterman Archief, box 10, in the bijzondere collecties of the Universiteit Leiden.

pure mathematics, but stretched to physics as well. In 1936-1937 Kloosterman lectured on mathematical tools in quantum mechanics, where he introduced linear algebra and, above all, group theory, which had been omitted by Kramers.74 The next year, Kloost- erman discussed linear operators in Hilbert space, the first mathematical introduction of the Hilbert space in Leiden.75 Through these capita selecta, modern and abstract mathematics, even with applications in physics, found its way to Leiden.76

74Notes Kloosterman in the Kloosterman Archief, box 3, in the bijzondere collecties of the Universiteit Leiden. The Dutch name of the course was: Wiskundige hulpmiddelen der quantenmechanica. 75De Bruijn, ‘remembering’, 132. The Hilbert space is not discussed at all by Kramers in his textbook. 76Only in 1947, when Kloosterman became professor, did these subjects also became a part of the curriculum. Kloosterman got the leeropdracht ‘mathematical analysis, more specifically algebra and number theory’, which can be seen as the take off of mathematics in leiden, both of pure and applied mathematics. Dijk, Hoogleraren wiskunde, 59-60.

74 These lectures were not part of the curriculum though and might have attracted mostly students with special interest in mathematics. Also, every subject was discussed only once and the courses could therefore not be seen as general knowledge. The second part of Kramers’s book did require some extra mathematical knowledge in the field of vector analysis, to understand for instance the theory of spinors. Yet it is clear that none of the mathematicians lectured on this subject. It is therefore not surprising that Kramers gave lectures on vector analysis himself. This course was followed by the stu- dents that also followed Kramers’s elementary quantum mechanics course, such as, again, Adriaan Blaauw.77 Kramers introduced vectors first in a geometrical, rather intuitive manner.78 Afterwards, he continued with an analytic introduction. After both introduc- tions, Kramers presented something extraordinary, exercises, shown in figure 3.14 and figure 3.15. Apparently, different from learning physics, exercises were necessary to learn mathematics!

Figure 3.14: Exercises in Kramers’s course on vector analysis. Kramers, Vectoranalyse, 11.

Figure 3.15: Kramers’s exercises, written down by Adriaan Blaauw. From the Blaauw Archief, section A), in the Groninger Archieven.

77Notes Adriaan Blaauw in the Blaauw Archief, section A), in the Groninger Archieven. 78Kramers, Vectoranalyse, 1-3.

75 After the exercises, Kramers discussed differentiation of vectors, vector fields, and in- tegration of vectors. All in all, this would grant the students the required prior knowledge for Kramers’s lectures on the applications of quantum mechanics. Kramers knew the subject matter of the available mathematics courses, since he followed them himself and the curriculum had not changed since he studied in leiden. Moreover, the professor of mathematics and mechanics, Van der Woude, was still in the same position. The mathematician in Leiden with the best knowledge of modern mathematics, Kloosterman, was not in a position to lecture on those subjects, he could discuss one modern subject each year at most. Fortunately, as professor of physics and mechanics, Kramers held a position where he could lecture on mathematical subjects himself. The lack of mathematical knowledge was thus solved by Kramers himself.

Conclusion

When Kramers returned at Leiden University, the curriculum had not changed. As a result, the students were trained to get the same mathematical background as they had received in the 1910s. Kramers introduced quantum theory in his lectures in a heuristic manner, starting off with classical concepts and theories. The mathematical background for the students was sufficient to follow these foundations of quantum theory. Kramers’s textbook on quantum mechanics resembles this. The theory is introduced in a similar manner, often with the correspondence principle as key concept. The mathe- matics used in the textbook is therefore the mathematics of classical physics. In the parts where other mathematical techniques are introduced, the pace of the book is lowered and mathematical definitions are given. Already in his early lectures, Kramers encountered a lack of prior knowledge when he considered applications of the quantum theory. In his early lectures, he took much more time to introduce spin, and the lectures seem better prepared. At Leiden University, Kramers set up a course on vector analysis, to prepare his students for the advanced courses of quantum theory. The course on vector analysis thus focused on the aspects of the mathematics necessary for physics. Kloosterman would have been the ideal mathematician to lecture on abstract math- ematics form a mathematical perspective. But due to his position as lecturer, he was unable to do so. All he could do was introduce one subject every year, in his capita selecta. The mathematical training given by mathematicians in Leiden thus remained unchanged.

76 Conclusion

The study of mathematics remains painstaking work. It demands discipline to attend lectures, work through textbooks, and, above all, to make exercises. But it also offers the opportunity to be creative, to come up with new ideas by yourself. Finding a solution for a problem you have been working on over and over again can be one of the most satisfying feelings in the world. The lecture notes of Hendrik Kloosterman show that he did have the discipline, that he attended lectures and made exercises. That he would become a successful mathematician might be considered unsurprising. Kramers’s lecture notes, on the other hand, do not present this discipline. But they do show mathematical talent and skill. Although Kramers did not always attend all lectures, he did follow lectures that were not mandatory for a physics student. In these he broadened his knowledge of mathematical analysis and mastered branches of mathematics that were uncommon for physicists. In this, one can see the influence of the professor of mathematics, professor Kluyver. It was Kluyver who was responsible to teach physics students the necessary prior knowledge to attend higher-level physics courses, and it was Kluyver who taught Kramers mathematical analysis. By mentioning Kluyver in his letter to Bohr, Kramers did acknowledge this himself. He emphasized his wish of becoming a mathematical physicist, rather than a theoretical physicist, as well. This remains curious, since Kramers’s study would later be called mathematical physics. And as it turned out, the extra mathematical knowledge obtained by Kramers was exactly the knowledge crucial in his dissertation. His work did impress the physics community and gave Kramers an important position next to Niels Bohr in the construction of concepts and theories. Kramers was able to make calculations of the ideas that Bohr came up with, which made them experimentally testable. The close collaboration between Bohr and Kramers would in turn have a lasting influence on Kramers interpretation of theories. He would always construct a theory in the style of Bohr, i.e. heuristic rather than axiomatic. This is demonstrated by Kramers constant use of Bohr’s correspondence principle, which would for Kramers almost become an axiom itself. During his stay in Copenhagen, Kramers gained experience in lecturing. Due to his mathematical skill, he would lecture on general relativity, rather than Bohr. Later on, he would even carry out all of Bohr’s teaching duties. And he would give public lectures to popularize science and show Bohr’s accomplishments in physics. Besides those (public) lectures, Kramers wrote a popular science book, together with Helge Holst. The book discussed the atomic model of Bohr and presents just the way Kramers wanted to explain atomic physics. This line of reasoning is later found in Kramers textbook on quantum

77 theory as well. There, he builds up the theory heuristic, with the ideas of Bohr constantly present. As a result, the mathematics in the textbook resembled the mathematics used by Kramers in his time in Copenhagen, with a focus mostly on mathematical analysis. There where the book would consider different mathematics, the pace of the book dropped, and the description became elementary. This is linked to the first courses Kramers gave in the Netherlands as well. Kramers’s notes for the first course offer the expectation of the treatment of more abstract mathe- matics, but in practice this is barely treated. Kramers also took much longer to introduce the parts where abstract mathematics was necessary, such as in the part on spin. It seems that the lectures in the end were adjusted to the demands of the students present in the lectures. Those students would not have had any background in abstract mathematics. The curriculum in Leiden had not changed during Kramers’s stay abroad and the math- ematics discussed had stayed the same. The mathematics staff had also not seen the same growth as the physics staff, even though the mathematicians were responsible for the mathematical education of all science students. To make sure his students had the necessary prior knowledge for more advanced courses, Kramers lectured in mathematics himself. After all, he was not the professor of theoretical physics, but of physics and me- chanics, where mechanics had formerly always been part of mathematics. This again fits with Kramers once expressed wish to study mathematical, instead of theoretical, physics. At the time Kramers was professor in Leiden, Kloosterman was lecturer there. He had also been abroad, studying abstract mathematics in G¨ottingenand Oxford. In fact, he would be the ideal mathematician to lecture on the subjects Kramers was lecturing about. But the curriculum in Leiden, together with Kloosterman’s position as lecturer rather than professor, made this impossible. Kloosterman had to lecture on mathemati- cal analysis. But he did not lecture solely on this subject. Every year, he would choose a subject and gave extra lectures, the so-called capita selecta. There, Kloosterman intro- duced many concepts of modern mathematics in Leiden. Only after Kloosterman became professor, in 1947, did those subjects become part of the official curriculum as well. All in all, mathematics education in Leiden as taught by professors and lecturers of mathematics did barely change between 1912 and 1940. The only difference was the introduction of Kloosterman’s capita selecta course, where he would introduce new mathematical fields. But the course was not part of the official curriculum and thus no part of the mathematical training. However, this does not imply that the mathematics in physics education in Leiden did barely change. The developments in physics demanded the prior knowledge of different fields of mathematics and the physics students had to gain this prior knowledge. Kramers, as professor of physics, lectured on those subject,

78 to prepare the students for advanced courses in quantum theory. These courses were therefore not aimed to be formal courses on mathematics, but focused on the aspects of the mathematical fields useful for physics. And what aspects were actually useful for physics, was decided by Kramers and his interpretation of quantum theory. Kramers did not consider the theory of groups necessary, since one could avoid it by using Kramers’s own symbolic method, thus no course on the theory of groups was available in Leiden. (Except of course for the one year in which Kloosterman discussed groups in his capita selecta.) In his inaugural address, Kramers discussed the different styles of lecturing on quan- tum theory of professor Palaeophilos and professor Neophilos. Kramers’s own style re- sembled that of Palaeophilos, he would constantly refer back to classical physics. In many cases the key concept used by Kramers was the correspondence principle and using this principle Kramers was able to stay close to classical physics. This presents the huge influence of Niels Bohr on Kramers’s interpretation of physical theories, which in turn makes it fair to say that Bohr even had an influence on the mathematics curriculum at Leiden University. Kramers symbolic method would not become very popular, nor would other methods of Kramers that avoided abstract mathematics. His restrain of abstract mathematics, and the old-fashioned curriculum in mathematics, resulted in a generation of Leiden physicists that were generally not trained in using abstract mathematics in physics. It might be interesting for further research to study whether Kramers’s ‘Palaeophilos’ style was a factor in the decline of status of Leiden University, and the end of the ‘second Golden Age’. This ‘love of the old’, this resistance to change at Leiden University, is sketched out by Casimir, who was asked to succeed Kramers in 1952. When he went to the Institute of Theoretical Physics in Leiden, he could not help feeling depressed.

“The old rooms, the old library, were almost exactly as they used to be, without new life, without an indication of growth. There was hardly any administrative assistance, there were only very limited funds for traveling and for inviting lecturers from abroad. It was returning to a past that I had liked but that should be changed, and I had been to much part of that past to be able to effect the change.”1

1Casimir, Haphazard Reality, 241.

79 Bibliography

Archives

American Institute of Physics, Niels Bohr Library & Archives.

Archief van de Faculteit der wis- en natuurkunde, AFA FA, bijzondere collecties of the Universiteit Leiden.

Archive for the History of Quantum Physics, Niels Bohr Archive.

Blaauw Archief, Groninger Archieven.

Collegedictaten Van Laer, bijzondere collecties of the Universiteit Leiden.

Kloosterman Archief, bijzondere collecties of the Universiteit Leiden.

Kramers Archief, wetenschapsarchieven of the Noord-Hollands Archief.

Kramers correspondence, Niels Bohr Archive.

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Heilbron, J.L., ‘Relativity’, in: J.L. Heilbron, The Oxford Companion to the History of Modern Science (2003) 711-713.

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81 Klein, M.J., Paul Ehrenfest (Amsterdam 1972).

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Kox, A.J. (ed.), The Scientific Correspondence of H.A. Lorentz Vol. 1. (New York 2008).

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Kragh, H., ‘Paul Dirac and The Principles of Quantum Mechanics’, in: M. Badino, and J. Navarro (eds), Research and Pedagogy (Berlin 2013) 249-264.

Kramers, H.A., and H. Holst, De Bouw der Atomen (Amsterdam, 1927).

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