Erwin Schrödinger, Berlin 1927 (Courtesy of the Österreichische Zentralbibliothek für Physik) Erwin Schrödinger – 50 Years After
Wolfgang L. Reiter Jakob Yngvason Editors Editors: Wolfgang L. Reiter Jakob Yngvason Faculty of Historical and Cultural Studies Faculty of Physics University of Vienna University of Vienna Spitalgasse 2 Boltzmanngasse 5 1090 Wien 1090 Vienna Austria Austria E-mail: [email protected] Email: [email protected]
2010 Mathematics Subject Classification: 01-02, 81-02, 81-03, 81P05, 81P15
ISBN 978-3-03719-121-7
The Swiss National Library lists this publication in The Swiss Book, the Swiss national bibliography, and the detailed bibliographic data are available on the Internet at http://www.helveticat.ch.
This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained.
© 2013 European Mathematical Society
Contact address:
European Mathematical Society Publishing House Seminar for Applied Mathematics ETH-Zentrum SEW A27 CH-8092 Zürich Switzerland
Phone: +41 (0)44 632 34 36 Email: [email protected] Homepage: www.ems-ph.org
Typeset using the authors’ TEX files: I. Zimmermann, Freiburg Cover picture: Erwin Schrödinger and Matter Waves at Rapperswil, ca. 1926 (Courtesy of the Österreichische Zentralbibliothek für Physik) Printing and binding: Beltz Bad Langensalza GmbH, Bad Langensalza, Germany ∞ Printed on acid free paper
9 8 7 6 5 4 3 2 1 Foreword
To commemorate the 50th anniversary of Erwin Schrödinger’s death inVienna the Erwin Schrödinger International Institute for Mathematical Physics (ESI) invited eminent scholars to meet in the city where Schrödinger was born and were he spent a good part of his life as a student and young scientist. In 2013 the ESI will celebrate its foundation 20 years ago. It was a happy coinci- dence that the institute found its first quarters at Pasteurgasse 4 in the very house where Schrödinger spent his last years in Vienna. In 1997 the ESI moved to its present more spacious premises in Boltzmanngasse 5, but like in the previous housing Schrödinger’s spirit continues to guide the institute’s advancement. The International Symposium “Erwin Schrödinger – 50 Years After”, held Jan- uary 13 to 15, 2011 in the Boltzmann Lecture Hall of the ESI, was dedicated to Er- win Schrödinger’s legacy in quantum theory and to his views on the interpretation of quantum mechanics from a contemporary perspective. The symposium brought to- gether scholars from all over the world whose scientific achievements bear evidence to Schrödinger’s fundamental contributions to the development of science. The equation that bears his name has become a trade mark and sign post far beyond the inner circles of physics, and the Gedankenexperiment “Schrödinger’s Cat” of his famous paper of 1935, where he coined the term “entanglement” (Verschränkung), is still central to the discussions of the appropriate interpretation of quantum physics. It plays also an important role in recent theoretical and experimental advancements in Quantum Information Science, some of which are discussed in this volume. Indeed, since Schrödinger’s paper of 1935 and the paper of Einstein, Podolsky and Rosen of that same year the question of the universality of quantum theory has not lost its pressing actuality.1 The simple question asked by Einstein and Schrödinger (and much later by John S. Bell) about the relation between quantum mechanics and reality, i.e., the world “out there”, may be disregarded as “metaphysical”, but this attitude does not answer the question. A significant part of the contributions to this volume has a bearing on the still ongoing discussions of foundational issues of quantum mechanics, a topic hunting Schrödinger during all his life. The series of lectures at the Symposium outlined recent experimental and theoretical developments related to Schrödinger’s work, as well as the early developments of quantum theory and his discussions with fellow scientists. Schrödinger, like Einstein or Bohr – to name only two outstanding representatives of a wide variety of “modern” scientists – was a genuine philosopher-scientist. Therefore it is more than justified to present this side of his work from a point of view of history and philosophy of science.
1Erwin Schrödinger, Die gegenwärtige Situation in der Quantenmechanik. Die Naturwissenschaften, 23. Jahrg., Heft 48, 807–812, 823–828, 844–849, 1935. A. Einstein, B. Podolsky, N. Rosen, Can quantum- mechanical description of physical reality be considered complete? Physical Review, Vol. 47, 777–780, 1935. vi Foreword
Clearly, Schrödinger’s scientific achievements are as important today as they had been when he shared the Nobel Prize in Physics for 1933. He belongs to the rare species of scholars whose ideas have became daily practice to many scientists and will continue to be part of the treasures of science. The contributions collected in this volume shed light on a few aspects of Schrödinger’s multifaceted legacy. The articles collected in this volume convey the spirit which guided the presentations and vivid discussions at the Symposium. The contribution “Erwin Schrödinger: personal reminiscences” is a verbatim tran- script of the oral presentation by Walter Thirring, where he, the only one of the speakers to have known Schrödinger personally, drew a colourful picture of Schrödinger as a person, scientist and poet. The article “Schrödinger and the genesis of wave mechanics” of Jürgen Renn contains a thorough review of the history of the Schrödinger equation together with many new insights from an ongoing research project on the history and foundations of quantum physics. Jürg Fröhlich and Baptiste Schubnel take the question “Do we understand quantum mechanics –finally?” as a starting point for a grand tour through a conceptual and mathematical edifice where quantum mechanics finds it proper place and the differences to “realistic” theories are clearly brought out. The contribution “Schrödinger’s cat and her laboratory cousins” by Anthony J. Leggett is a verbatim transcript of his “Erwin Schrödinger Distinguished Lecture” held in Vienna on March 18, 2011. Sir Anthony’s discussion of the Gedankenexperiment and its relation to real experiments and foundational questions is a lucid testimony of the topicality of Schrödinger’s ideas. The article “Digital and open system quantum simulation with trapped ions” by Markus Müller and Peter Zoller summarizes some recent theoretical and experimental achievements in the simulation of quantum systems that have become possible by spectacular advances in the last two decades in cooling and trapping atoms. In the article “Optomechnical Schrödinger cats – a case for space” by Rainer Kaltenbaek and Markus Aspelmeyer, that was written specially for the proceedings, the authors describe a scenario for an experiment in space with Schrödinger cat states of unprecedented size and mass. The article “A quantum discontinuity: the Bohr–Schrödinger dialogue” by Helge Kragh brings a lively account of the sometimes heated discussions between Niels Bohr and Schrödinger who, despite their high mutual respect for each other, had quite dif- ferent views on the foundational issues of quantum mechanics. In the contribution “The debate between Hendrik A. Lorentz and Schrödinger on wave mechanics” Anne J. Kox discusses the remarkable correspondence between the two physicists about Schrödinger’s first papers on quantum theory that made the latter aware of the non-classical character of the wave mechanics he had created. The last article of this volume, “A few reasons why Louis de Broglie discovered Broglie’s waves and yet did not discover Schrödinger’s equation” by Olivier Dar- Foreword vii rigol is concerned with de Broglie’s seminal work of 1923–24 that laid the ground for Schrödinger two years later. The editors are deeply indebted to the contributors of this volume, for their will- ingness to put in writing their presentations at the Symposium, as well as enduring the subsequent tedious interventions by the editors. Our special thanks goes to Sir Anthony Leggett for making his “Distinguished Schrödinger Lecture” available for publication in this volume. Our thanks to Markus Aspelmeyer and his team for arranging the transcription of Leggett’s oral presentation and designing the figures of his paper. Besides the contributors to this volume further speakers at the Symposium were Michel Bitbol, CREA, École Polytechnique, Paris (Schrödinger’s translation scheme between (abstract) representations and facts: a reflection on his late interpretation of quantum mechanics), Roberto Car, Princeton University (Quantum mechanics and hydrogen bonds), Moty Heiblum, Weizman Institute of Science (Entangled electrons in the solid state: quantum interference and controlled dephasing ), K. Birgitta Whaley, University of California, Berkeley (Quantum coherence and entanglement in biology) and Anton Zeilinger, University of Vienna Wien and IQOQI Vienna (The career of Schrödinger’s entanglement from philosophical curiosity to quantum information). As a final remark we note that neither Schrödinger’s late and frustrated contribu- tions to unified field theory, nor his writings on philosophy related to quantum physics influenced by the Vedic tradition, are covered in this volume. These topics still await an extensive discussion in the literature reflecting the philosophical thinking of this em- inent scholar of 20th century science and philosophy. In fact, Schrödinger developed quite early a strong inclination for philosophy, and, as mentioned in Walter Thirring’s contribution, he planned in 1918 to accept a post as professor for theoretical physics at the then Austrian University in Czernowitz (today Chernivtsi, Ukraine) intending to dedicate a good part of his work to philosophy. The fall of the Habsburg Empire ended this early dream. The complex life and intellectual carrier of Erwin Schrödinger certainly merits further study. We are indebted to Peter Graf of the University of Vienna’s Central Library for Physics for providing photos. Mrs. Ruth Braunizer, Schrödinger’s daughter, attended the Symposium as a guest of honour. We are grateful to Maga. Verena Tomasik, Schrödinger’s granddaughter, for her careful transcription of Walter Thirring’s oral presentation. The Symposium was generously supported by Professor Georg Winckler, the Rektor of the University of Vienna. The co-operation with the City of Vienna’s lecture series “Vienna Lecture” and the efforts of Professor Hubert Christian Ehalt in organising a public lecture for Jürgen Renn is gratefully acknowledged. We owe special thanks to Mrs. Irene Zimmermann for many valuable suggestions and her proofreading of all the papers during the final stage of the preparation of the volume. Finally, we wish to thank Manfred Karbe of the European Mathematical Society Publishing House for his support and constant help in producing this volume. The editors hope that this small volume by merging technical papers with those of historical research contributes to the memory of a great physicist, a scholar of incom- viii Foreword passing cultural passions and a man – repeatedly dislocated by the political turmoil of the short 20th century (E. Hobsbawn) – of outstanding human, scientific and philo- sophical sophisitication.
Vienna, March 2013 Wolfgang L. Reiter Jakob Yngvason Contents
Foreword...... v
Erwin Schrödinger – personal reminiscences by Walter Thirring...... 1
Schrödinger and the genesis of wave mechanics by Jürgen Renn ...... 9
Do we understand quantum mechanics – finally? by Jürg Fröhlich and Baptiste Schubnel ...... 37
Schrödinger’s cat and her laboratory cousins by Anthony J. Leggett ...... 87
Digital and open system quantum simulation with trapped ions by Markus Müller and Peter Zoller ...... 109
Optomechnical Schrödinger cats – a case for space by Rainer Kaltenbaek and Markus Aspelmeyer ...... 123
A quantum discontinuity: the Bohr–Schrödinger dialogue by Helge Kragh ...... 135
The debate between Hendrik A. Lorentz and Schrödinger on wave mechanics by Anne J. Kox ...... 153
A few reasons why Louis de Broglie discovered Broglie’s waves and yet did not discover Schrödinger’s equation by Olivier Darrigol ...... 165
Chronology ...... 175
List of contributors...... 179
Name index ...... 181
Subject index ...... 183 Erwin Schrödinger, ca. 1902 (Courtesy of the Österreichische Zentralbibliothek für Physik) Erwin Schrödinger – personal reminiscences
Walter Thirring
Ladies and gentlemen, compared to the previous presentations at this symposium this will be a totally different lecture. There will be no power point presentation, no trans- parencies, not even formulas. But then you may wonder, why do I need this book. Well, these are poems. And why do I need this sheet? That’s an unpublished poem of Schrödinger’s written in a letter. I will try to paint a picture of this man from my memories, which date from the year 1937 when I first met Schrödinger. Ever since then I met him on and off, so there are many memories that I have, personal and scientific, and anecdotal ones. Anecdotal ones will be more interesting for many because it’s clear that about such a man with so many faces there are many anecdotes, and what is true is another question, so I will mainly rely on my own memories. But to bring some order into this chaos I will try to follow a temporal order. So let me start with the year 1937. At that time Schrödinger was professor in Graz and he came to visit my father, Hans Thirring, inVienna in early summer and what I vividly remember is when I accompanied them on what they liked to do at the time namely to go swimming in the Danube. They went swimming to Kuchelau, this was their favourite place, and Schrödinger, he was fifty years at that time, was still sportive enough to go there, although there were no facilities. They just changed behind the bushes and then went to the Danube to swim. And well the Danube is a cold river, it has about 18 centigrades or so and it has a rather strong current, up to 10 meters per second, so to go swimming there is a kind of sportive enterprise, but Schrödinger did it. In fact we even did more, namely at that time there were these towing boats, which towed the cargo ships upstream and the front wave was rather strong. So we tried to jump into the water in such a way that we could reach the front wave while it was still very high, perhaps two meters or so, and dive through it. This was a nice sport. But this Schrödinger didn’t want to do. He was sportive to some extent but he didn’t want to overdo it. This kind of relaxed atmosphere was treacherous because both my father and Schrödinger knew that the political situation was very dangerous. In fact the next year came the annexation of Austria, the “Anschluss” and Schrödinger made one of his great mistakes, he made a declaration of solidarity with the Nazi regime. It didn’t do him any good because the Nazis in any case were determined to confiscate his passport. Perhaps one should not be too harsh and condemn him for writing this article because one has to keep in mind that he had been pushed around by the political forces, he left Berlin in 1933 in disgust for the Nazi regime although he had “the” job for a theoretical physicist, namely as the successor of Max Planck. So, finally he came to Graz in 1936 and he didn’t want to be pushed around once more. But nevertheless it happened. Dismissed from his chair in Graz he escaped to 2 W. Thirring
Italy and proceeded to Oxford and then as a guest professor to the University of Gent. In mid 1939 he arrived in Dublin. Then of course during the war there was a long gap when I couldn’t see Schrödinger. When I had my doctor’s degree, I had prepared a paper essentially on the question: how do we know that E D mc2 is correct. He was sort of pleasantly impressed by my inquisitive mind and said he could offer me a scholarship in Dublin, which for me was the chance of my lifetime to go to Schrödinger after the doctor’s degree. Of course, now it sounds trivial to go from Vienna to Dublin, you just take a jet plane and in three and a half hours you are there. At that time travelling was something totally different. First of all there were no planes, you could only take a train, trains were very slow and at every border you had to show your passport and the visa, and the visa question was even more difficult because to go to Ireland I needed a visa to go through England, a transit visa to England but to get a transit visa I first had to get a visa for Ireland. So it was very complicated, and in addition to that the Austrian citizens were legally forbidden to posses any foreign money. There was money forthcoming only once I was in Ireland. So there were some difficulties and after a journey of several days I arrived in Dublin. And then how do you find Schrödinger in Dublin? It wasn’t so easy because he didn’t live in Dublin, he lived in a small suburb called Clontarf, it took us about half an hour by bicycle, it’s a little bit outside. But still I succeeded in finding him, and he was really very charming and friendly and invited me to live with them until I found accommodation. He really did everything to make my stay as pleasant as possible. In fact they lived quite modestly, they didn’t have a permanent maid, and so Schrödinger washed the dishes after meals, and of course I felt I should help but he said no, no don’t worry, I’ll do that. Well, his life wasn’t that simple in Dublin because unfortunately his wife suf- fered from depressions and from time to time she had to go to hospital where they treated her with the terrible method of electroshocks which is something very brutal and Schrödinger had to run the whole place by himself, so he just asked me to take the laundry to the cleaners. It is said that he always had many girlfriends. They were noto- riously absent when he really needed help. I tried to be as useful to him as best I could, even if it was against my convictions. He was a notorious chain smoker. Once he found out during the weekend that he had run out of tobacco. This to him was a catastrophe, so he asked me whether I could cycle into town and get some tobacco for him. As I said it was against my convictions being a militant anti-smoker but nevertheless I felt I had to do it, so I did it and the weekend was saved for Schrödinger. Schrödinger’s easy-going way was reflected in the spirit of the Dublin Institute for Advanced Studies where they had some scholars, a worldwide selection, from India, America, Australia. Schrödinger arrived around 11 o’clock in the morning and then we had tea. And tea, of course, served only as a possibility to meet people and to discuss all sorts of things, among them also scientific issues. But Schrödinger somehow was hesitant to talk about what he himself was doing at the moment. He would just discuss what other people were doing and you could observe how he thought. But I think it was one of his defects that he did not say that he was dealing with some hot subjects. I will soon come to that. Erwin Schrödinger – personal reminiscences 3
The way he was thinking made it clear that he was a physicist, he was thinking in terms of physics and not mathematics. For instance when one writes the Schrödinger equation in the usual way in polar coordinates then there is this term `.` C 1/=r2,he said, well that’ s obviously the centrifugal force. He did not say how this term was arrived by calculating with connection forms. I mean he was thinking in terms of what is the physical meaning of this term. Nevertheless there were some instances where the physical intuition didn’t guide him properly. Let me take an example, namely quantum field theory. He somehow had some aversion against quantum field theory and he explained why. The reason, he said, is because the number of states is not enumerable. And the argument he gave was very simple. If you just think in terms of occupation of states, an infinite number of states, even if you have fermions where you have fewer states, there is the possibility of the state being occupied or unoccupied, and so to specify the state you have an infinite sequence of, say, zeros or ones. And this of course is the same as all numbers in the binary system and therefore the power of states is equal the power of the continuum. And he said this doesn’t make any sense because to verify, to measure such a state, you need a not denumerable number of measurements. And a not denumerable number of measurements doesn’t make any sense. Now he did not realize that this question had been clarified by von Neumann and Murray already earlier. When I was in Dublin this was 1949 to 1950. And this had been clarified by Murray and von Neumann, namely that with this infinite tensor product the states are not denumerable but nevertheless the representation of the observables is always in the denumerable subspaces. So this difficulty of Schrödinger does not arise. But this level of mathematics he did not reach. And another way of observing his mathematical skill was when he gave some lectures, which he then collected into little books and they were published by Cambridge University Press. I was fortunate to be present at two of his lectures. One was “Space-Time Structure” and the other was “Expanding Universes”.1 Well, “Space-Time Structure” was on general relativity, and he gave a very nice exposition of the subject, however, using the conventional formalism with the connections given by the Christoffel symbols and so on. Strangely enough, the more effective formulation of Elie Cartan was not mentioned by him. I don’t know if it was not known to him. Of course in German speaking countries it was not so well known, because Cartan published in French and French was not so common among physicists. But he had actually prepared all the ingredients. In his lectures he made a point that for totally anti-symmetric covariant vectors you could define differential operation without referring to the metric, which is one of the keystones of the Cartan formulation. But nevertheless he didn’t mention Cartan at all and he didn’t use that. Strangely enough Cartan was for a very long time unknown to physicists. Apparently there was a discussion, an exchange of letters, between Einstein and Cartan, which was friendly but not one of them tried to understand the other. The reason might have been that Cartan had visions, he never worked out the details. The details were worked out later on by people from the Bourbaki School, like Jean Dieudonné, but that was not
1Erwin Schrödinger, Space-Time Structure, Cambridge University Press 1950; Expanding Universes, Cambridge University Press 1956. 4 W. Thirring available at the time, and so perhaps because this was not done so explicitly it did not become so popular. So this was one of the little books which he then published, “Space-Time Structure”. I think it is still a good introduction even today. “Expanding Universes” is essentially all about the de Sitter universe. It gives you the different aspects of the de Sitter universe in different coordinate systems which is very amazing for the layman or the beginner to see that this expanding universe can also be in other coordinate systems a static universe which doesn’t expand because the coordinates run along with the expansion of the universe. However, he did not mention that he was finding something, which later could be identified as the Hawking radiation. Schrödinger had written a paper on the Dirac equation in the de Sitter universe and he found that the positive and negative frequencies in the course of time get mixed. And he even realized that this means that there is a particle acceleration. But this is where the story ends. It was of course a kind of Hawking radiation in the de Sitter universe which he had discovered but he did not pursue this any further. It is strange that he had in fact many missed opportunities partly because he did not have enough appropriate collaborators to follow up his ideas, to recognize their importance . In Dublin there was John Singh, who was a very good mathematician but interested in mathematics only in the classical sense. He had one disadvantage, he did not speak Bourbaki so if you expressed something in modern terminology not only did he not understand, but he did not want to have anything to do with that. Nevertheless he heard lectures of Schrödinger on “Space-Time Structure”. He made a remark that it would be nice if all these things could be transformed at least into existence and uniqueness theorems of the solutions of the Einstein equation. They were notoriously absent and also Schrödinger had nothing to offer on that. This was not Schrödinger’s fault it was the Zeitgeist of that time, people didn’t worry about existences or uniquenesses solutions. And Singh said, this will take about fifty years to be clarified and I think that was a pretty good guess, it took about fifty years until people were seriously thinking about this mess. The most famous of his little books on public lectures which took place a little earlier, in 1943, was “What is Life?”.2 Somehow he never mentioned it afterwards, and also strangely enough, although it certainly was one of the most influential books which got all this molecular biology going, there was some harsh criticism. It was unduly harsh, I don’t know why, but it seems that he had some enemies. They complained that Schrödinger talked all the time about entropy and wanted to illustrate things with entropy but entropy is irrelevant for living beings. It’s free energy, which counts because they are always in contact with the environment, so one should concentrate on free energy. Well, there is something to be said for that, after all our diet is more after energy than entropy, we eat sausages and not diamonds. I think from the point of view of a representation, however, it is quite legitimate to talk about entropy. Also it was criticised that there was nothing new in what Schrödinger said. It is true, in none of his
2Erwin Schrödinger, What is Life? The Physical Aspect of the Living Cell. Cambridge University Press 1944. Erwin Schrödinger – personal reminiscences 5 public lectures there was anything new. It was just that he represented things in such a way that he got people interested in the subject. I think there is great merit in that. In fact Crick and Watson admitted that they had been initiated by Schrödinger to move into this field. Another occasion which he did not talk about afterwards, was that he was on the verge of discovering supersymmetry because he was interested in factorising Hamil- tonians, to write a Hamiltonian as an object and its conjugate, which is the standard way for supersymmetry. He made some interesting contributions to this but he did not realize that this had anything to do with Einstein’s programme of unification of theories. Another thing he anticipated, and didn’t talk about it to us, didn’t talk to anyone about it, so it was unknown, was the Higgs mechanism for creating the mass of a gauge particle. He wrote this in a small letter to Nature and he observed that if you take a purely classically scalar field and you couple it to an electromagnetic field what you get, – and everybody knows this today – is that the electromagnetic field assumes a mass. He didn’t have anything, which depends on the data of the scalar fields, he didn’t have anything to fix the masses, so he didn’t have this Higgs mechanism, but nevertheless he had a good example for a system where a gauge particle gets a mass. This is something, which he always jokingly talked about, the holy gauge invariance, which he doesn’t believe in, but he never pressed this example. So, these were my impressions from the year in Dublin. Then I met Schrödinger in 1955 when he came to the Pisa Conference [on Elementary Particles] where he discussed the mass of the photon, which was one of his main occupations in his old days. As I said before his argument was rather simple, namely that if there was a mass and there were longitudinal modes then the nature of the black body radiation will change by a factor of 2/3 or so. And he finally saw that if the mass goes to zero the longitudinal modes get less and less interaction and they slowly die out. It’s an amusing fact but nobody was interested in it. It is certainly a fundamental question, but whether neutrinos have a mass or not is also a fundamental question. Now we are already approaching the time when Schrödinger returned to Vienna. This was a long process right after the war. Schrödinger was approached already in 1946 to come back and accept a professorship in Vienna. But he was very careful and didn’t want, under these political conditions, to come back. I think this was quite justified, as he had been pushed around in his life so often. So he didn’t want to go to a city which was surrounded by the Russian occupation troops. Only afterAustria existed again as a free state in 1955, he eventually wished to come back. Now apparently he had some great expectations. Maybe I will illustrate this with a little poem, which he wrote. I didn’t mention this before, but physics was only one of his occupations. Another occupation was poetry. In fact, even when he did not write poetry but scientific papers his style was always superb. He told me how to do this. When I was writing a paper I showed it to him. He said, no, that’s not good, you have to do it quite differently, you have to go to your room, lock the door and read it loudly, one sentence after the other. After each sentence you think a little bit about whether this is really the best way you can do it or whether by reshuffling the sentence you can make it a little bit clearer what you want to say. So apparently this is how he got to his style. 6 W. Thirring
The poem I want to read to you now is called “Mid September”.3
Mitte september
Wenn erst die gleiche vorbei wir müden schwäne kommt wieder sonnenglanz; zum nichtsein hinüber. leuchtender als im mai wirbelt der blättertanz So sagen andre. – durch die alleen. Glaub es nicht, wandre Bräutlicher neigen mutig und frei die überreichen durch all dies sterben. zweige sich zum spiegel der seen. Nur den vermessenen selbstvergessenen ist es verderben. Aber nein, nein, voüber Uns schafft es neu. ist neues grünen, vorüber die kühnen Schöner als einst im mai furchtlosen pläne. wirbelt im herbstesglanz Nur mit geliehnen leuchtender blättertanz – fittichen sühnen nichts ist vorbei.
Apparently these were his hopes. Unfortunately when he arrived in Austria it was not September but it was November. His health was exhausted. He really didn’t have the strength anymore to enjoy his life here in Vienna. Now, as I said there are many facets in Schrödinger, in fact, originally he wanted to become a philosopher. He had an offer as a professor in Czernowitz, and he said this was what he wanted to do, because in this provincial university he could fulfil his duties by giving his lectures in physics and this would leave ample time to pursue his philosophical ideas. About religion he was somewhat ambiguous but I never discussed this with him in detail. Anyway, he sometimes declared happily to be an atheist, on the other hand he spent his whole life trying to develop a notion of God, so it is difficult to position him. Once we had a controversy about religious instruction. This was when Mao came to power in China. I said this is bad because now even religious instruction is forbidden. He said, is this really so bad? I said, yes because once religious freedom is suppressed freedom of thought will erode afterwards. This is what actually happened in China, now there is religious freedom but freedom of thought is very much suppressed. He had some religious questions, which follows from a poem he wrote about his Indian philosophy. I will read it to you and you can judge for yourselves. It is called Upanishad.4
3Erwin Schrödinger, La mia visione del mondo. Garzanti Editore s.p.a., 1987, p. 182. 4Erwin Schrödinger, Commilitonen! Upanishad. Quo via fert? Wohin führt der Weg, 10, Wien: Matu- ranten des Döblinger Gymnasiums, Wien XIX. 1959 [Short letter and unpublished poem] Österreichische Zentralbibliothek für Physik. Erwin Schrödinger – personal reminiscences 7
Upanishad
Die ewige Frage: woher kommt, wohin geht dieses Ganze, drin ein Punkt ich bin und doch es ungeteilt und ganz und gar: nichts außer mir, das erst nicht in mir war? Soll nach mir ohne mich das All bestehn? Wird es durch mich, werd ich mit ihm vergehn? War eine Zeit, als es mich noch nicht gab? Wird’s weiter welten über jenem Grab – das jedem sicher ist, wie nun das Licht, so sich zu abertausend Farben bricht, und eine Welt von Wundern rings erbaut, im Baume grünt, im hohen Himmel blaut, vor edler Frauen reiner Lichtgestalt ins Knie dich zwingt mit siegender Gewalt. Ist’s wirklich Licht, was allen Glanz erschafft und nicht des eignen Auges Seherkraft? Kann Schönheit sterben? Hat sie Stund und Ort? Einmal geboren, rollt sie ewig fort. Hier, heut geschaut, wird es sie immer geben. Sie wird in einem – deinem Auge beben. Wer solches weiß, dem weitet sich das Leben.
I think this is my attempt to paint a picture of Erwin Schrödinger. Erwin Schrödinger 1904 (Courtesy of the Österreichische Zentralbibliothek für Physik) Schrödinger and the genesis of wave mechanics
Jürgen Renn
The story of Erwin Schrödinger and of his contributions to quantum physics has been told many times. They indeed represent one of the most fascinating subjects in the history of science. But is there anything new to say about this story? At the Max Planck Institute for the History of Science and at the Fritz Haber Institute, both in Berlin, we have begun, in the context of a joint research project on the history and foundations of quantum physics, to carefully study once again the notebooks in which Schrödinger left traces of his pathbreaking research. And parallel to the pursuit of a comprehensive review of the emergence of quantum physics, we are also investigating more broadly the contexts in which this development took place. Against this background, new questions arise and old questions appear in a new light. For instance, what exactly was the relation between Schrödinger’s breakthrough and the contemporary efforts by Werner Heisenberg and his colleagues to establish a new quantum mechanics? How can one explain, from a broader historical and epistemological perspective, the astonishing simultaneity and complementarity of these discoveries? Based on the work so far accomplished in our research project, it is to these questions that the present paper attempts to give new answers.1 But let us proceed in due order, beginning with a very short reminder of who Erwin Schrödinger was. Apart from being one of the most important scientists, he was probably also one of the most sensitive and educated persons of his time. He was born in 1887 in Vienna, studied there, and then was appointed to a professorship in Zurich in 1921. There he wrote his famous works on wave mechanics in 1926. In 1927 he succeeded Max Planck as the Chair for Physics in Berlin, but resigned his professorship in 1933 upon the National Socialists’ accession to power. In the same year he learned that he had been awarded the Nobel Prize for Physics. He spent his years of exile in Oxford and in Dublin. In the Irish capital he wrote the famous book What is Life?, which was to play a decisive role in the development of molecular biology. In 1955 Schrödinger returned to Vienna, where he died in 1961. He was buried in Alpbach, amidst the Tyrolean Alps, where the equation bearing his name still adorns his tombstone today (see Figure 2).2 The Schrödinger equation, postulated in 1926, is a key equation of quantum physics. Arnold Sommerfeld once referred to it as “the most remarkable of all remarkable dis- coveries of the 20th century.” When Max Planck held Schrödinger’s second publication in his hands, he related to Schrödinger on a post card:
1See, in particular, (Duncan and Janssen, 2007a,b; Joas and Lehner, 2009). 2There are several book-length biographies of Erwin Schrödinger (Hoffmann, 1984; Moore, 1989, 1994) and further works on his life and science (Scott, 1967; Mehra and Rechenberg, 1987a,b). Schrödinger’s letters concerning wave mechanics and the interpretation of quantum mechanics have recently been edited by Karl von Meyenn (2011). 10 J. Renn
Figure 1. Erwin Schrödinger (1887–1961), ca. 1926. By permission of R. Braunizer.
I am reading your communication in the way a curious child eagerly listens to the solution of a riddle with which he has struggled for a long time, and I rejoice over the beauties that my eye discovers, which I must study in much greater detail, however, in order to grasp them entirely.3 With reference to the origin of the Schrödinger equation, the American Nobel laureate Richard Feynman noted: Where did we get that [Schrödinger equation] from? Nowhere. It is not possible to derive it from anything you know. It came out of the mind of Schrödinger, invented in his struggle to find an understanding of the exper- imental observation of the real world (Feynman et al., 1965, Chapter 16, p. 12). For a historian, this answer is unsatisfying. What exactly was the knowledge on which the Schrödinger equation was based? If it were just some contemporary experimen- tal findings that it describes, how come then that this equation serves to this day in
3Planck to Schrödinger, 2 April 1926. Translation from (Meyenn, 2011, p. 206): “Ich lese Ihre Ab- handlung, wie ein neugieriges Kind die Auflösung eines Rätsels, mit dem es sich lange geplagt hat, voller Spannung anhört, und freue mich an den Schönheiten, die sich dem Auge enthüllen, die ich aber noch viel genauer im einzelnen studieren muß, um sie voll erfassen zu können.” Schrödinger and the genesis of wave mechanics 11 accounting for ever new phenomena that could not have been known to Schrödinger? And what was going on in Schrödinger’s mind, on which intellectual resources did he draw to formulate his consequential equation? In the following, we shall review, in a rather non-technical way, the genesis of the Schrödinger equation with the aim to contribute to a better understanding of one of the great upheavals in our scientific world view, as well as of the role that fell to Erwin Schrödinger in these events. The Schrödinger equation supplies the foundation for understanding atomic physics and the chemical bond. It significantly changed our view of the constitution of matter and describes basic properties of the world as we understand them today. At the same time, even today it still challenges our thinking, in terms of not only physics and mathematics, but also natural philosophy. And yet the Schrödinger equation is a simple equation. Admittedly, it is not quite as simple as Einstein’s famous formula E D mc2. But the physics behind it is, in principle, even simpler.
Figure 2. Tombstone of Annemarie and Erwin Schrödinger (1887–1961) with the Schrödinger equation i„ P D H in modern notation. Photo by C. Joas, 2008.
The Schrödinger equation is a wave equation that describes material processes as wave processes. Wave phenomena like sound waves or light waves had been known for a long time. What was surprising was that even matter itself would allow itself to be described as such a wave phenomenon, that states of matter could overlap each other like vibrational states, and that there could be something like diffraction, interference or standing waves in matter as well, in short, that matter behaves like light in many respects. But this thought was not entirely new, either. The development of quantum theory had begun a quarter of a century before with the discovery that electromagnetic radi- 12 J. Renn ation, like light, behaves under certain conditions as if it consisted of particles. Thus Einstein, with his light quantum hypothesis of 1905, had attempted to explain Planck’s law of radiation and the photoelectric effect (Einstein, 1905). Since then the indications in favor of such a wave-particle dualism had steadily increased. Of course, all of the experiments that had led to the acceptance of the wave theory of light at the beginning of the 19th century, namely the phenomena of diffraction and interference mentioned above, were still valid, and still spoke in favor of wave theory.4 But more recent experiments like the photoelectric effect, that is, the release of electrons from metals through irradiation with high-frequency light; or the so-called Compton effect, that is, the scattering of X-rays by electrons in solids, could best be explained by the assumption of indivisible energy packets, known as the quanta of radiation. These behave like small billiard balls that possess a certain energy and a certain momentum. Neither could the surprising fact that the effect of radiation does not depend on its intensity, but on its color, such that electrons can be released only by high-frequency light, while low-frequency light has no effect no matter how strong it is, be explained by classical wave theory. This relation between energy and the color or frequency of radiation is given instead by a formula dating back to Planck, according to which the radiation energy E is proportional to its frequency , where the proportionality constant is Planck’s quantum of action h, thus
E D h : (0.1)
Therefore, the higher the frequency of light, the greater the energy of the corresponding light quantum. If light exhibits such a duality of wave and particle properties, why should this not also be the case for matter? Einstein’s formula E D mc2, where m is the mass and c2 the square of the speed of light, states that mass and energy are merely dif- ferent manifestations of the same quantity: mass corresponds to energy and energy corresponds to mass. So if, according to Planck’s equation E D h , each energy is also associated with a certain frequency, then one need only combine both equations to happen upon the idea that every mass is also associated with a frequency, or more correctly, a wave phenomenon. The French physicist Louis de Broglie had arrived at this admittedly speculative thought in his dissertation of 1924 (de Broglie, 1924).5 In order to supplement the wave picture of matter, he used, in addition to Planck’s relation between energy and frequency, another relation that was just as simple: one between wavelength and momentum p: h D : p (0.2) Not only did this yield the possibility of translating completely the mechanical quantities energy and momentum into the wave quantities frequency and wavelength, it also offered a provisional solution to a fundamental puzzle of atomic physics.
4For a further discussion, see (Wheaton, 1983; Büttner et al., 2003). 5See (Kubli, 1970) and (Darrigol, 1986). Schrödinger and the genesis of wave mechanics 13
Figure 3. Louis de Broglie (1892–1987).
The spectra of the chemical elements had been known since the 19th century. Spectra are like fingerprints that allow elements to be recognized on the basis of which colors of light they emit and absorb. The frequencies of this characteristic light had to be linked with the internal structure of the atoms, but for a long time it was not understood how. Even when experiments allowed ever more conclusions about this inner structure to be drawn, and when it became clear that the small charged particles, the electrons, played an important role in this process, it remained a mystery how their movement could generate or absorb the radiation visible in the spectra.6 One could certainly imagine that the atom is structured like a small solar system, with the negatively charged electrons revolving on different orbits around a positively charged nucleus. But there was no way the orbital frequencies on these electron paths could have anything to do with the emitted or absorbed radiation. Against this backdrop, the Danish physicist Niels Bohr made a radical proposal in 1913 (Bohr, 1913).7 He simply assumed that the frequency of the radiation emitted by an atom had nothing to do with the mechanical motion of its electrons, but depended instead on the energy difference between the various orbits. An atom should emit or absorb light only if an electron jumps from one orbit to another. The frequency of the light emitted or absorbed should then be defined by Planck’s relation between energy and frequency, that is, by E D h . This may have been no more than wild speculation,
6See (Jungnickel and McCormmach, 1986). 7See also (Heilbron and Kuhn, 1969). 14 J. Renn but it was sufficient to explain key properties of the spectra. What Bohr could not explain, however, were the basic properties of his model. Why, for instance, should the electrons be able to move only on certain orbits with precisely determined energies, why are these orbits stable, and why should atoms emit or absorb radiation only when their electrons jump back and forth between these orbits? These assumptions were justified provisionally only by the success of their application, but otherwise remained a mystery. This is where the above-mentioned dissertation by de Broglie picked up around ten years later. De Broglie conceived matter to be linked with a wave phenomenon, arguing that the electron orbits in Bohr’s model were stable because they corresponded to periodic waves. Just as for a vibrating string, a resonance effect was involved for electron motion, too, such that the wave returns back to itself along the orbital path. Only orbits on which this happens are stable. With this interpretation de Broglie succeeded in explaining the discreteness of a quantum phenomenon, that is, the occurrence of isolated orbits with certain energy levels rather than a continuum of possible orbits and energies, by resorting back to a well-known phenomenon of classical wave physics, and without having to introduce any additional assumptions along the lines of quantum jumps. And thus the idea of wave mechanics was born. Einstein soon picked up on this idea, addressing it briefly in an entirely different context, the theory of gases (Einstein, 1924). This is where Schrödinger first encoun- tered the idea of a wave theory of matter in 1925. The first work in which he himself dealt with the subject, even before formulating his famous equation, is also dedicated to gas theory (Schrödinger, 1926a). As de Broglie before him, he also used the wave conception so that he could render unnecessary the presumption of a mysterious quan- tum property, in this case that of a mysterious statistics of quantum gases,8 and instead make the properties of such gases comprehensible on the basis of classical statistics.9 From here, after all, it was no longer far to the Schrödinger equation – or so it may seem in retrospect. As discussed, the conception of matter as a wave had already yielded considerable successes; in particular, it made it possible to explain the strange discreteness of quantum states, that is, the occurrence of only certain values of energy, as resonance phenomena of vibrations. What was still missing was – obviously – a wave equation. But wave equations were certainly well known in classical physics, too, especially in wave optics. Take, for instance, the simplest wave equation, which describes the spatial distribution of the amplitude of a wave, .x/, at a fixed point in time @2 4 2 .x/ C .x/ D 0: @x2 2 (0.3) Now replace the wavelength appearing in this equation with momentum p D h= , in accordance with the simple rule stated by de Broglie (Eq. (0.2)): @2 4 2p2 .x/ C .x/ D 0: @x2 h2 (0.4)
8That is, the so-called Bose–Einstein statistics. 9That is, Maxwell–Boltzmann statistics. Schrödinger and the genesis of wave mechanics 15
Figure 4. Erwin Schrödinger lecturing. By permission of R. Braunizer.
2 Finally, express the momentump as a function of kinetic energy using Ekin D p =2m D E V , such that p D 2m.E V/(E D Ekin C V is the total energy, where Ekin is the kinetic and V the potential energy): @2 8m 2.E V/ .x/ C .x/ D 0: @x2 h2 (0.5) And voilà – there it is, the famous Schrödinger equation!10 The quantity .x/ is called the wavefunction in wave mechanics. Thus, no new mathematics at all is necessary to formulate the Schrödinger equation, and no new physical assumptions aside from the known rule by de Broglie, which is in principle only an obvious extension to Planck’s relation between energy and frequency, that is, of E D h . “The idea of your work testifies to genuine ingenuity!” Einstein wrote in the margin of a letter to Schrödinger.11
10This is the time-independent (or stationary) Schrödinger equation. To our knowledge, this derivation was first presented by Born in July 1926 (Born, 1926b, p. 811-812). See also (Wünschmann, 2007; Ludwig, 1969). Note that Schrödinger’s own derivation proceeded along a different path (Joas and Lehner, 2009). See also (Gerber, 1969; Kragh, 1982; Wessels, 1983) and the discussion below. 11Einstein to Schrödinger, 16 April 1926. Translated after (Meyenn, 2011, p. 214): “Der Gedanke Ihrer Arbeit zeugt von ächter Genialität.” 16 J. Renn
One might respond: but that cannot have been the whole story, what is then so special about this equation that it has become the foundation of atomic physics and has earned Schrödinger a Nobel prize; if one is bold, one may even ask what is supposed to have been so special about its formulation? Anyone could have come up with that! What is more, the only physics problem that Schrödinger solved in his first work was the calculation of the spectrum of the hydrogen atom, that is, one of the problems for which quantum theory already offered various solutions. If one is skeptical about this achievement by Schrödinger, to the extent that I have described it so far, one finds oneself in the best of company. Even some of Schrödinger’s contemporaries questioned whether the new wave mechanics would last any longer than a salon perm. Werner Heisenberg, for instance, reacted dismissively. Just half a year before, he made a proposal of his own for the establishment of quantum mechanics, for which he, too, was awarded the Nobel Prize. In a letter to the physicist Wolfgang Pauli he passed the following judgement on Schrödinger: Many thanks for your wonderful book, […] reading it was real recreation after Schrödinger’s lectures here in Munich. As nice as Schrödinger is personally, I find his physics at least as strange: when you hear it you feel 26 years younger.12 Yet for Heisenberg, who was 25 at the time, feeling “26 years younger” in the face of Schrödinger’s theory meant that he must associate it with the dawn of quantum theory, at the end of the age of the classical physics of the 19th century. In the same letter he did, in fact, call Schrödinger’s physics “classicist” and accuse him of having made it too simple by “throwing overboard” all substantial quantum effects. Heisenberg, on the other hand, had come to his own theory, which later became known as matrix mechanics, by occupying himself laboriously and step by step with these quantum problems. A crucial issue, which ultimately showed Heisenberg the way, was the problem of optical dispersion, namely the fact that diffraction in a certain material was dependent on the color of light.13 For this an appropriate formula had been found back in the 19th century, using a model of the atom based on classical physics to explain dispersion as an effect of the incoming radiation on the electrons, a covibration that disturbs their natural orbits. It had just recently turned out, however, that the emission and absorption of radiation could not be described by a classical model, but only by a quantum theoretical model of the atom like the one by Bohr. Yet in such a model there could no longer be any covibrations of electrons nor any perturbation of their orbital paths, for in this case the frequency of the emitted and absorbed radiation is not at all dependent on the orbital motion of the electrons, but rather on their quantum jumps
12Heisenberg to Pauli, 28 July 1926. Translated from (Pauli, 1979, p. 336–337): “Haben Sie vielen Dank für Ihr schönes Buch, in dem ich zwar kritisch und unnachsichtig, aber doch mit viel Freude gelesen habe. Es ist eben eine exakte Darstellung der physikalischen Zusammenhänge, die vor dem Durcheinander des letzten Jahres bekannt waren, und seine Lektüre war mir eine wahre Erholung nach Schrödingers Vorträgen hier in München. So nett Schr[ödinger] persönlich ist, so merkwürdig find’ ich seine Physik: man kommt sich, wenn man sie hört, um 26 Jahre jünger vor.” 13See (Duncan and Janssen, 2007a,b). Schrödinger and the genesis of wave mechanics 17 between the various paths. The supposedly long solved problem of dispersion thus had become a genuine quantum puzzle, which somehow seemed to make it necessary to link the established classical views on the nature of diffraction of light with its undeniable quantum properties. Such transitional problems between classical and quantum physics existed at various junctures in the field of physics at the time, and they proved to be especially productive, for such problems could show the way to expand classical physics by suggesting special additional assumptions in order to solve the quantum riddles.
Figure 5. Werner Heisenberg (1901–1976).
Indeed, before 1925 the connection between classical physics and the new quantum physics was conceived rather generally as merely extending the former by adding sup- plementary quantum conditions.14 According to this view, not all of the theoretically possible solutions to a mechanical problem were possible any longer – only those that obeyed these additional quantum conditions, which typically boiled down to allowing certain physical quantities to equal only multiples of Planck’s quantum of action h. Therein lay the essence of the so-called “old quantum theory,” which was actually no theory at all, but an accumulation of calculation and translation rules to extend classical physics. Despite many obscurities and contradictions, this older quantum theory dom- inated the thinking of physicists between 1913 and 1925, simply because there was no alternative and, as it appeared to some, perhaps there never would be one.
14See, e.g., (Darrigol, 1992b). 18 J. Renn
This school of thought was so fruitful, for one, because it was constantly occupied with new problems, some of which had been raised by new experiments; and second, due to the fact that transitional problems kept arising, which made it possible to mod- ify results from classical physics in a targeted way so that they could yield solutions for quantum problems. For the emission of radiation with very low frequencies, for instance, Bohr’s atomic model and its classical counterpart, in which the frequency of radiation really does depend on the orbital frequency of the electrons, yielded prac- tically equivalent results. In 1916, Einstein was able to show that Planck’s radiation law could be derived if it was assumed that the intensity of the radiation emitted by many Bohr atoms corresponds exactly to the intensity of classical radiation (Einstein, 1916a,b). Niels Bohr subsumed such correspondences under the somewhat vague term “correspondence principle” and systematically applied this heuristic principle in his search for solutions to quantum problems.15 In a sense, the correspondence principle had the basic function of restoring the classical connection between the radiation prop- erties and periodicity properties of the atom, for which there was initially no natural place in Bohr’s model. The transitional problems that thus accumulated, some of which were true quan- tum problems, but others of which could be solved in part or in limiting cases using established concepts of classical physics, expanded the catalogue of calculation and translation rules ever further. But they initially did not lead to a comprehensive and consistent scheme for translating from classical to a new quantum physics. Many prob- lems, even quite central ones like the spectrum of the helium atom, proved inaccessible to any solution, no matter how elaborate the conversion. Thus there was talk of a cri- sis of the old quantum theory,16 which came to a head shortly before the time when Heisenberg and Schrödinger published their works. Nevertheless this was not a crisis from the creative confusion of which a new paradigm ultimately rose like a phoenix from the ashes. The birth of the new quantum mechanics, in short, was no scientific revolution according to Thomas Kuhn’s defi- nition. Instead, what ultimately led to the resolution of the crisis was the consistent pursuit of the established strategy of the old quantum theory, and an especially sophis- ticated modification of classical physics, developed in order to solve a quite concrete physical problem. The pursuit of the tried-and-true strategy of old quantum theory consisted above all in the search for possibilities to “sharpen” Bohr’s correspondence principle; this meant going beyond the use of this principle in establishing the classical limit of quantum-theoretical treatments, and rather to elaborate this principle into a translation rule that allowed the solution of a quantum problem to be found, starting from its classical formulation. Over and again this approach was surprisingly success- ful for individual quantum problems, giving hope from the outset that a comprehensive translation scheme ultimately could be achieved. However, the attempt to proceed from individual successful translations to divine such a generalized scheme experienced de- feats as well. For instance, the physicist Max Born of Göttingen proposed in 1924
15See (Darrigol, 1992b). 16This notion goes back to Kuhn, see (Kuhn, 1982, 1987). See (Darrigol, 1992b). See also (Büttner et al., 2003). Schrödinger and the genesis of wave mechanics 19
Figure 6. Max Born (1882–1970). that a generalized quantum mechanics could be achieved by replacing the differential equations of classical physics with difference equations (Born, 1924), but this attempt failed. In fact it was the concrete problem of dispersion that ultimately put Heisenberg on the right path. Back in 1924, Born had written clairvoyantly:
As long as one does not know the laws of how light affects atoms, and thus the connection between dispersion, atomic structure and quantum jumps, one will be all the more in the dark about the laws of interaction between multiple electrons in an atom.17
Indeed, we already discussed that dispersion represented a borderline problem that was best addressed by linking the considerations of classical physics with those based on quantum theory. At the same time it was important to do justice to the fact that while the radiation behavior of an atom had nothing to do with the classical orbital frequency of its electrons, an explanation of dispersion nevertheless made some kind of covibration necessary, for otherwise it was practically impossible to understand why the classical explanation based on this assumption was so successful.
17Translated from (Born, 1924, p. 379): “Solange man die Gesetze der Einwirkung des Lichtes auf Atome, also den Zusammenhang der Dispersion mit dem Atombau und den Quantensprüngen, nicht kennt, wird man erst recht über die Gesetze der Wechselwirkung zwischen mehreren Elektronen eines Atoms im Dunkeln sein.” 20 J. Renn
Rudolf Ladenburg, a physicist from Breslau, thus introduced the concept of “vir- tual oscillators,” the frequencies of which corresponded to those of Bohr’s quantum jumps, but which also were able to interact with incident radiation by resonating, that is, co-vibrating.18 At the same time, the success of introducing these auxiliary virtual oscillators, which, as we will see below, could potentially lead to the desired “sharpen- ing” of the correspondence principle, also meant that the interior of the atom could not be conceived simply as a miniature planetary system; entirely new physical concepts were required instead. In the history of the theory of relativity, Lorentz’s ad hoc as- sumption of a contraction of moving objects caused by the ether played a similar role. It, too, turned out to be the useful auxiliary construction of an invisible mechanism, which ultimately helped Einstein to find an explanation for the problems of the electro- dynamics of moving objects on the deeper level of new concepts of space and time.19
Figure 7. Albert Einstein (1879–1955).
But the key factor at this point was the possibility of expanding the theoretical ap- proach to dispersion by adding ever more findings from quantum theory to this ad hoc assumption of virtual oscillators, up to the point where it finally amounted to a stricter application of the correspondence principle. While no convincing theoretical model was achieved that could have taken the place of the planetary concept of the atom, the
18See, e.g., (Duncan and Janssen, 2007a,b). 19See (Renn, 2006). Schrödinger and the genesis of wave mechanics 21 sharpening of the correspondence principle in the case of dispersion did yield a formula that corresponded to all experimental findings and thus could be understood as a trans- lation, albeit a complicated one, of classical properties into quantum properties. But precisely the complicated character of this successful translation allowed far-reaching conclusions to be drawn, in the same way the famous Rosetta Stone’s elaborate, multi- lingual texts in the 19th century allowed more to be learned about the then undeciphered Egyptian hieroglyphs than the mere translation of individual signs, even though it was the names of the rulers that had revealed the first patterns to Champollion. It was Werner Heisenberg who took the last, decisive step toward translation in July 1925, in his famous work, “Quantum-Theoretical Re-interpretation of Kinematic and Mechanical Relations” (Heisenberg, 1925). His point of departure was the dispersion formula. A method of formulating it in classical physics proceeded by way of what was called the Fourier series representation of the electron trajectory and its perturbations, that is, its portrayal as a superposition of harmonic oscillations. This method of repre- sentation was an established technique in classical physics, the origins of which extend all the way back to ancient astronomy, which likewise conceived of the complicated trajectories of the planets in the heavens as a superposition of uniform circular motions. For the classical dispersion formula certain very specific aspects of this Fourier series representation were decisive. The dispersion formula of quantum theory, in turn, as we saw above, could be formulated using the concept of virtual oscillators. Accordingly, from this parallel between the classical and quantum theories of optical dispersion, a relation could be established between the partial aspects of classical Fourier series decisive for the dispersion formula and the virtual oscillators of quantum theory. More than a month before submitting his famous Umdeutung paper, Heisenberg wrote in a letter to Ralph Kronig:
The basic idea is this: In the classical theory, it suffices to know the Fourier series of the motion in order to calculate everything, not just the dipole moment (and the emission), but also the quadrupole moment and higher moments etc. […]. It thus seems likely that also in quantum theory knowl- edge of the transition probabilities, or the corresponding amplitudes, yields everything. […] The quintessence of this re-interpretation [Umdeutung]to me appears to be the fact that the arguments of the quantum-theoretical amplitudes have to be chosen such that they reflect the connection between the frequencies. […] What I like in this scheme, is that it allows one to reduce all interactions of the atom with its surroundings to the transition probabilities (barring questions of degeneracy). For now I am, however, unsatisfied by the mathematical formalism.20
20W. Heisenberg to R. Kronig, 5 June 1925, translated from (Kronig, 1960, p. 23-24): “Der Grundgedanke ist: In der klassischen Theorie genügt die Kenntnis der Fourierreihe der Bewegung um alles auszurechnen, nicht etwa nur das Dipolmoment (und die Ausstrahlung), sondern auch das Quadrupolmoment, höhere Pole u.s.w. […] Es liegt nun nahe, anzunehmen, dass auch in der Quantentheorie durch die Kenntnis der Übergangswahrscheinlichkeiten, oder der korrespondierenden Amplituden alles gegeben ist.[…] Das Wesentliche an dieser Umdeutung scheint mir, dass die Argumente der quantentheoretischen Amplituden so gewählt werden müssen, wie es dem Zusammenhang der Frequenzen entspricht. […] Was mir an diesem 22 J. Renn
Heisenberg’s insight into the relationship between the classical Fourier series repre- sentation and the virtual oscillators of quantum theory in the case of optical dispersion corresponded to a “sharpened” correspondence principle in the sense of a complete translation of the classical solution of a problem into the solution of the corresponding quantum problem. But using Fourier series expansion to solve the problem of dispersion had consequences that extended much further, for the Fourier series was a quite general mathematical instrument used to represent physical quantities. So if the translation of the partial aspects decisive for the dispersion formula could be successfully generalized into a Fourier series representation of any physical quantity, then the sought-for general scheme for translating classical physics into a new quantum mechanics would be found at long last. It turned out that this generalized translation, which Heisenberg called “re-interpre- tation”, resulted nearly inevitably from the demand that the classical Fourier series be modified so that the different oscillation frequencies occurring are composed in a way that corresponds to Planck’s relation E D h between energy and frequency. From this it follows, as Bohr’s atomic model illustrates, that the radiation of a certain frequency emitted due to a quantum jump from a higher orbital path to a lower one can be traced back to the different energies of these two orbital paths. It further follows that such opportunities to jump between the different orbits and the energy levels corresponding to these orbits can be composed such that only very specific sets of emission frequen- cies are possible and thus observable in atomic spectra. According to Heisenberg, these emission frequencies and their possible combinations, in turn, are incorporated into the modified quantum-theoretical Fourier series representation. These frequencies, how- ever, could not be related to the orbital frequencies of electrons in the atom. As it turned out, the physical interpretation of the rules of Heisenberg’s re-interpreted mechanics no longer hinged on such mechanical conceptions. Ultimately their justification was based on no descriptive physical model at all, but on their origination from a “sharpened” cor- respondence principle and its success in explaining concrete physical problems like that of optical dispersion. This brings us to the crucial epistemological challenge of the genesis of quantum and wave mechanics. It seems to be a mystery how such different paths of thought as those of Heisenberg and Schrödinger could lead to perfectly-matching solutions of the quantum crisis. Even the mathematical formulation of the two solutions was completely different in the end, with partial differential equations in Schrödinger’s case and matrix calculus for Heisenberg, as would soon become clear in the context of further elaborating his re-interpretation in collaboration with Max Born and Pascual Jordan (Born and Jordan, 1925; Born et al., 1926). For the physicist, this mystery, or at least the provocation it evokes, is resolved for the most part at the moment when the equivalence of these two approaches, that is Schrödinger’s wave mechanics and Heisenberg’s quantum mechanics, can be proven. This actually happened quite quickly. Unpublished calculations by Wolfgang Pauli and
Schema gefällt, ist, dass man wirklich alle Wechselwirkungen zwischen Atom und Aussenwelt dann auf die Übergangswahrscheinlichkeiten reduzieren kann (von Entartungsfragen abgesehen). Nicht zufrieden bin ich zunächst mit der mathematischen Seite […].” Schrödinger and the genesis of wave mechanics 23 a further publication by Schrödinger (1926d) from the same year showed that the two theories could be mapped upon each other mathematically and in principle yielded the same physical results.21 From a historical point of view, however, this equivalence proof merely increases the challenge posed by the birth of quantum mechanics as a set of fraternal twins. Why was it possible that Schrödinger’s wave mechanics led to the same conclusions as Heisenberg’s matrix mechanics, although the problem of dispersion, which had played such a key heuristic role in the latter, did not lie along Schrödinger’s route to his equation? And why could matrix mechanics yield the same results for the spectrum of the hydrogen atom, although this problem, which had been at the center of de Broglies’ and Schrödinger’s wave mechanics from the outset, did not play a significant role in its formulation? In short, why did these two very different paths of thought ultimately arrive at the same destination? To these questions there are a number of obvious answers, including some from the standard repertoire of the history and philosophy of science. But only when we recognize that these standard answers are unsatisfactory, and why, does the parallel emergence of quantum mechanics along both paths allow deeper insights into the nature of this scientific breakthrough. It is a fact that Schrödinger knew Heisenberg’s theory before he formulated wave mechanics. Does this perhaps mean that wave mechanics emerged as a re-formulation of a theory that was already known? Everything we know from the historical documents speaks against this version. Heisenberg’s theory was indeed known to Schrödinger, but he found it so unappealing that the only thing about this alien theory that could have motivated his approach to wave mechanics was vehement rejection. Schrödinger himself writes – and this is the counterpart to the previously cited disparaging judgement of Schrödinger’s solution by Heisenberg:
I am indeed unaware of a genetic connection to Heisenberg[’s theory]. I knew of his theory, of course, but I felt discouraged, not to say repelled, by the methods of transcendental algebra, which appeared difficult to me, and by the lack of visualizability [Anschaulichkeit].22
Were the two theories perhaps merely two unfinished, but complementary halves of a greater whole – modern quantum mechanics – still in the process of emerging, which came into being only through the further developments and mathematical formulations of Born, Jordan, Dirac, Hilbert and finally von Neumann in 1932? Certainly, a great deal of evidence speaks for this: besides the unfinished character of both theories, especially the circumstance that what was most important about Schrödinger’s theory, from today’s perspective, was that it made the quantum states understandable, while Heisenberg’s theory describes the physically observable quantities as mathematical
21For a discussion of the equivalence proofs between matrix and wave mechanics and their mathematical rigor, see (Muller, 1997a,b, 1999; Perovic, 2008). 22Translated from (Schrödinger, 1926d, p. 735, footnote 2): “Eines genetischen Zusammenhanges mit Heisenberg bin ich mir durchaus nicht bewußt. Ich hatte von seiner Theorie natürlich Kenntnis, fühlte mich aber durch die mir sehr schwierig scheinenden Methoden der transzendenten Algebra und durch den Mangel an Anschaulichkeit abgeschreckt, um nicht zu sagen abgestoßen.” 24 J. Renn operators to be applied to these states. On the other hand, from the very outset the two unfinished halves must have borne the potential for their integration, or at least for fitting them together, for otherwise Schrödinger’s proof of equivalence would have been impossible, and neither would it have been conceivable that each of the two approaches was able to solve key problems like the hydrogen atom on its own and more or less independently of the other.23 Was it perhaps just harsh reality and its quantum character that forced the con- vergence of the two theories? After all, they both had to do with the same empirical knowledge, resulting from the contemporary experiments as well as from other, older empirical evidence. What speaks against this naïve-realistic response is that each of the two theories reflected only parts of this reality, while many other quantum aspects remained hidden, particularly the role of spin, which was still unclear at the time; sta- tistical properties of quantum systems, relativistic effects, and the entire complicated aspect of the variety of manifestations of the physics of condensed matter, which was by no means entirely unknown at the time. How did it happen that both approaches, despite their essentially different points of departure, ultimately ended up recording the same part of reality, although the demarcation of this part, in the sense of a kind of non- relativistic quantum mechanics of single-particle systems without spin and statistics, could not even be formulated clearly without knowing the final outcome? In the end the spontaneous Platonism of the natural scientist remains as a conven- tional explanation, that is, the conception that behind the apparent variety of phenomena is hidden a transcendental reality of mathematical ideas that have always existed in- dependent of man. Was it perhaps the pre-established harmony of such mathematical structures that resulted in Schrödinger and Heisenberg, although they lifted different parts of the veil, ultimately having to discover the same secret, the Hilbert space struc- ture, which, according to today’s understanding, is what constitutes the mathematical essence of quantum mechanics? But even if that were the case, it would still remain astounding that this supposedly pre-existing mathematical structure had no essential heuristic function for either of the two theories and was not discovered at all until so long after they were formulated.24 The notion of a world of Platonic ideas, which guide knowledge so little in the real world, is not terribly convincing. Therefore a different explanation seems in order, which one could call a genetic explanation. It has already been tried and tested in historical studies on the emergence of the theory of relativity and will be briefly summarized in what follows.25 Indeed, this name seems quite fitting when one recalls the image of the fraternal twins used before. In the history of relativity theory there is a puzzle similar to that of the double birth of quantum mechanics, the paradox of missing knowledge. Today the general theory of relativity is the theoretical foundation of astrophysics and cosmology, and especially the explanation of gravitational lenses, black holes and the expanding universe, all of which
23Schrödinger explained the hydrogen spectrum and the Stark effect splitting in his first three communica- tions as the key example for the application of his wave mechanics (Schrödinger, 1926b,c,e). Within matrix mechanics, the explanation of the hydrogen spectrum and the Stark effect splitting is due to Pauli (1926). 24See (von Neumann, 1932). 25See (Renn, 2007). Schrödinger and the genesis of wave mechanics 25 are phenomena that were unknown back when the theory was formulated in the year 1915. At that time there were very few indications at all of deviations from Newton’s theory that would recommend the development of a new theory of gravitation. So what was the empirical foundation upon which Einstein was able to formulate a theory that has stood up to all of the new – occasionally dramatic – advances in the insights of observational astronomy up to the present? Historical studies have shown that this theory emerged from a transformation of the knowledge of classical physics. It started not with a new paradigm that was irrecon- cilable with old concepts, but with a reorganization of the store of knowledge already available. This reorganization had become necessary because of challenges, only some of which were conditioned directly by new empirical findings, but the majority of which arose through internal tensions in the body of knowledge of classical physics, which, for their part, were certainly the consequence of increased empirical knowledge. This can be illustrated best using the example of Einstein’s equivalence principle and the corresponding mental model of an accelerated box. An observer in a closed, accelerated box without windows – often described as an elevator (see Figure 8) – has a yo-yo in his hand, which he then drops. The yo-yo falls to the floor of the box. In principle, the
Figure 8. The indistinguishability of gravitational and inertial forces, illustrated by the elevator thought experiment. By Laurent Taudin. 26 J. Renn observer cannot tell whether this falling motion is a consequence of the acceleration of the box, such that the yo-yo is merely obeying its own inertia, or whether the box is not actually accelerated at all, but standing on firm ground such that the falling motion is caused by gravity. This simple thought experiment, which remains completely within the framework of classical physics, suggests that inertia and gravitation are similar in nature. In particular, Einstein’s equivalence principle suggests that inertial forces occurring in accelerated systems, as in a carousel, for instance, are an expression of one and the same interaction between masses as gravitational forces. This conclusion yields a new perspective on classical physics, in which such forces do not actually have anything to do with each other. But can classical physics be reformulated so that the similar nature of gravitation and inertia becomes the center of its conceptual foundation, rather than just appearing as a marginal and apparently coincidental by-product? Such a process of reorganization, in which a previously marginal element of a conceptual system becomes the center of a new one, may be called a “Copernicus process”.26 It turns out that, if one combines the reorganization of classical physics in the light of Einstein’s elevator thought experiment with the insights of special relativity about the relation of rulers and clocks in reference systems moving with respect to each other, basic insights of the theory of general relativity follow. In other words, this theory emerged from a reorganization of classical physics, which allowed the experiences stored in this field to be linked with new insights like those of the special theory of relativity, with as few losses and as little conflict as possible. The long and sustained stability of the theory of general relativity is rooted in the success of this interpretation of knowledge. As I shall argue in the following, quantum mechanics, too, emerged from such a transformation of the knowledge of classical physics. This knowledge incorporated not only a wealth of empirical knowledge, but also a wide variety of concepts and techniques that had accumulated over centuries. It is no coincidence that quantum mechanics still uses concepts like place, time, mass, momentum, and energy, and that its mathematical form still exhibits such a close relationship to the advanced formulations of classical mechanics like the Lagrangian and the Hamiltonian formalism. It emerged from the transformation of this mechanics. And its validity, too, is not conceivable without this genesis, for it rests, just as was the case for the general theory of relativity, not only on the specific observational and experimental knowledge that accompanied its emergence, but on the whole body of empirical knowledge that had supported the classical mechanics before it. Quantum mechanics’ parallels to the emergence of the general theory of relativity can be pursued even further, however, and ultimately even lead us to a solution to the twin paradox of its emergence. The new knowledge that was to be combined with the old mechanics expressed itself primarily through Planck’s relation between energy and frequency and through the de Broglian counterpart, the relation between momentum and wavelength. In the old quantum theory, as we saw, this relation was, in a sense, merely grafted on to old mechanics – as an additional, auxiliary condition, which meant that only certain classical solutions were permitted. The counterpart from the history 26See (Renn, 2006, 2007). Schrödinger and the genesis of wave mechanics 27 of relativity theory is the failed attempts to impose the demand of the special theory of relativity, that no physical effect may propagate faster than light, as an additional con- dition to the Newtonian theory of gravity. In contrast, Heisenberg’s and Schrödinger’s approaches can be regarded as successful variants of a reorganization of classical me- chanics, in which Planck’s relation makes direct contact with its fundamental concepts. For Heisenberg’s re-organization, this happened, as we saw, through the Fourier series representation of mechanical quantities, which includes frequencies whose be- havior is then determined through the combination principle of spectra, which, in turn, is a consequence of Planck’s relation.27 Schrödinger’s re-organization, in turn, suc- ceeded, as we also saw, through the translation of a classical wave equation into the basic equation of a new mechanics using de Broglie’s relation between momentum and wavelength, which also constitutes a consequence of Planck’s relation. Thus the two approaches essentially process the same knowledge and respond to the same challenge, that of integrating Planck’s non-classical relation into the basic concepts of classical mechanics. From this genetic perspective it is thus no wonder that both arrived at results that fit together like a pair of gloves. But why did Heisenberg and Schrödinger take such different paths, and what is the relationship between these paths? This question leads back to Schrödinger’s for- mulation of wave mechanics and the extraordinary originality that distinguishes his approach. In comparing the two different ways of addressing the subject, again, the example of the general theory of relativity is helpful. Einstein himself pursued two different strategies, between which he vacillated, but which finally turned out to be complementary to each other. The first can be called the physical strategy and the other the mathematical strategy.28 The physical strategy was disproportionately more laborious, proceeding from the familiar Newtonian law of gravity and attempting to link it with the insights of the equivalence principle through cautious generalizations, occasionally going astray. The mathematical strategy was the apparently more direct one, proceeding from a sophisticated mathematical formulation of these new insights and attempting, inversely, to build a bridge to the tried-and-true Newtonian theory. However, it initially proved quite difficult to introduce the new mathematical relations into the context of the familiar physical concepts. Overall the emergence of the general theory of relativity presents itself as the result of a conflict-ridden interaction between the development of a mathematical formalism and the formation of physical concepts. The different components of knowledge that flowed into the theory grew together on the substrate of this interaction. Initially it appears that Schrödinger, with his almost graphic image of waves, fol- lowed a physical strategy, while Heisenberg, with his impenetrable matrix mechanics, which consciously disregarded the idea of concrete atomic models, preferred a mathe- matical strategy. But upon closer examination, the opposite is true! The path Heisenberg chose, and which Bohr, Kramers, Born, van Vleck and others before him had blazed, more resembles the physical strategy of Einstein. What guided knowledge here was
27See the discussion of Ritz’s combination principle in (Hund, 1967, p. 54) and (Darrigol, 1992b, p. 122). 28See (Renn, 2007). 28 J. Renn the correspondence principle, for which, as we saw, great, occasionally fruitless efforts were undertaken to generalize and “sharpen” it step by step, until this path ultimately flowed into Heisenberg’s work of re-interpretation. The mathematical importance of the theory thus achieved was initially as unclear as the mathematical approaches Einstein had developed alongside his physical strategy. Max Born was the first to recognize that Heisenberg’s computational rules corresponded to the matrix operations long known to mathematics (Born and Jordan, 1925). The physical meaning of Heisenberg’s theory, in contrast, was secured from the outset by its origination in the generalization of the solution of specific physical problems like that of dispersion.29 In contrast to this, Schrödinger’s path to his wave mechanics was a solo home run, achieved apparently without attending to concrete physical problems in order to guide his discovery – with the exception of gas theory (Hanle, 1971, 1975) and the touch- stone of any acceptable comprehensive quantum theory, the spectrum of the hydrogen atom. Although Schrödinger’s wave equation awakened the hope that quantum prob- lems could be solved from that point on using descriptive concepts like oscillations and standing waves, the route to its discovery is rather reminiscent of Einstein’s mathemat- ical strategy.30 With one fell swoop, Schrödinger’s wave equation embodied the keen conceptual insight into the wave nature of matter. Similarly, the Riemann tensor written down by Einstein, assisted by his mathematician friend Marcel Grossmann, stood for the revolutionary insight that gravity could be conceived of as a curvature of time and space. But how such insights could be connected with the established knowledge of classical physics, be it about planetary or electron orbits, still remained to be seen. At the same time, the solution of these questions was also linked with the challenge of interpreting the surprising further consequences of such mathematical formulations in terms of physics, often at the cost of giving up or modifying established concepts. In the case of the Schrödinger equation, hopes were soon dashed that it could be interpreted within a descriptive approach. Instead it had to give way to the fully novel idea that the solutions of this equation were connected with statements about the probability of the results of certain operations to measure physical systems.31 But how was Schrödinger, in view of the unexpected consequences of his equation, and initially in the absence of further physical applications, at all able to presume that his wave equation would offer a foundation for the solution of quantum problems? His notebooks reveal that he was by no means satisfied with a brief derivation along the lines sketched above, which merely inserted the de Broglie relation into a classical wave equation. First of all, he had set himself the more ambitious goal of deriving a relativistic wave equation of matter. This was more than plausible, considering de Broglie’s approach, which was motivated by the theory of relativity and E D mc2, and also opened up perspectives that lay beyond the non-relativistic quantum mechanics of the agenda pursued by Heisenberg and his colleagues. Yet Schrödinger did not succeed in deriving such a relativistic wave equation that accounted for the hydrogen spectrum
29See (Duncan and Janssen, 2007a,b) 30For Schrödinger’s rooting in the descriptive tradition of Boltzmann and others, see (Wessels, 1983). 31This interpretation goes back to (Born, 1926a,b). For the history of the ensuing debates, see, e.g., (Jammer, 1966; Beller, 1999). For Schrödinger’s later stance in the ensuing debate, see (Bitbol, 1996). Schrödinger and the genesis of wave mechanics 29 including the Sommerfeld fine-structure. In the end, in (Schrödinger, 1926b,c)he reverted to a non-relativistic wave equation, which then did yield the correct spectrum in the appropriate approximation. Yet how could this equation be convincing if the relativistic generalization it sug- gested – it necessitated – led to untenable results? What is more, the derivation of the hydrogen spectrum remained an isolated result at first. Of course, before publishing his first paper Schrödinger attempted to derive other physical effects as well, like the Stark effect, that is, the shifting and splitting of spectral lines in an electrical field, but such attempts were not crowned with success until later.32 So how could he be sure that the correspondence of his result with the known spectrum of hydrogen was anything more than a fluke? As we saw, Heisenberg had his “sharpened” correspondence principle to secure the connection to the established results of classical physics. What could take the place of this reassurance for Schrödinger? Merely proposing a wave equation did not achieve the task of linking old and new knowledge. At this juncture Schrödinger was aided by an insight of classical physics that orig- inally seemed incidental, but which was to play a crucial role in linking his wave mechanics with classical mechanics similar to that of Einstein’s equivalence principle for connecting classical mechanics with the general theory of relativity. In the early 19th century Hamilton had shown that classical mechanics and ray optics could be formulated in analogous mathematical terms (Hamilton, 1833, 1837). Schrödinger had been familiar with this optical-mechanical analogy long before turning to the problem of wave mechanics.33 Now it became an indispensable instrument for him to establish a connection between his theory and classical mechanics as well as the older quantum theory. Since the early 19th century the wave theory of light had prevailed, as this was the only way to explain phenomena like diffraction and interference. Nevertheless, ray optics remained a good approximation of reality for certain circumstances, namely for wavelengths that are short with respect to the dimensions of optical instruments. Thus for Schrödinger it seemed logical to conceive of the relation between his wave mechan- ics and classical mechanics in a similar way to the relation between wave optics and ray optics: just as ray optics was simply an approximation to the “real” wave optics, corpuscular mechanics was a mere approximation to an underlying, more fundamental “wave” mechanics. At the same time, this approach solved the problem that had been covered by the correspondence principle in Heisenberg’s approach. In one of his notebooks (see Figure 9) Schrödinger attempted to derive his wave equation directly from an expression that plays a key role in connecting classical me- chanics and ray optics, and at the same time supplied the starting point of the old quantum theory.34 This derivation was doomed to failure, however, for deriving wave optics from ray optics is just as impossible as deriving wave mechanics from an equation of classical mechanics. But a minor change to the mathematical conditions imposed on this expression meant that the desired wave equation actually was yielded; from this, in turn, approximation could be applied to attain classical physics.
32This happened in May 1926, in (Schrödinger, 1926e). 33See (Joas and Lehner, 2009). 34Archive for the History of Quantum Physics, reel 40, section 5, item 3. 30 J. Renn
Figure 9. Double page from Schrödinger’s notebook that very probably served as the basis for his first communication on wave mechanics (Schrödinger, 1926b) and can thus be dated to late 1925 or early 1926. On an earlier page of this notebook, Schrödinger had proposed to reconsider “the old Hamiltonian analogy between optics and mechanics.” On the present double page, he claimed to have found the “somewhat astonishing connection between the two ‘quantum methods’,” probably referring to the Sommerfeld–Epstein procedure of the old quantum theory, which was based on the Hamilton–Jacobi equation, and his own. He attempted, in fact, to derive his wave equation from an ad hoc generalization of the Hamilton–Jacobi equation.
In this manner Schrödinger had discovered a route, albeit a somewhat bumpy one, that linked his wave equation with the knowledge of classical mechanics and the old quantum theory, and ultimately made it the basis for introducing the Schrödinger equa- tion in his first communication (Schrödinger, 1926b). Since the connection between this derivation and the optical-mechanical analogy was not made expressly there, its true purpose and meaning seemed mysterious,35 and probably would have remained concealed without close study of Schrödinger’s notebooks and other sources that had previously eluded consideration (Joas and Lehner, 2009). Not until his second paper (Schrödinger, 1926c) did Schrödinger then elaborate on the optical-mechanical anal- ogy, now that he himself, after completing the first paper, had comprehended how important it was for physics in establishing the connection between classical and wave mechanics. So what was Schrödinger’s path to wave mechanics? We started, somewhat provoca- tively, with the apparently simple character of the discovery of the Schrödinger equation,
35See, e.g., (Kragh, 1982; Wessels, 1983). Schrödinger and the genesis of wave mechanics 31 depicting it along the lines of the famous caricature in which Einstein stands thought- fully in front of a blackboard on which the formulae E D ma2 and E D mb2 have already been rejected and crossed out (Figure 10). Does it really take a genius now to arrive at the idea that the right solution is E D mc2? The wit of this caricature is apparently that the rejected solutions already contain the ingenious thought that there is
Figure 10. Cartoon by S. Harris. ©ScienceCartoonsPlus.com. a connection between energy and mass, whereby the question as to how this connection can be justified is entirely absent. By contrast, in the case of the Schrödinger equation, we saw that it was the recognition of the connection between his new theory and the old mechanics that made his solution so ingenious. Incidentally, that was Schrödinger’s own estimation even as early as 1926: 36 The power of the present attempt – if I may be permitted to pass judgement – lies in the guiding physical aspect which builds the bridge between the macroscopic and the microscopic mechanical events, and which makes comprehensible the formally different method of treatment they require. We further saw that the twin paradox of the history of quantum mechanics, its parallel emergence as matrix and as wave mechanics, can be explained genetically. 36Translated from (Schrödinger, 1926c, p. 514): “Die Stärke des vorliegenden Versuches – wenn es mir erlaubt ist, darüber ein Urteil zu sagen – liegt in dem leitenden physikalischen Gesichtspunkt, welcher die Brücke schlägt zwischen dem makroskopischen und dem mikroskopischen mechanischen Geschehen, und welcher die äußerlich verschiedene Behandlungsweise, die sie erfordern, verständlich macht.” 32 J. Renn
Both theories emerged from a transformation of the knowledge of classical physics, in which Planck’s relation between energy and frequency was to be integrated. Both theories include findings about the connection between classical and quantum physics. In Schrödinger’s formulation of wave mechanics, the optical-mechanical analogy plays the same role that the correspondence principle plays in leading to matrix mechanics. The two routes are complements of each other. Schrödinger proceeded from what was in principle a familiar mathematical formulation, whose consequences for physics were sounded out afterward, while Heisenberg obtained his theory from a generalization of the solution to concrete physical problems and then found a satisfactory mathematical formulation. The great historian of science Thomas S. Kuhn was one of the most knowledgeable experts of the history of quantum physics (Kuhn, 1962, 1982, 1987). Yetin his treatment of this history his concept of scientific revolution crumbles a bit, dissolving into many smaller paradigm shifts. The picture sketched here leads to a different understanding of this great breakthrough in the history of science. This breakthrough can be understood properly only if we keep in mind that the scientific knowledge of quantum theory is part of a more comprehensive knowledge system of physics, which already had been developing for a millennium at the time our story was taking place. There is probably no greater success for a physicist than recognition for having pointed out a new direction in this tremendous history of human development. Erwin Schrödinger deserves this recognition.
Acknowledgments
I am grateful to Professor Wolfgang L. Reiter for providing the opportunity to present these thoughts on the occasion of the Erwin Schrödinger-Symposium 2011 in Vienna. I am very grateful also to Ruth andArnulf Braunizer for sharing with me their memories of Erwin Schrödinger, as well as for their generosity and hospitality on several occasions. Susan Richter and Christian Joas have prepared the English version of this text. I want to express my particular gratitude to Christian Joas for the substantial investment of his time and effort to enrich and edit this text, as well as for the many productive discussions on Schrödinger, in Alpbach as well as in Berlin. This paper is an outcome of the joint work on Schrödinger’s notebooks with Christian Joas and Christoph Lehner who also provided many helpful suggestions. I am grateful also to Alexander S. Blum for helpful insights. The final editing of this paper is due to Lindy Divarci.
References
Beller, M. (1999). Quantum dialogue. Chicago: University of Chicago Press. 28 Bitbol, M. (1996). Schrödinger’s philosophy of quantum mechanics. Dordrecht: Kluwer. 28 Born, M. (1924). Über Quantemechanik. Zeitschrift für Physik 26, 379–395. 19 Born, M., and P. Jordan (1925). Zur Quantenmechanik. Zeitschrift für Physik 34, 858–888. 22, 28 Schrödinger and the genesis of wave mechanics 33
Born, M., W. Heisenberg, and P. Jordan (1926). Zur Quantenmechanik. II. Zeitschrift für Physik 35, 557–615. 22 Born, M. (1926a). Zur Quantenmechanik der Stoßvorgänge (Vorläufige Mitteilung). Zeitschrift für Physik 37, 863–867. 28 Born, M. (1926b). Quantenmechanik der Stoßvorgänge. Zeitschrift für Physik 38, 803–827. 15, 28 Bohr, N. (1913). On the constitution of atoms and molecules. Part I. Philosophical Magazine (6) 26, 1–25. 13 Broglie, L. de (1924). Recherche sur la théorie des quanta. Thèse. Paris: Masson & Cie. 12 Buchwald, J. Z. (1989). The rise of the wave theory of light: Optical theory and experiment in the early nineteenth century. Chicago: University of Chicago Press. Büttner, J., J. Renn, and M. Schemmel (2003). Exploring the limits of classical physics. Planck, Einstein and the structure of a scientific revolution. Studies in History and Philosophy of Science Part B: Studies In History and Philosophy of Modern Physics 34, no. 1, 37–59. 12, 18 Darrigol, O. (1986). The origin of quantized matter waves. Historical Studies in the Physical and Biological Sciences 16, 197–253. 12 Darrigol, O. (1992a). Schrödinger’s statistical physics and some related themes. In M. Bitbol and O. Darrigol (eds.), Erwin Schrödinger: philosophy and the birth of quantum mechanics, Gif-sur-Yvette: Editions Frontières, 237–276. Darrigol, O. (1992b). From c-numbers to q-numbers. the classical analogy in the history of quantum theory. Berkeley: University of California Press. 17, 18, 27 Duncan, T., and M. Janssen (2007). On the verge of Umdeutung in Minnesota: VanVleck and the correspondence principle. Part one. Archive for the History of the Exact Sciences 61, 553–624. 9, 16, 20, 28 Duncan, T., and M. Janssen (2007). On the verge of Umdeutung in Minnesota: VanVleck and the correspondence principle. Part two. Archive for the History of the Exact Sciences 61, 625–671. 9, 16, 20, 28 Einstein, A. (1905). Über einen die Erzeugung und die Verwandlung des Lichtes betreffenden heuristischen Gesichtspunkt. Annalen der Physik 17, 132–148. 12 Einstein, A. (1916a). Strahlungs-Emission und -Absorption nach der Quantentheorie. Deutsche Physikalische Gesellschaft. Verhandlungen 18, 318–323. 18 Einstein, A. (1916b). Zur Quantentheorie der Strahlung. Physikalische Gesellschaft Zürich. Mit- teilungen 18, 47–62. Also published in: Physikalische Zeitschrift 18 (1917), 121–128. 18 Einstein, A. (1924). Quantentheorie des einatomigen idealen Gases. Sitzungsberichte der Preussischen Akademie derWissenschaften. Physikalisch-Mathematische Klasse 22, 261–267. 14 Feynman, R. P., R. B. Leighton, and M. L. Sands (1965). The Feynman lectures on physics. Vol. III. Reading, Massachusetts: Addison-Wesley. 10 34 J. Renn
Gerber, J. (1969). Geschichte der Wellenmechanik. Archive for the History of the Exact Sciences 5, 349–416. 15 Hamilton, W. R. (1833). On a general method of expressing the paths of light, and of the planets, by the coefficients of a characteristic function. In Conway,A. W., Synge, J. L. (eds.), The math- ematical papers of Sir William Rowan Hamilton, Volume I: Geometrial optics (p. 311–332). Cambridge: Cambridge University Press. 29 Hamilton, W. R. (1837). Third supplement to an essay on the theory of systems of rays. In Conway, A. W., Synge, J. L. (eds.), The mathematical papers of Sir William Rowan Hamilton, Volume I: Geometrial optics (p. 164–293). Cambridge: Cambridge University Press. 29 Hanle, P. A. (1971). The coming of age of Erwin Schrödinger: His quantum statistics of ideal gases. Archive for the History of the Exact Sciences 17, 165–192. 28 Hanle, P. A. (1975). Erwin Schrödinger’s statistical mechanics, 1912–1925. PhD Dissertation, Stanford University. 28 Hanle, P. A. (1977). Erwin Schrödinger’s reaction to Louis de Broglie’s thesis on the quantum theory. Isis 68, 606–609. Heilbron, J. L., and T. S. Kuhn (1969). The Genesis of the Bohr atom. Historical Studies in the Physical Sciences 1, 211–290. 13 Heisenberg, W. (1925). Über quantentheoretische Umdeutung kinematischer und mechanischer Beziehungen. Zeitschrift für Physik 33, 879–893. 21 Hoffmann, D. (1984). Erwin Schrödinger. Biographien hervorragender Naturwissenschaftler, Techniker und Mediziner Vol. 66. Leipzig: Teubner. 9 Hund, F. (1967). Geschichte der Quantentheorie. Mannheim: Bibliographisches Institut. 27 Jammer, M. (1966). The conceptual development of quantum mechanics. New York: McGraw- Hill. 28 Joas, C., and C. Lehner (2009). The classical roots of wave mechanics: Schrödinger’s transforma- tions of the optical-mechanical analogy. Studies in History and Philosophy of Modern Physics 40, 338–351. 9, 15, 29, 30 Jungnickel, C., and R. McCormmach (1986). Intellectual mastery of nature: Theoretical physics from Ohm to Einstein, Volume 2. Chicago: University of Chicago Press. 13 Kragh, H. (1982). Erwin Schrödinger and the wave equation: The crucial phase. Centaurus 26, 154–197. 15, 30 Kronig, R. (1960). The turning point. In: M. Fierz and V. F. Weisskopf (eds.). Theoretical physics in the twentieth century. A memorial volume to Wolfgang Pauli. New York: Interscience. 21 Kubli, F. (1970). Louis de Broglie und die Entdeckung der Materiewellen. Archive for History of Exact Sciences 7, 26–68. 12 Kuhn, T. S. (1962). The structure of scientific revolutions. Chicago: University of Chicago Press. 32 Kuhn, T. S. (1982). Was sind wissenschaftliche Revolutionen? 10. Werner-Heisenberg-Vorlesung gehalten in München-Nymphenburg am 24. Febr. 1981. München-Nymphenburg: Carl- Friedrich-von-Siemens-Stiftung. 18, 32 Schrödinger and the genesis of wave mechanics 35
Kuhn, T. S. (1987). Black-body theory and the quantum discontinuity, 1894–1912. Chicago: University of Chicago Press. 18, 32 Ludwig, G. (1969). Wellenmechanik. Einführung und Originaltexte. Berlin: Akademie-Verlag. 15 McCormmach, R. (1970). H.A. Lorentz and the electromagnetic view of nature. Isis 61, 459–497. Mehra, J., and H. Rechenberg (1987a). The historical development of quantum theory, Volume 5, Erwin Schrödinger and the rise of wave mechanics, Part 1, Schrödinger in Vienna and Zürich, 1887–1925. New York: Springer. 9 Mehra, J., and H. Rechenberg (1987b). The historical development of quantum theory, Volume 5, Erwin Schrödinger and the rise of wave mechanics, Part 2, The creation of wave mechanics: Early response and applications, 1925–1926. New York: Springer. 9 von Meyenn, K. (2011). Eine Entdeckung von ganz außerordentlicher Tragweite: Schrödingers Briefwechsel zur Wellenmechanik und zum Katzenparadoxon. Heidelberg: Springer. 9, 10, 15 Moore, W. J. (1989). Schrödinger: life and thought. Cambridge: Cambridge University Press. 9 Moore, W. J. (1994). A life of Erwin Schrödinger. Cambridge: Cambridge University Press. 9 Muller, F. A. (1997a). The equivalence myth of quantum mechanics. Part I. Studies In History and Philosophy of Science Part B: Studies In History and Philosophy of Modern Physics 28, 35–61. 23 Muller, F. A. (1997b). The equivalence myth of quantum mechanics. Part II. Studies In History and Philosophy of Science Part B: Studies In History and Philosophy of Modern Physics 28, 219–247. 23 Muller, F.A. (1999). The equivalence myth of quntum mechanics (Addendum). Studies In History and Philosophy of Science Part B: Studies In History and Philosophy of Modern Physics 30, 543–545 23 von Neumann, J. (1932). Mathematische Grundlagen der Quantenmechanik. Berlin: Springer. 24 Pauli, W. (1926). Über das Wasserstoffspektrum vom Standpunkt der neuen Quantenmechanik. Zeitschrift für Physik 36, 336–363. 24 Pauli, W. (1979). Scientific correspondence with Bohr,Einstein, Heisenberg a.o., VolumeI: 1919– 1929. Edited by A. Hermann, K. v. Meyenn, V. F. Weisskopf. New York: Springer. 16 Perovic, S. (2008). Why were matrix mechanics and wave mechanics considered equivalent? Studies in History and Philosophy of Modern Physics 39, 444–461. 23 Renn, J. (2006). Auf den Schultern von Riesen und Zwergen. Einsteins unvollendete Revolution. Weinheim: Wiley. 20, 26 Renn, J. (ed.) (2007). The genesis of general relativity. 4 Volumes. Dordrecht: Springer. 24, 26, 27 Schrödinger, E. (1926a). Zur Einsteinschen Gastheorie. Physikalische Zeitschrift 27, 95–101. 14 Schrödinger, E. (1926b). Quantisierung als Eigenwertproblem (Erste Mitteilung). Annalen der Physik 79, 361–376. 24, 29, 30 36 J. Renn
Schrödinger, E. (1926c). Quantisierung als Eigenwertproblem (Zweite Mitteilung). Annalen der Physik 79, 489–527. 24, 29, 30, 31 Schrödinger, E. (1926d). Über das Verhältnis der Heisenberg-Born-Jordanschen Quanten- mechanik zu der meinen. Annalen der Physik 79, 734–756. 23 Schrödinger, E. (1926e). Quantisierung als Eigenwertproblem (Dritte Mitteilung). Annalen der Physik 80, 437–490. 24, 29 Schrödinger, E. (1926f). Quantisierung als Eigenwertproblem (Vierte Mitteilung). Annalen der Physik 81, 109–139. Scott, W. T. (1967). Erwin Schrödinger. An introduction to his writings. Amherst: University of Massachusetts Press. 9 Sommerfeld, A. (2004). Wissenschaftlicher Briefwechsel, Band 2: 1919–1951. Eckert, M. and Märker, K. (eds.). Berlin: GNT-Verlag. Wessels, L. (1979). Schrödinger’s route to wave mechanics. Studies in History and Philosophy of Science 10, 311–340. Wessels, L. (1983). Erwin Schrödinger and the descriptive tradition. In R. Aris, H. T. Davis, and R. Stuewer (eds.), Springs of scientific creativity : essays on founders of modern science. Minneapolis: University of Minnesota Press, 254–278. 15, 28, 30 Wheaton, B. R. (1983). The tiger and the shark: Empirical roots of wave-particle dualism. Cambridge: Cambridge University Press. 12 Wünschmann, A. (2007). Der Weg zur Quantenmechanik. Kirchheimbolanden: Studien-Verlag Wünschmann. 15 Do we understand quantum mechanics – finally?
Jürg Fröhlich and Baptiste Schubnel
“If someone tells you they understand quantum mechanics then all you’ve learned is that you’ve met a liar.” Richard P. Feynman
“Anyone who is not shocked by quantum theory has not understood it.” Niels Bohr
In these notes we present a short and necessarily rather rudimentary summary of some of our understanding of what kind of a physical theory of nature quantum mechanics is. They have grown out of a lecture the senior author presented at the Schrödinger memorial in Vienna, in January 2011. A more detailed and more pedagogical account of our view of quantum mechanics, attempting to close various gaps in the mathematics and physics of these notes, will be published elsewhere. We do not claim to offer any genuinely new or original thoughts on quantum mechanics. However, we have made the experience that there is still a fair amount of confusion about the deeper meaning of this theory – even among professional physicists. The intention behind these notes (and a more detailed version thereof) is to make a modest contribution towards alleviating some of this confusion. After a short introductory section on the history of Schrödinger’s wave mechanics, we will sketch a unified view of non-relativistic theories of physical systems comprising both classical and quantum theories. This will enable us to highlight the fundamental conceptual differences between these two classes of theories. Our goal is to sketch what it is that quantum mechanics predicts about the behavior of physical systems when appropriate experiments are made, and in what way it differs radically from classical theories. Incidentally, we hope to convince the reader that Bohr and Feynman may have been a little too pessimistic in their assessment of our understanding of this wonderful theory. Our main results may be found in Sections 2, 3.1, 3.2 and 4. Acknowledgements. The senior author has learnt most of what he understands about quantum mechanics from Markus Fierz, Klaus Hepp and Res Jost, many years ago when he was a student at ETH Zurich. This did not spare him sufferings through prolonged periods of confusion about the nature of the theory, later on. Apparently, there is no way around thinking about these things and trying to clarify one’s thoughts, all by oneself. He is especially grateful to Klaus Hepp and Norbert Straumann for plenty of hints that led him towards some understanding of various elements of the foundations of the theory. He acknowledges many useful discussions with Peter Pickl and Christian 38 J. Fröhlich and B. Schubnel
Schilling. He thanks his friends in Vienna for having invited him to present a lecture on quantum mechanics at the Schrödinger memorial and JakobYngvason for insightful comments. These notes were completed during a stay at the “Zentrum für interdisziplinäre Forschung” (ZiF) of the University of Bielefeld. The authors thank their colleague and friend Philippe Blanchard and the staff of the ZiF for their very friendly hospitality. These notes are dedicated to the memory of Ernst Specker (1920–2011), who set a brilliant example of a highly original and inspiring scientist and teacher for all those who had the privilege to have known him. His contributions to quantum mechanics will be remembered.
1 Schrödinger and Zurich
With Heisenberg and Dirac, Schrödinger is one of the fathers of (non-relativistic) quan- tum mechanics in its final form. Everyone has heard about his wave mechanics and about the Schrödinger Equation. He made his most important discoveries during a time when he held a professorship at the University of Zurich. With Berlin, Berne, Göttingen and Cambridge (UK), Zurich was one of the birth places of the new theories of 20th century Physics. It may thus be appropriate to begin with a short summary of some important facts about “Schrödinger and Zurich”. The sources underlying our summary are [55], [50]. Erwin (Rudolf Josef Alexander) Schrödinger was born in Vienna, on August 12, 1887. His father was catholic, his mother a Lutheran. His mother’s mother was English. German and English were spoken at home. Erwin started to study Mathematics and Physics at the University of Vienna in 1906. After only four years, he was promoted to Dr. phil., in 1910. Among his teachers were Franz S. Exner and Friedrich Hasenöhrl. The latter was killed in 1915, during World War I. Through Hasenöhrl’s influence, Schrödinger liked to think of himself as a student of Ludwig Boltzmann. He recognized Hasenöhrl’s scientific importance and influence on his own work in his Nobel lecture. In “Mein Leben, meine Weltsicht”, Schrödinger writes: “Ich möchte nicht den Eindruck hinterlassen, mich hätte nur die Wissenschaft interessiert. Tatsächlich war es mein früher Wunsch, Poet zu sein. Aber ich bemerkte bald, dass Poesie kein Geld einbringt. Die Wissenschaft dagegen offerierte mir eine Karriere.” Glancing through Schrödinger’s early work, one notices his talent for language and his pragmatism in choosing seemingly promising research topics. One also encounters many signs of his excellent mathematical education and his talent for mathematical reasoning. No wonder science offered him a rather smooth career. However, before his appointment as “Ordinarius für Theoretische Physik” at the University of Zurich, in the fall of 1921, to the chair previously held by Einstein and von Laue, there were only few signs of his extraordinary genius. Schrödinger’s years in Zurich constitute, undoubtedly, the most creative period in his life. His first important paper, which concerned an application to quantum theory of Weyl’s idea of the electromagnetic field as a gauge field, was submitted for publication Do we understand quantum mechanics – finally? 39 in 1922. His epochal papers on wave mechanics only followed a little more than three years later. The first one, “Quantisierung als Eigenwertproblem (Erste Mitteilung)”, was submitted for publication on January 27, 1926; the second one (same title – Zweite Mitteilung) on February 23, the third one (classical limit of wave mechanics) shortly thereafter, the fourth one (equivalence of wave- and matrix mechanics) on March 18, the fifth one (Dritte Mitteilung – perturbation theory and applications) on May 10, the sixth one (Vierte Mitteilung – time-dependent Schrödinger equation, time-dependent perturbation theory) on June 21, the seventh one – a summary of his new wave mechanics published in the Physical Review – on September 3, the eighth one (Compton effect) on December 10 – all during the year of 1926. In November 1926, he completes his “Vorwort zur ersten Auflage” of his “Abhandlungen zur Wellenmechanik”. The intensity of Schrödinger’s scientific creativity and productivity, during that one year, may well be without parallel in the history of the Natural Sciences, with the possible exception of Einstein’s “annus mirabilis”, 1905. Schrödinger discovers all the right equations, all the right concepts and all the right mathematical formalism. Mathematics-wise, he is well ahead of his competitors, except for Wolfgang Pauli. He talks about linear operators, introduces Hilbert space into his theory, addresses and solves many of the pressing concrete problems of the new quantum mechanics – and somehow misses its basic message. He is haunted by the philosophical prejudice that physical theory has to provide a realistic description of Nature that talks about what happens, rather than merely about what might happen. His goal is to find a description of phenomena in the microcosmos in the form of a classical relativistic wave-field theory somewhat analogous to Maxwell’s theory of the electromagnetic field – of course without succeeding. In spite of his philosophical prejudices, he is to unravel some of the most important concepts typical of the new theory, such as entanglement and decoherence, and to arrive at all the right conclusions – apparently without ever feeling comfortable with his own discoveries. On the first of October of 1927, Schrödinger was appointed as the successor of Max Planck in Berlin. In 1933, he shares the Nobel Prize in Physics with Paul (Adrien Maurice) Dirac. In 1935, in the middle of the turmoil around his emigration from Nazi Germany, he conceives his famous “Schrödinger’s cat” Gedankenexperiment, which introduces the idea of decoherence. Coming from Zurich, we feel we should ask why this city was the right place where Schrödinger could make his epochal discoveries. Berne and Zurich were the cities where Einstein had made his most essential discoveries in quantum theory. Thanks to the presence of many famous refugees of WorldWar I, Zurich was a rather cosmopolitan city with a liberal spirit. The scientific atmosphere created by Einstein, von Laue, Debye, Weyl and others must have been fertile for discoveries in quantum theory. In his work on wave mechanics, Schrödinger was, according to his own testimony, much influenced by Einstein’s work on ideal Bose gases (1924/25) and de Broglie’s work on matter waves. It is reported that Debye asked Schrödinger to report on de Broglie’s work and suggested to him to look for a wave equation describing matter waves. But, according to Felix Bloch, Schrödinger had already started to look into this problem. In Arosa, where Schrödinger repeatedly spent time to cure himself from tuberculosis, he 40 J. Fröhlich and B. Schubnel apparently discovered the right equation. Rumor has it that he pursued his ideas at the “Dolder Wellenbad”, a swimming pool above Zurich with artificial waves (and pretty women sun bathing on the lawn). It is appropriate to mention the roleWeyl played as Schrödinger’s mathematical men- tor, during their Zurich years. It was Weyl who apparently explained to Schrödinger that his time-independent wave equation represented an eigenvalue problem and di- rected him to the right mathematical literature. Schrödinger acknowledges this in his first paper on wave mechanics. Apart from his superb knowledge of mathematics, Weyl was intensely familiar with modern theoretical physics including quantum theory. He had prescient ideas on some of the radical implications of quantum theory (such as its intrinsically statistical nature and problems surrounding the notion of an “event” in the quantum world), quite some time before matrix- and wave mechanics were discovered. Weyl was Schrödinger’s senior by only two years. They were close friends. It is thus plausible that Weyl played a rather important role in the development of Schrödinger’s thinking. Their relationship is a model for the fruitfulness of interactions between mathematicians and theoretical physicists. While the physical arguments that led Heisenberg and, following his lead, Dirac to the discovery of quantum mechanics (in the form of matrix mechanics and transforma- tion theory) appear to us as relevant and fresh as ever, Schrödinger’s formal arguments based on an analogy with optics,
geometrical optics W wave optics Hamiltonian mechanics W wave mechanics; may nowadays seem to be of mainly historical interest. Although they initially misled him to an erroneous interpretation of wave mechanics, they made him discover very powerful mathematical methods from the theory of partial differential equations and of eigenvalue problems that his competitors did not immediately recognize behind their more abstract formulation of the theory. But it is time to leave science history and proceed to somewhat more technical matters.
2 What is a physical system, mathematically speaking?
In this section, we outline a mathematical formalism suitable for a unified description of classical and quantum-mechanical theories of physical systems. It is most conveniently formulated in the language of operator algebras; see, e.g., [53]or[5]. We suppose that there is an observer, O, who studies a physical system, S. To gather information on S, O performs series of experiments designed to measure various physical quantities pertaining to S, such as positions, momenta or spins of some particles belonging to S. No matter whether we speak of classical or quantum-mechanical theories of physical systems, physical quantities are always represented, mathematically, by (bounded) linear operators. For classical systems, they correspond to real-valued functions on a space of pure states (phase space, in the case of Hamiltonian systems) acting as multiplication operators on a space of half-densities over the space of pure states; for Do we understand quantum mechanics – finally? 41 quantum-mechanical systems with finitely many degrees of freedom, they correspond to (non-commuting) selfadjoint linear operators acting on a separable Hilbert space. In these notes, we study non-relativistic theories of physical systems, i.e., we assume that signals can be transmitted arbitrarily fast. It is then reasonable, in either case, to imagine that the physical quantities pertaining to S generate some -algebra of operators, denoted by AS , that does not depend on the observer O. (In contrast, in general relativistic theories, the algebra of physical quantities not only depends on the choice of a physical system but will also depend, in general, on the observer.) Given a physical quantity represented by a selfadjoint operator a 2 AS , a measure- ment of a results in an “event” [29] corresponding to some measured value of a. Since measurements have only a finite precision, one may associate with every such event a real interval, I , describing a range of possible outcomes in a particular measurement of a. It is natural to associate to this possible event the corresponding spectral pro- jection, Pa.I /, associated with the selfadjoint operator a and the interval I R via the spectral theorem. Thus, spectral projections associated with selfadjoint operators in AS corresponding to physical quantities of S represent possible events in S. Generally speaking, one may argue that there are events happening in S that are not necessarily triggered by an actual measurement (undertaken by an observer using experimental equipment) of some physical quantity represented by an operator in the algebra AS but rather by interactions of S with its environment. It is plausible to assume that any possible event in S can be represented by an orthogonal projection, P (or, more generally, by a positive operator-valued measure). The operator P is a mathematical representation of the acquisition of information about S; it does not represent a physical process. It is assumed that all possible events in S generate a C - or a von Neumann algebra, henceforth denoted by BS . For simplicity, it will always be assumed that the algebra BS contains an identity operator. The algebra AS is contained in or equal to the algebra BS . (In these notes, we will be somewhat sloppy about the right choices of these algebras. At various places, this will undermine their mathematical precision. Things will be rectified in a forthcoming essay.) States of the system S are identified with states on the algebra BS , i.e., with nor- malized, positive linear functionals on BS . The purpose of a theory of a physical system S is to enable theorists to predict the probabilities of (time-ordered) sequences of possible events in S – “histories” of S – to actually happen when S is coupled to another system, E, needed to carry out appropriate experiments, given that they know the state of the system corresponding to the composition of S with E. We emphasize that E is treated as a physical system, too, and that it plays an important role in associating “facts” with “possible events” in S (this being related to the mechanisms of “dephasing” and “decoherence”). Generally speaking, E can either correspond to some experimental equipment used to observe S, or to some environment S is coupled to. We think of E as “experimental equipment” if the initial state of E and its dynamics can be assumed to be controlled, to some extent, by an observer O (an experimentalist who can turn various knobs and tune various parameters). If the state and the dynamics of E are beyond the control of any observer we think of E as “environment”. Of course, the distinction is usually not sharp. It is 42 J. Fröhlich and B. Schubnel important to understand why probabilities of histories of S do not sensitively depend on precise knowledge of the state and the dynamics of E. (See Section 3.3 for a result going in this direction. We plan to return to these matters elsewhere.) A “realistic” (or “deterministic”) theory of a physical system S is characterized by the properties that any possible event in S has a complement, in the sense that either the event or its complement will happen, and that if the state of S _E is pure the probability of a possible history of S is either D 0 (meaning that it will never be observed) or D 1 (meaning that it will be observed with certainty). We say that pure states of a system described by a realistic theory give rise to 0-1 laws for the probabilities of its histories. In contrast, a quantum theory is characterized by the properties that, in general, there may be “interferences” between a possible event and its complement – meaning that they do not mutually exclude each other – and that there are pure states that predict strictly positive probabilities that are strictly smaller than 1 for certain histories. A quantum theory is therefore intrinsically non-deterministic. These remarks make it clear that a crucial point to be clarified is how one can prepare a physical system in a specific state (pure or mixed, in case the state is only partly specified) of interest in an experiment that theorists want to make predictions on. This point has been studied, in a rather satisfactory way, for a respectable class of quantum theories. However, the relevant results and the methods used to derive them go beyond the scope of this review. They will be treated in a forthcoming paper (but see, e.g., [20], [13]). We hope that the meaning of the notions and remarks just presented will become clear in the following discussion. John von Neumann initiated the creation of the theory of operator algebras, in order to have a convenient and precise mathematical language to think and talk about quantum physics and to clarify various mathematical aspects of the theory. There is simply no reason not to profit from his creation – no apologies! Thus, as announced, above, we will consider the rather vast class of theories of physical systems that can be formulated in the language of operator algebras. Such theories are further characterized by specifying the following data. Definition 1 (Mathematical data characterizing a theory of a physical system S). (I) A C -algebra, BS , generated by “all” possible events in S, containing the -al- gebra AS BS generated by physical quantities pertaining to S.
(II) The convex set of states, S , on the algebra BS .
(III) A group of symmetries, GS ,ofS, including time evolution. Elements g 2 GS are assumed to act as -automorphisms, ˛g ,onBS . The group of all -automor- phisms of BS is denoted by Aut.BS /. We remark that the algebra BS and the group of symmetries GS depend on the environment S is coupled to. 0 0 (IV) Subsystems: S is a subsystem of S , S S ,iffBS BS0 . Composition of systems:IfS, S 0 are two systems and Sx D S _ S 0 denotes their composition then BSx BS_S0 AS ˝ AS0 . Do we understand quantum mechanics – finally? 43
0 If S ' S then one must specify an embedding of the state space S_S S ˝ S . This is the issue of statistics, which plays a crucial role in quantum mechanics (Fermi–Dirac-, Bose–Einstein-, or fractional statistics).
The choice of (I) and (III) depends on the experimental equipment available to observers exploring the system S. To illustrate this point, think of the solar system, Sˇ. An astronomer in the times of Tycho Brahe would have chosen much fewer physical quantities to describe possible events in Sˇ than a modern astrophysicist equipped with the latest instruments. This results in drastically different choices of A B algebras Sˇ and Sˇ , in spite of the fact that the actual physical system remains the same. (Tycho Brahe would obviously have chosen much smaller such algebras than a contemporary astrophysicist, and, thus, their theoretical descriptions of the solar system would drastically differ from one another.) Well, our theories of physical systems are but images of such systems inside some mathematical structure. These images are never given by “isomorphisms”; they are more or less coarse-grained (depending on the experimental equipment and the precision of the data available to us), and our choice of mathematical structures as screens for images of physical systems need not be unique. It is remarkable that the new physical theories of the 20th century appear to arise from older (precursor) theories by “deformation” of the structures (I), (III) and (IV). For example, quantum mechanics can be obtained from classical Hamiltonian me- chanics by deforming the algebra AS from a commutative, associative algebra to a non-commutative, associative one – “quantization” – (theory of deformations of as- sociative algebras), the deformation parameter corresponding to Planck’s constant „. One can view atomistic theories of matter as arising from (Hamiltonian) theories of continuous media by a deformation of the algebra AS , the deformation parameter cor- responding (roughly speaking) to the inverse of Avogadro’s number NA. By deforming the Galilei symmetry of non-relativistic systems one is led to the Poincaré symmetry of (special) relativistic systems; the deformation parameter is the inverse of the speed of light c. This is an example of a deformation of (III) (deformation of Lie groups and -algebras) leading to new physical theories. Fractional statistics – a form of quantum statistics encountered in certain two-dimensional systems, in particular, in 2D electron gases exhibiting the fractional quantum Hall effect – which was overlooked by the pioneers of quantum theory, can arise as a deformation of ordinary Bose–Einstein or Fermi–Dirac statistics (deformation of braided tensor categories). The “deformation point of view” alluded to here was originally proposed by Moshe Flato [18] and taken up by Ludwig Faddeev. Some elements of it are sketched in Section 3.4 (see also [21], [22] and references given there).
3 Realistic theories versus quantum theories
In this section, we introduce two distinct classes of physical theories. A physical theory is called “realistic” if the algebra BS isAbelian (commutative). It is called “quantum” if BS is non-Abelian (non-commutative). We will see that there is an intimate connection 44 J. Fröhlich and B. Schubnel
between the commutativity of BS and determinism – determinism necessarily fails if BS is non-commutative.
3.1 Realistic theories. In this section, we summarize some of the most important features of realistic (or deterministic) theories.
3.1.1 Characterisation of BS and S in realistic theories. Realistic theories of physical systems are theories with an Abelian algebra BS of possible events. Impor- tant examples of realistic theories are Hamiltonian systems. For such systems, the algebra of possible events BS is given by the algebra of bounded (continuous or mea- surable) functions on the phase space, S , (some symplectic manifold) of the system S composed with the environment it is interacting with, and the algebra AS is some subalgebra contained in or equal to BS . Phase space S is equipped with a symplec- tic form, S (a closed, non-degenerate 2-form on S ), which gives rise to a Poisson bracket, ff; ggD S .Xf ;Xg /,onBS , with Xf denoting the Hamiltonian vector field corresponding to the function f 2 BS . This furnishes BS with the structure of a Lie algebra. States on BS are given by probability measures on S , pure states are given by Dirac measures (ı-functions). In this section, we wish to consider general realistic theories. Let BS be the Abelian C -algebra of possible events of a realistic theory of a physical system S. We denote the set of non-zero homomorphisms of BS into C by MS , called the spectrum of BS . One can prove that MS is locally compact in the .BS ; BS /- topology, and that it is a compact Hausdorff space if BS is unital, i.e., contains an identity, I. It is appropriate to recall a famous theorem due to I. M. Gel’fand.