The Lesser-Known Albert Einstein: Without a Trace of Relativity (History of Physics) 3031355679, 9783031355677

Table of contents :
Preface
Acknowledgments
Contents
1 The Sowing (Until 1905): The Annus Mirabilis
1.1 The Young Einstein Bursts onto the Scientific Scene
1.2 First Contributions: About Molecular Forces
1.2.1 Capillarity and Molecular Forces (1901)
1.2.2 Potential Differences and Molecular Forces (1902)
1.3 Trilogy on Thermostatistics
1.3.1 A Word on Probability
1.3.2 Thermal Equilibrium and the Second Law of Thermodynamics (1902)
1.3.3 Foundations of Thermodynamics (1903)
1.3.4 Relevance of Statistical Fluctuations (1904)
1.4 Gibbs’s Formulation Compared with Einstein’s
1.4.1 Background, Premises, and Objectives
1.4.2 Methods and Results
1.4.3 Impact and Diffusion
1.5 Energy Quanta and the Photoelectric Effect (1905)
1.5.1 Planck’s Energy Quanta
1.5.2 Einstein’s Energy Quanta: A Heuristic Point of View
1.6 Molecular Constitution and Brownian Motion (1905‒1906)
1.6.1 Avogadro’s Number and Brownian Motion
1.6.2 Kinetic Theory of the Motion of Particles in Suspension (1905)
1.6.3 Some Precisions
1.7 Experimental Confirmation of Einstein’s 1905 Predictions
1.7.1 Molecules Exist (Perrin, 1908‒1909)
1.7.2 Reality of Energy Quanta (Millikan, 1916)
2 The Flowering (1906–1913): Einstein Introduces Himself to the Scientific Community
2.1 From Bern to Zurich via Prague
2.2 The Critique of Planck (1906)
2.3 From Radiation to Matter: First Quantum Theory of Solids (1907)
2.4 Quantum Refinements (1909–1910)
2.4.1 A Highly Fruitful Gedankenexperiment
2.4.2 On the Possible Dual Structure of Radiation
2.4.3 Stochastic Fields and Quantum Theory
2.5 Interlude with Luminescence
2.5.1 Smoluchowski and His Interest in Fluctuations (1904–1908)
2.5.2 Einstein on Critical Opalescence (1910)
2.6 Einstein Before International Scientific Community: First Solvay Conference (1911)
2.6.1 Development of the Conference: Some Interventions
2.6.2 Einstein’s Communication
2.6.3 Conclusions of the Conference and Einstein’s Opinions
2.7 On the Necessity of Energy Quanta: Ehrenfest (1911) and Poincaré (1912)
2.7.1 Ehrenfest (1911): “Weight Function” and the Necessity of Quanta
2.7.2 Poincaré (1912): Mechanisms of Equilibrium and the Necessity for Quanta
2.7.3 Two Very Different Impacts
2.8 Law of the Photochemical Equivalent (Einstein, 1912)
2.9 Specific Heat of Gases and Quantum Theory (1911–1913)
2.9.1 Planck’s “Second Theory” and “Zero-Point Energy” (1911)
2.9.2 Einstein and Stern (1913): On the Specific Heat of Hydrogen
2.9.3 Ehrenfest’s Counterproposal (1913): Rotational Quantisation
3 The First Harvest (1914–1924): In Search of the Photon
3.1 Wartime and Peacetime in Berlin
3.2 Einstein, De Haas and Ampère’s Molecular Currents (1915–1916)
3.2.1 Einstein and De Haas Experiment
3.2.2 “Gyromagnetic Anomaly” and “Spinning Electrons”
3.3 Birth of the Photon (1916–1917)
3.3.1 New Elementary Processes (1912–1916)
3.3.2 Reality of Quanta (1916–1917)
3.3.3 On Directionality of Elementary Processes
3.4 Digression: A New Quantisation Rule (1917)
3.4.1 Quantisation Rules: Times of Splendour (1913–1916)
3.4.2 Einstein’s Quantisation Rule of 1917
3.4.3 First Impact: Epstein’s Criticism
3.4.4 Influence of Einstein’s Rule on Louis De Broglie
3.4.5 Einstein’s Rule and Wave Mechanics
3.5 First Reactions to the Photon Concept. Alternative BKS
4 The Last Collection (1924–1925): Formulation of The First Quantum Statistics
4.1 Fame and Unrest in Berlin
4.2 A Shooting Comet: Bose, 1924
4.3 From Photons to Molecules: Quantum Theory of Ideal Gases, 1924
4.4 Loss of Independence in the New Statistics, 1925
4.5 On the Association of a Wave Field to Molecules
4.6 Digression: The Case of a Master Formula
4.7 Additional Bases for the Quantum Theory of Ideal Gases, 1925
4.8 From Quantum Gas Theory to Wave Mechanics. Schrödinger, 1926
4.8.1 Schrödinger and the Quantum Theory of Ideal Monoatomic Gases
4.8.2 Analogy Between Optics and Mechanics: Formulation of Wave Mechanics
5 Heterodox to the End: EPR, Cats and Entanglements
5.1 A Long Lonely Road (1927–1955)
5.2 The Beginning of the “Bohr–Einstein Debate”
5.2.1 On the Fifth Solvay Conference, 1927
5.2.2 Einstein and Bohr at the Fifth Solvay Conference
5.2.3 First Phase of the Bohr–Einstein Debate, 1927–1930
5.3 Einstein, Podolsky and Rosen Paper (EPR) of 1935
5.3.1 Premises and Content
5.3.2 EPR Conclusions
5.4 First Reactions to EPR
5.4.1 Bohr’s Reaction: The Notion of Complementarity
5.4.2 Schrödinger Reaction: Entanglements and Cats, 1935
5.4.3 Other Disquisitions: “Many Worlds” and “Wigner’s Friend”
5.5 Last Approach: Einstein and “Hidden Variables”
5.5.1 Einstein’s Interpretation of the Quantum Formalism
5.5.2 A Theorem with Pedigree: Von Neumann, 1932
5.5.3 Bohm’s Surprising Hidden Variables Model, 1952
5.5.4 Last Word
6 An Unfinished Story: Bell’s Inequalities, Experiments and Loopholes
6.1 Bell’s Proposal, 1964
6.2 Aspect Experiment, 1982: Loopholes and Refinements
6.3 What if Einstein…?
A Einstein and Marić, an Ignored Collaboration?
A.1 A Statement of Objections
A.2 Defence Case
A.3 What Became of Mileva Marić?
B Albert Einstein (1879–1955). Brief Chronology
C The Collected Papers of Albert Einstein
References
Index

Citation preview

History of Physics

Luis Navarro Veguillas

The Lesser-Known Albert Einstein Without a Trace of Relativity

History of Physics Series Editors Arianna Borrelli, Institute of History and Philosophy of Science, Technology, and Literature, Technical University of Berlin, Berlin, Germany Olival Freire Junior, Instituto de Fisica, Federal University of Bahia, Campus de O, Salvador, Bahia, Brazil Bretislav Friedrich, Fritz Haber Institute of the Max Planck, Berlin, Berlin, Germany Dieter Hoffmann, Max Planck Institute for History of Science, Berlin, Germany Mary Jo Nye, College of Liberal Arts, Oregon State University, Corvallis, OR, USA Horst Schmidt-Böcking, Institut für Kernphysik, Goethe-Universität, Frankfurt am Main, Germany Alessandro De Angelis Padova, Italy

, Physics and Astronomy Department, University of Padua,

The Springer book series History of Physics publishes scholarly yet widely accessible books on all aspects of the history of physics. These cover the history and evolution of ideas and techniques, pioneers and their contributions, institutional history, as well as the interactions between physics research and society. Also included in the scope of the series are key historical works that are published or translated for the first time, or republished with annotation and analysis. As a whole, the series helps to demonstrate the key role of physics in shaping the modern world, as well as revealing the often meandering path that led to our current understanding of physics and the cosmos. It upholds the notion expressed by Gerald Holton that “science should treasure its history, that historical scholarship should treasure science, and that the full understanding of each is deficient without the other.” The series welcomes equally works by historians of science and contributions from practicing physicists. These books are aimed primarily at researchers and students in the sciences, history of science, and science studies; but they also provide stimulating reading for philosophers, sociologists and a broader public eager to discover how physics research – and the laws of physics themselves – came to be what they are today. All publications in the series are peer reviewed. Titles are published as both print- and eBooks. Proposals for publication should be submitted to Dr. Angela Lahee ([email protected]) or one of the series editors.

Luis Navarro Veguillas

The Lesser-Known Albert Einstein Without a Trace of Relativity

Luis Navarro Veguillas University of Barcelona Barcelona, Spain

ISSN 2730-7549 ISSN 2730-7557 (electronic) History of Physics ISBN 978-3-031-35567-7 ISBN 978-3-031-35568-4 (eBook) https://doi.org/10.1007/978-3-031-35568-4 This book is an updated translation of El desconocido Albert Einstein, published in Spanish by Tusquets Editores, Barcelona (2020). © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Einstein, 1920. Painting by Harm Kamerlingh Onnes, nephew of Heike Kamerlingh Onnes, Nobel Prize in Physics 1913 [Courtesy of the National Museum Boerhaave Leiden].

To Arnau and Laia, David and Laura, Patricia. Also, to Nuka and Floky.

Preface

In my opinion, Einstein would have been one of the greatest theoretical physicists of all time even if he had not written a single line about relativity. —Max Born, 19491

When a new book appears, readers appreciate it if the author explains as early as possible the contribution they believe their work will make to the themes to be addressed. Two points, at least, should also dictate both the conception of the text and its writing: the objectives pursued, and the average type of reader to whom the work is addressed. We aspire to offer a rigorous and accessible historical account of part of the scientific work of Albert Einstein (1879–1955), considering the context in which it was produced.2 Which part of his work? All of his scientific contributions, except those that refer to relativity. A seasoned reader might ask: why leave these out if they are precisely those that led him to become the scientific icon par excellence? Well, simply and sincerely, because we are not able to contribute anything novel to the abundant amount of literature—scientific, historical, and informative—existing on the subject. In contrast, in this book, we will try to show the amount and scope of the scientific contributions of a ‘nonrelativistic’ Einstein; that is, of the almost unknown Einstein to whom we refer in the title.3 In our exposition, we will try to maintain, as far as possible, the originality and the scientific level of his intuitions and reasoning, while at the same time showing the scope of his conclusions and always guarding against falling into whiggism, here understood as analyzing the past with our eyes fixed on the present.

1

In Schilpp (1970), 163. Albert Einstein was awarded the Nobel Prize in Physics 1921 “for his services to theoretical physics and especially for his discovery of the law of the photoelectric effect”. 3 A first task in this direction has already been undertaken in Navarro (2009). 2

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In our text, we try to address a reader with a background in physics interested in knowing first hand—through the faithful and contextualized disclosure of the original works—Einstein’s nonrelativistic contributions, essentially during the first third of the last century. The text includes reflections and details that, in our opinion, have sufficient substance to be of interest also to those who have a higher education in physics; all of this is complemented with an extensive and careful bibliography. Consistent with the above, the text is written in such a way as to be accessible to those who are not very interested in technical details, generally mathematical deductions, that can complicate the reading without being fully necessary for a global understanding of the content. By setting in a different typeface those fragments that, although essential for a specialist, could unnecessarily hinder the understanding of an average reader, we have sought to cater for readers with different levels of training in physics. Although we are not covering Einstein’s biography, the Appendix contains a minimal chronology to remind readers of significant dates and events. However, we will occasionally use data from his biography, but only to keep in mind his personal environment and the circumstances of the moment. Each chapter begins with a sketch of the social and personal circumstances that surrounded Einstein’s existence at the relevant time. Biographical aspects will thus not be included in a specific chapter but intermingled with the evolution of his scientific thought. We seek to justify the publication of a book such as this by referring to some relevant opinions about the quantity and importance of Einstein’s contributions outside his relativistic theories. The opinion of Max Born (1882–1970) which opens this Preface is already illustrative.4 Louis De Broglie (1892–1987) in 1949 and before Einstein himself, was also forceful in this respect5 : But, great as it was, this achievement [formulation of relativity] must not cause us to forget that Albert Einstein also rendered decisive contributions to other important advances in contemporary physics .... Thus, Einstein became the source of an entire movement of ideas which, as wave mechanics and quantum mechanics, was to cast so disturbing a light upon atomic phenomena twenty years later.

In 1926, the Austrian physicist Erwin Schrödinger (1887–1961) developed his theory of wave mechanics.6 On publishing this formulation, he addressed a letter to Einstein in which he expressly recognized Einstein’s decisive influence,

4

Nobel Prize in Physics 1954 “for his fundamental research in quantum mechanics, especially for his statistical interpretation of the wave function”. The prize was shared with W. Bothe (see Sect. 3.5). 5 De Broglie (1949). In Schilpp (1970), 109. De Broglie received the Nobel Prize in Physics 1929 “for his discovery of the wave nature of electrons”. 6 Nobel Prize in Physics 1933, shared with the British Paul Adrien Maurice Dirac (1902–1984), “for the discovery of new and productive forms of atomic theory”.

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through his theory of degenerate gases, in the development of the new quantum mechanics7 : Besides, this whole matter [Schrödinger’s own formulation of wave mechanics] would certainly not have been arisen now, and perhaps not at all (I mean, not on my part), if your second gas degeneracy paper had not shoved the importance of De Broglie’s ideas in front of my nose.

The work to which Schrödinger refers contains the formulation of what is known as the Bose–Einstein statistics, which turned out to be suitable for analyzing electromagnetic radiation in thermal equilibrium (Bose) as well as a number of problems associated with certain types of gases (Einstein). These topics will be addressed in detail in Chap. 4. Nevertheless, we want to emphasize here two aspects related to our interests in this book: to anticipate the enormous relevance of Einstein’s contribution, and to insist on the absence of any hint of relativity in this work, which only concerns statistical mechanics and quantum theory. Incidentally, Einstein’s contributions in the latter field were made over quite a prolonged period of time, as we shall see in the final chapter, concerning the interpretation of the quantum formalism. Throughout the book we will follow the authoritative views of Born, De Broglie and Schrödinger on the importance of Einstein’s nonrelativistic contributions with our own estimates. Thus, in Chap. 2, we will show how the analogy that he emphasizes—and will exploit on several occasions—between radiation and matter would lead him to formulate, in 1907, the first quantum theory of monoatomic solids, a first step in the direction of incorporating quantum hypotheses into the treatment of problems related to the behaviour of matter, especially at low temperature. In this sense, Einstein should be included among the founders of modern solid-state physics. In summary, Einstein’s decisive contributions to quantum physics, statistical mechanics, and solid-state physics, among others, seem to us sufficient reasons to justify the assumptions and contents of this book. Rather than the ‘context of justification’ associated with ordinary texts on physics, this book moves through the ‘context of discovery’—a context that leads, as a secondary effect, to the deepening of ideas and physical conceptions that we thought were completely assimilated. Let us take an example. Both physicists and other potential readers of this book will probably have read and learned something like this: since the classical statistical mechanics of the Austrian Ludwig Boltzmann (1844–1906) or the American Josiah Willard Gibbs (1839–1903) are based on Newtonian mechanics, when this was replaced by quantum mechanics in 1925–1926, it was necessary to look for a reformulation of statistics that would incorporate the ideas of the new mechanics. In other words, after the formulation of quantum mechanics, it was necessary to try to find a

7

Letter from E. Schrödinger to A. Einstein, 26 April 1926. In Nolar-James et al. (2018), 257.

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quantum statistical mechanics, although in the end, it was two quantum statistical mechanics that appeared. So far, everything seems clear. However, if we go deeper into the context of the discovery, and investigate the circumstances surrounding the formulation of quantum statistics, we will see that things did not happen precisely as suggested in the previous paragraph, but rather the other way around. This is not only because the Bose–Einstein statistics were formulated in 1924—that is, before any quantum mechanics, since matrix mechanics dates from 1925 and wave (or undulatory) mechanics from 1926—but also because of something that may certainly surprise us even more: to a great extent it was this new statistic that inspired Schrödinger’s later formulation of wave mechanics. For those who are not very interested in delving deeper into the physical meaning of the concepts involved, the mismatch we have just mentioned can remain a simple alteration in the order of events, without substantially modifying the result. However, those who are more sensitive to the importance of assimilating in depth the concepts underlying physical theories can hardly remain indifferent to the above ‘revelation’. One should, at least, ask this question: from a strictly logical and conceptual point of view, how can one arrive at a quantum statistical mechanics before any quantum mechanics was formulated? This seems as impossible as that Boltzmann or Gibbs could have established classical statistical mechanics before Isaac Newton (1643–1727) formulated his laws of mechanics in 1687. Although we will clarify this situation in Chap. 5, here are some hints for readers impatient to find a solution to the riddle. Quantum wave mechanics presents a formalism that carries as its flagship the well-known Schrödinger equation. Based on the interpretation of that formalism, we can justify certain conclusions that are not entirely independent of each other, which we then use to try to understand the behaviour of the atomic world. We thus obtain the quantization, under certain circumstances, of the energy and linear momentum of a molecule, for example. Additionally, the indistinguishability of identical particles and also the quantization of the molecular phase space, in which a state is no longer represented by a point but by a finite volume of value h3 , h being the Planck’s constant that is so prominent in quantum physics. However, it turns out that in the first formulation of quantum statistical mechanics—the Bose–Einstein one—Schrödinger’s equation plays no explicit role, as can be seen by reviewing the Einsteinian treatment. In contrast, the indistinguishability of molecules and the quantization of the corresponding phase space do play a role: two allowed and largely applied hypotheses, strange as this may seem today. However, the indistinguishability had been used by Boltzmann, although in a different context, which Bose mistakenly transferred to photons, and Einstein, with full awareness, to the molecules of an ideal gas. As for the quantization of the phase space of the molecules, we will see that as early as the first Solvay conference, in 1911, arguments were put forward to justify it. It thus becomes clear that it is unnecessary to resort to Schrödinger’s equation to formulate the new statistics. In due course (Chap. 5) we will go deeper into this subject.

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Chapter 1 of this book presents Einstein’s initial scientific interests and his first nonrelativistic publications, including those of 1905, his famous annus mirabilis. The chapter includes an account of Einstein’s formulation of a genuine statistical mechanics, in the tradition of Boltzmann and different from that of Gibbs. The chapter also deals with his treatment of the energy quanta of electromagnetic radiation, and his explanation of the photoelectric effect and other phenomena occurring in the interaction between matter and radiation. In Chap. 2, we will see how Einstein begins to be interested in transferring some ideas about radiation to matter, applying them on different occasions to solve problems that arose when comparing the theoretical predictions of classical physics with experimental results, especially with regard to the measurement of specific heats of solids and gases at low temperatures. We will also see how Einstein begins to take advantage of his ideas about fluctuations in statistical mechanics, sometimes to explain phenomena that resisted rigorous theoretical justifications, as in the case of critical opalescence, and at others to think about—and suggest—a dual structure of electromagnetic radiation: undulatory and corpuscular. All this is based on one of his famous Gedankenexperimente that would later lead him to important and unsuspected conclusions. Among these, perhaps the most important was the intuition of the dual nature of radiation, which would lead to the definitive incorporation of the photon into physics. As we will see in Chap. 3, Einstein reached this conclusion by deepening the analysis of the Gedankenexperiment. He realizes that, when studying the interaction between electromagnetic radiation and matter, to make everything fit together it is necessary to add a third elementary process to the two usual ones of (spontaneous) emission and absorption: the emission induced by the radiation itself, a process that would later play an important role in the construction of laser. He assigns a probability to each of these three possible processes; this is how ‘transition probabilities’ emerge in the development of atomic physics. Chapter 3 also discusses a rule of quantization formulated by Einstein that, in spite of its repercussions, was to have a considerable influence on the ideas that led De Broglie to establish the famous ‘universal duality’—undulatory and corpuscular—on the behaviour of radiation and matter. We will also refer here to a very little-known facet of Einstein: his experimental work on Ampère currents and gyromagnetic anomaly. Chapter 4 is devoted to the birth of Bose–Einstein statistics in 1924–1925 and its first applications. In 1924 Bose presented for the first time a completely quantum deduction—starting from the concept of the photon and without the slightest Maxwellian interference—of the law of black-body radiation that Max Planck (1858–1947) had proposed in 1900.8 Between 1924 and 1925 Einstein, following Bose’s method, applied the same statistical treatment to the ideal monoatomic gas, thus formulating the first quantum theory of ideal gases. These new statistical

8

Nobel Prize in Physics 1918 “in recognition of the services he rendered to the advancement of physics by his discovery of energy quanta”.

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conceptions had a clear influence on the formulation of wave mechanics in 1926 by Schrödinger, an influence that he immediately recognized. As soon as wave mechanics appeared, discrepancies arose over the interpretation of the new formalism. In 1927, after the fifth Solvay conference, the majority of physicists accepted Bohr’s views, which became the orthodox and official doctrine that would soon be known as the “Copenhagen interpretation”. Einstein’s heterodox views and especially his quarrels with Bohr are the subject of Chap. 5. In 1935 Einstein and two colleagues published a paper that became known by the acronym EPR, the authors’ initials. Reactions to the paper led to more rigorous consideration of the usual interpretation of the quantum formalism and the corresponding theory of measurement. The chapter ends with a reference to the first “hidden variable” theories, the emergence of which was in no small part due to the appearance of the EPR article. Finally, Chap. 6 includes a brief account of developments that arose after Einstein’s death but are closely linked to the ideas exposed in EPR, in particular the 1964 proposal by the Irish physicist John Stewart Bell (1928–1990) to rule experimentally on the feasibility of realistic and local hidden variable theories. We also report the now famous 1983 experiment by the French physicist Alain Aspect (1947–), together with some antecedents and later refinements.9 These experimental results suggest that the heterodoxy in relation to the interpretation of the quantum formalism is practically residual, although it has not been completely eliminated. The chapter concludes with a question: if Einstein had known the theoretical developments and experimental results that appeared after his death, how would his ideas about the feasibility of hidden variable theories have evolved? Appendix A covers the little-known controversy that arose in the late 1980s about the possibility that Einstein’s first wife co-authored at least some of the famous papers published during the annus mirabilis and which was fed by the publication of part of the correspondence between the two and with third parties, especially during their courtship. As we shall see, this sheds light on some of the pending questions, although not all of them. Two further appendices summarize the key dates in the life of our character, and outline the plans and the course of the publication of The Collected Papers of Albert Einstein. Finally, some details regarding the presentation of the text. Where a reproduction or translation is included in the final bibliography, our citations refer to these rather than to original versions. In the case of primary texts in German or French, we have chosen to cite their English translation, where available, as more accessible to the vast majority. Where the source of an English translation is not cited, it is by the author. Square brackets [ ] within a citation are reserved for clarifications and are used elsewhere in the text for references. Double quotation marks are used for direct quotations or to introduce for the first time concepts or names consecrated by

9

Aspect, together with John F. Clauser (1942–) and Anton Zeilingier (1945–) were awarded the Nobel Prize in Physics 2022. See Chap. 6.

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usage, such as “ergodic”, “hidden variables” or “Copenhagen interpretation”. Single quotation marks (‘ ’) are used for quotations inside a citation and also where terms are employed in a somewhat different sense than usual, including figurative or ironic usages, such as that Einstein ‘stole’ Bose’s statistical method. Words and names in their original language are printed in italic, for example Gedankenexperiment or Preussische Akademie der Wissenschaften. The first mention of historical characters indicates their full name and dates of birth and death; subsequently only the name by which they are usually known is used. Most of the book remains faithful to the original notation, although it is sometimes cumbersome, so that interested readers may more easily consult primary sources. In the formulae, and within the text, physical quantities and Greek letters are represented in italics, for example, energy E or the parameter τ . For other mathematical symbols—operators, functions, numbers, etc.—ordinary characters are used. It’s time to get down to work. Barcelona, Spain April 2023

Luis Navarro Veguillas

Acknowledgments

I want to thank Tusquets Editores for the facilities given so that the present English translation of El desconocido Albert Einstein, originally published in Spanish in 2020, could appear. I would also like to acknowledge Dr. Angela Lahee her encouraging attitude in the preparation of this English version.

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Contents

1 The Sowing (Until 1905): The Annus Mirabilis . . . . . . . . . . . . . . . . . . . . . . . 1.1 The Young Einstein Bursts onto the Scientific Scene . . . . . . . . . . . . . 1.2 First Contributions: About Molecular Forces . . . . . . . . . . . . . . . . . . . . . 1.2.1 Capillarity and Molecular Forces (1901) . . . . . . . . . . . . . . . . . . 1.2.2 Potential Differences and Molecular Forces (1902) . . . . . . . . 1.3 Trilogy on Thermostatistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 A Word on Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 Thermal Equilibrium and the Second Law of Thermodynamics (1902) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.3 Foundations of Thermodynamics (1903) . . . . . . . . . . . . . . . . . . 1.3.4 Relevance of Statistical Fluctuations (1904) . . . . . . . . . . . . . . . 1.4 Gibbs’s Formulation Compared with Einstein’s . . . . . . . . . . . . . . . . . . . 1.4.1 Background, Premises, and Objectives . . . . . . . . . . . . . . . . . . . . 1.4.2 Methods and Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.3 Impact and Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Energy Quanta and the Photoelectric Effect (1905) . . . . . . . . . . . . . . . 1.5.1 Planck’s Energy Quanta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.2 Einstein’s Energy Quanta: A Heuristic Point of View . . . . . 1.6 Molecular Constitution and Brownian Motion (1905–1906) . . . . . . 1.6.1 Avogadro’s Number and Brownian Motion . . . . . . . . . . . . . . . . 1.6.2 Kinetic Theory of the Motion of Particles in Suspension (1905) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6.3 Some Precisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7 Experimental Confirmation of Einstein’s 1905 Predictions . . . . . . . . 1.7.1 Molecules Exist (Perrin, 1908–1909) . . . . . . . . . . . . . . . . . . . . . 1.7.2 Reality of Energy Quanta (Millikan, 1916) . . . . . . . . . . . . . . . . 2 The Flowering (1906–1913): Einstein Introduces Himself to the Scientific Community . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 From Bern to Zurich via Prague . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 The Critique of Planck (1906) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 From Radiation to Matter: First Quantum Theory of Solids (1907) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2.4 Quantum Refinements (1909–1910) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 A Highly Fruitful Gedankenexperiment . . . . . . . . . . . . . . . . . . . 2.4.2 On the Possible Dual Structure of Radiation . . . . . . . . . . . . . . 2.4.3 Stochastic Fields and Quantum Theory . . . . . . . . . . . . . . . . . . . 2.5 Interlude with Luminescence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1 Smoluchowski and His Interest in Fluctuations (1904–1908) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.2 Einstein on Critical Opalescence (1910) . . . . . . . . . . . . . . . . . . 2.6 Einstein Before International Scientific Community: First Solvay Conference (1911) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.1 Development of the Conference: Some Interventions . . . . . . 2.6.2 Einstein’s Communication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.3 Conclusions of the Conference and Einstein’s Opinions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 On the Necessity of Energy Quanta: Ehrenfest (1911) and Poincaré (1912) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.1 Ehrenfest (1911): “Weight Function” and the Necessity of Quanta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.2 Poincaré (1912): Mechanisms of Equilibrium and the Necessity for Quanta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.3 Two Very Different Impacts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8 Law of the Photochemical Equivalent (Einstein, 1912) . . . . . . . . . . . 2.9 Specific Heat of Gases and Quantum Theory (1911–1913) . . . . . . . 2.9.1 Planck’s “Second Theory” and “Zero-Point Energy” (1911) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.9.2 Einstein and Stern (1913): On the Specific Heat of Hydrogen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.9.3 Ehrenfest’s Counterproposal (1913): Rotational Quantisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 The First Harvest (1914–1924): In Search of the Photon . . . . . . . . . . . . . 3.1 Wartime and Peacetime in Berlin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Einstein, De Haas and Ampère’s Molecular Currents (1915–1916) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Einstein and De Haas Experiment . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 “Gyromagnetic Anomaly” and “Spinning Electrons” . . . . . . 3.3 Birth of the Photon (1916–1917) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 New Elementary Processes (1912–1916) . . . . . . . . . . . . . . . . . . 3.3.2 Reality of Quanta (1916–1917) . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.3 On Directionality of Elementary Processes . . . . . . . . . . . . . . . . 3.4 Digression: A New Quantisation Rule (1917) . . . . . . . . . . . . . . . . . . . . 3.4.1 Quantisation Rules: Times of Splendour (1913–1916) . . . . . 3.4.2 Einstein’s Quantisation Rule of 1917 . . . . . . . . . . . . . . . . . . . . .

90 92 95 98 99 103 105 109 111 115 118 121 121 127 132 133 138 139 141 143 147 149 155 156 158 161 161 165 169 172 173 176

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3.4.3 First Impact: Epstein’s Criticism . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.4 Influence of Einstein’s Rule on Louis De Broglie . . . . . . . . . 3.4.5 Einstein’s Rule and Wave Mechanics . . . . . . . . . . . . . . . . . . . . . 3.5 First Reactions to the Photon Concept. Alternative BKS . . . . . . . . . . 4 The Last Collection (1924–1925): Formulation of The First Quantum Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Fame and Unrest in Berlin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 A Shooting Comet: Bose, 1924 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 From Photons to Molecules: Quantum Theory of Ideal Gases, 1924 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Loss of Independence in the New Statistics, 1925 . . . . . . . . . . . . . . . . 4.5 On the Association of a Wave Field to Molecules . . . . . . . . . . . . . . . . 4.6 Digression: The Case of a Master Formula . . . . . . . . . . . . . . . . . . . . . . . 4.7 Additional Bases for the Quantum Theory of Ideal Gases, 1925 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8 From Quantum Gas Theory to Wave Mechanics. Schrödinger, 1926 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8.1 Schrödinger and the Quantum Theory of Ideal Monoatomic Gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8.2 Analogy Between Optics and Mechanics: Formulation of Wave Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Heterodox to the End: EPR, Cats and Entanglements . . . . . . . . . . . . . . . 5.1 A Long Lonely Road (1927–1955) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 The Beginning of the “Bohr–Einstein Debate” . . . . . . . . . . . . . . . . . . . 5.2.1 On the Fifth Solvay Conference, 1927 . . . . . . . . . . . . . . . . . . . . 5.2.2 Einstein and Bohr at the Fifth Solvay Conference . . . . . . . . . 5.2.3 First Phase of the Bohr–Einstein Debate, 1927–1930 . . . . . . 5.3 Einstein, Podolsky and Rosen Paper (EPR) of 1935 . . . . . . . . . . . . . . 5.3.1 Premises and Content . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 EPR Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 First Reactions to EPR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 Bohr’s Reaction: The Notion of Complementarity . . . . . . . . . 5.4.2 Schrödinger Reaction: Entanglements and Cats, 1935 . . . . . 5.4.3 Other Disquisitions: “Many Worlds” and “Wigner’s Friend” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Last Approach: Einstein and “Hidden Variables” . . . . . . . . . . . . . . . . . 5.5.1 Einstein’s Interpretation of the Quantum Formalism . . . . . . . 5.5.2 A Theorem with Pedigree: Von Neumann, 1932 . . . . . . . . . . 5.5.3 Bohm’s Surprising Hidden Variables Model, 1952 . . . . . . . . 5.5.4 Last Word . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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179 180 181 183 191 193 194 205 215 221 225 227 230 231 235 239 241 246 247 250 253 259 261 268 269 270 272 275 278 278 281 285 291

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6 An Unfinished Story: Bell’s Inequalities, Experiments and Loopholes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Bell’s Proposal, 1964 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Aspect Experiment, 1982: Loopholes and Refinements . . . . . . . . . . . 6.3 What if Einstein…? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

295 298 303 307

Appendix A: Einstein and Mari´c, an Ignored Collaboration? . . . . . . . . . . . 311 Appendix B: Albert Einstein (1879–1955). Brief Chronology . . . . . . . . . . . . 323 Appendix C: The Collected Papers of Albert Einstein . . . . . . . . . . . . . . . . . . . 329 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343

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The Sowing (Until 1905): The Annus Mirabilis

History is the most fundamental science, for there is no human knowledge which cannot lose its scientific character when men forget the conditions under which it originated, the questions which it answered, and the function it was created to serve. A great part of the mysticism and superstition of educated men consists of knowledge which has broken loose from its historical moorings. Farrington (1949), 173

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 L. Navarro Veguillas, The Lesser-Known Albert Einstein, History of Physics, https://doi.org/10.1007/978-3-031-35568-4_1

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In 1903 in Bern, the Swiss mathematician Conrad Habicht, the Romanian philosopher Maurice Solovine and the physicist Albert Einstein founded what they called the “Akademie Olympia”. This group of close friends met regularly to debate books in physics and philosophy [Courtesy of ETH-Bibliothek Zürich]

1.1 The Young Einstein Bursts onto the Scientific Scene

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The Young Einstein Bursts onto the Scientific Scene

Albert Einstein was the elder of the two children of Hermann Einstein (1847– 1902) and Pauline Koch (1858–1920). His sister Maria Einstein (1881–1951)— whom he affectionately called Maja—was always very close to him. In 1924 she wrote a short biography of her brother, including family data from which many of the biographical notes on Einstein’s youth are usually taken. Albert was born in Ulm, in southern Germany, in a family that taught him the principles of Judaism in a very liberal way. He received a religious education—first Catholic and later Jewish—in the public school in Munich, as prescribed by Bavarian law. In the autumn of 1894, when he was still three years short of completing his secondary education, he left the Munich school and went to Milan to join his parents, who had moved to Italy for work. He decided to prepare himself for entrance exams at the prestigious Eidgenössische Technische Hochschule (ETH) in Zurich. A year later, in October 1895, Einstein travelled to Zurich to take entrance exams at ETH, as he could not be admitted directly because he had not obtained a high-school diploma and was not yet 18 years old. Despite his outstanding performance in mathematics and physics, his results in the other subjects were not good enough to gain direct admission, so he was recommended to enrol at the Cantonal School of Aarau in Switzerland and take the courses he needed to obtain the Matura (high-school graduation). The widespread legend that Einstein’s initial academic performance was poor is unfounded. In October 1896 he took up residence in Zurich and entered the ETH for a four-year course that essentially qualified him to teach mathematics and physics in secondary schools. This was the beginning of an important stage in his life, not only because of his enduring attachment to the customs and way of life of the Swiss people, but also because there he would meet three people who were to have a decisive influence on the course of his life: his two fellow students Marcel Grossman (1878–1936) and Mileva Mari´c (1875–1948)—Einstein’s future wife—and Michele Angelo Besso (1873–1955), a Swiss engineer who was to be his closest friend and confidant for the rest of his life. Little is known about Mileva Mari´c. She was four years older than Einstein. The daughter of a high-ranking Hungarian official, she was born in Titel, then in southern Hungary, today in Serbia, and was the only woman in her class at ETH. Of the 11 students who started the course, only five made it to the final exams, but only Mileva failed. Albert’s grades were the lowest among the four successful students; his grades were very similar to Mileva’s. Only in “Theory of functions” were they clearly different: 11 out of 12 for him and 5 out of 12 for her, which was ultimately responsible for her failure. Mari´c would try again a year later and fail again, despite Einstein’s help. The other three classmates who graduated with Einstein managed to stay at ETH as assistants but Einstein was unable to do so. He blamed his failure squarely on Professor Heinrich Friedrich Weber (1843–1912), director of his diploma thesis and of his first doctoral thesis project. In addition to privately accusing him of personal animosity, he considered him incompetent as a scientist and not up to

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date with the latest developments in physics. On several later occasions, however, he would refer in highly laudatory terms to other professors at the ETH, such as Hermann Minkowski (1864–1909). Einstein tried, but failed, to get a job at a university, among others with Friedrich Wilhelm Ostwald (1853–1932) of Leipzig, the physicist-chemist famous for his contributions to catalysis, who was to receive the Nobel Prize in Chemistry 1909 “in recognition of his work on catalysis and for his investigations into the fundamental principles governing chemical equilibria and rates of reaction”, and with Heike Kamerlingh Onnes (1853–1926) of Leiden, a renowned specialist in low temperatures and future recipient of the Nobel Prize in Physics 1913, “for his investigations on the properties of matter at low temperatures which led, inter alia, to the production of liquid helium”. To make ends meet he had to resort to temporary jobs at the secondary school, until through the good offices of Grossman’s father he obtained his first serious job at the Swiss Patent Office in Bern as a “third-class technical expert” in June 1902. He was to remain there for seven years. Shortly before leaving for Bern, Einstein received a letter from Mari´c—at the time she was at his parents’ home—informing him of the birth of Lieserl, their daughter, whose existence was unknown until recently and who, in view of the almost total lack of information after her birth, must have died very soon or, more probably, was given up for adoption in view of the difficulties facing an unmarried mother in those days. From his autobiography, from the letters written to Mileva during the period of their courtship when they were living in different cities, and from the documentation included in his Collected Papers, it is possible to gain an idea of Einstein’s scientific interests and of his physics training at this time, which were acquired essentially by direct reading. For example, he had a high degree of assimilation of the electromagnetic field theory of James Clerk Maxwell (1831–1879) with its attendant applications and difficulties. He was familiar with the main contributions of Hermann von Helmholtz (1821–1894), Gustav Robert Kirchhoff (1824–1887), and Heinrich Rudolph Hertz (1857–1894) to the physics of the time in general and electromagnetism in particular. He was deeply impressed by the ideas in Ernst Mach’s (1838–1916) work on the historical–critical development of mechanics [Mach (1883)]. He had also followed the first attempts to formulate a theory of the electrodynamics of moving bodies undertaken by Hendrik Antoon Lorentz (1853–1928), joint winner with Pieter Zeeman (1865–1943) of the Nobel Prize in Physics 1902 “in recognition of the extraordinary service they rendered by their researches into the influence of magnetism upon radiation phenomena”.1 As an admirer of Boltzmann, Einstein was well acquainted with his ideas about the statistical conception of the second law of thermodynamics as well as of his kinetic theory of gases.

1

That is, for the discovery (Zeeman) and the explanation (Lorentz) of what is known today as the Zeeman effect.

1.1 The Young Einstein Bursts onto the Scientific Scene

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At certain times Einstein was also interested in the experimental work of Philipp E. A. von Lenard (1862–1947) on the production of cathode rays by ultraviolet light—called the photoelectric effect—which would catapult Lenard into receiving the Nobel Prize in Physics 1905, “for his work on cathode rays”. He was also familiar with the studies by Wilhelm Wien (1864–1928) on the thermodynamics of radiation, leading to the award of the Nobel Prize in Physics 1911 “for his discoveries regarding the laws governing the radiation of heat”. Einstein was obviously aware of Planck’s latest ideas on the properties of black-body radiation. Despite such an exquisite and thorough training in physics, Einstein seems to have started from an initially unfavourable position. His lack of connection with the academic world after the completion of his studies at the ETH could have been a negative factor when it came to tackling major problems. However, this same factor may have had a positive effect on his research: the boldness of his youth and the lack of links with authorities to approve his work may have facilitated the full development of his intelligence and imagination. Nor should we overlook the benefits, later acknowledged by Einstein himself, that his work as a rigorous and conscientious analyst of patent applications—which required constant contact with the world of inventions and experimentation—may have brought to his scientific formation. The months between the completion of his studies at ETH Zurich and his move to Bern do not represent a lost time for Einstein in terms of scientific activity. At the end of 1900—five months after his graduation—he submitted to the prestigious Annalen der Physik his first scientific paper, published in March 1901 under the title “Conclusions drawn from the phenomena of capillarity” [Einstein (1901)]. At the end of the same year, he presented an expanded version of this paper as his doctoral thesis at the University of Zurich. Professor Alfred Kleiner (1849–1916), an experimental physicist who held the only chair of physics at that time at the University of Zurich, was the supervisor. It is recorded that the thesis was withdrawn weeks later, presumably by Einstein himself, due to the negative opinion of those in charge of the mandatory report. His second work, along the same lines as the previous one, was not long in coming. It was also received, in Annalen, in the spring of 1902, when Einstein was already settled in Bern, and it was published in July under the title “On the thermodynamic theory of the difference in potentials between metals and fully dissociated solutions of their salts and on an electrical method for investigating molecular forces” [Einstein (1902 b)]. Both works pursue basically the same objective: the discovery of the characteristics of intermolecular forces by means of the analysis of their effects on certain liquids. In 1901, the phenomenon to be considered was capillarity, and in 1902, some electrical properties of metals in saline solutions were considered. Einstein’s plans had included his marriage to Mari´c, despite strong family opposition, as soon as he achieved basic financial independence. His recent employment at the Patent Office—and perhaps the death of his father in Milan at that time— made it easier to fulfil his purpose. The wedding took place at the beginning of 1903. The monotonous school years, the frequent changes of residence, the

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search for a decent job and the loud arguments with his parents, especially with his mother, because of their oppressive interest in preventing the marriage with Mileva at all costs, though recent, were far distant in his memory. Mileva’s intellectual interests certainly did not fit the traditional model his parents thought best suited to make him happy, but Einstein trusted that with a steady job, tensions would ease and he would be spared any more unpleasant comments about his now wife. In the summer of 1900, Albert describes in a letter addressed to Mileva part of a conversation with his mother when he informed her that her future daughter-in-law had failed to pass the final exam at the ETH2 : We come home, I into Mama’s room (just the two of us). First I have to tell her about the examination, then she asks me quite innocently: ‘so, what will become of Dockerl?’, ‘My wife’, say I, equally innocently, but prepared for a real ‘scene’. This then ensued immediately. Mama threw herself on the bed, buried her head in the pillow, and cried like a child. After she had recovered from the initial shock, she immediately switched to a desperate offensive, ‘You are ruining your future and blocking your path through life’. ‘That woman cannot gain entrance to a decent family’. ‘If she gets a child, you’ll be in a pretty mess’ (...). Next day things were already better, and this, as she herself said, for the following reason: ‘If they have not yet had intimate relations (so much dreaded by her) and will wait so long, then ways and means were surely be found’. Only what is most terrible for her is that we want to stay together forever. Her attempts at converting me were based on speeches like: ‘She is a book like you—but you ought to have a wife’. ‘When you´ll be 30, she´ll be an old hag’, etc. But as she sees that in the meanwhile she is accomplishing nothing except to make me angry, she has given up the ‘treatment’ for the time being.

Now everything would be different. He would no longer have to put up with such outrages. In 1901—before he moved to Bern—he had been exempted from military service (flat feet, varicose veins and excessive sweating) and, in the same year, he was granted Swiss nationality, which he would never lose despite all the vicissitudes of his life. Thus ended the five years he had spent as a stateless person after renouncing his German nationality in 1896. The young couple’s happiness was enhanced by the birth of their first child, Hans Albert, on May 14, 1904. Hans went on to complete his studies in Zurich, where he graduated in civil engineering in 1926 and received his doctorate in technical sciences in 1936. Both degrees were obtained from the famous ETH where his father had studied. Hans Albert would emigrate to the United States in 1938, where he acquired great prestige as a hydraulic engineer and became a professor at the University of California. He died in 1973. Shortly after Einstein’s marriage, he met Maurice Solovine (1875–1958), a young Romanian philosopher eager for ideas about the physics of the time, whom he would later sometimes refer to as the “good Solo”, and Conrad Habicht (1876– 1958), a friend from Zurich who had come to broaden his studies in mathematics. Albert became the leader of the trio that, with great pomp, they baptized the Akademie Olympia. They met regularly to discuss philosophy, physics, and literature. Although the Akademie was dissolved three years later due to the physical

2

Letter from A. Einstein to M. Mari´c, 29 (?) July 1900. In Beck (1987), 141–142.

1.1 The Young Einstein Bursts onto the Scientific Scene

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separation of its members, it was never erased from the memory of its leader. Half a century later, in a letter addressed to Solovine, Einstein recalled the discussions and readings that took place there—of Sophocles, Plato, Spinoza, Hume, Mach, Poincaré, Racine and Cervantes, among others, referring to “our happy ‘Academy’, which after all was less childish than those respectable ones which I got to know later from close in”.3 His two papers on the nature of the molecular forces had hardly any impact, although Einstein took it upon himself to send offprints to some of the leading physicists of the time, in particular to those to whom he had applied for an academic position. However, he himself must not have valued these two contributions very highly either. On a later occasion he referred to them in almost derogatory terms, considering them simply as the first two works of an inexperienced beginner in a difficult field. Nor does he make any reference to them in his scientific autobiography. The same was not true of the works that followed immediately afterwards. Between September 1902 and June 1904, when Einstein was already settled in Bern, he published three articles in Annalen devoted to presenting his own formulation of the classical statistical mechanics of equilibrium [Einstein (1902 b), Einstein (1903) and Einstein (1904)]. Although with clear originality, he follows the traditional line of Boltzmann’s kinetic theory, completely different from that which Gibbs had just presented in his 1902 book Elementary principles in statistical mechanics [Gibbs (1902)], a book which Einstein surely did not know until 1905, when its German translation appeared. The aim of the works that make up this trilogy is clear: to obtain the principles of thermodynamics, specifically the second one, from the—at that time only supposed—molecular constitution of matter, admitting that molecules obey the laws of Newtonian mechanics and by resorting to the probabilistic description of thermodynamic states introduced by Boltzmann. In the first, entitled “Kinetic theory of thermal equilibrium and the second law of thermodynamics” (1902), Einstein devotes his efforts to arriving at a microscopic deduction of the second principle, and obtains a mechanistic mathematical expression for entropy. In the second (1903), entitled “A theory of the foundations of thermodynamics”, he again obtains this principle, but now, according to Einstein, in its more general form: the impossibility of constructing a perpetuum mobile of the second kind (a cyclic machine that totally converts the heat absorbed from a single source into work). The last article of this trio, which appeared in June 1904 under the title “On the general molecular theory of heat”, shows a much more refined style than the previous ones and a greater depth in the treatment, which Einstein makes compatible with simplicity in the exposition. It is hard to imagine that its author was a young man of 25, at that time far from the academic world. It is an essential work for deep analysis if one is interested in the evolution of the scientific thought of our character.

3

Letter from A. Einstein to M. Solovine, 25 November 1948; quoted in Pais (2005), 47.

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It is in this article that Einstein shows the reasons that will lead him to the field of black-body radiation. It was a subject that had hitherto been outside his scientific interests. It is also in this paper that, for the first time, a clear relationship appears between the world of statistical physics—until then Einstein’s favourite one—and that of quantum physics, which was about to begin to absorb a good part of his energies. This relationship will be essential to understanding the development of Einstein’s ideas in these fields for at least the next 20 years. In 1905—his annus mirabilis as it is usually known—Einstein makes a definitive breakthrough in the scientific scene. It is the right moment. At last, he has achieved not only professional stability, but also emotional stability, both of which he had been pursuing since he finished his studies at the ETH. His five previous papers published in Annalen had familiarized him with how to write up his research results. Now, in addition to examining patents, he could tackle the indepth analysis of major questions in theoretical physics that had been troubling him for some time and about which he had some ideas. For example, since he was a child, he had been seduced by the mystery of the behaviour of a compass, always obliged to face north, guided by a mysterious force. When he was about to begin his university studies, he often thought of somewhat enigmatic situations, usually leading to the need to reveal the true nature of light. For example, one of these recurring thoughts consisted of imagining what would happen if one could ride on the back of a ray of light: what, if anything, would one see? Or maybe the question was not well posed? In accordance with the objectives set out in the Preface, we will leave aside two papers that Einstein published—also in Annalen—in 1905, which constitute the basis of his special theory of relativity.4 Our chronological study of Einstein’s contributions in 1905 will therefore begin with the analysis of his article “On a heuristic point of view concerning the production and transformation of light” [Einstein (1905 a)]. In this paper he suggests a then mysterious property of light: the apparently discrete behaviour of energy in its emission and absorption. We then discuss his doctoral thesis project entitled “A new determination of molecular dimensions”, which contains an original method for obtaining numerical values for molecular dimensions—and for Avogadro’s number—by application of the kinetic theory of gases [Einstein (1905 b)]. Similar ideas, as we shall see, lead him shortly thereafter to provide the first adequate theoretical explanation for Brownian motion in his next Annalen paper entitled “On the movement of small particles suspended in stationary liquids required by the kinetic-molecular theory of heat” [Einstein (1905 c)].

4

These two works of Einstein, as well as the other three famous ones of 1905, are translated into English and slightly commented in Stachel (1998).

1.2 First Contributions: About Molecular Forces

1.2

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First Contributions: About Molecular Forces

On 3 October 1900 Einstein writes to Mari´c from Milan—where the young man was spending a few days with his parents after finishing university studies in Zurich—anticipating his forthcoming return to Switzerland. He also informs her of his interest in the phenomena of capillarity5 : The results on capillarity, which I recently found in Zurich, seem to be totally new despite their simplicity. When we come to Zurich, we shall seek to get empirical material on the subject through Kleiner. If a law of nature emerges from this, we will send it to Wiedemann’s Annalen [Annalen der Physik].

Since very soon afterwards—on 16 October—the article in question was received by the magazine, the “natural law” found must have been very positively valued by the author. In any case, Einstein’s interest in intermolecular forces predates this period. In a previous letter to Mari´c he had already referred to his interest in investigating the nature of intermolecular forces6 : The Boltzmann [he means the book by Boltzmann (1896–1898)] is magnificent. I have almost finished it. He is a masterly expounder. I am firmly convinced that the principles of the theory are right, which means that I am convinced that in the case of gases we are really dealing with discrete mass points of definite finite size, which are moving according to certain conditions. Boltzmann very correctly emphasizes that the hypothetical forces between the molecules are not an essential component of the theory, as the whole energy is of the kinetic kind. This is a step forward in the dynamic explanation of physical phenomena.

1.2.1

Capillarity and Molecular Forces (1901)

In his 1901 paper “Conclusions drawn from the phenomena of capillarity”, Einstein takes the phenomenon of capillarity—for which there was a considerable amount of experimental data at the time—as a starting point for obtaining results on the nature of intermolecular forces. The work begins with simple reasoning in which, considering experimental results on the variation of the surface tension of a fluid with absolute temperature, he arrives at the conclusion that “the energy of the surface is of potential nature”. In other words, the molecular configuration of the surface layer of liquids does not depend on temperature but on the interaction between molecules. Thus, the key to the understanding of the properties of those surfaces—and, therefore, to the theoretical description of the phenomena of capillarity—will be found in the nature of the intermolecular forces.

5 6

Letter from A. Einstein to M. Mari´c 3 October 1900. In Beck (1987), 152. Letter from A. Einstein to M. Mari´c 13 (?) September 1900. In Beck (1987), 149–150.

10

1 The Sowing (Until 1905): The Annus Mirabilis

Allowing himself to be guided, Einstein affirms, by the case of the gravitational forces, he adopts the following expression for the interaction potential between two molecules: P = P∞ − c1 · c2 · ϕ(r),

(1.1)

where ϕ (r) is a function—unknown at that time—of the distance between both centres of gravity, c1 is a constant characteristic of each molecule and P∞ is the arbitrary value of the potential energy at infinity, since ϕ ( ∞ ) = 0. If there are n equal molecules, the resulting potential will be: P = P∞ −

n n ) 1 2 Σ Σ ( ϕ rα , β c 2

(1.2)

α=1 β=1

Σ In the case of polyatomic molecules c = cα , where the constants cα are the characteristic constants of the different atoms in the molecule. Transforming the sums for all the molecules into integrals for the whole volume, and supposing—implicitly—that both the interactions between the atoms of a molecule and between the molecules of a fluid are determined by the same function ϕ, Einstein obtains for the potential energy per unit volume (which he also designates as P): ( Σ P = P∞ − K

)2 cα ,

V2

(1.3)

where K is a numerical value determined from ϕ, and V is the molecular volume (quotient between the volume occupied by the fluid and the number of molecules that make it up). Similar reasoning leads Einstein to the expression for potential energy per unit area of fluid (he also designates it with P!): ( P = K'

Σ

)2 cα

V2

,

(1.4)

where both the numerical value of K ' and K are determined from ϕ. However, simple thermodynamic considerations allow him to deduce that this potential energy per unit area can also be written in the form: P =γ −T

dγ , dT

(1.5)

where γ represents the surface tension coefficient and T is the absolute temperature of the liquid. Thus, the comparison between the last two expressions leads to: / Σ dγ 1 cα = V γ − T (1.6) ·√ dT K'

1.2 First Contributions: About Molecular Forces

11

The numerical value of the right-hand side can be obtained from experimental data. Einstein appeals to those of R. Schiff for several liquids at boiling temperature, the molecules of which consisted of atoms of hydrogen, carbon, and oxygen. Adjusting by the method of least squares he obtained the values—relative ones, since K’ at (1.6) has an unknown value—of the corresponding constants: cH = − 1.6

cC = 5 5.0

cO = 4 6.8

Oddly enough, Einstein did not write any comment on the negative value of the first of these constants, which might suggest a repulsion of the hydrogen atoms towards each other. Presumably, since the value of this constant was obtained by an adjustment method, Einstein did not consider it appropriate to draw major physical conclusions from such a crude procedure. Σ With the above values, he calculates cα for molecules other than those previously used and compares the result with the experimental data, arriving at the conclusion that “It can be seen that in almost all cases the deviations barely exceed the experimental errors and do not show any trend [Gesetzmässigkeit]” [Einstein (1901), in Beck (1989), 5]. After repeating the procedure with more complex molecules, he obtains values for the constants for chlorine, bromine, and iodine, finding that the greatest differences between his theoretical predictions and experimental results occur for liquids with molecules of large mass and small volume. However, these discrepancies do not seem to Einstein of sufficient importance to distort the starting hypotheses and, in particular, the one that he understands as fundamental: to each molecule—and to each atom—there corresponds an attraction field independent of the temperature and of the chemical bond between atoms. Although Einstein ends by stressing the provisional character of his theory, which will require further complementary investigations, he did not return to the problems raised in this work. The only exception is a short article of 1911 in which he clarified that the assumption of his 1901 paper—although without explicitly citing it—about the universal character of the function ϕ (r) was untenable in light of the latest experimental results on surface tension in liquids [Einstein (1911 a), in Beck (1993), 329–330]. Einstein sent copies and offprints of his 1901 paper to accompany his application for employment in different academic institutions. However, the impact of this work was very small. At a time when the very existence of molecules was being questioned, an incipient theory about the properties of molecular forces did not seem a suitable subject for a young beginner in the field of research. In November 1901 Einstein presented a doctoral thesis at the University of Zurich (ETH was not authorized to confer the degree of doctor until 1909), although there is no certainty about its content as the manuscript has not been preserved. His correspondence with Mari´c, among other hints, seems to indicate that it was a compendium of his ideas about molecular forces in liquids and their possible extension to gases. The thesis was withdrawn in February 1902, presumably by Einstein himself. It was probably rejected, more or less explicitly, because

12

1 The Sowing (Until 1905): The Annus Mirabilis

of the excessively theoretical and speculative character of the treatment—opposite to what was usual at that time at the University of Zurich, where laboratory research prevailed, together with the lack of experimental data to certify the results obtained in the work.7

1.2.2

Potential Differences and Molecular Forces (1902)

Einstein’s interest in those days in determining the properties of molecular forces is clearly manifested in the correspondence with Mari´c. For example, shortly after the publication of his first paper in March 1901—in the same letter in which he informs his fiancée that Grossman’s father’s efforts to obtain him a job at the Patent Office in Bern were going along the right lines—he tells her about a possible generalization of the previous work to the study of molecular forces in gases, as well as the reasons for limiting his analysis in the case of solutions to completely dilute ones.8 A year later, in April 1902, now settled in Bern, he sent to Annalen der Physik a manuscript entitled “On the thermodynamic theory of the difference in potentials between metals and fully dissociated solutions of their salts and on an electrical method for investigating molecular forces”. The article, which appeared in July of the same year, begins with a section devoted to the presentation of a “hypothetical extension” of the second principle of thermodynamics: he assumes that this law must also be valid for the case of mixtures whose individual components are separated by semipermeable walls, as was then generally accepted in the theoretical treatment of mixtures and solutions. Note that, a priori, the applicability of the second principle in such cases is not absolutely guaranteed, since this type of separation could imply that the transformation of heat into work does not now conform to the requirements of that law. However, says Einstein, since both the thermodynamic theory of the dissociation of gases and that of dilute solutions give results in agreement with experiments, it seems justified to admit the validity of the second principle of thermodynamics for these cases. In other words, it is permissible to extend the validity of the second law to the case of mixtures whose components are subject to arbitrary conservative forces, which would play the same ideal role as that of semipermeable membranes, by making the different components occupy certain regions of space. Einstein advances that such a hypothesis, although not necessary, will be assumed throughout his work. He considers a system consisting of two electrodes of the same metal as the solute ions of a fully dissociated salt solution, for simplicity assuming that electrons, ions and solvent molecules are all subject to external conservative forces,

7

For details on this unsuccessful attempt of Einstein to submit a dissertation, see Editorial note: Einstein’s dissertation on the determination of molecular dimensions, in Stachel (1989), 170–182; specifically, 173–175. 8 Letter from A. Einstein to M. Mari´ c, 15 April 1901. In Beck (1987), 166–167.

1.2 First Contributions: About Molecular Forces

13

gravitational, for example. The system is assumed to be in electrical, mechanical and thermal equilibrium, at the absolute temperature T. A reasoning, certainly not too easy to follow, in which repeated use is made of the equilibrium, allows Einstein to obtain an expression for the difference between (Δ Π)2 , which is the difference of potential cathode-solution, and (Δ Π)1 , the difference of potential anode-solution: ( ) nm V m nm R T ν2 − (1.7) log (Δ Π)2 − (Δ Π)1 = (p02 − p 01 ), nE ν1 nE where n m represents the number of metal ions per molecule gram of salt, n is the sum of the valences of the metal ions, E is the amount of electricity necessary to electrolytically separate one univalent ion from one molecule gram of salt, ν 1 and ν 2 are the respective ion concentrations (at the anode and at the cathode), V m is the volume of a metal ion (in the metallic state), p 0 1 and p 0 2 are the respective hydrostatic pressures (at the anode and at the cathode) and R is a constant “common to all classes of ions”. As can be seen, the initially assumed external forces do not appear in the final expression, which seems to be consistent with the hypothesis introduced about the extension of the validity of the second law of thermodynamics to systems subjected to external forces. The author emphasizes that when the concentration of ions at the anode and at the cathode is the same, the minuend of the second member of equality (1.7) cancels out, while it is the subtrahend that cancels out when the hydrostatic pressure is the same at both electrodes. Thus, in both experimental situations, which are quite common, the expression is remarkably simplified. Next, Einstein shows that the quantity Δ Π depends on the nature of the acid, reaching the following conclusion [Einstein (1902), in Beck (1989), 18]: The potential difference between a metal and a completely dissociated solution of a salt of this metal in a given solvent is independent of the nature of the electronegative component, and depends solely on the concentration of the metal ions. It is assumed, however, that the metal ion of these salts is charged with the same amount of electricity.

He tackles the topic of other possible dependencies of Δ Π by dealing with its relation with the nature of the solvent, for which he begins by recalling his ideas on the molecular forces set out in 1901, especially the expression (1.1) as well as the main conclusions derived from it. On that basis he deduces an expression relating Δ Π to the constant c associated with each molecule of the solvent. Thus, by measuring these potential differences, one can evaluate, in principle, these constants. Alternatively, in Einstein’s own words, “this dependence can be used as a basis for a method of exploring the molecular forces”. The last part of the paper is aimed specifically at proposing an experimental method to determine the constant c—both for metal ions and for the solvent molecules—whose results could serve as a check not only for the expressions obtained on potential differences between electrodes in different situations, but also for his basic ideas on molecular forces. However, the proposed experimental

14

1 The Sowing (Until 1905): The Annus Mirabilis

test, in which it was necessary to measure potential differences between electrodes immersed in solutions subjected to diffusion phenomena, in a cylindrical container with a solvent and several completely dissociated salts, was not within the reach of the experimental scientists of that time and much less within the reach of Einstein, who ends the article apologizing for the unaffordability of his proposal [Einstein (1902), in Beck (1989), 29]: In conclusion, I feel the need to apologize for outlining here a skimpy plan for a laborious investigation without contributing anything to its experimental solution; but I am not in the position to do so. All the same, this work will have achieved its goal if it motivates a researcher to tackle the problem of molecular forces from this direction.

This author’s expectations were not fulfilled, for no one, to our knowledge, was interested in the proposed experimental investigation. Not even Einstein ever again truly concerned himself with the determination of the nature of the molecular forces along the lines anticipated in his articles of 1901 and 1902.

1.3

Trilogy on Thermostatistics

Three days after taking up his position at the Patent Office in Bern, Einstein sent to Annalen his first manuscript on the foundation of thermodynamics on a statistical basis, following developments by Boltzmann in the late nineteenth century [Einstein (1902 b)]. This article, then, was written at the time when he was looking for an appropriate job with some guarantee of continuity. During the two short years that had elapsed between completing his studies at the ETH in Zurich and settling in Bern, he had been earning his living as a teacher in a minor school and giving private lessons. The article was followed shortly after by two others on thermostatistics; one was received on 26 January 1903 and the other on 29 March 1904 [Einstein (1903) and Einstein (1904), respectively]. They constitute a trilogy that has come to be accepted, more or less implicitly, as a proper formulation of equilibrium statistical mechanics independent of, but equivalent to, that developed by Gibbs in 1902. Einstein’s self-confessed aim in his trilogy is to fill in, as far as possible, the gaps left by Maxwell and Boltzmann by trying to deduce the foundations of thermodynamics from the general principles of classical mechanics and of an adequate introduction of statistics in the description of macroscopic systems, which at the time was thought to consist of a very high number of molecules. It should not be forgotten that his previous training in the field had been acquired in a self-taught way, since the kinetic theories developed by Maxwell and Boltzmann—and much less the polemics surrounding them—were not yet included in the syllabus at the ETH.9

9

Among the books Einstein had studied was Boltzmann (1896–1898), the most influential in his training in kinetic theory. For more details, see Editorial note: Einstein on the foundations of statistical physics, in Stachel (1989), 41–55; specifically, 42–47.

1.3 Trilogy on Thermostatistics

15

The impact and subsequent development of Einstein’s proposal will be discussed below, as well as its relationship—including its supposed equivalence—to Gibbs’s formulation. However, we first analyze the content and meaning of each of the three articles.

1.3.1

A Word on Probability

Although statistical methods in their various facets were already in widespread use in 19th-century science, their full consolidation in physics would have to wait until the next century. However, the probability that appears in physics was never a concept free of difficulties. Among other reasons, the same terminology— “probability”—was frequently associated with different theories, notions and uses, without this being reliably stated. Given the prominent role of statistical methods throughout this book, we begin by considering three different ways of using probability to describe and study radically different situations. In the last third of the nineteenth century, probability theory was intensively applied to the justification of thermodynamics in microscopic terms. It was a question of starting from molecular processes that obeyed deterministic laws, such as the laws of classical mechanics, although these did not appear in the analysis, not only because the forces and the initial conditions of the molecular universe were not known with exactitude but also because both were irrelevant to reaching the goal. It is this conception of probability that presided over the developments of kinetic theories in the nineteenth century, culminating in the formulation of modern statistical mechanics at the beginning of the twentieth century. We call it “methodological probability”, and it is part of a procedure that explains the behaviour of thermodynamic systems in terms of the molecules that make them up. On other occasions, scientists have resorted to probabilistic descriptions because it turns out to be the only skill available at the time. This is what happened, for example, in the first attempts to describe radioactive disintegrations. Pending a better understanding of the phenomenon, and as long as no deterministic explanation of it was found, a statistical description of the apparently spontaneous decay of certain nuclei was useful to obtain some theoretical predictions that fitted reasonably well with the experimental results. This represents a “phenomenological probability”, which is usually described, rather than explained, in the history of physics, meaning that theories in which this type of probability is involved have a markedly provisional character. Quantum mechanics, formalized in 1925 and 1926, incorporates a new conception of probability, as its basic laws state that statistical description is the only possible way to access knowledge of the atomic world. At this level, theoretical predictions are no more than statistical in nature. We are now confronting an “essential probability”. It is part of a theory—quantum mechanics—which, as we shall see in Chap. 6, does not seem to be compatible with any deterministic theory.

16

1 The Sowing (Until 1905): The Annus Mirabilis

Unfortunately, these different conceptions of the role that probability can play in physical theories do not usually appear clearly differentiated either in historiographical works or in physics textbooks. In short, probability is the imprecise name of a concept whose outstanding role in the history of physics can only be understood if one is aware of the surname it carries. For Einstein, only probabilities of a methodological or phenomenological nature had a place at the heart of physics.

1.3.2

Thermal Equilibrium and the Second Law of Thermodynamics (1902)

The aims of Einstein’s research in this field are clearly stated at the beginning of this 1902 paper [Einstein (1902 b), in Beck (1989), 30]: Great as the achievements of the kinetic theory of heat have been in the domain of gas theory, the science of mechanics has not yet been able to produce an adequate foundation for the general theory of heat, for one has not yet succeeded in deriving the laws of thermal equilibrium and the second law of thermodynamics using only the equations of mechanics and the probability calculus, though Maxwell’s and Boltzmann’s theories came close to this goal. The purpose of the following considerations is to close this gap. At the same time, they will yield an extension of the second law that is of importance for the application of thermodynamics. They will also yield the mathematical expression for entropy from the standpoint of mechanics.

The first section of the article is merely a reminder of the description of a mechanical system by means of Lagrangian formalism, including the possible existence of forces that do not derive from a potential, which would be responsible for heat flows. In the absence of these forces, an adiabatic process occurs. In the second section, mechanical-statistical concepts are introduced by considering “infinitely many (N) systems of the same kind whose energy content is continuously distributed between definite, very slightly differing values E and E+δ E. External forces that cannot be derived from a potential shall not be present, and the potential V a shall not contain the time explicitly, so that the system will be a conservative one. We examine the distribution of states, which we assume to be stationary.” Further assuming that the energy, or a function of it, is the only constant of motion of such systems, the distribution of stationary states, or equilibrium distribution as we would call it today, is determined by the value of their energy E, which allows the definition of a distribution function of states ψ by means of equality: ∫ d N = ψ(p 1 , ..., q n )

d p 1 ...d q n , G

(1.8)

1.3 Trilogy on Thermostatistics

17

where, always in Einstein’s own notation, (p1 , ..., pn ) represent the position variables and (q 1 , ..., q n ) are the corresponding momenta.10 Furthermore, G (from Gebiet, region) denotes an infinitely small region of phase space in which the energy E(p 1 , ..., q n ) acquires values between E and E + δ E, and d N is the number of systems that, at a given instant, are in the state determined by the values of variables (p 1 , ..., q n ) whose representative points in phase space are contained in G. A short reasoning, essentially based on Liouville’s theorem, allows him to conclude that ψ must be independent of (p 1 , ..., q n ): it can only depend on energy, and therefore (1.8) can be written as follows: ∫ dN=A dp1 · · · dqn , (1.9) G

where A is a constant, independent of variables (p 1 , ..., q n ). Using current terminology, it can be stated that what Einstein has done thus far is to introduce the “microcanonical ensemble” by considering an infinitely large number of analogous mechanical systems with the same constant energy. To introduce the temperature into this is necessarily to complicate the situation because the thermal equilibrium—consubstantial with the notion of temperature—requires energy exchange between the system and thermal bath, while in the microcanonical formalism action is taken on isolated systems. Thus, in the third, fourth and fifth sections of his article, Einstein must consider a more complex scenario. It returns to consider an infinitely large number N of analogous mechanical systems whose individual constant energy E (we continue with the awkward Einsteinian notation) is comprised between E and E + δ E. However, each specimen of this virtual collection consists of a system S of energy E and mechanical state variables (p 1 , ..., q n ) and another one Σ, interacting with S, with state variables (π1 , ..., χn ) and energy H which is assumed to be “infinitely large” compared to E. Moreover, the interaction between the two systems S and Σ is assumed to be negligible, so that the energy of the complex system is E = E + H and its state variables are (p1 , ..., q n , π1 , ..., χ n ). If in the phase space of a complex system an “infinitely small” region g is considered, in which the value of energy E is between E and E + δ E, the number of systems whose state variables correspond to points contained in g will be given, according to (1.9), by: ∫ dN=A

d p1 · · · d χn

(1.10)

g

After referring to the continuity of A, which can only be a function of energy, without further explanation Einstein chooses for it the following form:

10

This notation of Einstein, designating the generalized coordinates as p and the corresponding moments as q, is the same as in Boltzmann (1896–1898), 271–273.

18

1 The Sowing (Until 1905): The Annus Mirabilis

A = A' exp (−2 h E) = A' exp (−2 h E) · exp (−2 h H ), where h is a new constant whose characteristics he will determine later. It is then that Einstein poses the question that allows the introduction of the “canonical ensemble”, which is the appropriate term to describe thermal equilibrium [Einstein (1902 b), in Beck (1989), 34]: How many systems [of type S] are in states in which p1 is between p1 and p1 +dp1 , p2 between p2 and p2 +dp2 … qn between qn and qn +dqn , but π 1 ,…,χ n have arbitrary values compatible with the conditions of our system?

It is obvious that this number can be written as follows: ∫ dN' = A' exp(−2hE)dp1 · · · dqn exp(−2hH )dπ1 · · · dχn ,

(1.11)

where the integral is extended over values of the state variables for which H lies between E − E and E + δ E − E. After showing that the latter integral must be independent of E, Einstein shows that this implies a univocal determination of the constant h: h=

1 ω' (E) , 2 ω(E)

(1.12)

where ω (E) is the following “hypervolume”: ∫ ω (E) =

dπ1 · · · dχn ,

(1.13)

where the integral extends to all values of the variables compatible with the range of energy determined by E and E + δ E. It is easily verified that both ω (E) and ω' (E) represent positive quantities, so h will also be positive, which just translates the fact that the hypervolume of the phase space is increasing with energy. All of the above makes it possible to answer the question posed above, simply by rewriting the expression (1.11) in the form: d N ' = A'' exp (−2 h E ) d p 1 · · · d q n ,

(1.14)

where the energy and variables of systems S appear explicitly and the rest (variables and energy of systems Σ) are included in the constant A'' , which is independent of the energy E. This is the first occasion in which the expression of the canonical distribution appears explicitly written in the article. The rest of the paper is essentially devoted to establishing it and showing conclusions that can be deduced from it, starting by translating the above into the language of probability theory and establishing the corresponding normalizations. Einstein begins his analysis of thermal equilibrium considering a specific system of type S, which he calls a “thermometer”. It interacts with another of type Σ—which today we would call thermal bath or thermostat—whose energy is

1.3 Trilogy on Thermostatistics

19

infinitely large in relation to that of the first one. Assuming that the whole system is in a stationary state, the state of the thermometer can be defined by the distribution: d W = A exp (−2 h E ) d p 1 · · · d q n ,

(1.15)

where dW represents the probability that the variables p1 , ..., q n of the state of the thermometer are within the range bounded by p1 and p1 + dp1 , and so on up to qn and qn + dqn . The constants A and h are related by the normalization condition: ∫ 1 = A exp (−2h E ) d p1 · · · d qn , (1.16) where the integral extends to all possible values of the thermometer variables. Thus, the constant h, which Einstein calls the “temperature function”, determines the state of the thermometer S, although it is not a function of its energy, but is determined, as we saw earlier, by the energy of the system Σ and does not depend on how it is coupled to the system S. The usual temperature, similar to the rest of the observables of S, will depend on h. Thermal equilibrium between two systems is only possible if the respective temperature functions—and hence the usual temperatures—are equal. Two mechanical systems subjected to a weak interaction that leads them to equilibrium maintain equal temperatures if they are then separated. In addition, reciprocally, two systems with the same temperature can be coupled without appreciable changes in the distributions of their states. Furthermore, if two systems are in thermal equilibrium and one of them is in thermal equilibrium with a third, this one must also be in equilibrium with the remaining one. It is here that Einstein emphasizes that the mechanical systems to which he is referring must satisfy Liouville’s theorem and the principle of conservation of mechanical energy, both of which were used in the treatment. Other more general systems were, at least at that time, excluded from his analysis. The aim of the sixth section of Einstein’s article was to explore the physical meaning of h.11 Keeping the notation L for “living force” (from the Latin vis viva), for what today we call kinetic energy, Einstein considers the case in which this is a homogeneous quadratic function of the variables q i , which can be linearly transformed into another variable r n which Boltzmann called “momentoids” (Momentoiden), so that in these new variables, kinetic energy can simply be written as follows: L=

11

) 1( α 1 r 21 + α 2 r 22 + ... + α n r 2n 2

(1.17)

Einstein quotes paragraphs 33, 34 and 42 of the second part of Boltzmann (1896—1898). This book is the only reference Einstein includes in his paper.

20

1 The Sowing (Until 1905): The Annus Mirabilis

From (1.15) it is trivial to check that each momentoid contributes the same to the mean kinetic energy of the system: L 1 = n 4h

(1.18)

To finally find the relationship between the temperature function and the absolute temperature, Einstein now resorts to the ideal gases. Assuming that the Maxwell distribution is a particular case of (1.15) and taking into account some precepts of the kinetic theory of gases, he obtains: 1 = κ T, 4h

(1.19)

where κ is a universal constant, which is exactly half of what would later become known as the Boltzmann constant. The last two relations lead to L/n = κ T , that is, to the principle of equipartition of kinetic energy. It is curious, to say the least, that Einstein did not even refer to such a principle here. Einstein devotes the following section, which is undoubtedly the most ambitious, to obtaining the second principle of thermodynamics as a consequence of the laws of mechanics, but always with some additional assumptions. We present an outline of it. Deduction of the Second Principle of Thermodynamics (1902)

To this end, Einstein starts from a mechanical system ( S with generalized ) coordinates (p 1 , . . . , p n ) and their time derivatives p'1 , . . . , p 'n . He designates by (P 1 , . . . , P n ) the components of the external forces “tending to increase the coordinates of the system”. If V i represents the potential energy of the system, and L its kinetic energy (which is assumed to be a homogeneous quadratic function of the derivatives p 'ν ), the corresponding Lagrange equations take the following form: [ ] ∂(vi − L) d ∂L − Pv = 0, (v = 1, . . . , n), + ∂pv dt ∂pv'

(1.20)

where we continue to use, for the sake of fidelity, the tedious original notation. Einstein divides the external forces Pν into two different groups: one, which he designates Pν(1) , is associated with the system’s conditions (adiabatic walls, gravitational forces, etc.); and others, Pν( 2 ) = Π ν , are (1) responsible for heat exchange. Only the Pν derive from a certain potential Va , which is a function of the variables (p 1 , . . . , p n ). This potential may depend explicitly on time, but it is assumed that the derivatives of the

1.3 Trilogy on Thermostatistics

21

(1)

forces Pν with respect to time are “infinitely small” since, in the processes to be treated, the system can always be considered to be in a stationary state. By introducing potential V = V i + V a , associated with the set of all the forces that derive from a potential, Lagrange equations (1.20) can be rewritten in this form: [ ] d ∂L ∂(vi − L) , (v = 1, . . . , n) (1.21) + Πv = ∂pv dt ∂pv' The work done by these forces Π ν , in a small time interval dt, represents the corresponding amount of heat—which Einstein somewhat inappropriately designates as d Q—absorbed by the system S during that time. That is to say: dQ =

n Σ

Πν d p ν

(1.22)

ν=1

A simple calculation, which starts from this equality and takes into account expressions (1.19) and (1.21), leads to the following relation: n Σ ∂V dQ dL +4 κh d pν =nκ T ∂ pν L ν=1

(1.23)

Einstein goes on to deal with the summation in the above expression, beginning by pointing out that it represents the increase of the potential energy of the system in the time interval dt, assuming that V does not depend explicitly on time. The value of this time interval must be sufficiently large that the summation can be replaced by its average over an infinity of systems S with the same temperature T, and sufficiently small that the consequences of the possible explicit time variation of V and h are not appreciable. Einstein redefines potentials and energies, which he renames V* and E*, respectively, so that the constant A of (1.16) is equal to unity. Substituting the summation in the expression (1.23) by the indicated mean value, and after a calculation that is not excessively clear—it seems to be confirmed by the fact that Einstein modifies it in the following work, as we shall see—he arrives at the following expression: dQ =δ T

(

) E∗ , T

(1.24)

where the symbol δ is used to represent the variation of the corresponding quantity “during the transition of a system to a new state”.

22

1 The Sowing (Until 1905): The Annus Mirabilis

In view of the last expression, Einstein has arrived at a double result: d Q/T turns out to be “a complete differential”—total differential, or exact differential, as we say today—and, moreover, E ∗ /T = (V ∗ +L)/T is “the expression for the entropy of the system”. His final conclusion is categorical [Einstein (1902 b), in Beck (1989), 46]: “The second law [of thermodynamics] thus appears as a necessary consequence of the mechanistic world picture”. The short ninth section of the work is devoted to the calculation of entropy. Apparently, this had already been/calculated at the end of the previous section by means of the expression ε ≡ E∗ T , but it was necessary to determine E∗ from the quantities of the system; specifically, from its energy E. A simple calculation leads Einstein to the following final entropy expression: E∗ E ε= = + 2 κ log T T

[∫

] exp (−2h E) d p1 · · · d q n + const.

(1.25)

Thus, entropy depends only on E and T —something that Einstein considers “strange” (merkwürdig)—but not on the breakdown of energy into potential and kinetic terms, which leads him to the following consideration [Einstein (1902 b), in Beck (1989), 47]: This fact suggests that our results are more general than the mechanical model used, the more so as the expression for h found in §3 [our (1.12)] shows the same property.

One slip of the tongue—at the very least—on the part of the young Einstein, which the editors of his complete works have not overlooked either: both in the last expression and in his subsequent discussion, he should have written E, instead of E [Stachel (1989), 75, note 41]. He did not do so here, nor did he do so in the rest of the trilogy. Two years later, though, he wrote this expression correctly in his article on Brownian motion. Certainly, on this occasion, his particular method of calculation did not make it easy to make out the difference. The young man must not have been conceptually mature yet, or he would not have missed the fact that, in the canonical ensemble—the one suitable for describing thermal equilibrium—temperature is a datum but not the energy of the system, which is subject to fluctuations; only mean energy makes sense with regard to describing thermal equilibrium, as was the case. The bombastic title of the final brief section is nothing less than “Extension of the second principle of thermodynamics”, which suggests rather more substance than is developed in the ten lines it consists of, and also than Einstein anticipated in the introduction of the article [Einstein (1902 b), in Beck (1989), 47]: No assumptions had to be made about the nature of the forces that correspond to the potential V a [from which the external forces are derived], not even that such forces occur in

1.3 Trilogy on Thermostatistics

23

nature. Thus, the mechanical theory of heat requires that we arrive at correct results if we apply Carnot’s principle to ideal processes, which can be produced from the observed processes by introducing arbitrarily chosen V a ´s. Of course, the results obtained from the theoretical consideration of those processes have a real meaning only when the ideal auxiliary forces [which derive from] V a no longer appear in them.

Translated into clearer language, Einstein states that the applicability of the second principle of thermodynamics to systems whose physical conditions are not easily attainable is guaranteed independently of the feasibility of a realistic mechanical description. The main result of this 1902 paper is the step taken by Einstein, along the lines of Boltzmann, in deducing the canonical ensemble distribution, taking the microcanonical one as a starting point. The introduction of the constant h, together with the determination of its relation to temperature, led him to study thermal equilibrium in a more natural way than had been done before. The use of the canonical distribution allowed him to present a statistical justification of the second law of thermodynamics and to deduce the equipartition of kinetic energy as well as obtain the corresponding mechanical expressions for constant h and entropy ε.

1.3.3

Foundations of Thermodynamics (1903)

This manuscript was received at Annalen on 26 January—seven months after the first of the trilogy—and published on 16 April. Einstein’s assessment of this work, as well as the confirmation of his physical and emotional settlement—he had already been working at the Patent Office for seven months and had been married to Mari´c for 20 months—are clearly expressed in his letter to Besso on 22 January from Bern12 : Thank you so much for your letter. Well, now I am a married man and am living a very pleasant, cosy life with my wife. She takes excellent care of everything, cooks well, and is always cheerful. I am very curious about the note about your work, and pleased with the gentle flattery that you have joined with it. (The day before yesterday) Monday, after many revisions and corrections, I finally sent off my paper. But now the paper is perfectly clear and simple, so that I am quite satisfied with it. The concepts of temperature and entropy follow from the assumption of the [conservation of] energy principle and the atomistic theory, and so does also the second law [of thermodynamics] in its most general form, namely the impossibility of a perpetuum mobile of the second kind, if one uses the hypothesis that state distributions of iso [abbreviation for isolated] systems never evolve into more improbable ones.

In the introduction Einstein states the aims of the new work, as well as its relation to the 1902 paper [Einstein (1902 b), in Beck (1989), 48, emphasis added]:

12

Letter from A. Einstein to M. Besso, 22 (?) January 1903. In Beck (1994), 7. Emphasis in original.

24

1 The Sowing (Until 1905): The Annus Mirabilis In a recently published paper I showed that the laws of thermal equilibrium and the concept of entropy can be derived with the help of the kinetic theory of heat. The question that then arises naturally is whether the kinetic theory is really necessary for the derivation of the above foundations of the theory of heat, or whether perhaps assumptions of a more general nature may suffice. In this article it shall be demonstrated that the latter is the case, and it shall be shown by what kind of reasoning one can reach the goal.

These “assumptions of a more general nature” included hypotheses additional to those of a strictly mechanical nature, as could not be otherwise, while at the same time giving full meaning to the statistical description of thermodynamic systems, the calculation of averages and the identification of these with the results of measurements of macroscopic physical quantities. In short, these new assumptions are an essential part of the conceptual bases on which the classical statistical mechanics of equilibrium are based. The first section of this paper is devoted to the mechanical description of isolated systems with a large number of degrees of freedom. The corresponding mechanical variables are denoted by (p1 , ..., pn )—thus simplifying the notation of the previous work by eliminating the q variables, which now are included by including them in the set of ps—and their evolution in time is governed by the mechanical equations of motion. Without explicit justification, Einstein assumes that there is no constant of motion other than energy or functions of it. In the second section, once the stationary states have been defined, hypotheses are established, and methods are presented that are key to properly introducing the mechanical-statistical formulation of the problem. Stationary states are those that, in light of experience, any isolated system ends up reaching, following its temporal evolution. Once one of these states has been reached, the observable physical properties of the system do not change in the course of time. Regarding these properties, Einstein writes [Einstein (1903), in Beck (1989), 49]: If we now assume that a perceptible quantity is always represented by a time average of a certain function of the state variables p1 ,…,pn , and that these state variables p1 ,…,pn keep on assuming the same systems of values with always the same unchanging frequency, then it necessarily follows from this condition, which we shall elevate to a postulate, that the averages of all functions of the quantities p1 ,…,pn must be constant; hence, in accordance with the above, all perceptible quantities must also be constant.

The previous paragraph is crucial as far as the foundation of statistical mechanics is concerned because it introduces—albeit in a rudimentary way—the “ergodic hypothesis” and the need to resort to mean values of physical quantities. We will analyze these assumptions in more depth. However, it should be noted that these are not real novelties, since both Maxwell and Boltzmann formulated similar hypotheses, although not totally equivalent to Einsteinian ergodicity, for example. In fact, Boltzmann, in 1871, extrapolating the results of his analysis of Lissajous

1.3 Trilogy on Thermostatistics

25

figures corresponding to pairs of incommensurable frequencies, had formulated the following hypothesis about the atoms of a gaseous system13 : The great irregularity of the thermal motion, and the multiplicity of forces that act on the body from outside, make it probable that the atoms themselves, by virtue of the motion that we call heat, pass through all possible positions and velocities consistent with the equation of [conservation of] kinetic energy.

And in 1879, the year of his death, Maxwell had stated [Maxwell (1879), in Brush (1976), vol. 2, 366]: The only assumption which is necessary for the direct proof [of the energy equipartition theorem] is that the system, if left to itself in its actual state of motion, will, sooner or later, pass through every phase which is consistent with the equation of [conservation of] energy.

Without claiming a detailed analysis of the evolution of ideas about ergodicity and its relation to the foundations of statistical mechanics, it is worth noting here that neither Maxwell nor Boltzmann, contrary to what is often asserted, believed that ergodicity was truly a property guaranteed by the laws of statistical mechanics. Nothing more clarifying in this respect than the words of Maxwell himself [Maxwell (1879), in Brush (1976), vol. 2, 367]: But if we suppose that the material particles, or some of them, occasionally encounter a fixed obstacle such as the sides of the vessel containing the particles, then, except for special forms of the surface of this obstacle, each encounter will introduce a disturbance into the motion of the system, so that it will pass from one undisturbed path into another. The two paths must both satisfy equations of energy (...), but they are not subject to the equations of momentum. It is difficult in a case of such extreme complexity to arrive at a thoroughly satisfactory conclusion, but we may with considerable confidence assert that except for particular forms of the surface of the fixed obstacle, the system will sooner or later, after a sufficient number of encounters, pass through every phase consistent with the equation of [conservation of] energy.

Thus, according to the introducers themselves, the validity of this type of hypothesis is not guaranteed only by the mechanical laws of movement. Moreover, the very notion of ergodicity is not uniform, because while for Boltzmann it seems that one single trajectory passes through all the phases compatible with the conservation of energy, Maxwell suggests that the incessant variation in the trajectory caused by the walls of the vessel is necessary to reach the same conclusions. Although the innovation did not prove decisive, let us see how Einstein introduces a different kind of ergodicity that he extends to an arbitrary set of independent trajectories.

13

Boltzmann (1871). Reprinted in Brush (1976), vol. 2, 366. For the origins of the ergodic problem see sections 10.10 and 10.11 of the latter; on pages 367—372 there appears a brief history of the term “ergodic”, introduced by Boltzmann in 1884, and of its various meanings.

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1 The Sowing (Until 1905): The Annus Mirabilis

He starts from a system in equilibrium for which the values of the variables (p1 , ..., pn ), i.e., its trajectory in phase space, are known over a certain time interval T, and Γ is a region of phase space in which the system ‘remains’ for a time τ.14 Einstein now introduces the following postulate: “… if a system assumes a stationary state, then for each region Γ, the quantity τ/T takes for T = ∞ a definite limiting value. For any infinitely small region this limiting value is infinitesimally small…” [Einstein (1903), in Beck (1989), 50]. The previous reasoning allows Einstein to introduce the probabilistic description that he had neglected in the first article of the trilogy. After considering a large number N of independent analogous systems, all with the same energy E—a value in the infinitesimal interval (E ∗ , E ∗ + δ E ∗ )—and governed by the same equations of motion but with possibly different initial conditions, the question arises: what is the probability that one of these N systems, chosen at random at an instant t, has its state variables in that region Γ? Einstein considers that such a probability is “immediately deduced” from the previous postulate and writes its value as follows: lim

T=∞

τ = const. T

Thus, the number of systems whose state variables are in region Γ is given by: N lim

T=∞

τ , T

an expression that, Einstein recalls, does not depend on the arbitrarily chosen instant t. This allows him to conclude that the number of systems whose state variables are found, at any instant, in the interior of an infinitely small region g around the point (p1 , ..., pn ) will be given by the expression: ∫ d N = ε (p 1 , ..., p n )

d p 1 · · · d p n,

(1.26)

g

where the function ε (p 1 , ..., p n ) will be determined from our information on the system. The stationary character of the distribution represented by the last expression, together with a suitable transformation of the state variables (p1 , ..., pn ) into the more appropriate ones (π1 , ..., π n ), allows Einstein to rewrite the latter expression as follows: ∫ d N = const.

dπ1 · · · dπn ,

(1.27)

g

14

In this article, Einstein employs—and let us remember that we are following his own notation— the letter T to designate both the absolute temperature and also a certain time interval; but certainly, he only uses the latter option in brief considerations on his ergodic hypothesis.

1.3 Trilogy on Thermostatistics

27

assuming from now on that, for the sake of convenience, the state variables employed will always be those that allow writing the latter expression. Thus, the ergodic hypothesis has led to the equiprobability of regions of phase space with equal volume if they are compatible with the energy of the system. In language more in keeping with the present day, it could be said that, in a given mechanical system with unknown initial conditions, all those compatible with the energy are equally probable. This is usually understood as the fundamental principle on which rests the microcanonical distribution expressed by (1.27), an expression that was already contained in the 1902 paper—see our (1.9), but then it was practically postulated, whereas now it has been deduced from an ergodic hypothesis with which Einstein gives more solidity to his probabilistic description of isolated systems. In the third section of the paper, entitled “On the distribution of states of a system in contact with a system of relatively infinitely large energy”, Einstein introduces a new notation. Each of the N isolated systems, all with energy E within the interval (E ∗ , E ∗ + δ E ∗ ), is composed of two subsystems, Σ and σ, with respective energies H and η. The state variables are (Π 1 ,..., Π λ ) for Σ systems, and (π1 ,...,πl ) for σ systems. As usual in these treatments, the interaction between the two subsystems is still assumed to be negligible, so that E = H + η is satisfied. It is also admitted that the energy η of each subsystem of type σ (the thermometers of 1902) is infinitely small compared to the energy H of those of kind Σ (the thermostats). An essential point concerns the exponential form of the state distribution associated with thermometers. However, in this second article of the trilogy there are no appreciable novelties in relation to the previous one. Einstein again obtains the expressions (1.12) and (1.14), although now written with the new notation. On the other hand, the relation (1.19)—between the constant h and the absolute temperature T —which in 1902 had been justified by appealing to the kinetic theory of gases, is now introduced as a definition of absolute temperature, thus freeing Einsteinian formulation from any dependence on the kinetic theory, a precursor discipline of statistical mechanics but distinct and independent from it. Einstein summarizes the result thus [Einstein (1903), in Beck (1989), 55]: Thus, the state of the system σ depends only on the quantity h, and the latter only on the state of the system Σ. We call the quantity 1/4hκ = T the absolute temperature of the system Σ, where κ denotes a universal constant.

The adoption of this definition allows Einstein to consider the case in which the subsystem σ is constituted by an ideal gas molecule and the rest of the gas is identified with Σ. By comparison with the kinetic theory of gases he deduces that the definition of absolute temperature introduced is consistent with the experimentally contrasted results of that theory. Thus, the kinetic theory does not properly form part of the body of Einstein’s formulation but is only used to test the reliability of the new theory. Of course, if equilibrium exists, h has the same value in both subsystems Σ and σ, so the rules of thermal equilibrium are obtained along the lines of the previous work of 1902.

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1 The Sowing (Until 1905): The Annus Mirabilis

It should be emphasized that in this second article of the trilogy, Einstein follows practically to the letter the methodology set out in the first one, although now the hypotheses are clearly specified, and more rigour is used in the treatment. In particular, he again obtains a deduction of the second law of thermodynamics, starting by making the concept of entropy more precise. Let us look at it. A New Deduction of the Second Principle of Thermodynamics (1903)

Einstein is now interested in systems out of equilibrium, limiting himself to modifications of equilibrium states under “infinitely slow processes”: those that occur “so slowly that the distribution of states existing at an arbitrary instant differs only infinitesimally from the stationary [equilibrium] distribution”. In such a process it is convenient to consider certain additional parameters λi , to characterize the interaction of the system with external agents capable of modifying the value of its energy. The total variation of the energy of a thermometer, in an infinitely slow process of infinitesimal duration dt, is given by: dE=

Σ ∂E Σ ∂E d λi + d pν ∂ λi ∂ pν

(1.28)

In adiabatic processes the second summand of this expression is null because, as there is no external influence on the system, its energy must be conserved (remember that variables pν now represent all the mechanical variables). Einstein called “isopycnic processes” those in which all the parameters λi are constant, so that in these processes it is the first summand of (1.28) that cancels out.15 The value of d E in such a process, Einstein argues, must be identified with the heat absorbed by the system in the time interval dt. Thus, in any infinitely slow process the equality is verified: Σ∂E dE= d λi + d Q (1.29) ∂λ i

15

Boltzmann had already used the term “isopycnic” to designate changes of state at constant volume; see, for example, Boltzmann (1896–1898), 259.

1.3 Trilogy on Thermostatistics

29

If in an infinitely slow process the system is kept in contact with another which has larger energy than the first, the distribution of states of this, according to (1.15), will be given by: d W = const. exp (−2 h E ) d p 1 · · · d p n = exp ( c−2 h E ) d p 1 · · · d p n , (1.30) where the constant c is determined by the normalization condition: ∫ exp (c−2 h E ) d p 1 · · · d p n = 1

(1.31)

Certain considerations concerning initial and final situations in an infinitely slow process, together with a simple calculation using the last three expressions and (1.19), allow Einstein to obtain the relation: dQ =d T

(

) E − 2 κ c = dS T

(1.32)

Thus, he demonstrates again—as he had already done in 1902—that “dQ/ T is a total differential of a quantity that we will call the entropy S of the system”. In addition, taking into account (1.32) and (1.31) he obtains the following final expression for this entropy: S=

E + 2κ log T

∫ exp(−2hE)dp1 · · · dpn ,

(1.33)

where the integral extends to all accessible values for the variables of the system. Of course, we must reiterate here our comment of the previous section: in the last two expressions Einstein should also have written E, instead of E. However, he failed to do so and continued to carry over the 1902 inaccuracy. It was all Einstein needed to reach the objectives pointed out in his letter to Besso quoted in footnote 12. To achieve this, he had to introduce an additional hypothesis, which forms part of the “assumptions of a more general nature” mentioned at the beginning of this section, with the aim of avoiding recourse to kinetic theory. Even if only briefly, we will dwell on the exposition of the problem and of the proposed solution. Let us consider a very large number N of isolated systems described by the same mechanical equations and with the same energy, except for very small amounts of this. We have seen that the stationary distribution of states is given by the relation (1.27), which leads to the following expression for the probability dW that the values of the state variables of a system, arbitrarily chosen among N,

30

1 The Sowing (Until 1905): The Annus Mirabilis

are in a certain region g, compatible with the energy: ∫ d W = const. d p1 · · · d pn

(1.34)

g

The above implies that, if one subdivides the whole accessible region of the phase space into enclosures g1 , g2 , …, gμ , all with the same hypervolume, and the respective associated probabilities are represented by W1 , W2 , …, Wμ , the following is verified: W 1 = W 2 = ...= W μ ,

(1.35)

which expresses the equiprobability of regions of phase space with the same volume, provided that the values of the state variables of the system are compatible with the total energy of the system.16 Thus, the probability that of the N systems considered at any instant, ε1 are in g1 , ε2 in g2 and so on up to εμ in gμ , is given by: ( )N N! 1 W= μ ε1 ! ε2 ! · · · εμ !

(1.36)

Since numbers ε1 , ε2 , …, εμ are much larger than unity, it is permissible to apply the Stirling approximation, which leads to the following expression: ε = εμ

log W = const. −

Σ

ε log ε

(1.37)

ε = ε1

Finally, if the subdivision is made into a very large number of infinitesimal enclosures—that is, if μ is much larger than 1—it is possible to replace the summation of the last expression by the corresponding integral, without sensible errors: ∫ log W = const. − ε log ε dp1 · · · d pn (1.38) In this latter equality the numbers ε are determined by the distribution of the state variables. When considering isolated systems and stationary distributions, the appropriate distribution is the microcanonical one expressed by (1.27) according to which all the numbers ε are equal (they do not depend on the variables of the system) and, as can be easily demonstrated, log W reaches its maximum value. If ε depended on the variables (p1 , ..., p n ), it is easy to verify that log W would not present any minimum; that is, given a distribution of these characteristics, there

For typographical reasons, we have adopted here the subscript μ to designate the number of enclosures in the phase space. In the original this number is represented by the letter l.

16

1.3 Trilogy on Thermostatistics

31

would always exist distributions that would differ infinitesimally from it and for which log W is larger. It is here that Einstein explicitly introduces the additional hypothesis to which we have referred a few lines above: the distribution of states—and therefore also log W—varies with time continuously, but it does so in such a way that “we will have to assume that always more probable distributions of states will follow upon improbable ones, i.e., that W increases until the distribution of states has become constant and W [reaches] a maximum” [Einstein (1903), in Beck (1989), 63]. This allows Einstein to conclude that, if ε and ε’—functions of the variables (p1 , ..., p n )—are associated with distributions at instants t and t’,respectively, it is verified that: ( ) − log ε' ≥ − log ε for all t ' > t (1.39) The last expression represents a form of temporal irreversibility, one of the essential aims of Einstein’s paper and a great novelty in relation to that of 1902, in which no reference was made to the universal tendency of thermodynamic systems towards equilibrium, but it was assumed from the outset that this was the state the system was always in. The Einsteinian hypothesis according to which the “most probable distributions” always follow the least probable ones is certainly difficult to admit without further ado unless one accepts in advance some form of temporal irreversibility outside Newtonian mechanics. The necessity of justifying such a hypothesis was denounced by Paul Hertz (1881–1940) in 1910 and was followed by verbal discussions between the two parties involved. We will return to this point later on, although we can anticipate that Einstein settled the controversy with a very brief article—only four paragraphs—in which he limited himself to recognizing the validity of Hertz’s criticism, referring to Gibbs’s book [Gibbs (1902)] as the right place for a rigorous treatment of thermal equilibrium starting from the canonical ensemble and not from the microcanonical ensemble, as Einstein had done in his papers [Einstein (1911 b)]. Of course, expression (1.39) is, in a certain way, equivalent to the Boltzmann H-theorem, which he had deduced 30 years earlier on the basis of kinetic theory.17 However, contrary to what might be thought at first sight, we do not think that this implies any demerit for Einstein’s presentation. Among other advantages, the role of collisions—always difficult to include properly in microscopic treatments—has been reduced to a minimum in Einstein’s formulation. Einstein devotes the eighth and penultimate section of his article to deducing an important property of entropy, as defined in (1.32). To do so, he applies the results found to the case of an isolated system constituted of subsystems in adiabatic interaction, which allows him to reach the following conclusion: “The sum of the

17

Boltzmann’s H-theorem appeared in a long article of 1872, the English translation of which can be found in Brush (2003), 262–349. It is the same article in which Boltzmann first derives the equation that today bears his name. The deduction of the theorem is on pages 280–291.

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1 The Sowing (Until 1905): The Annus Mirabilis

entropies of the partial systems of an isolated system after an arbitrary process is equal to or larger than the sum of the entropies of the partial systems before the process”. If this is applied to the case of a single subsystem, it follows that the entropy of an isolated system cannot decrease in any process. Finally, in the last section of his 1903 article, Einstein deduces the second principle of thermodynamics as the impossibility of the existence of a perpetuum mobile of the second kind. For this purpose, he considers an isolated global system composed of a heat reservoir W, an engine (Maschine) M with energy infinitely smaller than that of W, and a set of subsystems Σ1 , Σ2 ,…, which are adiabatically related to each other. It is assumed that the individual energy of these is infinitely larger than that of the machine and that the state of each subsystem W, M, Σ1 , Σ2 ,…, is stationary. Machine M now undergoes an arbitrary cyclic process that modifies the respective state distributions of the systems Σ1 , Σ2 ,…, through an infinitely slow adiabatic influence. Thus, during the process, the machine produces work and receives a quantity of heat Q from the reservoir W. The calculation of the entropy variation of each subsystem in the process is simple. The variation in the entropy of the bath is given, according to (1.32), by − Q /T . The entropy of the machine does not vary, since it is a cyclic process, and neither does that of the systems Σ1 , Σ2 ,…, since they are subject to infinitely slow adiabatic influences. For all these reasons, the variation in the entropy of the global system will be: S' − S = −

Q T

(1.40)

In the previous section, Einstein showed that in every isolated system, it must be verified that S ' ≥ S, which in view of (1.40) leads to the fact that the quantity of heat absorbed by the machine must verify Q ≤ 0. “This equation expresses the impossibility of the existence of a perpetuum mobile of the second kind.” This is the sentence with which Einstein ends this 1903 article, since the last inequality implies the impossibility of the existence of processes such as the one assumed, in which the machine—which, let us remember, operates by cycles—only extracts heat from the reservoir and transforms it entirely into work.

1.3.4

Relevance of Statistical Fluctuations (1904)

In 1904 Einstein published an article with which he concluded his trilogy on the foundations of statistical mechanics [Einstein (1904)]. The paper starts as follows: “In the following I present a few addenda to an article I published last year.” He thus anticipates the aims of the work, which is devoted exclusively to the clarification of some points treated in his previous article of 1903. The addenda are treated in five sections:

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1. Justification of Boltzmann’s expression for entropy, which relates this to the logarithm of the number of microstates accessible to the system. 2. Deduction—for the third time—of the second principle of thermodynamics. 3. Meaning of the constant κ in the framework of the kinetic theory of gases. 4. The constant κ and energy fluctuations in the situation of thermal equilibrium. 5. Energy fluctuations and thermal radiation. The treatment of these points does not offer anything new of importance relative to the 1902 and 1903 papers as far as the foundation of statistical mechanics is concerned. The topics are, however, of special interest for understanding the evolution of Einstein’s thinking about the quantum theory of radiation, which is one of the principal topics of the present book. Einstein begins by investigating the relation between energy E and absolute temperature T “for a system that can absorb energy only in the form of heat, or, in other words, for a system not affected adiabatically by other systems”. Varying the notation slightly, he rewrites expressions (1.12), (1.13) and (1.19) as follows: h=

1 ω' (E) 1 = ; ω(E) · δE = 2 ω(E) 4κT

∫

E+δE

dp1 · · · dpn

(1.41)

E

The two equalities, with the help of (1.33), allow Einstein to obtain the following expression: ∫ dE S= = 2κ log[ω(E)] + const. (1.42) T He ends the paragraph by emphasizing the generality of this expression for entropy, although his justification of the absence of an explicit proof of (1.42) is surprising: “I omit it [the proof] because here I do not intend to present any application of the law in its general significance”. In the next section, Einstein returns to the second principle of thermodynamics. In 1902, he deduced it from a situation of thermodynamic equilibrium, while in 1903, he not only presented a more rigorous proof but also extended it to nonequilibrium situations. Therefore, what, if anything, is new in the 1904 treatment? We think that simplicity and a somewhat more mature form of exposing his reasoning are the only novelties in the paper. Einstein’s 1903 scheme—the reservoir, the machine, and the subsystems capable of undergoing only adiabatic transformations—has now been replaced by a different and clearer scheme: a heat reservoir and a machine working in cycles interact in a situation of thermal equilibrium at a certain temperature. The reasoning and the hypotheses used—conservation of energy and that states of higher probability never spontaneously evolve into states of lower probability—are those of 1903. For this reason, we will not dwell on their consideration or on the consequent deduction of the second principle, which Einstein understands in this section as a law that ensures the non-decreasing of entropy in the infinitely slow processes undergone by an isolated system.

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1 The Sowing (Until 1905): The Annus Mirabilis

It is in the following section that Einstein addresses, for the first time, an aspect that was to prove crucial in the long road, first towards black-body radiation, and then to quantum theory: that of the physical meaning of the constant κ. For a system—a gas, for example—at thermal equilibrium at the absolute temperature T0 , the distribution of states is given by: d W = const. exp

( −

E 2 κ T0

) d p1 · · · d pn ,

(1.43)

where κ is the universal constant whose meaning is to be elucidated. With this distribution Einstein calculates the mean kinetic energy per molecule, after idealizing the latter as a point particle, and he finds: L ν = 3 κ T0 ,

(1.44)

where the subscript ν is used to characterize each of the individual particles. Taking one mole as a reference, the well-known law for ideal gases allows him to write: p V = R T0 , being R = 8.31 · 107 erg · mol −1 · K−1

(1.45)

If L is the mean value of the kinetic energy of the centre of gravity of a molecule, then the kinetic theory of gases gives the following relation: pV =

2 N L, 3

(1.46)

where N stands for Avogadro’s number. Considering that L = L ν , one obtains: N ·2κ = R

(1.47)

Einstein does not explicitly mention Avogadro’s number in his treatment but refers to the “number of molecules contained in one equivalent” and assigns to it the value N = 6.4 · 10 2 3 mol −1 , which he claims to take from O. E. Meyer. With this, he obtains the following numerical determination of the constant κ: κ = 6.5 · 10 − 1 7 erg · K −1

(1.48)

These are the values currently accepted for these constants, with three significant digits, in the centimetre–gram–second (CGS) system of units: N = 6.02 · 10 23 ; R = 8.31 · 10 7 ; 2 κ = 1.38 · 10 − 16 . Let us remember that 2κ is equivalent to the Boltzmann constant. However, Einstein does not seem fully satisfied with this numerical determination of κ because it depends entirely on the validity of the kinetic theory of gases. Therefore, in the next section of his paper, he tries to obtain the real physical meaning of κ, now by means of a more general discourse. Thus, he goes on to deal with

1.3 Trilogy on Thermostatistics

35

the topic of energy fluctuations in systems in thermal equilibrium at a certain temperature T. We will not reproduce here the simple calculation—it appears in any manual of statistical mechanics when studying energy fluctuations in the canonical ensemble—which leads to [see, for instance, Pathria (1986), 69–70)]: ε 2 ≡ ( E − E) 2 = 2 κ T 2

dE , dT

(1.49)

where all energy mean values are obtained from the canonical distribution (1.43). Since ε 2 represents a measure of the thermal stability of the system—the greater the fluctuation, the lesser the stability—this last equality allows Einstein to reach an important conclusion expressing the true physical meaning of κ [Einstein (1904), in Beck (1989), 75]: Thus the absolute constant κ determines the thermal stability of the system. The relationship just found [(1.49)] is interesting because it no longer contains any quantity reminiscent of the assumptions on which the theory is based.

Einstein is not the first to deal with fluctuations. Boltzmann, and later Gibbs, had already shown their existence—it could not be otherwise in a pure statistical treatment—and had evaluated them in different situations. Concretely we can find expression (1.49) in Gibbs’s book of 1902 [Gibbs (1902), 81]. Both of them considered fluctuations as something inherent to the statistical description itself, but with such undesirable effects as the poor representativeness of the mean values when fluctuations are high, that it was necessary to justify the irrelevance of fluctuations in gases composed of molecules. Along these lines, they proved that for systems with a very high number of degrees of freedom, as is the case for thermodynamic systems, the value of fluctuations is so small that they are practically unobservable and, therefore, irrelevant in equilibrium statistical mechanics. It is here that Einstein shows one of his first flashes of boldness and creativity: since fluctuations are characteristic of every statistical description, and since in the usual thermodynamic systems they are unobservable, why not look specifically for a physical system in which they are observable and in which, therefore, the validity of (1.49) can be tested? If such a system were found, and if the result of the test were positive, this would represent a validation of the statistical description, a description which, although seriously justified by Boltzmann, Gibbs and Einstein, not all physicists of the time acknowledged, because probability and statistics did not yet occupy a respectable place in the edifice of physics. The finding of a system in which energy fluctuations were relevant and measurable not only enabled a contrast with the validity of (1.49), but also made possible the numerical determination of the constant κ and, from the relation (1.47), could even contrast the then accepted value for Avogadro’s number N. In the final section of his 1904 paper, Einstein proposes a surprising candidate: black-body radiation. This is surprising not only because it differs from the usual systems treated within the kinetic theory but also because electromagnetic radiation had been explicitly

36

1 The Sowing (Until 1905): The Annus Mirabilis

rejected on different occasions by Boltzmann and Gibbs as a system suitable for analysis by means of statistical methods.18 Einstein argues his choice thus [Einstein (1904), in Beck (1989), 75–76]: The last-found equation [our (1.49)] would allow an exact determination of the universal constant κ if it were possible to determine the mean value of the square of the energy fluctuation of a system; however, at the present state of our knowledge this is not the case. In fact, there is only a single kind of physical system for which we can surmise from experience that it possesses energy fluctuation: this is empty space filled with temperature radiation. That is, if the linear dimensions of a space filled with temperature radiation are very large in comparison with the wavelength corresponding to the maximum energy of the radiation at the temperature in question, then the mean energy fluctuation will obviously be very small in comparison with the mean radiation energy of that space. In contrast, if the radiation space is of the same order of magnitude as that wavelength, then the energy fluctuation will be of the same order of magnitude as the energy of the radiation of the radiation space.

With these premises, Einstein writes: ( )2 ε2 = E ,

(1.50)

where the value of E can be obtained from the Stefan–Boltzmann law: E = c V T 4,

(1.51)

where V is the volume of the cavity and c is a universal constant. From the last three expressions, the following relationship can be immediately deduced: / 3 κ 2 √ c 0. 42 3 V = = , (1.52) T T where the constants have been replaced by their respective numerical values: κ by the one “obtained from the kinetic theory of gases” and c—without Einstein providing any additional justification—by 7. 0 6 · 10 − 1 5 e rg · c m − 3 · K − 4 . On the other hand, if λm represents the wavelength corresponding to the maximum energy, the results of experiments carried out to determine its dependence on temperature led to what is known as Wien’s displacement law: λm =

0. 293 T

(1.53)

A comparison of the last two expressions serves to confirm Einstein’s intuition on the lawfulness of applying statistical methods to thermal radiation, enabling him

18

Both Boltzmann and Gibbs required the systems studied by their respective statistical methods to have a finite number, however large, of degrees of freedom, which did not seem compatible with the Maxwellian electromagnetic field theory for radiation.

1.4 Gibbs’s Formulation Compared with Einstein’s

37

to present, for the first time, a certain microscopic justification of the displacement law. There is nothing more appropriate to close this section than Einstein’s own backdrop to his 1904 article [Einstein (1904), in Beck (1989), 77]: One can see that both the kind of dependence on the temperature and the order of magnitude of λm can be correctly determined from the general molecular theory of heat, and considering the broad generality of our assumptions, I believe that this agreement must not be ascribed to chance.

We think that the importance, and the consequences, of the way in which Einstein seems to have been impelled to turn his attention to thermal radiation problems—that is, to black-body radiation—have not yet been sufficiently appreciated. Some of the results of our research on this issue is presented in later chapters.

1.4

Gibbs’s Formulation Compared with Einstein’s

We have seen that Einstein’s trilogy, published in 1902–1904, contains his own formulation of equilibrium statistical mechanics, independent of that developed by Gibbs in his 1902 book Elementary principles in statistical mechanics. Einstein’s foundational period came to its close in 1905 with the publication of the result of his research on Brownian motion [Einstein (1905 b)]. In addition to solving a crucial problem of his time, Einstein theoretically demonstrates the reality of molecules, essential to his formulation of statistical mechanics, which starts from the molecular constitution of the thermodynamic systems. In his scientific autobiography we can read [Einstein (1949 a), in Schilpp (1970), 47]: Not acquainted with the earlier investigations of Boltzmann and Gibbs, which had appeared earlier and actually exhausted the subject, I developed the statistical mechanics and the molecular-kinetic theory of thermodynamics which was based on the former. My major aim in this was to find facts which would guarantee as much as possible the existence of atoms of definite finite size. In the midst of this I discovered that, according to atomistic theory, there would have to be a movement of suspended microscopic particles open to observation, without knowing that observations concerning the Brownian motion were already long familiar.

Gibbs’s formulation is clearly and rigorously stated in his 1902 book, but no version of Einstein’s formulation exists. Although the analysis of his trilogy allows, in principle, a formal reconstruction of the underlying theory—a task that is not without serious difficulties—the fact is that such a reconstruction has not been effected. Einstein himself may have been partly responsible for this, with some comments that certainly undervalued his contribution. For example, in his short article replying to P. Hertz’s accurate criticisms in 1910 of certain of the assumptions made by Einstein about the properties of temperature in the contact and

38

1 The Sowing (Until 1905): The Annus Mirabilis

separation of systems in thermal equilibrium, he attributes such criticisms to “a misunderstanding caused by an all-too terse and insufficiently careful formulation” (he is referring here to his 1903 paper), and concludes his reply paper as follows [Einstein (1911 b), in Beck (1993), 250]: I only wish to add that the road taken by Gibbs in his book [of 1902], which consists in one’s starting directly from the canonical ensemble, is in my opinion preferable to the road I took. Had I been familiar with Gibbs’s book at that time, I would not have published those papers at all, but would have limited myself to the discussion of just a few points.

Despite the disparity between the respective versions, they have usually been considered two equivalent formulations of the same theory: equilibrium statistical mechanics. Gibbs’s is usually considered the rigorous, closed, and definitive form, while Einstein’s—virtually ignored—has rather been understood as a not entirely successful attempt to reach Gibbs’s conclusions through a more intuitive, although not so rigorous, route. However, neither the independence nor the equivalence between the two formulations has ever been questioned. Nevertheless, we will point out here that this supposed equivalence must be revisited, since premises, objectives, methods and even the respective results of Gibbs and Einstein differ clearly, and in some respects radically.

1.4.1

Background, Premises, and Objectives

Gibbs published Elementary principles in 1902, a year before his death. He was already internationally known then, especially for his contributions to thermodynamics 25 years earlier. These contributions found an early and positive reception in Great Britain, largely due to Maxwell’s followers, and later in Europe as a consequence of the translations of his book into German and French by two scientists of great prestige: W. Ostwald in 1889 and H. Le Châtelier in 1892, respectively. In 1901 he was awarded the Royal Society Copley Medal, one of the highest scientific distinctions of the time. Thus, Gibbs presented his formulation after more than 30 years of high-level research, not only in thermodynamics but also in other fields, such as the electromagnetic theory of light, vector analysis and the algebra of quaternions.19 Einstein formulated his first ideas on statistical mechanics also in 1902, when he was only 23 years old, with hardly any experience in the presentation of scientific papers and with no connections to the academic world, so he had no opportunity to discuss his concepts with prominent members of the scientific community. There is thus a quite remarkable difference between Gibbs and Einstein, as far as their initial status is concerned. However, this is not the only difference; in their respective quests for a mechanistic explanation of thermodynamics they followed very different paths.

19

For more details on Gibbs’s life and work, see his perhaps most prestigious biographer, Wheeler (1970).

1.4 Gibbs’s Formulation Compared with Einstein’s

39

Although Gibbs’s formulation of statistical mechanics has certain analogies with the kinetic theory of gases, there are very notable differences, largely for reasons of principle. For example, in Gibbs’s book, neither are molecules a starting hypothesis, nor is the mechanistic explanation of thermodynamics the main purpose. Statistical mechanics—as Gibbs baptizes his approach—is born with a very different pretension: the generalization of Newtonian mechanics to systems with a large number of degrees of freedom and with initial conditions given by probability distribution law. And this was all without making any direct assumption about the ultimate constitution of matter. If, finally, some relations between mean values coincide in his form with some of the fundamental equations of thermodynamics, it is simply a formal analogy. Einstein, in contrast, follows the route opened by kinetic theory, trying to solve the difficulties that had halted the development of this theory and setting as his ultimate goal the reduction of thermodynamics to mechanics, while including some additional assumptions, among which the molecular constitution of matter played an essential role. However, in either case it is worth asking: how do probability and statistics arise in the respective formulations, if in both cases Newtonian mechanics is the starting point? Here, another of the great differences between Gibbs and Einstein appears. We have seen in the previous sections that for Einstein, the statistical description is ideal for analyzing thermodynamic systems in microscopic terms. Each situation, i.e., each macroscopic state or thermodynamic state, is characterized by a distribution function that provides the probability that a certain arbitrary microscopic state represents the concrete realization of the given macroscopic state. The rigorous definition of this thermodynamic probability—as a starting point—and its resulting computation for isolated systems in thermal equilibrium are central issues in the Einsteinian formulation. In contrast, in Gibbs’s formulation, probabilistic ideas do not arise as hypotheses or as results but as an original way of presenting the data—as a distribution function giving the probability for the different possible initial conditions—of a certain mechanical problem. In another way, usual mechanical problems are solved, although now supposing that the initial conditions are not perfectly known, and that only statistical information about them is available. Gibbs initially proposes the statistical study of a very large number of independent mechanical systems of the same nature—ensemble in his original terminology and in ours— which is why he calls the new discipline “statistical mechanics”. Later, he will also demand that the mechanical systems under consideration have a very large number of degrees of freedom, which allows him to practically eliminate the effect of fluctuations in his statistical forecasts. Einstein explains the behaviour of thermodynamic systems, which he assumes to be composed of molecules. Thus, thermodynamics takes centre stage from the very beginning. For Gibbs, on the other hand, the primary objective is the analysis of the statistical behaviour of ensembles of mechanical systems. The relation with thermodynamics comes later; in fact of the 15 chapters of his book only the penultimate chapter, “Discussion of thermodynamic analogies”, is devoted to the

40

1 The Sowing (Until 1905): The Annus Mirabilis

issue. As this title indicates, the connection is described in terms of some marked analogies that Gibbs detects between the equations that govern the behaviour of his mechanical models and the main relations between thermodynamic quantities, which allows him to arrive at the following conclusion [Gibbs (1902), viii–ix]: The laws of thermodynamics, as empirically determined, express the approximate and probable behaviour of systems of a great number of particles, or, more precisely, they express the laws of mechanics for such systems as they appear to beings who have not the fineness of perception to enable them to appreciate quantities of the order of magnitude of those which relate to single particles, and who cannot repeat their experiments often enough to obtain any but the most probable results. The laws of statistical mechanics apply to conservative systems of any number of degrees of freedom, and are exact. This does not make them more difficult to establish than the approximate laws for systems of a great many degrees of freedom, or for limited classes of such systems (...). The laws of thermodynamics may be easily obtained from the principles of statistical mechanics, of which they are the incomplete expression [since they are approximate laws, valid only for the case of systems with a very large number of degrees of freedom], but they make a somewhat blind guide in our search for those laws. This is perhaps the principal cause of the slow progress of rational thermodynamics, as contrasted with the rapid deduction of the consequences of its laws as empirically established. To this must be added that the rational foundation of thermodynamics lay in a branch of mechanics of which the fundamental notions and principles, and the characteristic operations, were alike unfamiliar to students of mechanics.

1.4.2

Methods and Results

Thus, to a large extent, not only are the profiles of Gibbs and Einstein different but so too are their premises, objectives, and results, so it is not surprising that the two formulations are developed with different methodologies. We will pause here to highlight some notable differences between them, starting with the statistical ensemble to which each resorts as his respective starting point. For reasons that we have just pointed out, Gibbs does not require the existence of ultimate constituents of matter that would obey the laws of mechanics, because he is concerned with the study of mechanical systems with very many degrees of freedom, without taking into account their possible molecular constitution, which is obviously much less demanding. The validity of the molecular hypothesis, foundation stone of the Einsteinian edifice, would only be fully admitted from 1910 onwards, as we shall see later, when we deal with the experimental confirmation of Einstein’s 1905 theoretical predictions on Brownian motion. Gibbs’s formulation rests on the canonical ensemble, almost the only one used in his book, which is suitable for describing systems in thermal equilibrium. He made very limited use of the microcanonical ensemble, which is appropriate for analyzing isolated systems in equilibrium, and although he also introduces it in

1.4 Gibbs’s Formulation Compared with Einstein’s

41

his book, he hardly uses the grand canonical ensemble, which is particularly suitable for describing systems in thermal equilibrium in which, in addition, there is equilibrium with respect to possible exchanges of matter.20 Specifically, almost at the beginning of Chap. 4 of his book, Gibbs introduces the canonical distribution as follows: ) ( ψ −ε , (1.54) P = exp θ where P is the “coefficient of probability”, log P is the “index of probability”, ψ is a constant to be determined from the condition that the integral of P over the whole space must be equal to 1, ε represents the energy of the system and θ is a characteristic constant called the “modulus” of the distribution and must be positive to allow the normalization of the probability P since the energy ε is always positive due to the usual adoption of zero as the minimum value of energy. Contrary to what one might think, expression (1.54) is not reached by considerations of thermal equilibrium, but by its privileged mathematical properties. Moreover, as an exponential function, each member of a set of independent systems will conform to the same law of distribution as the whole [Gibbs (1902), 33]: The distribution [represented by P] (...) seems to represent the most simple case conceivable, since it has the property that when the system consists of parts with separate energies, the laws of the distribution in phase of the separate parts are of the same nature, a property which enormously simplifies the discussion, and is the foundation of extremely important relations to thermodynamics.

In the formulation that Gibbs presents in his book we can distinguish four clearly differentiated parts. He begins by demonstrating a number of general properties of the phase space and of the distributions that can be associated with statistical equilibrium (Chapers I, II, III, IV, IV, VI and XI). Then, taking these properties into account, he obtains expressions for certain mean values in the canonical ensemble, as well as relations between them (Chapters V, VII, VIII and IX). Some of the above results are extended to the microcanonical ensemble in Chapter X, and to the grand canonical ensemble in Chapter XV. It is in Chapter XIV that he shows the existence of a clear formal analogy between certain relations—which he has obtained for the mechanical systems that he has discussed previously—and the main thermodynamic formulae. And this is basically the mechanistic justification of thermodynamics, according to Gibbs’s original version. Thus, we might say that Gibbs presents a rigorously constructed theory of models. Hence his predilection for the canonical ensemble, represented by a distribution function of such exquisite mathematical behaviour as the exponential

20

In the original contribution the grand canonical ensemble was called by Gibbs “grand ensemble canonically distributed” and only appears in the last chapter of his book. See Gibbs (1902), 187– 207.

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1 The Sowing (Until 1905): The Annus Mirabilis

function. The microcanonical ensemble appears, very ephemerally, as a limit case of the canonical ensemble, but without physical interest and with mathematical difficulties to be adequately treated. On the other hand, in this formulation the justification of the equality between mean values over ensembles and thermodynamic quantities is not needed, but their equivalence is inferred from the formal analogy between the two (mean values and thermodynamic quantities). Nor is there any need for the molecular hypothesis since the construction of extremely general mechanical models is pursued exclusively. It is a posteriori when it is verified that these models present characteristics and properties that correspond to those of thermodynamic systems so that, for practical purposes, both (mean values and thermodynamic quantities) can be identified. As usual in the case of models characterized by a certain probability distribution, fluctuations are expressly considered and evaluated, but they are immediately relegated because of their irrelevance in relation to the observable properties of mechanical systems with a finite, but very high, number of degrees of freedom.21 It is precisely this finiteness that makes Gibbs clearly sceptical about the possible application of his formulation to thermal radiation [Gibbs (1902)], 167]: Although our only assumption is that we are considering conservative systems of a finite number of degrees of freedom, it would seem that this is assuming far too much, so far as the bodies of nature are concerned. The phenomena of radiant heat, which certainly should not be neglected in any complete system of thermodynamics, and the electrical phenomena associated with the combination of atoms, seem to show that the hypothesis of systems of a finite number of degrees of freedom is inadequate for the explanation of the properties of bodies.

Importantly, Einstein did not relegate fluctuations to oblivion: we have shown that it was his interest in finding a suitable system to evaluate fluctuations and to test certain predictions of his statistical formulation that first led him to become interested in the black-body radiation problem, and later to focus on it, developing for more than a decade his own quantum theory of radiation, on which fluctuations and statistical methods constantly left their mark, as we shall see. Not only the premises but also the method employed by Einstein in his 1902– 1904 papers deviates notably from Gibbs’s ideas. Einstein takes as a starting point something that, we insist, deserved little attention from Gibbs: the physical abstraction that assumes an isolated system for which no constant of movement different from and independent of energy exists. In other words, Einstein began his career via the microcanonical ensemble, introducing the canonical ensemble later on by considering parts within an isolated system; all of them are in thermal equilibrium with each other.

21

Gibbs (1902), 71–75. It is worth noting the curious terminology used there: the deviation of a quantity from its mean value is called “anomaly” and its significance is measured by the corresponding mean square deviation.

1.4 Gibbs’s Formulation Compared with Einstein’s

43

It is precisely in this leading role of the microcanonical ensemble that Einstein’s ergodic hypothesis can be framed, which allows him a certain justification of the equality between observed values for thermodynamic quantities (time averages along trajectories in the phases space of mechanical systems) and mean values calculated from the microcanonical distribution. Of course, that does not represent a true problem in Gibbs’s formulation, which is much less pretentious, merely showing the complete analogy between thermodynamic quantities and mean values on the canonical ensemble, suggesting the identification of both values, although without strictly proving it. Although other more precise and technical details could be considered to corroborate the differences between Gibbs’s and Einstein’s respective treatments, we will confine ourselves here to two highly significant aspects.22 First, the meaning of the concept “ensemble” does not coincide in the two treatments: while it is little more than a mathematical abstraction for Gibbs, Einstein conceives it as a collection of real physical systems. The other point to be highlighted concerns the radically different way in which they introduce absolute temperature. Gibbs infers this concept from the formal analogy existing between the modulus of the canonical distribution and the thermodynamic temperature; in this context, absolute temperature must be positive by hypothesis; otherwise, the normalization of the canonical distribution (1.54)—the basis for his treatment—would not be guaranteed. Einstein, on the other hand, introduces absolute temperature by comparing his results with those of the kinetic theory of gases. Logically, it implies assuming the validity of this theory; now temperature always turns out to be positive as a consequence of the corresponding demonstration—not as convincing and rigorous as one might expect—which is of a mechanistic nature.

1.4.3

Impact and Diffusion

Although the German translation of Gibbs’s book did not appear until 1905, after its author’s death, the favourable impact of Elementary principles was immediate and grew with the passage of time. The importance of the contribution seemed to be grasped at once, at least by the leading scientific minds of the day. A glance at the papers presented to the 1904 Congress of Arts and Science held in St. Louis, United States, for example, clearly demonstrates the generalized high appreciation felt for the new statistical treatment by renowned and influential figures such as Henri Poincaré (1854–1912) or Boltzmann himself.23 The reading of the proceedings of the first Solvay conference, held in Brussels in the autumn of 1911 under the title The theory of radiation and quanta, shows that

22

For a broader and more rigorous comparative analysis of the respective versions of Gibbs and Einstein, see Navarro (1998). 23 The papers on physics presented at the St. Louis Congress are reprinted in Sopka (1986). The laudatory remarks of Boltzmann and Poincaré are on pages 277–278 and 286, respectively.

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1 The Sowing (Until 1905): The Annus Mirabilis

Gibbs’s statistical mechanics was at that time highly esteemed by those investigating the main issues in the field.24 It is possible that Einstein’s explicit recognition in 1911 of the superiority of Gibbs’s method to his own also contributed to such a rapid and wide diffusion (see the discussion of the Hertz–Einstein dispute in Sect. 1.4). As for the impact of the Einsteinian formulation at the time, there is not much to report. Paul Hertz—not to be confused with Heinrich Rudolf Hertz (1857– 1894), the discoverer of electromagnetic waves in 1886—was one of the few who paid attention to these contributions of Einstein: in 1910, Hertz published a couple of papers devoted to a critical analysis of the foundations of statistical mechanics, both according to Gibbs’s and Einstein’s formulations [Hertz (1910)]. The undoubtedly greater consistency and generality of the former must have considerably influenced his later attitude. Indeed, six years later, Hertz himself wrote a volume on statistical mechanics for the prestigious Repertorium der Physik [Hertz (1916)]. Its 164 pages analyze Gibbs’s formulation. Only in the first chapter, devoted mainly to the basic contributions of Maxwell, Boltzmann and Gibbs, do we find a passing mention of Einstein—a paragraph in which Hertz recalls a point of the criticism he addressed to him in 1910, concerning the so-called “temporal ensemble” (Zeitgesamtheit), as well as Einstein’s brief reply [in Einstein (1911 b)]. In 1912, a monograph by Paul (1880–1933) and Tatiana Ehrenfest (1876–1964) entitled The conceptual foundations of the statistical method in mechanics appeared. It was prepared on behalf of the prestigious Encyklopädie der mathematischen Wissenschaften and was intended to provide a modern and rigorous update of the kinetic theory and statistical mechanics [Ehrenfest (1990)]. It is a work that, with the passage of time, would achieve great prestige and diffusion, especially because of the depth and rigour of the treatment. For example, only four years after it appeared, Hertz placed it on the same level as Gibbs’s own work by including it in the only three general references he cited in the Repertorium [Hertz (1916), 436]. In addition to its scientific value, The conceptual foundations already presents the essential characteristics, maintained up to the present day, of the analyses— and even of the value judgements—of the respective formulations of Gibbs and Einstein. The Ehrenfests’ book has 30 sections. Only in the 25th, entitled “Articles following or related to Gibbs’s treatment”, is a brief mention made of Einstein’s 1902 and 1903 papers—the first two of his trilogy—but only because of his use there of the ergodic hypothesis and of microcanonical and canonical ensembles. His work on Brownian motion is also cited in the same section. The other references to Einstein are few and far between, do not connect him with the foundations of

24

The proceedings of the first Solvay conference are in Langevin; De Broglie (1912). See, in particular, the papers by Lorentz, 12–39, and Planck, 93–144; in both, Gibbs’s statistical mechanics is not only mentioned but also used.

1.4 Gibbs’s Formulation Compared with Einstein’s

45

statistical mechanics, and do not even cite his 1904 paper, which opened the door to the application of the new statistical methods to black-body radiation. It is impossible to conceive that the omission of Einstein’s contributions to statistical mechanics was due to ignorance or improvization. Although the personal friendship between Einstein and the Ehrenfests only became strong after 1912— when the couple decided to end their residence in Tatiana’s Russia and move to Leiden so that Paul could fill the vacancy left there by Lorentz—there is wide evidence of Paul’s extensive knowledge of Einstein’s work at the time. For instance, Martin J. Klein (1924–2009), the prestigious historian of quantum physics and specialist in the life and work of Paul Ehrenfest, writes [Klein (1985), 175]: Ehrenfest had been reading Einstein’s papers for almost a decade [in 1912]; he shared so many of Einstein’s scientific interests that he could well appreciate the unprecedented boldness and depth of Einstein’s ideas.

The English translation of the Ehrenfests’ book to which we are referring appeared in 1959, with an additional foreword by Tatiana (Paul had committed suicide in 1933), which includes some remarks that “should have been included in the original version of the article, and for the omission of which I feel personally responsible”, but here too there is no mention of Einstein. This attitude of the Ehrenfests is difficult to understand. Certainly, Paul knew of Boltzmann’s high esteem for Gibbs’s book. In 1904 Boltzmann gave credit to Gibbs for having “brought this science into systematic form, set it forth in a major book, and given it a characteristic name, [statistical mechanics]”.25 The Ehrenfests’ predilection for Boltzmann’s statistical conceptions does not prevent them from appreciating the methodological rigour of Gibbs’ procedure, while clearly noting his inability to provide a Boltzmann-like physical picture of the behaviour of a thermodynamic system. Thus, for example, the Ehrenfests go so far as to describe Gibbs’s introduction of the canonical ensemble as an “analytical trick”, emphasizing, on the other hand, the greater physical sense of the microcanonical ensemble, even pointing out a hierarchical order between both ensembles. All along the lines followed by Einstein, only without any mention of him! The Ehrenfests’ ‘forgetfulness’ is even more incomprehensible when other details are considered. For example, when they wrote the article we are referring to, Paul knew that, contrary to what Gibbs had anticipated in his book—scepticism about the possible application of statistical mechanics to thermal radiation, not only had Einstein already applied statistical mechanics to the analysis of the structure of electromagnetic radiation but these issues were also playing an important role in the development of quantum theory from 1905 onwards. Moreover, in 1911,

25

See Klein (1985), 133, note 58. It is also worth bearing in mind that Boltzmann had supervised Paul Ehrenfest’s dissertation in 1904 at the University of Vienna.

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1 The Sowing (Until 1905): The Annus Mirabilis

Paul resorted to statistical mechanics to show the need to admit energy quantization to coherently explain some experimental results on black-body radiation.26 In view of these points, it seems appropriate to at least reconsider the extended idea that Gibbs’s and Einstein’s formulations of statistical mechanics are totally equivalent, while giving absolute primacy to the former. From a logical-formal point of view, Gibbs’s formulation is undoubtedly more rigorous and general than Einstein’s. On the other hand, Einstein’s is much more intuitive and makes more physical sense. In some sense, Gibbs’s version could be appropriately catalogued as axiomatic, while Einstein’s is much more pedestrian, although with notable advantages from an intuitive and didactic point of view, because in this case, abstraction is replaced by concrete mechanical systems and real situations, which certainly reduces generality but facilitates the assimilation of the ideas involved. Continuing with the didactic aspect, a simple consultation of most classical texts on statistical mechanics allows us to verify that, although Gibbs is often credited with the paternity of what is set out in them, the treatment of statistical ensembles is actually closer to Einstein’s formulation than to Gibbs’ axiomatic treatment. However, when it is a question of accentuating rigour, one tends to resort to Gibbs’s method, even at the cost of giving up Einstein’s intuitive physical image of the situation. A reconsideration of the admitted equivalence between Gibbs’s and Einstein’s versions leads of necessity to a re-evaluation of the latter’s scarcely recognized contributions. The mea culpa intoned by Einstein in his dispute with Hertz in 1911, already quoted in Sect. 1.4, does not diminish the soundness of our vindication, since the cited polemic only referred to a specific issue—intimately linked to the fact that absolute temperature is introduced in a more natural way if one starts from the canonical rather than the microcanonical ensemble, which does not imply depreciation of the relevance of Einstein’s contributions. Nor have we found any other later statements or writings by Einstein in which he insists on giving Gibbs’s treatment precedence over his own. On the contrary, we have verified that in his later courses on statistical mechanics, he was always faithful, in essence, to his own approaches. On the other hand, the deep admiration that Einstein felt for Gibbs is well reflected in an anecdote from 1954, a year before Einstein’s death and half a century after Gibbs’s death. When asked who the greatest men were, the most powerful thinkers whom he had known, Einstein answered without hesitation: “Lorentz”. However, then he added: “I never met Willard Gibbs; perhaps, had I done so, I might have to place him beside Lorentz” [Douglas (1956), 102].

26

For a detailed analysis of the real significance of Paul Ehrenfest’s 1911 work, see Navarro; Pérez (2004).

1.5 Energy Quanta and the Photoelectric Effect (1905)

1.5

47

Energy Quanta and the Photoelectric Effect (1905)

We have anticipated the reasons, derived from his research in statistical mechanics, why Einstein turned his attention in 1904 to thermal radiation, immersing himself in the problems related to the absorption and emission of light by a black body. The name black body refers to its characteristic property: by absorbing all the incident radiation, the body is not susceptible to being observed by the reflection of light on its surface. In addition, to remain at a constant temperature, the black body must emit the same energy that it absorbs. From the existing documentation, in particular from the letters written to his fiancée Mileva Mari´c, it is possible to state with certainty that Einstein knew the basic ideas of thermal radiation, at least through the two chapters devoted to them in the then well-known book by Mach, The principles of the theory of heat [Mach (1896)]. As Kirchhoff had established in 1859, the material of which the black body is formed does not influence the properties of its emission in any way. Thus, the energy of the radiation emitted by a black body at equilibrium at a certain temperature is a universal function; it does not depend on the wavelength of the radiation or on this temperature. Einstein was also aware of the main attempts to find this universal function because Weber—his underappreciated professor of physics at the ETH—had devoted some lectures to the subject; in fact, he was so interested that Weber himself had even proposed a candidate for such a function. In particular, Einstein knew that the function proposed by Wien in 1896, which we will detail later, was the one that best fitted the experimental data existing at the end of the nineteenth century; so much so that it was considered definitive in the days when the young man was finishing his higher studies.

1.5.1

Planck’s Energy Quanta

Einstein was also aware of the recent contributions of Planck, who seemed at that time convinced that thermal radiation could constitute an adequate system to justify, at a microscopic level, the second principle of thermodynamics. All this, however, without necessarily having to resort to statistical elements such as those introduced by Boltzmann, since from a methodological point of view, Planck deeply abhorred the idea that statistical description should be the support for the mechanical foundations of the second law of thermodynamics. Between 1897 and 1900, Planck devoted several works to analyzing these issues. Since, as established by Kirchhoff, in the interaction between radiation and matter the specific nature of matter is irrelevant, Planck devised a theoretical model that made it possible to enter into the world of the then mysterious relations between electromagnetic radiation and matter: the walls of the cavity containing the “black radiation” were assimilated to an aggregate of electrically charged particles, each of which oscillated permanently around a point, emitting and absorbing the corresponding amount of electromagnetic radiation. However, he soon realized

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that his model of oscillators, or “resonators”, as they were called, would not go very far without certain statistical aspects, such as those considered by Boltzmann. Planck introduced “natural radiation”—an idea that was soon to attract Einstein’s attention—as a state of maximum electromagnetic disorder; a notion very similar to Boltzmann’s “molecular chaos”, associated with a state of maximum disorder for particles of a gas. Along these lines, and with the help of Maxwellian electromagnetism, Planck obtained a formula that admirably resisted the passage of time to relate the spectral distribution of the energy density of thermal radiation ρ ν —energy emitted per unit volume and per unit of frequency interval—with the mean energy per oscillator E ν (T ) of a set of monochromatic resonators of frequency ν in equilibrium at absolute temperature T: ρ ν (T ) =

8 π ν2 E ν (T ), c3

(1.55)

where c represents the speed of light in vacuum. The above expression, together with the use of part of the Boltzmann statistical methods and some additional hypothesis on the behaviour of the resonators, allowed Planck to deduce Wien’s formula for radiation, which, as we have anticipated, was the one that best explained the experimental results of the time. A short digression must be introduced here. Towards the end of the nineteenth century, the optics laboratory Die Physikalisch-Technische Reichsanstalt (Imperial Physical-Technical Institute) in Berlin was one of the best equipped in its field at the time.27 Special research was carried out in photometry; in particular, several teams were dedicated to the search for suitable materials to obtain a standard for measuring luminous intensity, which would allow the implementation of an internationally accepted unit. To this end, the radiation emitted by a black body has been the subject of numerous measurements and studies, considering both the relative ease of building a black body and the universal nature of its emissions. One of these working groups was composed of Otto Lummer (1860–1925), the laboratory director, and Ernst Pringsheim (1859–1917). They refined the usual photometric techniques until they were able to measure for the first time in the infrared region—in the wavelength range corresponding to wavelengths of approximately 15 microns—over a wide temperature range. Their results at the beginning of 1900 point to a clear disagreement with the predictions of Wien’s law for this region. Another group from the same laboratory was formed by Ferdinand Kurlbaum (1857–1927) and Heinrich Rubens (1865–1922), Planck’s personal friend. They further refined the measurement techniques to the point that at the end of October of the same year, they presented results for wavelengths between 30 and 60 microns, which already corresponds to the far infrared, and for temperatures

27

It was universally known by the acronym PTR. In 1953 it was renamed Physikalisch-Technische Bundesanstalt (Federal Physical-Technical Institute), or PTB.

1.5 Energy Quanta and the Photoelectric Effect (1905)

49

between 200 and 1500 °C. These measurements confirmed and extended the mismatches previously detected in relation to Wien’s law. Approximately a fortnight before the new results were officially presented to the Berlin Academy of Sciences, Planck had already been informed by Rubens of all this. It is here that we pick up the thread of Planck’s attempts to explain the behaviour of thermal radiation. Having been informed of the divergence of the new data from Wien’s formula, Planck soon found a simple modification of that formula that seemed to fit the new results, which had been immediately confirmed to him by Rubens himself after the appropriate verifications. It all happened over the course of a few days in October 1900. Thus it was that on the 19th of that month, a week before Rubens and Kurlbaum made their results public, Planck presented to the German Physical Society his new formula to explain the spectral energy distribution of radiation emitted by a black body. This can be written, using current notation, as follows: ρ ν (T ) =

hν 8π ν 2 ( ) , 3 c exp kh Tν − 1

(1.56)

where k stands for Boltzmann constant, which, in those days, was written as the quotient between the universal gas constant R and Avogadro’s number N, and h was a new universal constant, later called “Planck’s constant”. On 14 December 1900, on the same stage as a few weeks before, Planck read his new memoir “On the theory of the energy distribution law of the normal spectrum”, in which he gave the first theoretical justification of his law for black-body radiation [Planck (1900)]. The procedure chosen was to determine the mean entropy per resonator S ν (E ) for a set of monochromatic resonators of frequency ν and total energy E, introducing the absolute temperature by means of the thermodynamic relation28 : 1 ∂ S ν (E) = T ∂ E ν (T )

(1.57)

Now it only remained to calculate the mean entropy S ν (E), which, replaced in (1.57), would enable the mean energy E ν (T ) to be determined. This, substituted in turn in (1.55), would lead to Planck’s formula (1.56). Unable to find a convincing way to obtain the entropy that would lead to that formula, Planck, in what he later called “an act of despair”, went so far as to apply Boltzmann’s statistical methods to the problem of radiation—methods that were not at all to his liking, for they were based on the statistical character of the laws of thermodynamics, against his general adherence to the principle of the absolute character of the laws of physics. In addition, Planck’s theoretical justification led to a surprising and unexpected

28

This is the usual way of introducing absolute temperature when the problem is initially posed in the microcanonical ensemble, which is the appropriate way of describing systems with constant total energy.

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1 The Sowing (Until 1905): The Annus Mirabilis

restriction: Planckian resonators of frequency ν could not absorb and emit any amount of energy, as was to be expected according to a purely classical treatment, but only multiples of an elementary unit—“energy quantum”—of value ε = h ν.29 This was certainly a strange property of Planckian resonators, but it was sufficient to obtain the new radiation law.

1.5.2

Einstein’s Energy Quanta: A Heuristic Point of View

On 9 June 1905, Einstein’s paper “On a heuristic point of view concerning the production and transformation of light” was published in Annalen [Einstein (1905 a)]. The term heuristic in some sciences can be understood as a way of seeking the solution of a problem by non-rigorous methods, such as trial and error, empirical rules, etc. In fact, Einstein does not proceed with strict rigour, although he deploys his creativity in a direction with a long tradition in the history of physics: the use of analogies. He begins by delimiting the scope of his conclusions while anticipating the original and revolutionary character of his ideas. Despite its length, we believe it is worth reproducing the introduction to the article verbatim: There exists a profound formal difference between the theoretical conceptions physicists have formed about gases and other ponderable bodies, and Maxwell’s theory of electromagnetic processes in so-called empty space. While we conceive of the state of a body as being completely determined by the positions and velocities of a very large but nevertheless finite number of atoms and electrons, we use continuous spatial functions to determine the electromagnetic state of a space, so that a finite number of quantities cannot be considered as sufficient for the complete description of the electromagnetic state of a space. According to Maxwell’s theory, energy is to be considered as a continuous spatial function for all purely electromagnetic phenomena, hence also for light, while according to the current conceptions of physicists the energy of a ponderable body is to be described as a sum extending over the atoms and electrons. The energy of a ponderable body cannot be broken up into arbitrarily many, arbitrarily small parts, while according to Maxwell’s theory (or, more generally, according to any wave theory) the energy of a light ray emitted from a point source of light spreads continuously over a steadily increasing volume. The wave theory of light, which operates with continuous spatial functions, has proved itself splendidly in describing purely optical phenomena and will probably never be replaced by another theory. One should keep in mind, however, that optical observations apply to time averages and not to momentary values, and it is conceivable that despite the complete confirmation of the theories of diffraction, reflection, refraction, dispersion, etc., by experiment, the theory of light, which operates with continuous spatial functions, may lead to contradictions with experience when it is applied to the phenomena of production and transformation of light.

29

The German term quantum, of Latin origin, was soon incorporated—without being translated— into other languages, specifically after the first Solvay conference in 1911. In fact, its initial use by Planck and Einstein was in accordance with the meaning it had at that time in the scientific sphere, which was “quantity”, without any special connotation referring to physical ideas. In that original sense it had been used before by Boltzmann, among others. For more details about the term quantum and its introduction into 20th-century physics, see Klein (1985), 253–254.

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Indeed, it seems to me that the observations regarding “black-body radiation”, photoluminescence, production of cathode rays by ultraviolet light, and other groups of phenomena associated with the production or conversion of light can be understood better if one assumes that the energy of light is discontinuously distributed in space. According to the assumption to be contemplated here, when a light ray is spreading from a point, the energy is not distributed continuously over ever increasing spaces, but consists of a finite number of energy quanta [Energiequanten] that are localized in points in space, move without dividing, and can be absorbed or generated only as a whole. In this paper I wish to communicate my train of thought and present the facts that led me to this course, in the hope that the point of view to be elaborated may prove of use to some researchers in their investigations.

Einstein himself called this work “revolutionary”, a term he did not use to describe any other, not even those dedicated to relativity. The anticipated conclusion is truly revolutionary: the possible discontinuity in both the spatial distribution of energy and in the energy exchange between matter and radiation (emission and absorption). A notable difference with Planck’s view of his own quantum hypothesis30 : I knew the formula that reproduces the energy distribution in the normal spectrum; a theoretical interpretation had to be found at any cost, no matter how high (…) [The hypothesis of energy quanta was] a purely formal assumption, and I did not give it much thought except for this: that I had to obtain a positive result, under any circumstances and at whatever cost.

In the first section of his 1905 paper, Einstein begins by highlighting a difficulty relating to a classical treatment of radiation, which was that employed by Planck, who claimed to have always moved within a body of doctrine as usual as that made up of mechanics, thermodynamics, kinetic theory, and Maxwell’s electromagnetism. For this purpose, Einstein considers electromagnetic radiation enclosed in a cavity with totally reflecting inner walls, containing molecules of a certain ideal gas, free electrons, and electrons harmonically bound to certain fixed points, able to undergo collisions and to emit and/or absorb electromagnetic radiation. Following Planck, Einstein admits that if there is “dynamic equilibrium” and no limitation is placed on the frequency of the harmonic motions, the radiation in the cavity must be identical to that emitted by a black body. Admitted thermal equilibrium between the ideal gas and the set of harmonic oscillators, at a certain absolute temperature T, the principle of equipartition of energy imposes that the mean energy of an oscillating electron, or of a Planckian resonator, is given by the following familiar expression: E=

30

R T, N

(1.58)

Letter from M. Planck to R. W. Wood, 7 October 1931. In this same letter Planck refers to the “act of despair” by virtue of which he adopted, at least partially, Boltzmann’s statistical methods. Cited in Klein (1966), 298–299. Emphasis in original.

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1 The Sowing (Until 1905): The Annus Mirabilis

where R is the universal gas constant and N is Avogadro’s number, which Einstein refers to here as the “number of real molecules contained in one gram-equivalent”. On the other hand, let us recall—see (1.55)—that Planck had obtained the following expression for the mean energy E ν of a monochromatic resonator of frequency ν: Eν =

L3 ρ ν, 8π ν 2

(1.59)

where, following the original notation, L represents the speed of light in vacuum and ρ ν is the spectral energy density, so that ρ ν d ν represents the energy per unit volume of radiation with frequency in the range (ν, ν + dν). Complete equilibrium imposes, Einstein affirms, E = E ν , which leads to: ρν =

R 8π ν2 T, N L3

(1.60)

known as the Rayleigh–Jeans law. Einstein considers this result unacceptable, not only because of its disagreement with the experimental data of those days, but also because the divergence to which it leads—in relation to the total energy for the whole frequency range—is conceptually inadmissible: ∫ 0

∞

R 8π ν2 T ρν dν = N L3

∫

∞

ν 2d ν = ∞

(1.61)

0

We have seen how Einstein had already resorted to the principle of equipartition, a result with a clear mechanical-statistical flavour, in his 1902 paper. Planck, on the other hand, had made no mention of this principle in the theoretical justification of his new formula for radiation, so that he did not write the relation (1.60), nor did he ever detect the fatal consequence (1.61) which, in 1911, Ehrenfest would baptize as the “Rayleigh–Jeans catastrophe in the ultraviolet” [Ehrenfest (1911), in Klein (1959), 186]. Einstein does not address—presumably, he was not yet in a position to do so— an issue that would turn out to be transcendental and which we will address later: if Planck affirmed that he had not departed from the classical ground, it is that, at least implicitly, he assumed the validity of (1.58). On the other hand, expression (1.59) was fully justified by Maxwell’s electromagnetism. One could, then, ask the following question: with such premises, how is it that Planck did not end up arriving at the undesirable equality (1.60) or some equivalent expression, but instead managed to obtain the correct law for black-body radiation, free of the difficulty expressed in (1.61)? Instead of seeking the answer to this question by analyzing Planck’s reasoning in depth, Einstein confines himself, in the second section of his paper, to a minor but sufficient criticism that shows how little credibility he gave to Planckian treatment: Planck’s determination of certain universal constants from his formula

1.5 Energy Quanta and the Photoelectric Effect (1905)

53

for the spectral energy density for radiation could not be used as a validation criterion for the new formula, contrary to what Planck seemed to suggest, because that determination could also be made from (1.60) by using experimental data for long wavelengths (i.e., for low frequencies). However, if, for example, the same values for Avogadro’s number are predicted by (1.60) or by Planck’s new formula, Planck’s contribution was deprived of part of its success, leaving the door open to the search for alternatives. This was the path initially chosen by Einstein. From here on, following his 1905 work, Einstein develops an original thermostatistical treatment of the problem, since he uses statistical mechanics in the opposite direction to the usual one: instead of starting from a probabilistic description of the radiation to obtain macroscopic relations, here he started from these to deduce the corresponding values for probabilities. Let us look at it. Einstein’s starting point was, oddly enough, not the new Planck law for radiation but the preceding phenomenological Wien law: ρν = α ν 3 e x p

(

−β

ν) , T

(1.62)

where α and β are constants. The tone of Einstein’s justification of his choice was extremely modest [Einstein (1905 a), in Beck (1989), 93]: Though the existing observations of ‘black-body radiation’ show that the law (1.62) postulated by Mr. W. Wien for ‘black-body radiation’ is not strictly valid, the law has been fully confirmed by experiment for large values of ν/T [properly β ν/T ]. We shall base our calculations on this formula, but will keep in mind that our results are valid within certain limits only.

The absence of any justification of this non-use of Planck’s law is surprising, since it did not present the limitations of Wien’s law and had already been fully confirmed by experimental results. On the one hand, it is easy to verify that if one repeats the calculations presented by Einstein in his paper but now using Planck’s law instead of Wien’s, one does not arrive at any result in connection with the quantization of the energy of radiation. However, on the other hand, it is not surprising that Einstein obtained this quantization from Wien’s law, since, as Ehrenfest proved in 1911, the quantization of the energy of radiation is implicit both in Planck’s law and in Wien’s law.31 The fourth, fifth and sixth sections of the work are the most technical from a formal point of view. Starting from (1.62) and introducing entropy by means of arguments based on classical radiation thermodynamics, Einstein calculated the

31

Ehrenfest (1911), in Klein (1959), 202–203. We anticipate here Ehrenfest’s conclusions in this contribution: whether Wien’s or Planck’s law is taken as the experimental result, the same discrete possible values are obtained for the radiation energy; we shall see later that the only difference lies in the fact that while in Planck’s case all levels are equiprobable, in Wien’s case the probability of the n-th level turns out to be proportional to 1/n!

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1 The Sowing (Until 1905): The Annus Mirabilis

variation in entropy of monochromatic radiation of frequency ν for a change in the volume of the cavity (from V 0 to V ) at constant energy E: S − S0 =

E log βν

(

V V0

) (1.63)

Here Einstein introduces the terminology “Boltzmann principle” to refer to the following expression: S=

R log W, N

(1.64)

which relates thermodynamic entropy S to the logarithm of the probability W— actually the relative probability—of the corresponding macroscopic state. The last two expressions are now used to move to the language of probabilities. In particular, if monochromatic radiation of frequency ν and energy E is enclosed in a cavity of volume V 0 , with totally reflecting walls, the probability W that, at an arbitrary instant, all the energy of the radiation is contained in a part of volume V, inside V 0 , will be given by: ( W=

V V0

)

N E R β ν

(1.65)

Next, Einstein deals with another system that, in principle, has nothing to do with radiation: an ideal gas of n molecules in a volume V 0 . He wonders about the value of the probability W that, at an arbitrary instant, the n molecules are in a volume V, inside V 0 . This is what Einstein calls “statistical probability”—a concept not previously defined and to which he will not resort again—to which he assigns the “evident” (offenbar) value: ( W=

V V0

)n (1.66)

It is the comparison between the last two expressions that leads Einstein to state the fundamental result of his work [Einstein (1905 a), in Beck (1989), 97]: Monochromatic radiation of low density (within the range of validity of Wien’s radiation formula) behaves thermodynamically as if it consisted of mutually independent energy quanta of magnitude R β ν/N .

Obviously, R β /N is also a universal constant, which would soon be named “Planck’s constant” and represented by h, so that the value of one energy quantum could be expressed as h ν. At no point in the paper does Einstein conceptually compare his energy quanta with those introduced by Planck five years earlier. The introduction of a concept of the calibre of the Einsteinian energy quantum necessarily required some experimental endorsement for its acceptance. Einstein

1.5 Energy Quanta and the Photoelectric Effect (1905)

55

begins to address this task by including in the paper suggestions of some experimental contrast of his proposal, applying it to some phenomena related to the emission and absorption of light. These were three phenomena that resisted an explanation adjusted to the framework of the current physics at that time and that Einstein tries to explain, totally or partially, by resorting to energy quanta: (i) Stokes’s rule. This was the name given to the experimentally observed fact that the frequency of the light emitted by luminescence was less than the frequency of incident light.32 For Einstein, once the discrete structure of the radiation energy is admitted—though this should not be understood as the discrete structure of radiation, which will take a further ten years and more to be established—the conservation of energy does the rest by prescribing that the energy of the emitted quantum can never be higher than that of the quantum that originated the process: ) ( ) ( R R (1.67) β ν2 ≤ β ν1 , N N equivalent to ν2 ≤ ν1 , which is the expression of the aforementioned Stokes’ rule. In the paper, it is made clear that each initial energy quantum initiates an independent process, so there is no lower limit to the intensity of the light that triggers the process. By virtue of this same description, it is possible to think of violations of the rule. This could occur both in the case of large illumination—it would then be feasible that several quanta would give rise to the emission of a single quantum—and when the frequency of light is not within the range of validity of Wien’s law, the only range of the spectrum for which Einstein’s energy quanta had been properly introduced. (ii) “On the generation of cathode rays by illumination of solid bodies”. In this best-known and most popular section of his work, Einstein presents an original and extremely simple theoretical explanation of the photoelectric effect—the release of electrons of metallic surfaces on which ultraviolet light is incident. Anyone who is at least minimally familiar with quantum physics knows the explanation given by Einstein, which is based on a simple energy balance between principle and end of the process. However, some other points are perhaps less well known. The photoelectric effect was originally detected by Heinrich R. Hertz as a secondary phenomenon in experiments to confirm the validity of the undulatory Maxwell’s theory of the electromagnetic field. In one of the ironies of history, in trying to establish a theory of an essentially continuous nature, such as Maxwell’s description, Hertz provided, in passing, an experimental basis, as we shall see later, for revealing the sometimes discrete behaviour of radiation. In his paper, Einstein does not refer to Hertz but to Lenard, who had proved three years earlier that the energy of the electrons emitted in the effect is independent of the intensity of the incident radiation, a property that is impossible to fit

32

Remember that in luminescence the effect occurs as long as energy is absorbed from a source of excitation, whereas in phosphorescence the phenomenon continues even in the absence of excitation.

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1 The Sowing (Until 1905): The Annus Mirabilis

into the framework of Maxwellian electromagnetism. In contrast, the explanation given by Einstein is simple: the increase in intensity translates into a greater number of electrons released, but not into an increase in the energy of each one, since, in accordance with the idea set out in the treatment of Stokes’ rule, a quantum gives up its energy to a single electron. Of course, this is a basic additional hypothesis, although it is not often seen as such. The subtlety certainly did not escape Einstein, although he could think of no other justification for this one-against-one hypothesis than its great simplicity. To achieve energy balance, it is also assumed that part of the energy of an incident quantum is invested in releasing an electron and the rest in communicating kinetic energy to the electron. The electrons that leave the metal with “the maximum perpendicular velocity” will be those that are “located directly on the surface and excited perpendicularly to it”. Thus, one arrives at what is sometimes called “Einstein’s equation”: maximum kinetic energy of each emitted electron =

R β ν − P, N

(1.68)

where P represents the work required to free the electron and is characteristic of each metal. After comparing (1.68) with the scarce experimental results of the time, Einstein cautiously affirms: “As far as I can see, our conception does not conflict with the properties of the photoelectric effect observed by Mr. Lenard” [Einstein (1905 a), in Beck (1989), 101]. The contents of this paper should not be presented as a corroboration of Einstein’s quantum ideas about radiation. It would be a distorted reconstruction of the emergence of his first quantum notions. The photoelectric effect had at that time very little entity and there were hardly any experimental data to test Einstein’s predictions. For example, there were no reliable measurements of the variation in the effect with the frequency of the incident light. As we shall see later on, the verification of the predictions based on (1.68) had to wait until 1916, when the American physicist Robert A. Millikan (1868–1953) fully confirmed them.33 We shall have to return to this topic. (iii) Ionization of gases by ultraviolet light. Einstein devotes the last section of the paper to this phenomenon, insisting at first on the additional character of his hypothesis on the individuality of the interaction between quanta and, in this case, molecules of the gas; that interaction which we referred to as the one-against-one hypothesis. Thus, if there was such ionization, it would be because the energy of the absorbed quantum is greater than the ionization energy of the molecule. The section concludes with an estimate of the upper limit for the value of the ionization energy in the case of air, which allows Einstein to show that his estimate

33

Nobel Prize in Physics 1923 “for his work on the elementary charge of electricity and on the photoelectric effect”.

1.6 Molecular Constitution and Brownian Motion (1905–1906)

57

is almost coincident with the experimental data of the time, due to Johannes Stark (1874–1957).34 So much for the 1905 article. Einstein’s energy quanta, however, are not the radiation quanta, the photons; it would be more than ten years before these were incorporated into the physics cast. A clear distinction must be made between discontinuity in the emission and absorption of radiant energy and the existence of elementary constituents—particles—for radiation.35 The following reasoning, which straddles physics and history, may serve to justify this distinction. If Einstein had truly thought in terms of particles, the creator of the theory of relativity would not have missed the fact that a quantum of energy h ν necessarily had to be endowed with a momentum hv/c. However, Einstein did not reason thus in 1905. If he had done so, he would probably have predicted, almost 20 years in advance, the Compton effect, the theoretical explanation of which involves imposing the corresponding conservation of energy and momentum to an elastic collision between one radiation quantum and a weakly bound atomic electron. This ‘forgetfulness’ of Einstein seems to be an irrefutable proof that in 1905, there was no place in his mind for the association of the discontinuity in the emission and absorption of radiant energy with the discrete structure of electromagnetic radiation itself. Or, in plainer language, that his 1905 quanta should not be identified with the later photons.

1.6

Molecular Constitution and Brownian Motion (1905–1906)

We referred in Sect. 1.2.1 to the doctoral thesis submitted by Einstein at the end of 1901, shortly after finishing his studies at ETH, and withdrawn by him at the beginning of the following year. We have also commented on the trilogy he published between 1902 and 1904 containing his own formulation—in the line of Boltzmann and different from that of Gibbs—of the equilibrium statistical mechanics. In his scientific autobiography he clearly describes the connection between his investigations in statistical mechanics and the research we shall deal with in this section [Einstein (1949), in Schilpp (1970), 46–47]: Not acquainted with the earlier investigations of Boltzmann and Gibbs, which had appeared earlier and actually exhausted the subject, I developed the statistical mechanics and the molecular-kinetic theory of thermodynamics which was based on the former. My major aim in this was to find facts which would guarantee as much as possible the existence of atoms of definite finite size. In the midst of this I discovered that, according to atomistic theory,

34

Nobel Prize in Physics 1919 “for his discovery of the Doppler effect in canal rays and the splitting of spectral lines in electric fields”. 35 For a more technical and detailed analysis of the differences between discontinuity, quantization and corpuscularity, we refer the interested reader to Navarro; Pérez (2004).

58

1 The Sowing (Until 1905): The Annus Mirabilis there would have to be a movement of suspended microscopic particles open to observation, without knowing that observations concerning the Brownian motion were already long familiar.

It is in this line that Einstein’s new 1905 research on statistical mechanics is framed. His doctoral thesis—dated from Bern on 30 April and presented at the University of Zurich, as in his first attempt—was unanimously approved by the Faculty of Physics and Mathematics on 27 July and published in 1906 in Annalen with some slight modifications [Einstein (1906 a)]. In his dissertation he presents some original developments, in which he skilfully combines the hydrodynamic methods of Stokes (continuum physics) with the theory of Jacobus Henricus van’t Hoff (1852–1911) on dissolutions (discrete physics) reformulated by Einstein, in accordance with his conceptions of the molecular theory of solutions.36 The method used by Einstein allows him to obtain values, not only for the size of the molecules but also for Avogadro’s number, which he still refers to as the number of real molecules contained in one gram-equivalent or similar expressions.37

1.6.1

Avogadro’s Number and Brownian Motion

In 1811, Amadeo Avogadro (1776–1856) introduced the following assumption: under the same conditions of pressure and temperature, equal volumes of any gases contain the same number of molecules. This hypothesis soon acquired prestige, especially through its diffusion by French physicist André Marie Ampère (1775–1836), to whom it has sometimes even been attributed. It was Stanislao Cannizzaro (1826–1910) who clarified the question at the first International Congress of Chemistry (Karlsruhe 1860), both as to the true authorship of the hypothesis and as to its importance for the understanding of gas chemistry. The terminology “Avogadro’s number”—originally “Avogadro’s constant”—to refer to the number of molecules contained in one mole (or gram molecule) of any chemical substance was proposed in 1909 by French physicist Jean Baptiste Perrin (1870–1942).38 When he published the results of his experiments specifically contrasting Einstein’s theoretical predictions about Brownian motion (see Sect. 1.7.1) he stated, after referring to Avogadro’s hypothesis [Perrin (1909), 16]:

36

Van’t Hoff received the Nobel Prize in Chemistry 1901 “in recognition of the extraordinary services he has rendered by the discovery of the laws of chemical dynamics and osmotic pressure in solutions”. 37 Towards the end of the nineteenth century, the distinction between atoms and molecules was not clear in physics. Here we will use the term ‘molecule’ in a generic sense unless the respective authors explicitly refer to atoms. Although the molecular constitution of matter was only accepted after 1910, as we will see later, many physicists and chemists before that time reasoned as if the existence of molecules was guaranteed. 38 He received the Nobel Prize in Physics 1926 “for his work on the discontinuous structure of matter, and especially for his discovery of the sedimentation equilibrium”.

1.6 Molecular Constitution and Brownian Motion (1905–1906)

59

This invariable number N is a universal constant which it seems fair to call Avogadro’s constant. If this constant were known, the mass of any molecule would be known; the mass of any atom would also be known, since we can know, by the various means which lead to chemical formulae, how many atoms of each kind there are in each molecule.

This quotation illustrates very well the importance, not only for chemistry but also for kinetic theory, of the proper determination of this universal constant, which is actually a numerical bridge between the microscopic and macroscopic worlds. It seems that the Austrian chemist Joseph Loschmidt (1821–1895) was the first to calculate the number of molecules contained in a particular gaseous substance. From the incipient kinetic theory of gases, he obtained in 1865 a value for the number of molecules contained in a cubic centimetre of any gas, under normal conditions of pressure and temperature, a value that is known as the Loschmidt number.39 As this is univocally related to Avogadro’s number, the determination of either of these two constants is of utmost importance since it allows us to relate the number of molecules to the physical properties of a macroscopic system; for example, it can be used to relate molecular energy to thermodynamic energy. Although Loschmidt’s calculations are full of errors, if these are corrected, the corresponding value for Avogadro’s number turns out to be approximately 0.43 × 1023 molecules per mole.40 Shortly afterwards Maxwell, reasoning according to his own developments of kinetic theory, found a value on the order of ten times greater. The disparity between the proposals was such that, towards the end of the nineteenth century, one could find determinations of Avogadro’s number ranging between 1022 and 1024 . These results were reached by generalizing the application of the kinetic theory of gases to fields as diverse as contact electricity between metals, scattering of light, black-body radiation, and certain phenomena in liquids. As for Einstein’s interest in the determination of Avogadro’s number, it should be noted that, over a short period of time, he approached the problem from different perspectives. For example, in his famous 1905 paper on the energy quanta of radiation, he proposes the numerical value 6.17 × 1023 , obtained from radiation law (1.60), which is a valid approximation of Planck’s law (1.56) for large values of the quotient h ν / k T . At the end of his dissertation concerning suspensions in liquids, he writes the value 2.1 × 1023 . In addition, in a supplement accompanying the publication of his 1905 article in Annalen, Einstein proposes the value 4.15 × 1023 as that most in agreement with the experimental data of the day on dissolutions of sugar in water, which is the case treated in his thesis [Einstein (1906 a), in Beck (1989), 191]. He obtained this value with a calculation that contained a mathematical error discovered years later by his collaborator Ludwig Hopf (1884– 1939). The appropriate correction led Einstein in 1911 to propose the numerical value 6.56 × 1023 [Einstein (1911 c), in Beck (1993), 337].

The currently accepted numerical value, to four significant figures, is 2.686 × 1019 . In German scientific literature it was customary to refer to Avogadro’s number as “Loschmidt’s number”, which may eventually cause some confusion. 40 Recall the current value to four significant digits: 6.022 × 1023 . 39

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1 The Sowing (Until 1905): The Annus Mirabilis

As we shall see later, in his famous 1905 paper on Brownian motion Einstein does not arrive at obtaining a value for Avogadro’s number, but proposes an expression to it, whose application requires the experimental determination of the mean free paths travelled by the molecules. The results of this kind of experiment were obtained and published later by Perrin. After a long series of precise experiments, he proposed in 1909 the value 7.05 × 1023 as “the most probable value” for Avogadro’s number, in view of the different results obtained according to the type of solutions used [Perrin (1909), 111]. Another subject closely related to Einstein’s paper that we are going to comment on is that of the attempts to achieve a theoretical explanation of Brownian motion. Thus, it is not superfluous to refer to the state of the problem before 1905. It was the outstanding Scottish botanist Robert Brown (1773–1858) who described the phenomenon for the first time in 1828 in a pamphlet intended to informally make his discovery known, the explicit title of which reads: “A brief description of the microscopic observations made during the months of June, July and August, 1827, on the particles contained in the pollen of plants; and on the existence of active molecules in organic and inorganic bodies”. By depositing the pollen of different plants in water, Brown detected a dispersion of the pollen molecules, each of which appeared to undergo irregular zigzag movements directly observable under the microscope. However, the meaning that biologists gave to the term molecule at the beginning of the nineteenth century must be clarified. It was commonly admitted that animals and plants developed a certain number of “organic molecules”, whose gestation was completely unknown, that had their own movements and that constituted a primary material for all living beings.41 It is in this context that Brown’s contribution must be placed: he did not limit himself to revealing the movement of microscopic particles in solutions—which, moreover, had already been observed on numerous occasions—but he dissociated such movements from their previous organic connotations. What Brown certainly proved is that both organic and inorganic matter can be divided into microscopic corpuscles that, immersed in fluids, undergo this curious zigzag, so the problem was transferred to the field of physics. The long history of attempts to explain the nature and properties of Brownian motion, as it has since become known, is closely linked to the development of the kinetic theory of gases during the second half of the nineteenth century. It could not be otherwise, since it was soon found that in the motion of particles, in addition to their size, heat played an important role. We refer readers interested in this history to the appropriate bibliography.42 As a summary of the state of the problem when Einstein approached it, we reproduce the words that Poincaré

41

Of course, the cell theory, which was to appear in 1839, radically ended these ideas about “organic molecules”. 42 See, for example, Chapter 15 in Volume 2 of Brush (1976).

1.6 Molecular Constitution and Brownian Motion (1905–1906)

61

dedicated to this question in his speech at the St. Louis Congress of Arts and Science in September 190443 : The biologist, armed with his microscope, long ago detected in his preparations disordered movements of small particles in suspension: this is Brownian motion; he first thought it was a vital phenomenon, but soon saw that inanimate bodies danced with no less ardour than the others; then he passed the problem to the physicists. Unfortunately, the physicists remained too long disinterested in this question; light is focused to illuminate the microscopic preparation, they thought; with light goes heat; hence inequalities of temperature and internal currents produce in the liquid the motions to which we are referring. Mr. Gouy [about 1888], however, looked more closely into the problem and saw, or thought he saw, that the former explanation was untenable, that the motions are more energetic when the particles are more minute, but that they are not influenced by the mode of illumination. So, if these movements never cease, or rather are incessantly reborn, without any collaboration from an external source of energy, what are we to think? Of course, we should not give up our belief in the conservation of energy, but we see before our eyes how sometimes motion is transformed into heat by friction but we see before our eyes how sometimes the movement is transformed into heat by friction and other times how, inversely, the heat is transformed into heat by friction. and other times how inversely the heat is transformed into movement, and this without loss, since the movement lasts indefinitely. This is the opposite of Carnot’s principle.

Between the contributions of Louis George Gouy (1854–1926) mentioned by Poincaré and Einstein’s foray into the field, mention should be made of Felix Exner (1876–1930), who in 1900, on the basis of his own experiments, added to the already widespread belief that the speed of Brownian particles decreased with increasing particle size, the fact that this speed increased with increasing liquid temperature. Einstein published his explanation of Brownian motion in Annalen in 1905 before his doctoral thesis. Since the theoretical formalism is practically the same in both papers, it has sometimes been implied that in his theory of dissolutions, Einstein’s early ideas on the conjunction of diffusion (discrete treatment) and viscosity (continuous treatment) already appear in his explanation of Brownian motion, and that in his doctoral thesis he merely repeats the same arguments. This is not entirely true: we need to clarify the chronology. The thesis is dated 30 April 1905. However, bureaucratic formalities, together with minor editing, meant that its publication in Annalen was delayed until February 1906. The famous article explaining Brownian motion, although it appeared in Annalen in July 1905, is dated May. Thus, all indications are that Einstein’s ideas on the physics of molecules suspended in liquids were embodied first in his doctoral thesis and then in his paper on Brownian motion, and not the other way around.

43

The memoir presented by Poincaré can be found in Sopka (1986), 281–299. The text we have quoted is on page 287; it is also reproduced in Brush (1976), Vol. 2, 670–671.

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1.6.2

1 The Sowing (Until 1905): The Annus Mirabilis

Kinetic Theory of the Motion of Particles in Suspension (1905)

In 1905, Einstein applied his formulation of statistical mechanics to study the motion of particles suspended in liquids [Einstein (1905 b)]. He begins by calculating the relationship between the pressure p exerted on these particles and the concentration ν, which he assumes to be low. After justifying that particles in suspension behave like dissolved particles under these conditions and that p is the osmotic pressure he arrives at: p=

RT ν, N

(1.69)

where T is the absolute temperature, R is the universal constant of gases and N Avogadro’s number. Einstein associates the motion of the suspended particles with a force K, which is the force exerted on each of them and which depends on the position of the particle but not on time. After considering, for simplicity, that the force K is deployed in the x direction, the imposition of the condition of thermodynamic equilibrium leads to the following equality: Kν −

∂p = 0, ∂x

(1.70)

which indicates that “force K will be balanced by the forces of the osmotic pressure”. The last two expressions allow us to relate this force to the concentration of suspended particles and to absolute temperature: Kν=

RT ∂ ν N ∂x

(1.71)

On the other hand, Stokes’s theory on the motion of a spherical particle of radius P (using Einstein’s notation) moving in a liquid with viscosity coefficient k and subjected to force K, gives an expression for the number of suspended particles passing through the unit section perpendicular to the direction of motion, per unit time. Equating this expression with the number of particles diffused under the same conditions, the following result is obtained: νK ∂ν = D , 6π k P ∂x

(1.72)

where D represents the diffusion coefficient of the suspended substance. Eliminating the concentration gradient between the last two equalities, the following is obtained: D=

RT 1 N 6π kP

(1.73)

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63

Thus, the diffusion coefficient, if gravity is not considered, depends on universal constants, absolute temperature, viscosity of the liquid and size of the suspended particle. Einstein’s Treatment of Brownian Motion (1905)

Einstein introduces some original considerations on the possible statistical treatment of the previous phenomenon, which up to now he had analyzed according to the deterministic laws of classical physics.44 These considerations represented the starting point for more general analyses of Brownian motion, and were decisive for the modern treatment of stochastic processes.45 In his own words [Einstein (1905 b), in Beck (1987), 88]46 : Obviously, we must assume that each individual particle performs a motion that is independent of the motions of all the other particles; similarly, the motions of one and the same particle in different time intervals will have to be conceived as mutually independent processes as long as we think of these time intervals as chosen not to be too small. We now introduce into the consideration a time interval τ , which shall be very small compared with observable time intervals but still so large that the motions performed by a particle during two consecutive time intervals τ may be considered as mutually independent events.

Since there are no correlations between consecutive time intervals, and supposing n particles suspended in the liquid, Einstein writes the following expression for the number dn of particles that undergo, in a time interval τ , a displacement of the x-coordinate between Δ and Δ + dΔ: d n = n ϕ (Δ) d Δ,

(1.74)

where ϕ (Δ) d Δ represents the probability that the x-coordinate of any particle suffers a displacement between Δ and Δ + dΔ, in this time interval τ. This is the moment when Einstein introduces concentration as a continuous function of space and time: ν = f(x, t) which represents the number of suspended particles per unit volume. The value of f(x, t + τ ) can be deduced

44

By “classical physics” throughout this book we understand, not its meaning according to today’s terminology, but the set of physical theories commonly accepted at the beginning of the twentieth century. For a critical analysis of the use of the term “classical” in the history of physics, see Needell (1988), xxii–xxiv. 45 This can be checked by consulting, for example, the collection contained in Wax (1954). 46 Einstein (1905 b), in Beck (1987), 88.

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from f(x + Δ , t). By assigning to each displacement Δ its corresponding statistical weight he obtains: + ∫∞

f(x, t + τ ) =

f(x + Δ, t)· ϕ (Δ)d Δ

(1.75)

−∞

From here, using the above expressions, other plausible assumptions and a simple reasoning, Einstein obtains the by then well-known differential equation of diffusion: ∂ f(x, t) ∂ 2 f (x, t) , = D ∂t ∂ x2

(1.76)

where now the diffusion coefficient is given by the following expression: 1 D= τ

+ ∫∞ −∞

Δ2 ϕ (Δ) d Δ 2

(1.77)

The solution of differential equation (1.76), taking into account the corresponding initial and boundary conditions, turns out to be

f(x, t) = √

n 4π D

( exp − · √

x2 4D t

t

) (1.78)

The last equality already allows us to obtain, between other results, expressions for different mean values. In particular Einstein uses it to deduce the mean square displacement of a suspended particle: / x2 =

λx =

√

2 D t,

(1.79)

an expression which, by simply substituting the diffusion coefficient by its value according to (1.73) leads to the final result of his work: λx =

√

/ t·

1 RT N 3π kP

(1.80)

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65

In the case of water at 17 °C, which is the example chosen by Einstein to illustrate this result, the particles of one micron in diameter would experience a mean displacement of approximately six microns in one minute.47 As has since been pointed out, this last expression could be taken as the first case in which a fluctuation–dissipation relationship appears explicitly in physics because it relates a fluctuation, represented here by the mean free path traversed by the particle, to a dissipation effect associated with the viscosity of the liquid. Einstein does not fail to point out that this last expression could also be used to determine the value of N (he still does not refer to Avogadro’s number), once the displacements of particles suspended in liquids can be measured. And he ends his article by expressing his wish that this will happen soon [Einstein (1905 b), in Beck (1987), 134]: Let us hope that a researcher will soon succeed in solving the problem posed here [the determination of N], which is of such importance in the theory of heat!

1.6.3

Some Precisions

Einstein was not the only researcher to address the theoretical explanation of Brownian motion. For instance, as early as 1906, the Polish-born physicist Marian Smoluchowski (1872–1917) published a paper, also in Annalen, in which, after quoting Einstein’s 1905 paper, he presented an alternative method (see Chap. 2). Einstein was later to publish other articles on Brownian motion that are basically refinements and generalizations of his 1905 paper.48 It is difficult to overstate the importance and subsequent influence of his ideas on the topic. For example, Einstein’s treatment is considered to be an essential starting point for further development of stochastic processes, a very important issue not only for physics but also for other fields, including what is known as “econophysics” and several other issues related to the social sciences.49 A very interesting clarification concerning the limits of application of the renowned formula (1.80) is found in a paper by Einstein that appeared as early as 1906 and is mainly devoted to generalizing some of his results of 1905, now taking into account a possible rotational movement of the suspended particles. Einstein explicitly now states that his expression for the mean square displacement is not valid for arbitrarily small time intervals, since, as he himself justifies, this

In his calculations Einstein takes as the viscosity of water the numerical value k = 1.35 · 10–2 and for Avogadro’s number N he adopts the value 6 · 1023 , “in accordance with the results of the kinetic theory of gases”. For more details on the choice of these data, which is not entirely correct, see the explanatory notes 20, 21 and 22, in Stachel (1989), 234–236. 48 Fürth (1956) contains a compilation in English of Einstein’s five major papers on Brownian motion which appeared between 1905 and 1908, including his famous 1905 paper and dissertation. 49 See, for example, Bartholomew (1982). 47

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would lead to infinitely large instantaneous velocities [Einstein (1906 c), in Beck (1989), 189–190]: The reason for this [his formula for the mean square displacement is not valid for arbitrarily small time intervals] is that we have implicitly assumed in our derivation that the process occurring during time t is to be conceived as an event that is independent of the process occurring during the times immediately preceding it. But the shorter the times t chosen, the less this assumption applies.

A sharp observation in this respect is found in a 1907 article by Einstein devoted to clarifying some aspects of his earlier work [Einstein (1907 a)]. According to the molecular theory of heat, the value of the root mean square speed of a Brownian particle of mass m is determined by the principle of equipartition of energy: m

v2 3 RT = 2 2 N

(1.81)

The application of this formula to colloidal solutions of platinum provided a speed (square root of v 2 ) of 8.6 cm/sec, a result far superior to those provided by previous experiments. How can this enigma be solved? The solution proposed by Einstein is simple, but highly significant, illustrating the complex relationships that have occurred throughout history between theories and experiments. Let us look at his reasoning. According to Einstein, the speed appearing in (1.81) is not observable, in any form, due to the rapid damping of the particle by the viscous liquid. Stokes’s theory foresees for this case (platinum particles suspended in water) a reduction in speed to 1/10 of its value in only 3.3 · 10–7 s. However, at the same time the particle is receiving new impulses from the particles of that viscous liquid. Thus, it is precisely the speed that can be deduced from (1.80), not the one in (1.81), that encompasses the two effects, so it is suitable for predicting experimental results. This explains why the efforts of experimental physicists, who at that time reflected their knowledge of statistical mechanics almost solely by unconditionally applying the principle of energy equipartition, failed to reconcile predictions and observations: they confused the two velocities by making forecasts from (1.81), instead of (1.80). This is an illustrative example—interesting for the study of the relationship between theory and experiment—of how a notable disagreement between theory and experiment can be motivated by a mismatch between a quantity treated theoretically and its experimental image. Einstein came to deal, although without great success, with the problem of the practical measurement of the fluctuation represented by mean free path λ x at (1.80) and other fluctuations. For example, he tried to extrapolate his treatment of Brownian motion to apply it to certain phenomena related to fluctuations of electric voltage in capacitors and thus to study the possibility of measuring very small quantities of electricity. The subject attracted him to the point that he

1.7 Experimental Confirmation of Einstein’s 1905 Predictions

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thought of patenting a device to carry out this mission. However, for various reasons “Einstein’s little machine”, as it has sometimes been called, never came to fruition.50

1.7

Experimental Confirmation of Einstein’s 1905 Predictions

Einstein’s two 1905 papers on energy quanta and Brownian motion were not given equal consideration by the physicists of the time. Even after the experimental confirmation of the respective theoretical predictions, the hypotheses and supporting ideas were not equally accepted. This example provides a good opportunity to reflect on the characteristics and complexities of what has been called the “scientific method”. Let us analyze the two situations in chronological order.

1.7.1

Molecules Exist (Perrin, 1908–1909)

Experimental confirmation of Einstein’s theoretical predictions about the motion of particles suspended in liquids was a definitive endorsement for the general admission of the molecular constitution of matter. Perrin carried out convincing experiments in 1908 and 1909, strongly attracted by the possibility of collaborating decisively in demonstrating the validity of the molecular hypothesis, of which he was a strong supporter, possibly influenced by his previous research on cathode rays, which seemed to confirm the existence of “atoms of electricity”. Perrin always intuited, on the basis of his experimental investigations, a material reality whose apparent homogeneity and continuity were only gross aspects that could not mask, to the eyes of a fine observer, the ultimate discontinuity underlying microscopic behaviour. It is easy, with such a premise, to form an idea of the great attraction that the possibility of subjecting Einstein’s theory of particles suspended in liquids to a solid experimental test must have had for him from the outset. A long series of experiments carried out by Perrin in 1908 with particles of certain resin dyes led him to the first confirmation of Einstein’s theory. In fact, he proved the validity of the distribution given by the expression (1.78), albeit in a form more suitable for experimental tests, proposed in 1906 by Einstein himself, and which took into account the effect of gravity: (

N d W = const. exp − V (ρ − ρ 0 ) g x RT

50

) · dx

(1.82)

For more details see the editorial note “Einstein’s ‘Maschinchen’ for the measurement of small quantities of electricity”, in Klein et al. (1993 b), 51–55.

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1 The Sowing (Until 1905): The Annus Mirabilis

This formula expresses the probability distribution W for the vertical distance x, counted from the bottom of the container, in the case of Brownian particles of volume V and density ρ, suspended in a liquid of density ρ 0 , with gravity acceleration g [Einstein (1906 c), in Beck (1989), 184–185]. Perrin’s experiments, which played a decisive role in his award of the Nobel Prize in Physics 1926 (see Sect. 1.6.1), supplied the kinetic theory, already renamed statistical mechanics, with its basic main experimental support, which was that of the generalized molecular constitution of matter that from then on came to be considered as experimentally proven. Around 1909, the conversion to atomism was practically general. An illustrative example comes from two eloquent texts, written only three years apart, by the energeticist Ostwald, one of the most illustrious activists against the new doctrine [quoted in Brush (1976), Vol. 2, 698–699, emphasis in the original]: As I have been maintaining for the last ten years, the matter-and-motion theory (or scientific materialism) has outgrown itself and must be replaced by another theory, to which the name Energetics has been given (...). The question as to the identity or non-identity of the different portions of water is without meaning, since there is no means of singling out the individual parts of the water and identifying them (...) atoms are only hypothetical things. [Harvard, 1906]. I have convinced myself that we have recently come into possession of experimental proof of the discrete or grainy nature of matter, for which the atomic hypothesis had vainly sought for centuries, even millennia. The isolation and counting of gas ions on the one hand— which the exhaustive and excellent work of J. J. Thomson has crowned with complete success—and the agreement of Brownian movements with the predictions of the kinetic hypothesis on the other hand, which has been shown by a series of researchers—most completely by J. Perrin—this evidence now justifies even the most cautious scientist in speaking of the experimental proof of the atomistic nature of space-filling matter. What has up to now been called the atomistic hypothesis is thereby raised to the level of a well-founded theory, which therefore deserves its place in any textbook intended as an introduction to the scientific subject of general chemistry. [Leipzig, 1909].

Referring specifically to this generalized conversion to atomism, Einstein includes a reflection in his autobiography, written 40 years after these events, which is of great interest to those interested not only in the history of science but also in the role of the mind in the process of creating scientific knowledge [Einstein (1949 a), in Schilpp (1970), 48–49]: The agreement of these considerations with experience together with Planck’s determination of the true molecular size from the law of radiation (for high temperatures) convinced the sceptics, who were quite numerous at that time (Ostwald, Mach) of the reality of atoms. The antipathy of these scholars towards atomic theory can indubitably be traced back to their positivistic philosophical attitude. This is an interesting example of the fact that even scholars of audacious spirit and fine instinct can be obstructed in the interpretation of facts by philosophical prejudices. The prejudice—which has by no means died out in the meantime—consists in the faith that facts by themselves can and should yield scientific

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knowledge without free conceptual construction. Such a misconception is possible only because one does not easily become aware of the free choice of such concepts, which, through verification and long usage, appear to be immediately connected with the empirical material.

Some additional data help to gauge the importance that researchers have assigned from the outset to the Einsteinian approach to the motion of particles suspended in liquids. For example, four of the 12 scientific papers published before 1912 by any author that have the highest number of citations between 1961 and 1975 are by Einstein. The version of Einstein’s dissertation that appeared in Annalen in 1906 ranks first, and his 1905 paper on particles in suspension comes in third place; his papers on relativity and quantum theory do not appear in this ranking [Data quoted in Pais (2005), 89–90]. A few words by Born referring in 1949 to Einstein’s research on Brownian motion constitute a good epilogue to this section. Not only did he emphasize that “by its simplicity and clarity this paper [of 1905] is a classic of our science”, but he completed his statement with the following comment [Born (1949), in Schilpp (1970), 166]: I think that these investigations of Einstein [on Brownian motion] have done more than any other work to convince physicists of the reality of atoms and molecules, of the kinetic theory of heat [statistical mechanics], and of the fundamental part of probability in the natural laws. Reading these papers, one is inclined to believe that, at that time [around 1905-1906], the statistical aspect of physics was preponderant in Einstein’s mind.

1.7.2

Reality of Energy Quanta (Millikan, 1916)

The initial impact of Einstein’s 1905 energy quantum hypothesis can be quite accurately inferred from the speech that Planck delivered as early as 1913 to the Prussian Academy of Sciences, presenting the candidate for a new academician51 : In sum, it can be said that among the important problems, which are so abundant in modern physics, there is hardly one in which Einstein did not take a position in a remarkable manner. That he might sometimes have overshot the target in his speculations, as for example in his light quantum hypothesis, should not be counted against him too much. Because without taking a risk from time to time it is impossible, even in the most exact natural science, to introduce real innovations.

In principle, one might think that the generalized objection in the face of such a new and problematic notion as energy quanta was justified as long as reliable experimental results about the photoelectric effect had not been obtained

51

Part of the proposal by Planck and other academicians for Einstein’s membership of the Prussian Academy of Sciences. In Beck (1995), 337–338.

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1 The Sowing (Until 1905): The Annus Mirabilis

and its agreement with Einstein’s theoretical predictions of 1905 roundly verified. However, nothing could be further from the truth. Although, as we have already anticipated in Sect. 1.5.2, the experimental task was satisfactorily carried out by Millikan in 1916, even he did not think that the results of his experiments represented the confirmation of the quanta hypothesis. In an article where he presents the results of the various experiments he carried out to rigorously test the validity of Einstein’s equation (1.68) for the photoelectric effect, Millikan concludes [Millikan (1916 a), 31]: Einstein’s photoelectric equation, whatever may be said of its origin, seems to stand up accurately under all of the tests to which it has been subjected.

However, the recognition of the validity of Einstein’s equation does not imply that of the theory on which it is based, as Millikan makes clear in another article of the same year, 1916, devoted to experimentally determining the value of Planck’s constant h = R β/N [Millikan (1916 b), 384]: Despite then the apparently complete success of the Einstein equation [our (1.68)], the physical theory of which it was designed to be the symbolic expression is found so untenable that Einstein himself, I believe, no longer holds to it. But how else can be the equation be obtained?

This perception of Einstein’s 1905 quantum theory lasted for more than 20 years. Let us recall that he was awarded the Nobel Prize in Physics 1921 “for his services to theoretical physics, and especially for his discovery of the law of the photoelectric effect [our (1.68)]”, a citation that consciously avoids any reference to his quantum hypothesis, which had served as a theoretical underpinning, in view of the insecurity still existing in 1921 about its validity. It must be admitted that Einstein himself did not seem very interested in clarifying his position in this respect; in all probability he was not in a position to do so. However, we must not confuse the indispensable essential caution over such new and problematic concepts with the doubt or rejection of his ideas about the light quanta. An illustration of this prudence can be found in a clarification that he had to make in the debate that followed his intervention in the famous first Solvay conference (1911) [Einstein (1912 a), in Beck (1993), 431, emphasis added]: The quantum hypothesis is a provisional attempt to interpret the expression for the statistical probability W of the radiation. By conceiving radiation as consisting of small complexes [Komplexen] of energy h ν, one found an intuitive interpretation of the probability law for low-intensity radiation. I emphasize the provisional character of this auxiliary idea, which does not seem compatible with the experimentally verified conclusions of the wave theory.

He had written to his friend Besso a few months earlier expressing similar feelings about the quantum hypothesis, although Einstein seemed to be pondering

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the idea of turning his research around and trying to draw conclusions from the hypothesis rather than seeking arguments for its rigorous justification52 : I no longer ask whether these quanta really exist. Nor do I try to construct them any longer, for I now know that my brain cannot get through in this way. But I rummage through the consequences [of their existence] as carefully as possible so as to learn about the range of applicability of this conception.

Contrary to what one might naively think, Planck’s quantum ideas were more widely accepted—or, at least, not as explicitly rejected—than those of Einstein, despite the caution and limitations that the latter had included in his 1905 article. In the opinion of the prestigious historian Abraham Pais (1918–2000), Einstein’s colleague at the Institute for Advanced Study in Princeton, there were two strong reasons for this: one of a theoretical nature and the other related to experimentation [Pais (2005), 382–386]. We will add a third, this one of a sociological nature. Planck’s quantum hypothesis concerned the quantisation of energy of material oscillators (Planckian resonators), which did not seem, in principle, to infringe upon Maxwell’s electromagnetic theory, which incorporated the notion of the continuous field of radiation. Moreover, it should be borne in mind that nor was the quantization of certain quantities a phenomenon completely alien to classical physics. Suffice it to recall the frequencies of the modes of vibration of strings. Taking all this into account, and given the reigning obscurity surrounding the topic of the interaction between matter and radiation, taking on Planck’s hypothesis did not seem an exaggerated obligation if one wished to advance in that field, since it did not seem to imply important side effects. In connection with experiments, Planck’s theoretical contributions provided a reliable and highly accurate explanation for the abundant experimental results that were being obtained about the black-body energy density distribution. Planck’s personal prestige at the beginning of the century—since 1892 he had occupied the chair at the University of Berlin that had fallen vacant five years earlier on the death of Kirchhoff—may also have been an important additional factor in the positive acceptance of his ideas. Einstein, on the other hand, who was scarcely known in the scientific field around 1905, presented a strange quantum hypothesis that seemed to suggest, at the very least, the need for a thorough revision of Maxwell’s equations for the radiation field. Moreover, hardly anything was offered in exchange because, as we have seen, the references to experimental data in his 1905 article could not be adequately assessed due to the lack of rigorous results at that time on the three phenomena he analyzed there (Stokes’ rule, the photoelectric effect and ionization of gases). For example, something as apparently simple, although fundamental for the confirmation of Einstein’s theory, as the linearity between the energy of the released electron and the frequency of the incident radiation in the photoelectric effect, had to wait another decade before it was regarded as ratified by experiments.

52

Letter from A. Einstein to M. Besso 13 May 1911. In Beck (1995), 187.

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As we shall see, the acceptance of Einstein’s quantum ideas that emerged from 1905 onwards would not improve substantially during the following decade, in spite of the growing confirmation of the experimental predictions elaborated with his incipient quantum theory.

2

The Flowering (1906–1913): Einstein Introduces Himself to the Scientific Community

My own interest in those years was less concerned with the detailed consequences of Planck’s results, however important these might be. My major question was: What general conclusions can be drawn from the radiation-formula concerning the structure of radiation and even more generally concerning the electro-magnetic foundation of physics. Albert Einstein, 1949.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 L. Navarro Veguillas, The Lesser-Known Albert Einstein, History of Physics, https://doi.org/10.1007/978-3-031-35568-4_2

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2 The Flowering (1906–1913): Einstein Introduces Himself to the Scientific …

Memorial plaque on the house of Mrs. Berta Fanta in Prague, where Einstein often went to talk with friends and play violin. The Einstein family lived in that city during 1911. [Photo: Oriol Navarro]

2.1 From Bern to Zurich via Prague

2.1

75

From Bern to Zurich via Prague

We had ‘abandoned’ young Albert in Bern, with his stable position at the Swiss Patent Office and a comfortable family life with his wife Mileva and his little son Hans Albert. The year is 1905, and he has just published papers on seemingly unconnected subjects—quantum physics and statistical physics, apart from the theory of relativity—which he had broken into the field of high-level research. On April 1, 1906, he was promoted to “second-class technical expert”, with a corresponding increase in salary: 600 Swiss francs, making him 4500 francs a year. The promotion was due—according to Friedrich Haller, then Director of the Patent Office—to the fact that Einstein has become “increasingly familiarized himself with technical problems, so that he now handles the most difficult technical patent applications with the best success and belongs to the most prized experts in the agency” [In Klein et al. (1993b), 39].1 Soon Einstein begins to hear about the impact of his 1905 publications. For example, Max von Laue (1879–1960)—Planck’s assistant in Berlin and later winner of the Nobel Prize in Physics 1914 “for his discovery of the diffraction of X-rays by crystals”—visited him at the Patent Office to consult some questions about his theory of relativity.2 His meeting was the beginning of a long and deep friendship between them. Although the work at the office kept Einstein away from the academic world, he never ruled out joining it one day. Therefore, in June 1907, he submitted a formal application for Privatdozent in theoretical physics by the University of Bern. This did not imply a specific teaching post or a salary; only the recognition of his ability to teach at that university if he were to be entrusted with a specialized course. In such a case, the few students interested in the particular issue would have to pay a small fee that was directly assigned to the professor. Einstein is unsuccessful in his attempt. The rejection of his application is because he does not enclose the required original unpublished research paper— Habilitationsschrift—, which he had to prepare specifically for the occasion. He did so in January 1908, submitting a memoir that has not been preserved, entitled “Consequences of the law of distribution of energy of black body radiation in relation to the constitution of radiation”. It is quite possible, because of the title and the dates, that the contents of this report have several points in common with articles that he was to publish on this subject in 1909, which we shall address in due course (Sect. 2.4). Credited as Privatdozent in February 1908, Einstein lectures two courses at the University of Bern—summer and winter semesters—on statistical mechanics and radiation theory, respectively. Since he has to combine the new assignment with his work at the Patent Office, the lectures are to be given at extreme hours of the day. The untimely timetable and the extremely theoretical nature of the

1

In Schilpp (1970), 47. Max von Laue was to publish in 1911 the first book devoted exclusively to the theory of relativity: Das Relativitätsprinzip.

2

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2 The Flowering (1906–1913): Einstein Introduces Himself to the Scientific …

courses must have been factors that influenced the shortage of students: three in the first semester, including his friend Besso, and four in the second. Thus, the new professor only receives symbolic remuneration in those semesters. Kleiner, the Zurich professor who had been involved in the rejection of the first doctoral thesis project and in the positive evaluation of the second one, had followed not only the young Einstein’s research, but also his first—by no means promising—step as university professor in Bern. Everything suggests that Kleiner intended to offer him a professorship at the University of Zurich as soon as there became available a vacancy of the appropriate academic rank. Einstein would later show signs that he understood his connection with the Patent Office as something temporary, always thinking that his future should be on the academic side, preferably in Zurich. An opportunity arose at the beginning of 1909, when Kleiner succeeded in getting the University of Zurich to endow a post in theoretical physics. The job initially seemed to have been allocated to Friedrich Adler (1879–1960), Kleiner’s assistant.3 However, when Einstein showed interest in applying for the position, both Kleiner and Adler himself—friend and admirer of Einstein since their student days—unequivocally supported Einstein’s candidacy. Thus, in May 1909, at the age of thirty, Einstein is appointed Ausserordentlichen Professor für Theoretische Physik, something like assistant professor of theoretical physics at the University of Zurich. Now he has a fixed salary and can supervise students’ work, in addition to his own research. In the autumn of 1909, the three members of the Einstein family move to Zurich, the city that Mileva had so longed for, in love with its streets and its people since her student days. It was in the summer of 1909 that important public signs of recognition of Einstein’s early contributions to physics began to appear. For example, in July, he was awarded honoris causa doctorate from the University of Geneva, the first of the nearly twenty that he was to receive during his life.4 In September of that year he was invited to take part in the meeting of the GDNÄ (Gesellschaft Deutscher Naturforscher und Ärzte)—German Society of Natural Scientists and Physicians— in Salzburg. Einstein presents a paper in which he analyses the behaviour and nature of electromagnetic radiation from a perspective that contains the germ that would later be known as “wave-particle duality”.

3

Friedrich Adler would soon abandon physics to devote himself to politics. Between 1911 and 1916 he was secretary of the Austrian Social Democratic Party, of which his father, Victor Adler, had been a founder and high leader. In 1916 Friedrich Adler—not to be confused with Alfred Adler, the Austrian psychologist co-founder of psychoanalysis—was charged with and convicted of the murder of the Austrian Prime Minister, Count Stürgkh. His death sentence was not carried out, and he was released in 1919, after the end of the First World War. 4 Einstein was one of the one hundred and ten recipients of a honoris causa doctorate by the University of Geneva in the same ceremony, which was held at this university to commemorate the tercentenary of its foundation by Calvin. Other prestigious personalities invested at the same ceremony were Marie Curie, Wilhelm Ostwald and Ernest Solvay.

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Einstein begins his lectures at the University of Zurich in October 1909, in the winter semester, with a course in thermodynamics and another in mechanics. In March 1910, summer semester, in addition to the second part of the mechanics course, he begins another one on kinetic theory of heat—today we would say on statistical mechanics—, being in charge also, together with Kleiner, of maintaining a laboratory for advanced students, as well as directing a physics seminar. One of the few students enrolled in his courses in this summer semester is Hopf—to whom we have already referred in Sect. 1.6.1—who came from taking courses with Arnold Sommerfeld (1868–1951) in Munich, and who was to collaborate with Einstein in two papers that would be published that same year. During the winter semester of 1910, the last one he will spend at the University of Zurich, Einstein teaches a course on electricity and magnetism, in addition to maintaining the seminar and the laboratory. There is no doubt that his courses and seminars on mechanics, thermodynamics, statistical mechanics and electromagnetism will be of great interest to him, contributing decisively to Einstein’s knowledge of these subjects. Einstein’s starting salary in Zurich is 4,500 Swiss francs, which is almost the same as what he was earning at the Patent Office when he left, and which did not provide the family with a minimum of financial relief. His scientific prestige, on the other hand, had risen to the point where he was proposed by Ostwald in October 1909 as a candidate for the Nobel Prize in Physics for his theory of relativity. The economic needs of the family increased with the birth of Eduard in July 1910, Albert and Mileva’s second son. In such circumstances, and in spite of the great attachment they both feel for Zurich, Einstein is forced to consider the proposal that came to him in the spring of 1910 for a professorship in theoretical physics in Prague. The salary increase offered to him in Zurich—he would earn 5,500 Swiss francs from October 1910—and other promises of better pay were not enough to keep Einstein in Switzerland. In January 1911 he formally accepts his appointment as Ordentlicher Professor für Theoretische Physik—full professor—at the German University of Prague.5 The starting salary of almost 8,672 crowns allowed him to improve his family’s standard of living in Zurich.6 Thus, at the end of March 1911, after the end of the winter semester, the family moved from Zurich to Prague, where Einstein begins in April 1911 the summer semester, by lecturing on mechanics and thermodynamics, and conducting a physics seminar. During the winter semester he will continue to teach the same subjects. It cannot be said that the stay in Prague was very pleasant for the Einstein family. Albert sometimes felt the more or less disguised enmity of some Czechs,

5

The German University of Prague —actually Karl-Ferdinand Universität— had resulted from the splitting, in 1882, of the former University of Prague (founded in 1349) into two: one in Czechlanguage and another in German-language. 6 Einstein’s starting wage in Prague, which included salary and various allowances, amounted to 9,100 Swiss francs, which was far more than the 5,500 francs he earned in Zurich. For details of Einstein’s financial retributions in Prague, see Beck (1995), 163.

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because he was fully regarded as a German, although his feelings were not in accordance with this. As for Mileva, it can be said that she never really integrated into the atmosphere of life in Prague; it seems that the Bohemian capital did not appeal to her at all. At the end of October 1911 Einstein travels to Brussels to participate in the first Solvay conference. It is the time for him to make personal contact with the international leaders of physics gathered to discuss the quantum ideas that have emerged during the first decade of the twentieth century. The paper presented by Einstein, and his interventions in the corresponding discussions, will be dealt with later on. Einstein’s prestige increases considerably after his participation in Brussels and the publication in 1912 of the proceedings of the conference [Langevin & De Broglie (1912)]. He receives tempting offers from various academic institutions, and soon he opts for the ETH in Zurich, the city that Mari´c had longed for and where they had both studied. Marie Curie (1867–1934)—née Maria Sklodowska, awarded the Nobel Prize in Physics 1903 and Nobel Prize in Chemistry 1911—7 and Poincaré, among others, wrote reports favourable to Einstein addressed to Pierre Weiss (1865–1940)—author, in 1907, of the “hypothesis of the molecular field”, basis of the theoretical explanation of ferromagnetism—as responsible for proposing the name of the candidate chosen to occupy the new chair in theoretical physics. We reproduce here a part of Poincaré’s letter because, although highly laudatory as a whole, it leaves a hint of doubt as to the French scholar’s real opinion of Einstein, of whom he does not expressly mention any contribution8 : Mr. Einstein is one of the most original minds I have ever known. In spite of his youth, he has already attained a very high reputation among the leading scholars of his time. What we must admire most in him is the ease with which he adapts himself to new conceptions and knows how to draw all the consequences from them. He does not remain anchored to classical principles and, faced with a problem of physics, he is ready to consider all possibilities. This translates immediately into his mind by predicting new phenomena which may one day be experimentally verified. I do not mean to say that all his forecasts must resist experimental control on the day when such control is possible. On the contrary, since he investigates in all directions, it is to be expected that most of the paths he follows will lead to dead ends; but at the same time it is to be expected that one of the directions he has indicated will be the right one; and that is enough. This is exactly how one should proceed. The role of theoretical physics is to pose the questions correctly, and only experimentation

7

Nobel Prize in Physics 1903 was awarded half a prize money for Henri Becquerel (1852–1908), “in recognition of the extraordinary services he has rendered by his discovery of spontaneous radioactivity”, and half for Curie couple, “in recognition of the extraordinary services they have rendered by their joint researches on the radiation phenomena discovered by Professor Henri Becquerel”. Nobel Prize in Chemistry 1911 was awarded “in recognition of her services to the advancement of chemistry by the discovery of the elements radium and polonium, by the isolation of radium and the study of the nature and compounds of this remarkable element”. 8 Reprinted (in French) in Seelig (2005), 228–229.

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79

can solve them. The future will show more and more what Mr. Einstein’s value is, and the university that knows how to get hold of this young master is sure to be highly honoured.

Einstein accepts the official offer of the ETH in January 1912 to take up a professorship in theoretical physics at a starting salary of 11,000 Swiss francs, twice what he earned at the University of Zurich when he left a year earlier. He is asked to finish the course in Prague, to which Einstein agrees. Therefore, the move to Zurich is delayed until August of the same year. Before leaving Prague Einstein had been visited by Ehrenfest, a young Viennese physicist who had received his doctorate in 1904 under Boltzmann’s supervision, and who was travelling through Europe in search of an academic position after finishing his five-year stay in St. Petersburg, in the homeland of his wife Tatiana, a mathematician he had met in Göttingen. It was the beginning of a long and intimate friendship. In fact, Einstein proposed Ehrenfest as his successor in Prague. Additionally, Sommerfeld offered him a position in Munich, but in the end Ehrenfest opted for the advantageous offer of the University of Leiden to fill the vacancy left by Lorentz at the end of 1912, who had prematurely retired—not yet reached the age of sixty—so that he could devote himself entirely to his work as director of research at Teylers Museum in Haarlem. Thus, it was in the winter semester of 1912 that Einstein incorporated at ETH lecturing courses on mechanics and thermodynamics at ETH. In the following summer semester, he teaches on mechanics of continuous media and molecular theory of heat (today, we would say statistical mechanics). Finally, in the winter semester of 1913–1914, his courses were on optics, electricity and magnetism. The teaching always included seminars and additional practical classes, although without ever exceeding a total of ten hours of teaching per week, as stated in the conditions of the proposal. The return to Zurich is a breath of fresh air for the Einsteins. Mileva returns to her longed for city and Albert is reunited with some of his former classmates who are now, like himself, professors at the ETH, among them Grossman, who helped him so much with his class notes and whose father helped him to get into the Patent Office. The scientific collaboration with Grossman—to whom, by the way, Einstein dedicated his doctoral thesis in 1905—was soon to become very intense.9 Einstein’s scientific reputation, although it did not reach the general public, was constantly growing among scientists. With some of the most famous scientists of the time, he even established personal ties—not only with German-speaking people. For example, Albert and Mileva visit Marie Curie—already awarded two Nobel prices—at Paris in March 1913, taking advantage of an invitation to deliver a lecture at the French Physics Society. Reciprocally, the Curies—mother and two daughters, Irène and Ève—travelled later to Zurich, in August of the same year, and shared with the Einsteins some excursions in the Swiss Alps.

9

For a more detailed account of the relations between Einstein and Grossman, especially as far as their collaboration in general relativity, see, for instance Pais (2005), 208–227.

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However, the stay in Zurich was much shorter than they would have wished; Mileva above all. Although, as will be seen later, Planck did not accept Einstein’s quantum theory of electromagnetic radiation at the time, among his plans then included to do his best to convince Einstein to move to Berlin. Thus, Planck—at the age of fifty-five and widely known at the time—travelled to Zurich in the spring of 1913, accompanied by Walter Nernst (1864–1941) to feel Einstein out.10 He was told that, if accepted, he would be proposed as professor at the University of Berlin and membership of the prestigious Preussische Akademie der Wissenschaften— Prussian Academy of Sciences—and later, he would be appointed director of research at the future section of the Kaiser Wilhelm Institute. As far as teaching was concerned—this was not a strong point in Einstein’s interests—he could freely choose the courses he wanted to lecture, or even devote himself exclusively to research.11 On 12 June 1913, Planck proposes Einstein as a member of the Prussian Academy of Sciences in its Physics-Mathematical Section. He is elected on July 24, with 21 votes for and 1 against, after which the offer becomes firm, although the appointment is delayed until November 12. Einstein could not let this opportunity pass him by, despite his attachment to Zurich, his discomfort with the German way of life and the more or less firm opposition of his wife, with the marriage already in the midst of crisis. Thus, a maxim of the time is fulfilled: Switzerland was a first-class waiting room (Wartesaal 1. Klasse) for German scientists until they were called to Germany [Klein et al. (1993b), xxxvii]. Einstein’s investigations between 1906 and 1913 show his first steps in directions that would bear fruit later on. His concern about the discrete behaviour of radiation energy pushed him in different directions. In this sense, it is worth mentioning his incorporation of quantum assumptions to explain the behaviour of matter, formulating in 1907 a quantum theory of monoatomic solids that can be understood as the origin of modern solid-state theory. Precisely this will be the subject chosen by Einstein for his presentation to the international community of physicists at the first Solvay conference in 1911. Their concerns about the nature of radiation do not cease, trying to find arguments to justify the necessity of incorporating into physics some quantum hypothesis. In this direction, he proposes a Gedankenexperiment, which, on different occasions later, will allow him to obtain results of great significance; among them, he suggests, in 1909, the possible necessity of having to admit a dual structure—undulatory and corpuscular—for radiation. And his deduction of the photochemical equivalent law, in 1912, is yet another episode in his struggle to deduce the need to admit the reality of quantum jumps.

10

Walter Nernst would be awarded the Nobel Prize in Chemistry 1920, “in recognition of his work in thermochemistry”. 11 Einstein, like many others in those days, sometimes referred to it as Berliner Akademie der Wissenschaften —Berlin Academy of Sciences—. The institution traces its origins back to 1700, when it was founded by the Elector of Brandenburg Frederick III—later King Frederick I of Prussia (1657–1713)— under the auspices of Gottfried Leibniz (1646–1716), who was its first president.

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Two new fronts appear in Einstein’s work at that time. On the one hand, statistical fluctuations acquire a relevance that they will never lose in future research. Fluctuations allow him to obtain a theoretical explanation of critical opalescence in 1910, although certainly with some weaknesses, as we will see. On the other hand, we shall end the present chapter by alluding to another attempt by Einstein to tackle problems different from the nature of radiation. We refer in particular to his incursion in 1913, together with Otto Stern (1888–1969), on the topic of the specific heat of gases.12 This is a subject in which, as we shall see, the “second Planck theory” and the “zero-point energy” will play an essential role.

2.2

The Critique of Planck (1906)

Let us return to an essential point of quantum theory, which is the relation between Planck’s quanta and Einstein’s quanta. We noted in Sect. 1.5.2 how Einstein showed in 1905 that Planck had departed from the classical orthodoxy by introducing energy quanta in 1900. However, he argues this on the basis of the result obtained by Planck, who had succeeded in avoiding the ultraviolet catastrophe, without this being possible—according to Einstein—within a pure classical treatment. We also anticipated that in 1905 Einstein did not submit Planck’s ideas to rigorous criticism—presumably because he did not know how to do it then—but only presents his own ideas on the issue. It was not long after, in an article that appeared in 1906, that Einstein analysed Planck’s hypothesis in depth [Einstein (1906d)]. It is a paper as generally ignored as it is interesting from different perspectives. We find Klein’s considerations on this paper highly clarifying [(Klein 1980, 171)]: He [Einstein] had at last penetrated the obscurities of Planck’s theory of the spectrum of black body radiation, and was evidently surprised at what he found. Anyone who still thinks that Einstein’s use of light quanta in 1905 was a generalization or extension of Planck’s theory need only read this paper of 1906 to be disabused of that idea. For Einstein began by remarking that when he wrote his paper of the previous year it had seemed to him that Planck’s work was a ‘contrast’ to his own. Planck had indicated that his theory was built on Maxwell’s theory, the electron theory, and statistical mechanics, and yet it had led to the distribution in Eq. (12.10) [our (1.56)]. Einstein was convinced, however, that if one started with precisely those basic theories the only possible outcome was the unacceptable distribution given by Eq. (12.9) [our (1.60)] how could Planck have arrived at a different answer [which leads to the catastrophe in the ultraviolet] if he had argued from the same premises? That was what puzzled Einstein, even as it puzzled Lord Rayleigh, who asked the same question in Nature in the spring of 1905, and Paul Ehrenfest, who asked it again a year later.

12

Stern had submitted his doctoral dissertation in physical chemistry in Breslau. He moved to Prague to work with Einstein as an assistant and would later accompany him on his move to Zurich in August 1912.

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On the first page of his article, Einstein already anticipates his conclusion: Planck, contrary to what he preached, not only departed from the strict framework defined by statistical mechanics, Maxwellian electromagnetism and electron theory, but also made implicit use of the hypothesis of the quanta of light. To highlight these incongruities in Planck’s work, Einstein begins by recalling that the application of statistical mechanics, in particular the principle of equipartition of energy, to a system of classical harmonic oscillators leads inexorably to classical harmonic oscillators leads inexorably to (1.60), with ultraviolet catastrophe included, and not to (1.56), which is Planck’s formula. However, this is reached if one imposes on the oscillators the constraint that their energy can only acquire values that are integer multiples of the following elementary quantity: ε = (R/N )βν,

(2.1)

where R is the universal constant of gases, N is Avogadro’s number and β is the constant appearing in Wien’s phenomenological law (1.62).13 This limitation is equivalent, as we will see later on, to switching from a ‘counting’ of points—to represent states—to another of cells in the phase space of the system. That is, one obtained (1.60) or (1.56) with analogous reasoning and methods, depending on whether a continuous or a discrete nature was admitted for the values of the resonator energy. More precisely, admitting this discrete nature of energy spectrum—the quantum hypothesis— is what deviates Planck from the strict classical procedure, according to Einstein. Thus, the ‘mystery’ associated with Planck’s theoretical deduction in 1900 of the formula that today bears his name was partly unravelled. However, we say ‘partly’ because there were still some obscure points. For example, was it possible to reconcile the discrete spectrum energy with the continuum of the electromagnetic field? It should not be forgotten that expression (1.55), one of the pillars on which Planck’s deduction is sustained, can only be justified on the basis of Maxwell’s electromagnetic field theory. In addition, how is it that such an apparently incoherent calculation had led him to the expression (1.56), so in agreement with the experimental results? Einstein does not clarify the question completely, but he gets out of the impasse with dignity, at the same time that he once again reveals his mechanical-statistical side [Einstein (1906d). In Beck (1989), 196. Emphasis in original]: Although Maxwell’s theory is not applicable to elementary resonators, nevertheless the mean energy of an elementary resonator in a radiation space is equal to the energy calculated by means of Maxwell’s theory of electricity.

13

It was not until 1909 that Einstein would use the notation k (“Boltzmann constant”) for the R/N quotient.

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Moreover, this consideration of Einstein shows a characteristic that will always be present in the evolution of his quantum ideas: the pretension to solve the difficulties arising when analysing the thermodynamic behaviour of radiation, starting from a microscopic vision of the situation, that is, from the interaction between the electromagnetic field and charged oscillators. Averages defined on the relevant microscopic structures would allow, according to the usual methods of statistical mechanics, to obtain values of macroscopic quantities that could present characteristics very different from the usual microscopic properties. Of course, Einstein cannot stop at the difficulty of having to consider apparently incompatible aspects in the same theory, as is the case of the microscopic discrete (energy quantisation of oscillators) and the macroscopic continuum (the radiation field). He had already gone through a similar trance when he had to reconcile mechanical time reversibility (microscopic level) with thermodynamic irreversibility (macroscopic level): statistical mechanics had taught him to reconcile such apparently contradictory properties. However, it is fair to say that this position represents for many physicists of those times an insuperable incoherence: it is not acceptable to introduce discrete and continuous aspects of the radiation field in the same theory. Difficulties of this type—the supposed incompatibility between the quantum hypothesis and Maxwellian electromagnetism—considerably delayed the acceptance of quantum ideas, given the solidity that the theory of the electromagnetic field had already acquired. The second and last section of this 1906 paper is devoted by Einstein to the application of the quantum hypothesis to the analysis of certain aspects of the “Volta effect”, a topic that is not particularly relevant for our purposes here.

2.3

From Radiation to Matter: First Quantum Theory of Solids (1907)

Soon, the quantum ideas expressed by Einstein in 1905 went beyond the issue of radiation to enter the topic of matter. His first paper in the new field was published in Annalen at the beginning of 1907 and is entitled “Planck’s theory of radiation and the theory of specific heats” [(Einstein 1907b)]. In connection with the above title, it is legitimate to ask the following question: what does Planck’s theory (on the properties of radiation in emission and absorption phenomena) have to do with the theory of specific heats (on nonconducting solids) in light of the radically different nature of the respective physical realities involved? We begin by clarifying this question. Einstein, once he was convinced that the quantisation of energy introduced by Planck is a property of resonators, and not only of the behaviour of energy in the emission and absorption of electromagnetic radiation, he tackles a problem outside that of the black body radiation, as is the theoretical calculation of the specific heat of solids, in the hope of achieving a good theoretical approximation to the experimental values of the moment which, as we shall see later on, presented clear

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discrepancies with the forecasts based on the traditional “Dulong-Petit rule”.14 He approaches it on the basis of certain experimental results obtained by Paul Drude (1863–1906), who is expressly cited, according to whom there were sufficient reasons—supported by his own experiments on optical dispersion—to think that the eigenfrequencies in the infrared dispersion were related to ion oscillations, while those in the ultraviolet were related to electron vibrations [Einstein (1907b). In Beck (1989), 219–220]. Einstein begins by recalling how the rigorous application of kinetic theory to an aggregate of linear oscillators monochromatic in thermal equilibrium at a certain absolute temperature T makes it possible to use the expression (1.58) to obtain the energy mean value of one of those oscillators. This expression—as we have already seen in Sect. 1.5.2—leads to the catastrophe in the ultraviolet and not to the desired Planck’s law of radiation (1.56), which Einstein now writes in the following form: ρν =

ν3 8π Rβ _ ( L 3 N exp βν − 1 T

(2.2)

Recall that in this notation, L represents the speed of light in vacuum, R is the universal constant of gases, N is Avogadro’s number and β is the constant appearing in Wien’s phenomenological law (1.62). In contrast—Einstein continues—to arrive at the latter expression, it is enough to substitute in (1.59) the expression obtained for the mean energy of a resonator E ν on the basis of Planck’s quantum hypothesis, according to which the energy of a monochromatic linear oscillator can only take on values that are integer multiples of the elementary quantity introduced by (2.1). This procedure leads to the following expression: Eν =

R βν (N _ βν exp T − 1

(2.3)

which, substituted in (1.59) leads to Planck’s law It is then that Einstein explicitly asks whether Planck’s quantum hypothesis should be applied exclusively to the analysis of energy exchange between matter and radiation or, on the contrary, whether it affects any theory involving linear oscillators. He is fully convinced that the latter option is the correct one, so that the usual kinetic theory must be modified in the case that it is applied to the analysis of properties of macroscopic systems for which it is justified to adopt a microscopic model based on elementary structures formed by harmonic oscillators such as in the case of the specific heat of solids.

14

Proposed in 1819 by the Frenchmen Pierre Louis Dulong (1785–1836) and Alexis Thérèse Petit (1791–1820).

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To study the thermal properties of a solid, the simplest picture, writes Einstein, is to suppose that its individual atoms perform linear harmonic oscillations around equilibrium positions. Thus, by simply deriving with respect to temperature the expression for the average energy of the aggregate of atoms—which one can be deduced from (1.58)—the kinetic theory gives the following value for what Einstein calls the “specific heat of an equivalent gram of substance”15 : ) ) d 3R c= N nT = 3Rn, (2.4) dT N where the factor 3 is given by the three independent oscillation directions that exist and n represents the number of atoms contained in the molecule. Substituting R by its numerical value admitted at that time, it results in: c = 5.94n

cal. , mol · K

(2.5)

where K represents the unit of absolute temperature (Kelvin). Of course, the atomic specific heat will be c/n, which is 5.94 in the same units. This result can be understood as a theoretical deduction of the Dulong-Petit law, which was widely used despite its initial lack of theoretical foundation. The law was established empirically in the early nineteenth century; almost a century later, most of the experimental results on the specific heat of solids seemed to conform acceptably well.16 However, a finer analysis of the experimental situation allows us to find some difficulties for the above application of the kinetic theory, which, as we have just seen, leads to the theoretical justification of the Dulong-Petit law. Einstein lists two: 1. There were some elements (he mentions carbon, boron and silicon) which, in the solid state and at ordinary temperatures, have an atomic specific heat much less than 5.94; always using the units employed in (2.5). Moreover, the value of c is less than 5.94n in all solid compounds containing oxygen or hydrogen. 2. Experiments on optical dispersion carried out by Drude allowed to reach the conclusion that the eigenfrequencies of solids in the infrared region must be associated with vibrations of ions—Atomionen, Einstein writes—while those responsible for the eigenfrequencies in the ultraviolet regions are electrons. This implies that, contrary to what had been observed, the specific heat of solids should significantly exceed the value 5.94n, since the number of mobile

15

This quantity c is now often referred to as the molar heat capacity —or, also, molar heat— at constant volume. We will abbreviate it, as Einstein also usually does, by simply calling it “specific heat”. 16 Petit and Dulong enunciated their phenomenological law in 1819. The first theoretical justification of the law is due to Boltzmann, who deduced it in 1876 from the principle of equipartition of energy. For more details on specific heats in the nineteenth century, see Pais (2005), 389–397.

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masses per molecule is greater than the number of atoms if the electrons are considered. Einstein finds a possible solution to eliminate such difficulties by a new approach to the situation [Einstein (1907 b). In Beck (1989), 220]: If we conceive of the carriers [Träger] of heat in solids as periodically oscillating structures whose frequency is independent of their oscillation energy, then according to Planck’s theory of radiation we should not expect the value of the specific heat always to be 5.94 n.

Accordingly, and admitting that those ultimately responsible for the transport of heat are elementary structures that oscillate with a certain frequency characteristic of the solid in question, the Planckian hypothesis leads to the mean energy per oscillator being given by the triple (three independent directions of vibration) of the value expressed by expression (2.3). Its derivative with respect to time gives the following value for the specific heat, following with the units we have been using: ( _ · exp βν T 3R ( ( _ _2 βν exp T − 1 (

βν T

_2

(2.6)

This allows Einstein to propose the following quantum expression in place of the classical law (2.5) for the specific heat of solids:

c = 5, 94

⎲

( _ · exp βν T _2 , ( ( _ − 1 exp βν T

(

βν T

_2

(2.7)

where the summation extends to “all the species of oscillating elementary structures” that concur in the solid. In the case of the diamond the graphical representation of c is the dashed curve in Fig. 2.1, where the small circles represent the experimental values of the epoch, which we will refer to shortly. The quantum expression (2.7) for the specific heat of solids allows interesting conclusions to be drawn while simultaneously resolving the two difficulties mentioned above. For example, for values of T /βν higher than 0.9, the value predicted by the new theory does not differ excessively from the classical value of 5.94. At the other extreme, for values of T /βν lower than 0.1 the elementary oscillators do not contribute sensibly to the specific heat, which shows a clear tendency to cancel out at absolute zero temperatures, in accordance with the prescription of the “heat theorem”—later labelled as the “third principle of thermodynamics”—, which was newly established [Nernst (1906)]. For intermediate values of T /βν, the specific heat is an increasing function of x: first rapidly and then more smoothly.

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87

Fig. 2.1 In this graph, included in his 1907 paper, Einstein compares—for diamond—the function obtained for the specific heat using the quantum hypothesis and the experimental data (the small circles) obtained by Weber in 1875. [In abscissae, the dimensionless variable x = T /βν]

We have already noted that the frequency ν is a characteristic of the solid. From here onwards, Einstein continues the discussion in terms of the corresponding wavelength λ, since this quantity lends itself more naturally to experimental discussion. He goes on to answer the two questions posed above, on the basis of his new results. 1. Taking into account the admitted value for β in those times, as well as the responsibility assigned by Drude to the electrons as far as the ultraviolet range is concerned, it is easy to establish that these electrons cannot contribute significantly to the value of the specific heat at ordinary temperatures (T ≈ 300 K). In fact, the inequality T /βν < 0.1—which characterizes the almost null contribution—is equivalent in this case to λ < 4.8 μ, a condition that is clearly fulfilled in the ultraviolet region of the electromagnetic spectrum.17 Thus, at ordinary temperatures, the electrons do not contribute to the specific heat because the vibrations in the ultraviolet, which are associated with the electrons, do not contribute to it. It is reasonable, therefore, that at ordinary temperatures the specific heat of solids is clearly lower than 5.94n.

17

At that time, it was accepted that the wavelengths corresponding to the ultraviolet region were assumed to be between 0.10 μ and 0.36 μ, and the range of the infrared region to be between 0.81 μ and 61.1 μ; see notes 23 and 24 in Stachel (1989), 390.

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Fig. 2.2 In this table, also included in his 1907 paper, Einstein presents Weber’s 1875 experimental data for the specific heat of diamond at different temperatures (and therefore at different values of the dimensionless variable x). The pairs (c, x) allow him to compare the experimental results (the twelve small circles in Fig. 2.1) with the predictions of his new theory (the curve in Fig. 2.1), given by the expression (2.7)

T

c

x

222.4

0.762

0.1679

262.4

1.146

0.1980

283.7

1.354

0.2141

306.4

1.582

0.2312

331.3

1.838

0.2500

358.5

2.118

0.2705

413.0

2.661

0.3117

479.2

3.280

0.3615

520.0

3.631

0.3924

897.7

5.290

0.6638

1,097.7

5.387

0.8147

1,258.0

5.507

0.9493

2. The relevant contributions at ordinary temperatures are those for which T /βν > 0.1 is fulfilled, i.e., λ > 4.8 μ. These vibrations clearly correspond to the infrared region and are those that that contribute significantly to the specific heat: all the more so, the longer the wavelength (i.e., the higher the value of x). Since these vibrations are to be attributed—still according to Drude—to the positive atomic ions, these are the only ones which, at ordinary temperatures, contribute significantly to the specific heat. Under these conditions, therefore, the electronic contribution is nonexistent for all effects, and the second difficulty is solved. Einstein ends the article by comparing his conclusions—obtained from (2.7)—with the experimental data. The last table, of the three that he includes in his paper, is the one reproduced in Fig. 2.2 and contains the values of the specific heat of diamond measured at different temperatures.18 To each of these corresponds a value of the dimensionless variable x = (T /βν) = (T λ/β L),19 where L represents, as hitherto, the speed of light and the constant β of Wien’s law has been assigned the value 4.86 ·10–11 K·s, according to the determination of that time. At the characteristic wavelength of the diamond is assigned the value λ = 11.0 μ, which Einstein deduces from his own theory applied to the specific case c = 1.838 for T = 331.3 K, that he must have considered highly reliable.

18

These are data published by H. F. Weber, in 1875, which Einstein claims to have obtained from the tables of H. Landolt and R. Börnstein, published in 1905. See notes 31 and 32 in Stachel (1989), 391. 19 In Einstein’s original paper, and in the English translation, it appears x = (T L/βλ), instead of x = (T λ/β L), which is correct. Given the meticulousness of the editors, it is surprising that this slip of the tongue should have escaped their notice.

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In view of the good agreement shown in Fig. 2.1, Einstein believes that the initial hypotheses are justified. In particular, he notes that positive ions, subjected to monochromatic vibrations, are the only carriers of heat in the diamond. Moreover, as he has obtained the value λ = 11.0μ for its characteristic vibration, he ends up predicting that, according to the new theory—from which he has used to calculate the previous value—, it is to be expected that diamond would show an absorption peak for this value of λ. Soon Einstein retracts both propositions. In a short paper, which appears months later, he opens the door to the possibility that atoms themselves, all vibrating with a frequency characteristic of the substance, were carriers of heat [Einstein (1907c). In Beck (1989), 234]: This proposition [positive ions are the only carriers of heat] does not hold up in two respects: First, one must assume not only positively, but also negatively charged atom ions. Second— and this is the essential point— Drude’s investigations do not justify the assumption that every elementary structure capable of oscillation that acts as a carrier of heat has always an electric charge.

By admitting the possibility that (neutral) atoms are responsible for heat propagation, he implicitly admits the possibility of the existence of other excitations, in addition to electromagnetic ones. Consequently, in this same paper, he also dismisses his earlier prediction about the existence of a possible absorption peak in diamond. In fact, he ends his article by explicitly acknowledging the possibility that such a characteristic wavelength of 11 μ does not necessarily correspond to a radiation absorption peak. As we shall see when referring to the first Solvay conference, Einstein was soon to find that the values deduced from his expression (2.7) for specific heat differed markedly from the results obtained by Nernst in Berlin, in 1911, for very low temperatures. Nernst himself attributed the discrepancy to the fact that he had assigned the same frequency to the vibrations of all elementary structures. Einstein in 1911, and Peter Debye (1884–1966) in 1912, were to take fundamental steps in this direction towards the solution of the problem. Einstein’s 1907 paper on the specific heat of solids represents the first step in the direction of extending the incipient quantum theory to systems other than electromagnetic radiation, as is the case of an aggregate of particles that oscillate, all with the same frequency, around their respective equilibrium positions. The quantum hypothesis applied to radiation seems at first sight incompatible—we insist—with the Maxwellian theory of the electromagnetic field. Not so if applied to mechanical systems, because the quantisation of certain properties of matter was already inscribed in physics; it is the case, for example, of the oscillation modes of a string fixed at its ends. Thus, it can be said that Einstein’s 1907 work had

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highly positive effects, if not for a general acceptance of the quantum hypothesis, at least not for an outright rejection.20

2.4

Quantum Refinements (1909–1910)

Whereas in Planck’s approach the quantum hypothesis is proposed as an option— although certainly the only one he can think—to obtain the law of black body emission, Einstein arrives at the energy quantum as a consequence of the analysis of this emission. In other words, Planck introduces the “sufficient quantum” to explain certain aspects of the behaviour of black body radiation, while Einstein arrives at the “necessary quantum” to account for such behaviour. This difference is noteworthy. Particularly as far as Einstein is concerned, the introduction of such a strange and problematic concept led him to a permanent search for arguments to justify the necessity of resorting to such an extravagant idea. In fact, as we shall see, Einstein had to take about a decade to justify the necessity for the quantum and thus be able to start with a proper quantum hypothesis in his investigations. The next milestone in the evolution of Einstein’s quantum can be located in 1909. After their contributions to the quantum theory of the specific heats, it follows some as a time of ‘rest’ appears, as far as his publications on quantum physics are concerned. However, a letter addressed on 17 May 1908 to his friend Jakob Johann Laub (1882–1962)—Einstein’s first scientific collaborator—shows, in addition to some curiosities, that his preoccupation with quantum questions was constant and intense [In Beck (1995), 119]: I busy myself incessantly with the question of the constitution of radiation, and am conducting a wide-ranging correspondence about this question withLorentz and Planck. The former is an amazingly profound and at the same time lovable man. Planck is also very nice in his letters. His only failing is that it is hard for him to follow other people’s trains of thought. This might explain how he could have made totally wrong-headed objections against my last radiation paper [(Einstein, 1906d)]. But he did not adduce anything against my criticism. So I hope that he has read it and accepted it. This quantum question is so extraordinarily important and difficult that everybody should take the trouble to work on it. I already succeeded in thinking up something that is formally more or less adequate, but I have good reasons for regarding it as “garbage” nonetheless.

Einstein’s new ideas about the nature of electromagnetic radiation appear in two articles published in 1909 [Einstein (1909a, 1909b)]. Their interest does not lie in the new results but in the suggestions made by Einstein after the analysis of

20

For more details on this 1907 Einstein’s work, as well as on its relevance for the development of quantum physics, see Klein (1965).

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an original Gedankenexperiment.21 They would turn out to be key ideas to understand the evolution of Einstein’s thinking about quantum physics because in them he introduced and handled resources that, as we shall see later on, were to be extremely fruitful in his hands. In the second of these articles appears the most elaborated justification of the necessity to introduce definitively in physics the quantum of radiation, as a consequence of the experimental results of the moment. The paper is a transcript of the memoir he presented at the LXXXI annual meeting of the GDNÄ in Salzburg on 21 September 1909 [Einstein (1909b)]. At the age of thirty, Einstein participates for the first time as a speaker at a high-level scientific meeting. It is a paper that Pauli would not hesitate to qualify in 1949 as “one of the landmarks in the development of theoretical physics”.22 This contribution has been analysed from different perspectives. For example, in relation to the Einsteinian conception of wave-particle duality [Klein (1964), 5– 15], and also because of the multipurpose that Einstein makes of the fluctuations of linear momentum in various contexts [Bergia & Navarro, (1988), 80–83)]. We must refer here to another aspect. We will do so not only because of the clarity with which he raises the question of the possible necessity of the quantum, but also because, in addition, it proposes a hint for a possible solution. In Einstein’s own words [Einstein (1909b). In Beck (1989), 390–391]: Planck’s theory leads to the following conjecture. If it is really true that a radiation resonator can only assume energy values that are a multiple of hν, then it is logical to assume that emission and absorption of radiation can take place only in quanta of this energy value. On the basis of this hypothesis, the hypothesis of light quanta, one can answer the questions raised above regarding the absorption and emission of radiation. As far as we know, the quantitative consequences of this hypothesis of light quanta are also being confirmed. The following question arises then. Isn’t it conceivable that Planck’s formula is correct, but that nevertheless a derivation of it can be given that is not based on an assumption as horrendouslooking like Planck’s theory? Would it not be possible to replace the hypothesis of light quanta by another assumption that would also fit the known phenomena? If it is necessary to modify the elements of the theory, would it not be possible to retain at least the equations for the propagation of radiation and conceive only the elementary processes of emission and absorption differently than they have been until now?

21 Gedankenexperiment (thought experiment) is a terminology widely used to refer to an ideal experiment proposed to critique a theory or some particular aspect of a it. The experiment is not designed to be performed, but to be subjected to rigorous logical analysis. Therefore, the experiment does not need to be feasible; in general, they are not, but the possible consequences that its logical analysis may bring to light do not lose their value. This sort of “experiments”, although not very abundant, have played a relevant role at certain times in the history of physics. Some famous examples: “Galileo’s boat”, “Maxwell’s demon”, “Schrödinger’s cat", “Wigner’s friend”, etc. 22 In Schilpp (1970), 154. Wolfgang Pauli (1900-1958) received the Nobel Prize in Physics 1945 “for the discovery of the Exclusion Principle, also called the Pauli Principle”.

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2.4.1

A Highly Fruitful Gedankenexperiment

To answer these questions, in his Salzburg paper, Einstein starts from Planck’s formula—which was by then almost unanimously considered to be an accurate representation of the experimental results obtained thus far on black body radiation—and tries to draw conclusions about the nature of electromagnetic radiation. To this end, he considers a Gedankenexperiment to which he will return on several occasions, which allows Einstein to display his intuition and use his statistical methods. A mirror can move freely in the direction perpendicular to its own plane. It totally reflects the radiation in the frequency range(ν, ν + dν) and is perfectly transparent to the rest. The mirror is located inside a cavity containing a monoatomic ideal gas and electromagnetic radiation; all in thermal equilibrium at a certain absolute temperature T.23 Irregular collisions of the gas molecules with the mirror cause the mirror to undergo a kind of Brownian motion. According to the well-worn principle of equipartition of energy, it is possible to assign to the mirror an average kinetic energy which will be the third part of that corresponding—under the same conditions—to one gas molecule, since the movement of the mirror is unidirectional. The forces due to the radiation pressure are not the same on one side as on the other; they would be equal if the mirror were at rest. Thus, there will be a nonzero resultant force that would slow down the mirror’s movement, forcing it to give up energy to the radiation field. As a consequence, the energy that the mirror absorbs in the collisions with the molecules of the gas would end up being transformed into electromagnetic radiation energy until everything was radiant energy, which would make impossible the supposed equilibrium between gas and radiation. For such an equilibrium to be reached—Einstein concludes—it is necessary to include in the reasoning a phenomenon that has not been considered in previous discussions, which is the existence of fluctuations in the radiation pressure [Einstein (1909b), 392]: This consideration [the fact that it is not possible to reach equilibrium between gas and radiation in the above Gedankenexperiment] is faulty because one cannot consider the forces of pressure exerted on the plate by radiation as constant in time and free of random fluctuations, just like the forces of pressure exerted on the plate by the gas. For thermal equilibrium to be possible, the fluctuations of the radiation pressure must be such that on the average they compensate for the velocity losses of the plate caused by radiation friction, where the average kinetic energy of the plate equals one-third of the average kinetic energy of a monoatomic gas molecule. If the law of radiation is known, one can calculate the radiation friction, and from this one can calculate the average value of the [linear] momenta imparted to the plate due to fluctuations of the radiation pressure so that statistical equilibrium can exist.

23

Einstein (1909b), 392. A linguistic detail: in this paper Einstein calls plate [Platte] what in Einstein (1909a) and in later papers he would call mirror [Spiegel]. For the sake of simplicity, we shall use the word mirror in both situations.

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Recall that the equilibrium of forces, in the case of the Brownian motion of a particle suspended in a fluid, is permanently guaranteed. In the case of Brownian motion, the fluid plays, simultaneity, the role of the gas and of the electromagnetic radiation in the case of the mirror. As the molecules of the fluid collide with the suspended particle, the latter experiences an “osmotic force”, which tends to direct it towards regions of lower concentration; on the other hand, the fluid, now considered a continuous medium, offers resistance to the movement of the Brownian particle by means of a “viscosity force”. It is precisely the imposition of equality between the two forces that forms the basis of Einstein’s famous 1905 treatment of Brownian motion, as we saw in Sect. 1.6.2. Let us now see how, to a large extent, Einstein follows his treatment of Brownian motion in 1905 for the analysis of his Gedankenexperiment of 1909. Analysis of the 1909 Gedankenexperiment

Einstein begins by evaluating the fluctuations for radiation in his Gedankenexperiment and establishes the balance that guarantees statistical equilibrium. A minimal sketch of his calculations is essential for a more proper evaluation of the scope of the analysis. If m is the mass of the mirror, v its velocity at instant t, and there were no fluctuations, the linear momentum at instant t + τ would be the following: mv − Pτ v,

(2.8)

where P denotes the friction force per unit velocity. Representing by ∆ the variation in the linear momentum of the mirror—during those τ seconds— due to the random fluctuations of the radiation pressure, its linear momentum at that instant would be: mv − Pτ v + ∆

(2.9)

If we now impose the condition that, on average, the linear momentum of the mirror does not change after τ seconds, the following expression is obtained: (mv − Pτ v + ∆)2 = (mv)2

(2.10)

Considering a time τ sufficiently small so that terms of order τ 2 can be neglected, and considering that v∆ = 0, due to the totally random character of fluctuations, the above expression leads to: ∆2 = 2m P τ v2

(2.11)

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The mean kinetic energy of the mirror is given by the principle of equipartition of energy: m v2 1R T = , 2 2 N

(2.12)

where T represents the absolute temperature of the cavity. Eliminating the mean square speed between these last two relations one arrives at the following equality: ∆2 = 2

R T P τ N

(2.13)

Classical electrodynamics allows to evaluate P—although Einstein does not include the corresponding calculations here—as a function of the energy density of the radiationρ(ν, T ) contained in the cavity: ) ) 1 ∂ρ 1 3 ρ− ν S dν, P= m2 2c 3 ∂ν

(2.14)

where c represents the speed of light in vacuum and S is the mirror surface.24 Expressions (2.13) y (2.14)—or analogous expressions—would later be used by Einstein for different purposes, as we have shown in another work.25 In 1909 Einstein uses them in the way one would expect: he uses Planck’s law (2.2)—which is thus taken as an experimental datum—to be substituted in (2.14) and thus obtain the following result: ) ) c3 2 ∆2 1 hνρ + ρ S dν = τ c 8π ν 2

(2.15)

This equality, “which is striking for its simplicity” is the one to which Einstein resorts to establish his conclusions. He justifies the appearance of the second summand as the consequence of a process of interferences—without going into further nuances—based on the usual wave interpretation of

24

From 1907 Einstein already uses the letter c (from Latin celeritas, velocity?) to designate the speed of light in vacuum, instead of the previous L. For details of the origin of the notation c— although it is usually attributed to Drude in 1894, it is a not uncomplicated assignment—, see Mendelson (2006). 25 See Bergia and Navarro (1988), paragraphs 2 and 3. Pay attention to the notation when comparing works: we designate with ∆ —for convenience and because Einstein will do it later— the variation in the linear momentum of the mirror, while in 1909 he represents with ∆ the variation in the velocity of the mirror.

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electromagnetic radiation. However, the first summand, which can be much more relevant than the second in certain situations, suggests some novelties in relation to Maxwellian electromagnetism (Einstein, 1909b, 393–394): Thus, in my opinion, the following must be concluded from the above formula, which is, in turn a consequence of Planck’s radiation formula. In addition to the nonuniformities in the spatial distribution of the radiation momentum which arise from the wave theory [he refers to the second summand], there also exist other nonuniformities in the spatial distribution of the momentum of the radiation [first summand], which at low energy density of the radiation have a far greater influence than the first mentioned nonuniformities mentioned. I shall add that another consideration concerning the spatial distribution of the energy yields that agree quite well with the above consideration concerning the spatial distribution of momentum.

Einstein’s “other consideration” refers to an expression obtained in his earlier work of 1909 for the fluctuation of the energy of the electromagnetic radiation contained in a cavity of volume V, which he estimates to be in accordance with (2.15) and which reads as follows [Einstein (1909a), 363–369]: ) ε2 = hνρ +

) c3 2 V dν ρ 8π V 2

(2.16)

So much for Einstein’s considerations on his Gedankenexperiment. One of the conclusions that he obtains from the analysis of (2.15) and (2.16) is very daring at the time: the corpuscular and wave aspects may not be as absolutely incompatible as then thought. Such a corollary must have been firmly installed in Einstein’s mind, judging from the evolution of his thinking about the nature of radiation. This matter deserves further attention.

2.4.2

On the Possible Dual Structure of Radiation

Indeed, a fine analysis by Einstein of the two summands of the respective expressions (2.15) y (2.16) leads him to intuit the necessity of having to incorporate, in the same theory, a certain dualism for the better understanding of the behaviour of black body radiation (governed by Planck’s law), given the different nature of these summands: the first is of quantal origin (discrete)—the constant h gives it away—while the second is of wave origin (continuous), since it contains the

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distribution of energy electromagnetic radiation ρ and the speed of light c—. This serves to justify the surprising start of his speech in Salzburg26 : It is even undeniable that there is an extensive group of facts concerning radiation that shows that light possesses certain fundamental properties that can be understood far more readily from the standpoint of Newton’s emission theory of light than from the standpoint of the wave theory. It is therefore my opinion that the next stage in the development of theoretical physics will bring us a theory of light that can be understood as a kind of fusion of the wave and emission theories of light. To give reasons for this opinion and to show that a profound change in our views on the nature and constitution of light is imperative is the purpose of the following remarks.

After noting the nonexistence of a theory of radiation involving the two aspects— quantum and undulatory—that he has inferred for both energy and momentum behaviour, Einstein ends by suggesting that the singularities represented by the quanta in the electromagnetic field might well have certain analogies with the electrons in the electrostatic field. His convictions in this respect should already be firm, since the last paragraph of his Salzburg speech is devoted to insisting on this kind of prediction [Einstein (1909b), 394]: All I wanted is briefly to indicate with its help that the two structural properties (the undulatory structure and the quantum structure) simultaneously displayed by radiation according to the Planck formula should not be considered as mutually incompatible.

Thus, by means of mechanical statistical ideas—with an outstanding role for the principle of equipartition of energy—, using Maxwell’s electrodynamics and assuming the validity of Planck’s law, already considered the mathematical expression of the experimental results on black body radiation, Einstein arrives at the necessity of conceiving electromagnetic radiation under a double aspect: quantal and undulatory. That is, discrete and continuous simultaneously. He no longer considers both aspects to be incompatible, although—and this is his great problem—at that time he does not foresee a coherent way of including both aspects in the same theory.27 Einstein’s analysis of the Gedankenexperiment is not entirely rigorous. It gives the impression, as happens not infrequently with great creators, that intuition clearly anticipates reasoning. He analyses an experimental situation, in which emission and absorption of radiation are the protagonists, on the basis of a theory of the interaction between matter and radiation—including Maxwellian ideas such as radiation pressure, for example—which is precisely part of the representation that is called into question.

26

Einstein (1909b), 379. Incidentally: a paragraph like the above serves to dismiss outright the so-called “wave-particle duality” as the cause of Einstein’s estrangement from later quantum mechanics. 27 For a detailed analysis of Einstein’s thought —around this time— about the validity of the principle of equipartition energy, see Bergia; Navarro (1997), especially, 195–200.

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This incoherence of principle was exposed by Planck in the discussion following Einstein’s presentation at Salzburg. Planck states that he too is convinced of the necessity of introducing certain quantum ideas, but that he does not believe that the analysis of fluctuations developed by Einstein could be taken as definitive proof of such necessity, given the weak foundation of the theory of matter-radiation interaction used by Einstein [Planck in Einstein (1909b), 395–396]. Einstein’s reply does not show additional arguments because he does not yet have them. However, the difficulties exposed by Planck—and, in fact, shared by Einstein himself, judging by his later attitude—did not make him doubt the absolute necessity of introducing radiation quanta, as well as the need to reconcile undulatory and quantal aspects in the same theory. All that, as well as the explicit rejection of certain objections of Lorentz and Planck to Einsteinian reasoning on the necessity of the quantum hypothesis, is clearly manifested, for instance, in his correspondence with Laub and Sommerfeld in those days.28 In the following months, Einstein does not seem to have made substantial progress in his attempts to deduce the necessity of quanta and to show their compatibility with Maxwell’s electromagnetism. Aware of his inability to do so, he set out to seek progress in a different direction. Instead of persisting in the quest for the necessary quantum, he devoted himself temporarily to the pursuit of possible consequences of the quantum hypothesis, that is, of the supposed existence of quanta. This change of attitude is clearly shown in a letter addressed to Besso from Prague29 : Just now I am trying to derive the law of heat conduction in solid insulators from the quantum hypothesis. I no longer ask whether these quanta really exist. Nor do I try to construct them any longer, for I now know that my brain cannot get through in this way. But I rummage through the consequences as carefully as possible so as to learn about the range of applicability of this conception. The theory of specific heats has celebrated a real victory, for Nernst found in his experiments that everything behaves more or less the way I had predicted. To be sure, the shape of the curve deviates systematically from the one resulting from Planck’s law. But these deviations can be explained in a quite natural way through the assumption that the atomic oscillations deviate very strongly from monochromatic oscillations.

Finally, we must admit that it is surprising, to say the least, that in the light of the expressions (2.15) and (2.16), Einstein did not consider the possible value of the momentum of the radiation quantum. In this respect, we quote Pais’ opinion [Pais (2005), 409]: One might therefore expect that the first term in (2.15) would lead Einstein to state, in 1909, the ‘momentum quantum postulate’: monochromatic radiation of low density behaves in

28

Letters from Einstein to Laub (March 1910) and to Sommerfeld (July 1910). In Beck (1995), 148–149 and 157–158, respectively. 29 Letter from A. Einstein to M. Besso, 13 May 1911. In Beck (1995), 187. Pay attention to Einstein’s reflection on the convenience of introducing oscillations with different frequencies.

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2 The Flowering (1906–1913): Einstein Introduces Himself to the Scientific … regard to pressure fluctuations as if it consists of mutually independent momentum quanta of magnitude h ν / c. . It is unthinkable to me that Einstein did not think so. But he did not quite say so (…). He did not refer explicitly to momentum quanta or to the relativistic connection between E = h ν and p = h ν / c. Yet a particle concept (the photon) was clearly on his mind, since he went on to conjecture that the electromagnetic fields of light are linked to singular points similar to the occurrence of electrostatic fields in the theory of electrons. It seems fair to paraphrase this statement as follows: light-quanta may well be particles in the same sense that electrons are particles.

Fully sharing the previous opinion, it only remains to be added that Einstein’s precautions, before proclaiming the corpuscular behaviour of light in some situations, give an idea of the conceptual difficulties existing at that time to reconcile quantum hypothesis with Maxwell’s electromagnetism, although the aforementioned compatibility between electrons and electrostatic fields provided support and guidance to continue searching for a definitive harmonization between quanta and electromagnetic fields.

2.4.3

Stochastic Fields and Quantum Theory

From 1909 onwards, Einstein lavishes the use, in various situations and contexts, of the expression (2.13) for momentum fluctuation, where the friction P is given by (2.14), in the case of the mirror. The first equality had been arrived at, we recall, by the analysis of a Gedankenexperiment. Even today, it is tempting not to consider ρ as a datum to evaluate first P and then the fluctuation, but to reverse the path: calculate the latter by means of some independent procedure and then deduce the expression for ρ. This is, precisely, what Einstein and Hopf did in 1910, in a work with interesting implications [Einstein and Hopf (1910)]. Einstein is fully convinced, from the very beginning, that strict application of classical physics leads inexorably to Rayleigh-Jeans law (1.60) which, in turn, implies the ultraviolet catastrophe. This law can be arrived at, as he had shown as early as 1905, by invoking the principle of equipartition of energy. However, it is known—and then it was known by Einstein—that classical physics does not guarantee the universal validity of this principle, so Einstein and Hopf propose a classical deduction of the law of black body radiation without resorting to equipartition. In the new deduction, the mirror in the Gedankenexperiment is replaced by an electrically charged oscillator interacting with the electromagnetic field. These oscillators are endowed with a translational motion in a fixed direction, inside a cavity containing radiation and a monoatomic ideal gas, all in thermal equilibrium at the absolute temperature T. After calculating ∆2 and P from classical electromagnetism, and substituting them in (2.13), the authors to obtain a differential equation whose only physically admissible solution turns out to be the RayleighJeans law (1.60). Thus, they considered that classical physics—even freed from its submission to the principle of equipartition of energy—does not lead to Planck’s law but to the ultraviolet catastrophe.

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The details of the calculation are too technical and dispensable for our purposes.30 However, it is necessary to emphasize a novelty with great later influence: Einstein and Hopf analyse the influence of the radiation field on a charged oscillator by introducing a stochastic field. For this purpose, they develop an original elementary theory for this type of process, which has been considered a forerunner of later stochastic electrodynamics, a discipline that has sometimes been proposed as a candidate for an alternative to quantum electrodynamics, although without much success.31 Just one detail. For the calculation of friction P—that is, of the force with which the radiation opposes the oscillator’s motion—they now introduce an electric field whose modulus developed in Fourier series is as follows: ) ) } { ⎲ αx + β y + γ z 2π n t− − θn , E= (2.17) An cos T c n where α, β and γ , represent the direction cosines of the field direction, T denotes “a very long time period” and θn are the random phases that make the electric field associated with a stochastic process. [Einstein and Hopf (1910). In Beck (1993), 222]. In their paper, Einstein and Hopf develop their own theory for this kind of process. Later, we will refer to this article because it is a clear precedent of another that Einstein, together with Stern, would sign in 1913. In any case, it should be pointed out that some of the ideas disseminated in Einstein and Hopf’s paper had already been anticipated, at least partially, by Planck.32

2.5

Interlude with Luminescence

There is a naive question that, throughout history, many thinkers have tried in vain to resolve satisfactorily: Why is the sky blue? Although some physicists had previously dealt with the problem and even provided theoretical explanations for it—the most prominent among these being John William Strutt (Lord Rayleigh) (1842– 1919), at the end of nineteenth century—Smoluchowski and Einstein, already in the twentieth century, were the first to solve the problem independently in a rigorous way, attributing the phenomenon to density fluctuations in a transparent medium that is traversed by light. These same fluctuations are responsible for the “critical opalescence”.

30

For those who are interested in following the calculations, see the commented English translation of this paper by Einstein and Hopf in Bergia et al. (1979). 31 For a detailed exposition of the origin, development and prospects of stochastic electrodynamics see De la Peña; Cetto (1996). 32 In Chapter III of Kuhn (1978), entitled “Planck and the electromagnetic H-theorem, 1897– 1899”, it can be found some of Planck’s ideas and developments about stochastic fields, which appeared in his early work on black body radiation. In particular, on pages 73–74, there appear developments with random phases, very similar to (2.17).

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The discovery of critical opalescence is usually attributed, albeit with some doubt, to the Ukrainian physicist Milkhail Avenarius (1835–1895), who published a paper in 1874 in which he described some curious phenomena associated with the passage of white light through a fluid just above its critical point [Avenarius (1874)]. The critical point is the limit point of the coexistence curve of the vapourliquid phases, in the pressure–temperature phase diagram. At this point, these two quantities reach the maximum value within the coexistence curve: they are the “critical temperature” and the “critical pressure”. At the critical point, the densities of liquid and vapour are equal. For values higher than the critical ones, the system behaves as a “supercritical fluid”, simultaneously presenting properties of gases and liquids: a supercritical fluid can diffuse like a gas and at the same time it can dissolve substances like a liquid. In the neighbourhood of its critical point, the properties of a fluid often undergo large changes in response to small perturbations. The phenomenon of critical opalescence is observed when light is passed through a transparent fluid that is in the neighbourhood of its critical point. Under these circumstances, its density and its refractive index undergo large fluctuations, and the fluid appears to become more opaque and cloudy with iridescence. However, back to the blue of sky. Traditionally, the theories that tried to explain it usually involved particles in suspension. For example, we could go back to Newton, who, in his Opticks (1704)—specifically, in proposition VII of part III of book II—, and on the basis of his own corpuscular theory of light, he suggests that this colour is due to very light aqueous particles suspended in the atmosphere. However, it was John Tyndall (1820–1893) who, in the 1860s, carried out numerous experiments that led him to discover, in 1869, what later came to be known as the “Tyndall effect”: when light passes through a transparent fluid containing particles in suspension, the small wavelengths (bluish tones of the visible spectrum) suffer greater deviations from their rectilinear trajectory than the longer wavelengths (reddish tones of the visible spectrum), with the phenomenon being more pronounced the larger the size of the suspended particles.33 Subsequently, the Tyndall effect has been held responsible, not only for the blue colour of the sky but also for many other common colourings such as, for example, the critical opalescence of the sky, the blue colour of the eyes—which is not due, contrary to what might be believed, to pigmentation—, the bluish tones of lights in fog or in dark enclosures with dust particles, the smoke of a cigarette, etc. However, although Tyndall’s experiments were convincing, his theoretical explanations lacked rigour. For example, he blamed the aforementioned effect, without

33

Tyndall effect occurs in colloids, or colloidal suspensions, but not in dissolutions. That is why it is currently used —among many other applications— to distinguish between colloids and solutions. In a colloidal suspension, the size of the suspended particles usually ranges between one micron and one thousandth of a micron, whereas in a solution the solute particles do not reach a thousandth of a micron.

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providing the corresponding justification, on a hypothetical “sky matter”: submicroscopic particles contained in the atmosphere, which are larger than molecules. Tyndall speculated on different occasions with several candidates: dust particles, small drops of water vapour or even organic germs. The first rigorous theoretical analysis of the Tyndall effect is due to Lord Rayleigh who, in 1899, published a paper summarizing his research since 1871 on the properties of light transmission through a medium containing small particles [Rayleigh (1899)]. In the paper, he applied his results to provide an explanation of the blue colour of sky. Rayleigh’s attention was drawn to this topic, as he acknowledges at the beginning of the article, by some epistolary observations addressed to him by Maxwell in the summer of 1873. Rayleigh’s investigations were based on the wave nature of light—by then firmly established—, by analysing the deviation that the direction of propagation of waves underwent when encountering obstacles (particles or molecules) on their way through a transparent medium. The theory required approximations, and these were all the more legitimate the more justified the following hypotheses were: – The dimensions of such particles are small compared to the incident wavelength. – Particles are randomly distributed in the medium. – Collisions with the particles modify the phase of the incident waves but not their respective amplitudes. – The phases of the waves, after these change their direction of propagation due to the presence of particles in the medium, are distributed completely at random. With such assumptions, Rayleigh succeeded in finding an expression for the intensity of the emerging light, which turned out to be inversely proportional to the fourth power of the wavelength of the incident light. Thus, he considered justified the blue colour of the sky during the day (the light that falls on our eyes corresponds to the wavelength that is most deviated by the atmosphere) and the reddish tone of the sunset (the light that falls on our eyes corresponds to the wavelength that is less deviated, since the sun is now seen more directly). This classical explanation would be further refined with the advent of quantum mechanics and quantum field theory, where it is called “Rayleigh scattering” the elastic collision between a photon and an atomic electron, provided that the binding energy of the electron is much higher than the energy of the photon. Application of perturbation theory to first order, leads also to the fact that the intensity of the phenomenon—determined by its “cross section”—is inversely proportional to the fourth power of the wavelength of the incident photon. [See, for example, Sakurai (1987), 50–51]. Incidentally, and in light of the contributions of Tyndall and Rayleigh, if the small wavelengths are the ones that suffer the greatest deviation and, therefore, are responsible for the colour of the sky, shouldn’t we see this violet, instead of blue? There is no contradiction. In addition to the fact that sunlight registers greater intensity for the blue tone than for violet, it turns out that our visual perception is

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much higher for blue than for violet. Therefore, this colour is strongly masked for our vision to the point that we see the sky as blue, with hardly any violet tones. Perhaps this is a good moment to comment on a terminological question that sometimes hinders the understanding of the physical problem. We are referring, concretely, to the terms scattering and dispersion as well as the phenomena to which they apply. Scattering refers to a phenomenon in which light (or other forms of electromagnetic radiation), sound, or beams of moving particles in inhomogeneous media are deflected from their rectilinear trajectory by the irregularities of the medium, for example, when the medium contains particles that are different and larger than the molecules of the medium itself. In this case, regardless of the nature of the incident beam, the rays are redirected in different directions in space because of the obstacle. Effect of a typical scattering phenomenon: the blue colour of the sky. The blue light is deflected much more than the red light when it encounters obstacles (particles and molecules) in its path. Dispersion refers to a phenomenon in which light (or other forms of electromagnetic radiation) is separated in terms of the different frequencies of which it is composed, as a consequence of the fact that the phase velocity of a wave depends on its frequency. In vacuum, the components of the different frequencies that make up white light propagate at the same speed—the so-called “speed of light”—but not so in dense transparent media, where components with different frequencies propagate at different velocities and, consequently, each has its own index of refraction. Typical phenomenon of dispersion: the splitting occurring among different colours when sunlight passes through a prism, since each colour (frequency) is refracted in a different way, determined by the corresponding refractive index. Thus, scattering of light involves changing the direction of light propagation when passing through a transparent medium due to interposed obstacles (particles and molecules). Dispersion of light, on the other hand, implies a change in its propagation velocity when passing through a transparent medium. Alternatively, in the case of dispersion light passes through a homogeneous medium (solid, liquid or gas), whereas in the case of scattering, light passes through a medium with material inhomogeneities, which are precisely responsible for the effect. In both cases, although as a consequence of different phenomena, each component of white light undergoes different angular deviations, and its theoretical description necessarily involves the study of light-matter interaction. Returning to the blue colour of the sky, it is worth noting that both Tyndall and Rayleigh attributed it to the scattering of light by small particles, e.g., dust and water vapour droplets in the atmosphere. However, it soon became clear that, if this were truly the case, humidity and haze should increase the blue hue of the sky, contrary to what was observed. Since, moreover, Rayleigh’s theory was only approximate, it cannot be said that Rayleigh had completely solved the problem of the theoretical explanation of the blue colour of the sky. In the twentieth century, Smoluchowski and Einstein took the next important steps in the search for such an explanation.

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2.5.1

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Smoluchowski and His Interest in Fluctuations (1904–1908)

We have commented in Sect. 1.6.3 how the scientific trajectories of Einstein and Smoluchowski intersected around 1906, concerning their respective theories of Brownian motion. Earlier, in Sect. 1.3.4, we referred to Einstein’s interest, in 1904, in energy fluctuations. At that time, Smoluchowski was also concerned with fluctuations; only in his case were they fluctuations of the number of particles in a gas of molecules, which were the subject of his research, as seen from his contribution—“On irregularities in the distribution of the number of gas molecules and their influence on the entropy and the equation of state”—in the volume edited in Leipzig (1904) to commemorate the sixtieth anniversary of Boltzmann’s birth [Smoluchowski (1904)]. Using more modern terminology one could say that, around 1904, while Einstein was interested in fluctuations in the canonical ensemble, the Smoluchowski treatment rests on the grancanonical one. In the aforementioned work of Smoluchowski in 1904, fluctuations in the number of molecules of a gas are calculated for the first time, and some effects of these fluctuations are analysed. For this purpose, he considers a small part v of a volume V that contains a gas in thermal equilibrium consisting of N molecules. In principle, the number n of these molecules contained in the volume v is not fixed, but will fluctuate around a certain mean value n such that vn = VN . The fluctuation of the number of molecules can be characterised by the value of the (local) “compression” (“Verdichtung”), which Smoluchowski defines as: n = n(1 + δ) or δ =

n-n n

(2.18)

After evaluating the probability that the compression takes a value between δ and δ + d δ, Smoluchowski calculates the mean value of δ—in the case of ideal gases—, which turns out to be, in absolute value, inversely proportional to the square root of the volume v. This means that the fluctuation of the number of particles in a volume is irrelevant as long as this volume is sufficiently large. A result currently well known to those with a minimum background in statistical mechanics. Smoluchowski is concerned in this same work with analysing the influence of fluctuations in the number of molecules on the equation of state of a real gas, but now considering the intermolecular forces. Subsequently, in a 1906 paper—“On the kinetic theory of molecular Brownian motion and of suspensions” [Smoluchowski (1906)]—he again used fluctuations; but that time his interest was in Brownian motion. After referring to Einstein’s two previous contributions to the topic—[Einstein (1905 b) and Einstein (1906 c)]— and acknowledging Einstein’s paternity in relation to the publication of the first results obtained by applying the kinetic theory to Brownian motion, Smoluchowski presents a method which, in his opinion, allows a better understanding of the inner mechanism of the phenomenon, in addition that “… my method seems to me to be more direct, simpler and therefore perhaps also more convincing than Einstein’s method” [Smoluchowski (1906), 756]. The new method, unlike Einstein’s, did not resort to the always problematic physics of collisions but introduced a simple

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combinatorial calculus, which led to the same result that Einstein had obtained for the mean displacement, except for a numerical factor to which we will refer later. In fact, Smoluchowski was the first to apply a “random walk” model to the analysis of Brownian motion, although Louis Bachelier (1870–1946) had already introduced the model six years earlier (1900) in his dissertation on speculation theory under the supervision of Poincaré [Bachelier (1900)]. Remember that a random walk can be imagined, for example, as the jumping movement, from vertex to contiguous vertex of the square tiles of a horizontal floor, in which we assign to each of the four possible jumps (forwards, backwards, right and left) the same probability: 1/4. Well: it turns out that a random walk like the previous one, in the limiting case in which the side of the tiles is extremely small—mathematically we would say in which the elementary length tends to zero—, adequately represents the properties of Brownian motion in two dimensions. These are essentially the basis of the method introduced by Smoluchowski in 1906. However, the final result of this for the mean squared displacement exceeded that predicted by Einstein—see (1.80)—in the numerical factor 64/27. A factor that was later shown to be wrong and that Smoluchowski himself corrected in later works. It was precisely in trying to clarify the nature of the difference between the two results that led the French physicist Paul Langevin (1872–1946) to take up the topic, proposing in 1908 an alternative and original treatment for Brownian motion: he also deduced the validity of (1.80) but this time from what was later known as “Langevin equation”. [Langevin (1908)]. In this same work Langevin made it clear that the respective methods of Einstein and Smoluchowski led to the same results if both were applied correctly. In 1908, the Polish physicist, following his 1904 treatment on the fluctuation of the number of particles, returns to the subject in an article entitled “Kineticmolecular theory of opalescence in gases in the critical state and some related phenomena”. [Smoluchowski (1908)]. He derives the following expression for absolute value of the mean compression, which is a measure of particle number fluctuation, in the case of an ideal gas: / δ=

2 ωp β , V π

(2.19)

where ω represents the molecular volume, understood here as the quotient between the volume V and the number of particles N, p is the pressure and β is the compressibility. isotherm —not to be confused with compression (2.18)—, defined by: 1 β=− V

)

∂V ∂p

) (2.20) T

From the last two expressions and taking into account vn = VN , it is again obtained that, for an ideal gas, δ turns out to be inversely proportional to the square root of the number of particles N. Thus, in general, the fluctuation of the number

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of particles contained in a finite volume of the gas turns out to be practically zero. However, this conclusion is not valid if the gas is(in a_ state close to the critical state. Recall that in the critical state the condition ∂∂Vp = 0 is fulfilled, so that, T in neighbouring states to this one, the compressibility β can acquire immensely large values and, then, the mean compression δ is not only not cancelled, but it could even acquire relevant values. In other words, in an ideal gas at equilibrium, the molecules are uniformly distributed in the volume of the gas, but if the gas is in a state close to the critical state, the number of molecules can fluctuate locally—that is, the number of molecules contained in a small volume can vary significantly—causing the appearance of clumps (high concentrations in relation to the mean concentration) and bubbles (low concentrations), which would break the usual homogeneity of the gas in such a way that light, on passing through it and encountering these inhomogeneities, would undergo the corresponding scattering effects. In Smoluchowski’s own words [excerpt quoted in Pais (2005), 102]: These agglomerations and rarefactions must give rise to corresponding local density fluctuations of the index of refraction from its mean value and thus the coarsegrainedness of the substance must reveal itself by Tyndall’s phenomenon, with a very pronounced maximal value at the critical point. In this way, the critical opalescence explains itself very simply as the result of a phenomenon the existence of which cannot be denied by anybody accepting the principles of kinetic theory.

Certainly, Smoluchowski saw the critical opalescence and the blue of sky as two manifestations of density fluctuations in the air. In fact, until 1911, he thought that these effects were basically due to two different causes: Rayleigh scattering, due to the presence of particles in the air and density fluctuations in the medium. It was Einstein who convinced him that, in the end, the latter was the sole cause of both phenomena.34

2.5.2

Einstein on Critical Opalescence (1910)

In 1910, Einstein publishes a paper entitled “Theory of the opalescence of homogeneous fluids and fluid mixtures in the proximity of the critical state”, in which he presents an original mechanical-statistical study of the optical phenomena appearing near the critical point of a gas or of a binary mixture of liquids. [Einstein (1910)]. There is no doubt about the high degree of knowledge—and appreciation—that Einstein had at that time of Smoluchowski’s research on density fluctuations. It is deduced not only because already in the first line of his article he refers to the 1908 paper of the Polish physicist, which we have just commented

34

For further information on the relationship between Smoluchowski and Einstein respective investigations on critical opalescence, see the note “Einstein on critical opalescence”, in Klein et al. (1993a), 283–285.

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but also because in the correspondence between them there is a request for the exchange of papers, at Einstein’s demand, on their respective contributions to the field35 : Along with this postcard I am sending you those of my papers that might still be of interest. At the same time, I wish to ask you to send me your papers, for I would like to study them more thoroughly.

In Einstein’s opinion, Smoluchowski’s research showed the existence of significant density fluctuations in the neighbourhood of the critical point of a fluid, which in turn justified the appearance of the phenomenon of opalescence in such circumstances. However, and we continue with Einstein’s opinion— Smoluchowski had not succeeded in obtaining a formula for the intensity of the light after light scattering due to the effect of such density fluctuations. Obtaining such an expression is the avowed aim of Einstein’s 1910 paper. This article begins with the analysis and discussion of the Boltzmann principle, which Einstein writes in the following form: S=

R log W + const. , N

(2.21)

where R represents the universal constant of gases, N is the number of molecules in a gram molecule (i.e., Avogadro’s number) and W indicates the number of microscopic states—mechanically described—compatible with the characteristics of the thermodynamic state considered. It is common to refer to W, following Boltzmann, as the “probability of the thermodynamic state” in question, although such a denomination is not very appropriate since the ordinary probability cannot have a value greater than one.36 After a schematic presentation of the fundamentals of statistical mechanics, density fluctuations are evaluated, both in the case of a homogeneous fluid and in the case of a binary mixture of liquids. The results obtained allow Einstein to calculate the influence of these fluctuations on the propagation of light through a medium. He begins by studying the propagation of a monochromatic beam of polarized light. He does this by resorting to electromagnetic optics, which takes Maxwell equations as the starting point for the analysis of optical phenomena. Concretely, for the case of a monochromatic ray passing through an ideal gas, Einstein deduces the following expression: ) ) Φ J0 R T (ε − 1)2 2 π 4 = cos2 ϕ, Je N p λ (4π D)2

35

(2.22)

Letter from A. Einstein to M. Smoluchowski June 11, 1908. In Beck (1995), 76. By this time, and especially in connection with his first quantum ideas, Einstein had already resorted on several occasions to Boltzmann’s principle and had dealt critically with its analysis. For more details on these aspects, one can see Navarro; Pérez (2002b), in Spanish.

36

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where the following physical quantities are involved: – – – – – – – –

J 0 is the intensity of emerging light (“opalescent light”, according to Einstein), J e is the intensity of incident light (“exciting light”, according to Einstein), T is the absolute temperature of the gas, ε is the square of the refractive index corresponding to the wavelength λ, p is the pressure, Φ is the volume of the gas traversed by the radiation, D is the distance travelled by the light beam in the gaseous medium, ϕ is the scattering angle (angle between incidence and emergence rays).

After recognizing that the above formula had been deduced by Rayleigh in 1899, “by summing the radiations of the individual gas molecules, which are considered to be completely randomly distributed”, Einstein concludes [Einstein (1910). In Beck (1994), 247]: As a rough calculation shows, this formula might very well explain why the light given off by the irradiated atmosphere is predominantly blue. In this connection it is worth noting that our theory does not make any direct use of the assumption of the discrete distribution of matter.

The article ends with the application of analogous reasoning to the propagation of a monochromatic light beam through a mixture of liquids, obtaining an expression with the same dependence on λ as in formula (2.22), although logically now involving some other physical quantities. In the last paragraph he comments that the two expressions obtained—one for gases and other for mixture of liquids—can lead to a new experimental determination of Avogadro’s number, whose precise value was still being elucidated in those days. Although, as the title of his 1910 paper anticipates, Einstein aimed at finding valid results for the neighbourhood of the critical point, he soon found that approximate methods, to which he had resorted, were no longer applicable to the critical point and its neighbourhood, while they were perfectly valid for the rest of the situations, particularly for the case of the ordinary atmosphere. That is precisely why these results can be considered as a theoretical explanation of the blue colour of sky which confirms Rayleigh’s conclusion, while Rayleigh blamed the effect on the existence of suspended particles in the atmosphere, Einstein blames it on the existence of density fluctuations in the molecules in the medium, without the need to resort to foreign particles. However, there is a weak point in Einstein’s reasoning—and in Smoluchowski’s—about density fluctuations, which can be revealed by a simple consideration. If, by virtue of such fluctuations, molecules accumulate, for example, in a certain small volume increasing the density of the medium there, it is obvious that some other region will be diminished with molecules, since the total number of these must be constant. In other words, density fluctuations in different regions of the medium cannot be considered completely independent. The admission, as

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Einstein does in his work, of the independence of such fluctuations invalidates, at least to some extent, his explanation of critical opalescence and the rest of the results derived from such a hypothesis. This remarkable lapse in Einstein’s treatment—assumption of independence of density fluctuations in different regions of the medium—was clearly denounced three years later when the Dutchmen Leonard S. Ornstein (1880–1914) and Frits Zernike (1888–1966) published a paper entitled “Accidental deviations of density and opalescence at the critical point of a simple substance”, which clearly showed the inadequacy of Einstein’s hypotheses.37 Now, without the hypothesis of the independence of density fluctuations, they obtained a more general formula than Einstein’s expression (2.22), but leading to this one in the case of “high temperatures”. However, in the neighbourhood of the critical point, the dependence of the former expression was not as λ−4 but as λ−2 . This result does not change in any way Einstein’s conclusions about the blue colour of the sky, since his explanation was based on the formula (2.22), which, we insist, is valid for regions far from the critical zone. Precisely in this article, the authors derive an integral equation for “the correlation function” in a homogeneous medium: it is the latter so-called “Ornstein–Zernike equation”, key to the later development of statistical physics of fluids. [Ornstein and Zernike (1914), 796]. It may be worth recalling that the correlation function of two particles—in other words, their distribution function—contains all the information about mutual influences between the molecules in the medium, and the Ornstein–Zernike equation serves precisely to determine that function. More precisely, the correlation between molecules 1 and 2, according to Ornstein and Zernike, can be considered divided into two parts: a direct part, which measures the mutual influence between 1 and 2, and an indirect part, which evaluates the mutual influence between 1 and the remaining molecules (3, 4, 5…), each of which, in turn, is related to 2, both directly and indirectly. The equation can be written using a slightly simpler notation than the original one as follows: ∫ g (r1, 2 ) = f (r1,2 ) + b d→ r3 g(r2,3 ) f (r1,3 ), (2.23) where g represents the correlation function (total) between two particles, f is the corresponding direct correlation function, b is a constant determined by the density of the medium, r→ 3 indicates the position of molecule 3 and ri, j is the distance between molecules i and j. Note that, by virtue of integration, molecule 3 actually represents the rest of the molecules. It should be noted that although the approximation used by Einstein in his 1910 paper is not lawful in the neighbourhood of the critical point, his evaluation

37

Ornstein and Zernike (1914). Ornstein had received his doctorate in 1908 under Lorentz supervision. Zernike would be awarded the Nobel Prize in Physics 1953 “for his demonstration of the phase contrast method, especially for his invention of the phase contrast microscope”.

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of molecular density fluctuations, together with his treatment of the corresponding scattering of the light passing through a transparent medium, represent important milestones on the long road to the generalized admission of the molecular constitution of matter and, also, in the consolidation of the then still novel statistical mechanics.

2.6

Einstein Before International Scientific Community: First Solvay Conference (1911)

At the end of Sect. 2.3, we referred to Nernst’s measurements in Berlin, around 1910, of the specific heats of solids at very low temperatures. The lower these were—the technique was progressing rapidly—the clearer the discrepancy between experimental results and theoretical predictions became, not only in relation to the predictions provided by the empirical law of Dulong-Petit but even with those of Einstein from his 1907 quantum theory. As soon as he learned about this theory, Nernst became deeply interested in the new quantum ideas, sensing that perhaps they could be appropriate to find an explanation for the growing number of anomalies he had been observing in the measurements of the specific heat of solids at low temperatures. Even these new ideas could perhaps serve to clarify the true meaning of the third law of thermodynamics, established by Nernst himself in 1906. He had visited Einstein in Zurich in 1910 and was therefore familiar with many of his ideas concerning specific heats at low temperatures. The Einsteinian quantum approach could perhaps represent, according to Nernst’s intuitions, an attractive hope in the search for theoretical reasons that would help to reveal the origin of his surprising experimental results. A personal friend of Ernest Solvay (1838–1922), a famous and wealthy industrial chemist, Nernst convinced him to finance the organization of an international meeting of a select group of scientists who would discuss at the highest level the nature of electromagnetic radiation and its relation with quanta.38 Thus, the first of the Solvay conferences—for many scholars, the most important of all—was forged.39 Eighteen scientists of the highest prestige took part, meeting between 30 October and 3 November 1911 at the Hotel Métropole in Brussels. The presidency of the conference fell to Lorentz, recognised as the foremost authority on physics at the time. He had to make real efforts to achieve the desired communication between the participants, given their different specialities and interests, as well as other differences, including linguistic differences. Although he largely achieved most of his objectives, Lorentz could not avoid a sour tone in some of the discussions, as can be clearly perceived by reading the

38

Solvay’s industrial and economic success was largely due to his invention of the ingenious “Solvay process” for the industrial production of soda, which he had patented in 1861. 39 For a comprehensive summary of the development —including attendees, papers and discussions— of the first sixteen conferences, between 1911 and 1973, see Mehra (1975).

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proceedings published the following year, in French, under the editorial tutelage of Langevin and Maurice de Broglie (1875–1960), the elder brother of Louis de Broglie. [Langevin and De Broglie (1912)]. Proceedings that, by the way, were soon widely disseminated in the physics nerve centres of the time. The official list of participants, all by invitation, including their nationality and place of origin, as recorded in the minutes of the meeting, is as follows: President:

H. A. Lorentz, of Leiden

For Germany:

W. Nernst (Berlin), M. Planck (Berlin), H. Rubens (Berlin), A. Sommerfeld (Munich), E. Warburg (Berlin-Charlottenburg), W. Wien (Würzburg)

For England:

J. H. Jeans (Cambridge), E. Rutherford (Manchester)40

For France:

M. Brillouin, Madame Curie [sic], P. Langevin, J. Perrin, H. Poincaré (all from Paris)

For Austria:

A. Einstein (Prague), F. Hasenöhrl (Vienna)

For Holland:

H. Kamerlingh Onnes (Leiden)

For Denmark:

Martin Knudsen (Copenhagen)

Additionally, included in the minutes are Lord Rayleigh, of London and van der Waals (1837–1923), of Amsterdam, both as official participants in the meeting, although without attending.41 After the initial speeches by Solvay, Lorentz and Nernst—in that order— the scientific meeting proper begins with the presentation of the invited papers, followed by the corresponding discussions, which are also transcribed in the proceedings. List of communications42 :

40

– H. A. Lorentz

“Application of the energy equipartition theorem to radiation”

– J. H. Jeans

“Kinetic theory of specific heat according to Maxwell and Boltzmann”

– E. Warburg

“Experimental verification of Planck’s formula for black body radiation”

– H. Rubens

“Verification of Planck’s radiation formula in the long wavelength domain”

Ernest Rutherford (1871–1937) has been already awarded the Nobel Prize in Chemistry 1908 “for his investigations into the disintegration of the elements, and the chemistry of radioactive substances”. 41 Lord Rayleigh sent a short communication to Nernst, to be read and discussed at the meeting. Van der Waals has been awarded the Nobel Prize in Physics 1910 “for his work on the equation of state for gases and liquids”. 42 The papers presented were commissioned to the respective authors by the organization of the conference, among topics previously proposed by Nernst; see Mehra (1975), 6–7. Further details on the first Solvay conference can be found in Barkan (1993). For a discussion of various aspects related to the role of the Solvay conferences in the development of modern physics, see Marage; Wallenborn (1995).

2.6 Einstein Before International Scientific Community: First Solvay Conference …

– M. Planck

“The law of black body radiation and the hypothesis of the elementary quantum of action”

– M. Knudsen

“Kinetic theory and the experimental properties of perfect gases”

– J. Perrin

“The proof of molecular reality (special study of emulsions)”

– W. Nernst

“Application of the quantum theory to several physico-chemical problems”

– H. Kamerlingh Onnes:

“Electrical resistances”

– A. Sommerfeld

“Application of the theory of action quantum to non-periodic molecular phenomena”

– P. Langevin

“Kinetic theory of magnetism and the magnetons”

– A. Einstein

“On the present state of the problem of specific heats”

111

Not all the invited scientists presented a paper at the conference. Among other reasons, some of them were not strictly specialists in subjects related to the first quantum notions, around which the interventions had to be essentially centred. This is the case, for example, with Poincaré and M. Curie. However, this did not prevent them from taking part in the lively discussions that followed the presentations of the invited papers. Precisely this was the objective pursued by Nernst in organizing the meeting: that, since they were prestigious research figures in different fields of physics, their objections and approaches would contribute to the establishment of incipient—and imprecise—new quantum ideas, as well as to the opening of future avenues of research in this respect.

2.6.1

Development of the Conference: Some Interventions

In the opening speech Lorentz —in his capacity as chairman— clearly states the purpose of the Brussels meeting [Langevin; De Broglie (1912), 7]: In such a state of things, the beautiful hypothesis of the energy elements, first launched by Mr. Planck and applied to numerous phenomena by Mr. Einstein, Mr. Nernst and others, has been a precious ray of light. It has opened up unexpected perspectives for us, and even those who regard it with a certain distrust must recognize its importance and its fruitfulness. Thus, it well deserves to be the main subject of our discussions, and certainly the author of this new hypothesis and those who have contributed to its development deserve our sincere tribute.

Lorentz recalls the main objections to quantum ideas: not only do they seem to contradict long-standing and well-established theories, but—above all—the corresponding theoretical formulations still lack the precision and rigour needed. For these reasons, it is urgent to raise the new conceptions and to debate them. In particular, the meeting seems to be an appropriate place to discuss the nature of

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quanta, how to incorporate them into physics and to look at possible applications of the new ideas. In Lorentz’s opinion, it is of the utmost interest to discuss the possible need to introduce quantum hypotheses in physics [Langevin; De Broglie (1912), 8]: First of all, we need to make ourselves clearly aware of the imperfections of the old theories by specifying the nature and causes of their defects. Then we shall examine this idea of units of energy in the various forms it has been given; we shall also deal with the cautious and systematic expositions, and the bold strokes that have sometimes been attempted. We shall endeavour to distinguish the incidental from the essential and to get as clear an idea as possible of the necessity and the degree of probability of the hypotheses. Finally, we would be very happy if we could get a little closer to this future Mechanics of which I have just spoken.

Notwithstanding the above, none of the submissions is expressly devoted to the issue of the necessity of quanta. It is surprising—at least to us—that it was not even among the topics proposed to the participants by the organizers. Nevertheless, it is possible to detect, through the interventions and discussions, the different positions of the assistants—a very representative sample of the scientific leaders of the time—about the possibility of necessarily having to resort to quantum hypotheses in the very near future. The contributions of Emil Warburg (1846–1931), Martin Knudsen (1871–1949), Rubens, Perrin, Nernst and Kamerlingh Onnes are mainly devoted to confirming known experimental data and presenting new data on various topics related to the emergence of the first quantum notions: black body radiation, ideal gases, properties of solids at low temperatures, etc. We prefer to move on to other interventions that attempt to go deeper into the nature of quanta and to present new applications of them. In his speech—“Application of the energy equipartition theorem to radiation”— Lorentz does not directly deal with the problem of the nature of the quantum hypothesis, although he tackles such an interesting topic as the analysis of the difficulties arising from the application of statistical mechanics of equilibrium— with the equipartition theorem as flagship—to radiation.43 In the letter sent for the occasion, Lord Rayleigh states that the quantisation introduced by Planck must be understood as “a simple method which has achieved interesting results thanks to the skill of those who have employed it”, but which in no way provides “a picture of reality”. [Langevin; De Broglie (1912), 49–50]. The experimental data of the moment, in Lord Rayleigh’s opinion, do not necessarily imply the abandonment of Boltzmann’s preceding ideas —and the more recent ones of the British James H. Jeans (1877–1946), present at the Brussels meeting— suggesting that ordinary experimental situations may not conform to the conditions

43

Langevin and De Broglie (1912), 12–39. In his paper Lorentz suggested ways of avoiding these difficulties, although without reaching definitive conclusions. For some clarifications, see Bergia; Navarro (1997), 201–202.

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required to represent a true statistical equilibrium state. [See, for instance, Bergia; Navarro (1997), 203]. Ultimately, both Lorentz’s intervention and that of Lord Rayleigh are aimed at trying to reconcile the indisputable experimental validity of Planck’s law and the nonnecessity of having to resort to quantum hypotheses for its theoretical justification. This is also the line followed by Jeans when in his intervention—“Kinetic theory of specific heat according to Maxwell and Boltzmann”—he introduces the notion of “effective terms” in the Hamiltonian: an ad hoc way of limiting the number of states allowed to a classical system, to avoid the introduction of quantum hypotheses. The procedure was so highly artificial and unjustified that it was strongly criticized by Poincaré in subsequent discussions.44 In his communication—“The law of black body radiation and the hypothesis of the elementary quantum of action”— Planck introduces what would become known as his “second theory”: the absorption of radiation is a continuous process that conforms to the prescriptions of Maxwellian electromagnetism, while emission is a discrete process described in terms of quanta.45 However, even with such a conceptual downgrading, the quantum hypothesis for emission that Planck presents in Brussels is no longer that of 1900. He now calls it the “hypothesis of the elementary quantum of action”: the elementary domains—infinitely small—in phase space are now to be replaced by finite domains of the following extension: ¨ d q · d p = h, (2.24) where q and p represent conjugate canonical variables in the Hamiltonian sense. Now it is not a hypothesis of energy quanta, and it is not applicable to radiation; only to matter (atoms and electrons). Planck seems to understand this quantisation as necessary, although he has no conclusive evidence for it. He states that the satisfactory implementation of any quantum hypothesis needs to be in harmony with the principle of least action [Langevin and De Broglie (1912), 114]: A complete understanding of the physical meaning of the action element h can only be obtained through the principle of least action, which seems to govern all fundamental phenomena and whose importance has been affirmed in the theory of relativity. The quantum theory must, in my opinion, harmonise with the principle of least action. It will only be necessary to give this principle a more general form which makes it applicable to discontinuous phenomena.

Thus, it seems that, for Planck, quantum discontinuities have come to mechanics to stay forever, even if it still requires some refinements.

44

Langevin; De Broglie (1912), 53–73. In relation to Poincaré’s critique see, Bergia; Navarro (1997), 204. 45 Langevin; De Broglie (1912), 93–114. For a detailed exposition in context of “Planck’s second theory”, see Kuhn (1978), Chap. 10.

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Sommerfeld presents a report—“Application of the theory of action quantum to non-periodic molecular phenomena”—in which he establishes a new quantum hypothesis applicable to general mechanical systems, not necessarily periodic or one-dimensional ones. [Langevin and De Broglie (1912), 313–372]. To this end, he recovers the notion of “action element” introduced by Planck to refer to the universal constant h and establishes a new quantum hypothesis [Langevin and De Broglie (1912), 315–316]. Emphasis in the original]: In any pure molecular phenomenon, the atom takes or loses a universally determined quantity of action and of magnitude of ∫ τ h (2.25) L dt = , 2π 0 where τ represents the duration of the process and L will usually be considered as a simple abbreviation for the difference between kinetic and potential energy of the system.

Here, the absorption and emission of radiation are treated with symmetry, contrary to Planck’s proposal in his second theory, which was criticized by Sommerfeld at the meeting—as also he had made before with 1900 Planck’s first hypothesis—both for its dark physical meaning and for the results provided by some of its applications. [Langevin and De Broglie (1912), 129 and 367–372]. However, although he shows that his new hypothesis conforms to the requirements of relativity, Sommerfeld explicitly recognizes the incompleteness of his hypothesis of the elements of action [Langevin; De Broglie (1912), 320–321]: The numerical agreement of our calculations with the value of h is generally surprising, but not completely satisfactory. I would have preferred to be able to let my ideas on this subject to mature further ideas, if the present meeting had not caused their premature publication by providing me with the opportunity to submit them to the criticism of the most competent men. I feel all the more obliged to acknowledge in advance the unsoundness of my reasonings, as I have perhaps not introduced the necessary restrictions everywhere. As regards the general comparison, with which I began this paragraph, between the element of energy and the element of action, I only wish to oppose my conception, with all reservations, to that of other scholars who have occupied themselves much longer and more deeply with these questions and who have obtained such important results by adopting only the point of view of the elements of energy.

In spite of the immaturity he assigns to his quantum ideas, Sommerfeld applies the hypothesis expressed in (2.25) to three phenomena: Roentgen’s rays, photoelectric effect and ionization of gases. In some cases—as in the case of the photoelectric effect—he obtains conclusions that lead him to affirm that his conceptions fit the experiments better than Einstein’s theory of energy quanta [Langevin and De Broglie (1912), 356]. Moreover, Sommerfeld emphasizes that his hypothesis offers, in addition, a remarkable advantage in relation to Planck’s primitive hypothesis and Einstein’s: while these are irreconcilable with Maxwellian electromagnetism, the new hypothesis not only does not contradict classical electrodynamics but—in a certain way—completes some aspect of it. Nevertheless,

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given the simplicity and the success obtained through the application of those hypotheses of Planck and Einstein, Sommerfeld believes that, in due course, these hypotheses can be deduced from the more general one—which he claims to have introduced—when it is properly applied to simple periodic motions. In the sharp and passionate discussions that follow the expositions, the incompleteness and provisional character of the new hypothesis becomes even clearer; especially after the profound remarks of Einstein and Poincaré in the debate following Sommerfeld’s exposition. Another clear distinction between Planck’s and Sommerfeld’s respective hypotheses on the action elements is also worth noting, as noted by Langevin in the same discussion: while the generalization of the former is of a statistical nature, since expression (2.24) refers to the phase space of the system, Sommerfeld’s proposal is of a mechanical nature, since expression (2.25) refers exclusively to the motion of such a system. [Langevin and De Broglie (1912), 375–376]. As anticipated in the title—“Kinetic theory of magnetism and the magnetons”— Langevin devotes the core of his speech to the mechanical-statistical study of magnetism and, in particular, to the deduction of “Curie’s law”, according to which the magnetic susceptibility of a paramagnetic material is inversely proportional to its absolute temperature. In an updated version of his microscopic theory of 1905, he associates to each molecule a magnetic moment as a function of which the paramagnetic susceptibility is obtained.46 He also includes an analysis of ferromagnetism by involving the interaction between molecules. In this connection, Langevin states that Weiss has extended the theory with own ideas—such as, for example, the introduction of his hypothesis about the “molecular field”—to explain ferromagnetism and using the latest experimental results of the time to test the theory. He has even formulated a quantum hypothesis of magnetic nature; according to this, the molecular magnetic moments of different substances must always be multiples of an elementary magnitude: the “Weiss magneton”. Langevin clarifies that if, as it seems reasonable, the molecular magnetic moment is due to the circulation of electrons, Sommerfeld’s hypothesis of the action elements provides a plausible explanation for the existence of the magneton, which is not provided by the hypothesis of energy elements.47

2.6.2

Einstein’s Communication

The first part of Einstein’s memoir—“On the present state of the problem of specific heats”, [Einstein (1912a)]—contains an updated version of his 1907 paper on the specific heats of solids, now taking into account the new experimental results

46

Langevin and De Broglie (1912), 393–404. Langevin’s theory, as we have pointed out on other occasions, is of a classical nature, without the slightest contamination of quantum ideas. See Navarro and Olivella (1997). 47 Langevin and De Broglie (1912), 402–404. Langevin’s remark was made in response to a question from Wien.

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obtained by Nernst at his Berlin laboratory. Einstein’s paper also includes some novelties contained in two of his papers then newly appeared in Annalen.48 Among these novelties, perhaps the most remarkable one is that of suggesting a new model of crystalline solid, not with a unique oscillation frequency for all the atoms of the lattice, but associating to the crystal a range of characteristic frequencies49 : Now we ask again: Why does the observed temperature dependence of specific heat deviate from the theoretically determined dependence? In my opinion, the cause for this deviation must be sought in the fact that the thermal oscillations of the atoms deviate markedly from monochromatic oscillations, and therefore do not actually have a definite frequency but rather a range of frequencies.

In the second section, Einstein states a clue question: “How is mechanics to be reformulated so that it does justice to the radiation formula [Planck’s law] as well as the thermal properties of matter [specific heats]?”. However, Einstein’s basic objective continues to lie, not in introducing a quantum hypothesis, but in deducing it from experimental results and from ‘something else’, we add. Only by bringing out this ‘something else’ is it possible to grasp with propriety what is truly to be understood by deducing the necessity of the quantum hypothesis.50 Einstein starts from something then so admitted as it was Boltzmann’s principle, which relates entropy (macroscopic magnitude) with statistical probability, i.e., with the number of accessible states (microscopic description). However, in his analysis Einstein uses something more debatable, which is the expression (2.7) obtained with his model of 1907 for the calculation of the specific heat of solids, although he now recognizes that perhaps it would be more appropriate to replace it by another with two summands of the type (2.6) one for the characteristic frequency ν and an analogous one for the frequency ν/2, since he believes that there are sufficient experimental reasons to refine his model in that direction. [Einstein (1912 a). In Beck (1993), 409]. The skilful use of his model allows him to evaluate the fluctuations of the energy of atoms in solids, obtaining an expression with two summands of very different nature. One of them is the one that could be expected—following usual calculations in statistical mechanics—associated to the number of degrees of freedom. The other—unexpected according to ordinary statistical mechanics—is the one that leads Einstein to track the quanta of energy in solids. In addition, without clarifying the reasoning too much, he concludes [Einstein (1912a). In Beck (1993), 415]:

48

Einstein (1911 a and 1911d). For a discussion of the role played by specific heats in the evolution of Einstein’s thinking about quantum ideas, see Klein (1965). 49 Einstein (1912 a). In Beck (1993), 408. This idea had already appeared in Einstein (1911b), 365. In spite of the opposition of some scholars—Nernst among them, as Einstein recognizes in his Solvay memoir—, it would soon be adopted and refined in the face of mismatches with certain experimental data; see Debye (1912). 50 Let us insist that this incessant search for the necessary quantum—against the Planck’s sufficient one—is a clearly detectable characteristic in Einstein’s research in those days.

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The magnitude of this fluctuation shows an exact agreement with the quantum hypothesis, according to which energy consists of quanta of magnitude h ν if, which change their location independently of each other.

We are particularly interested here in emphasizing that this procedure will be recurrently used by Einstein in his subsequent research: by means of an original analysis of statistical fluctuations—of energy, in this case—he draws conclusions on key aspects of quantum physics. Let us remember that it was precisely considerations of this kind in his 1909 Gedankenexperiment that led him to suggest a possible dual structure (wave and particle) for radiation. Similarly, now in 1911, they allow him to confirm his assumptions about the energy of oscillating atoms in solids. We shall see later how, in 1916, an analysis of the momentum fluctuations will lead him to the photon. In 1925, the study of the fluctuations in the number of molecules in another ingenious Gedankenexperiment, will allow him to suggest the extension of duality (wave and particle) to the case of gas molecules. In short, in Einstein’s hands, the sharp analysis of statistical fluctuations of certain physical quantities, in another original Gedankenexperiment, turned out to be a recurrent methodological element of enormous creative potential. The discussion following Einstein’s exposition focuses on a question that Einstein himself raises to begin the debate. Given the inability of classical dynamics to explain certain phenomena, it is essential to ask what general principles of classical physics should be preserved to lay the foundations for new developments. In Einstein’s opinion, the principle of conservation of energy must be preserved. Additionally, Boltzmann’s principle—so probability appearing therein—must be established on solid foundations. In particular, it is necessary to rigorously define the “probability of a configuration” so that the time evolution of an isolated system always leads to “more probable states”. In another way, Einstein believes that the statistical conception of the second principle of thermodynamics must also be preserved. After Einstein had given a summary of his ideas on probability in usual statistical mechanics, and after Lorentz had pointed out—in the face of an intervention by Poincaré—that this notion of probability should not be confused with the one introduced by Gibbs in 1902,51 the discussion centre on the role of probability in physics; a role that, as is well known, was to be radically modified by the later development of quantum theory. Finally, and although the issue requires more depth, it is worth reproducing Einstein’s last intervention in the meeting, since it represents a first connection between adiabatic invariants and quantisation rules, a topic that, especially in Ehrenfest’s hands, was soon to play a very important role in the development of quantum ideas52 :

51

For a detailed analysis of the differences between Einstein’s and Gibbs’ conceptions, not only in relation to probability but also to statistical mechanics itself, see Navarro (1998). 52 Einstein (1912a). In Beck (1993), 437. Although we shall return to this topic later, a summary of the formulation and applications of the adiabatic principle by Ehrenfest—its introducer—can be found in Klein (1985), Chap. 11.

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If one changes the length of the pendulum infinitely slowly in a continuous manner, then the oscillation energy remains equal to ν if it was h ν in the beginning; the energy of oscillation varies as ν. The same applies to an oscillatory electrical circuit without resistance and to free radiation.

2.6.3

Conclusions of the Conference and Einstein’s Opinions

The proceedings close with a section on “General conclusions”, in which Poincaré, in view of the development of the meeting, questions not only the validity of the fundamental principles of mechanics, but also something that—until then—seemed inherent to the very notion of natural law itself: the possibility of continuing to express the laws of nature by means of differential equations, if discontinuities were to invade the realms of physics. [Langevin and De Broglie (1912), 451]. In the same intervention. Poincaré highlights a characteristic common to the incipient quantum developments and which—in his opinion—represents a weakness of these: although they try to substitute theories that were well established until then, they rely on certain results of these, while introducing hypotheses that represent negations of basic points of those classical theories. Incidentally, it should be pointed out that this intended liberation of classical physics was not completely achieved even with the advent of quantum mechanics. Nernst and Poincaré emphasize that the proven experimental validity of Planck’s law does not imply, for that moment, the absolute necessity of resorting to quantum hypotheses. Nernst suggests that it might be enough to modify certain aspects of mechanics, such as, for example, making mass dependent on acceleration. However, the problem then lies in developing a new mechanics consistent with these ideas, which does not seem to resolve the question given the scope of the mission. Most of the participants share the conviction that quantum discontinuities will definitely be installed in physics, although we will have to be cautious when dismantling parts of classical physics with proven successes. At the same time, a profound disorientation can be perceived in relation to the path to follow: will some slight adjustments suffice or, rather, will drastic changes have to be made in mechanics and electromagnetism? In summary, it is worthwhile to reproduce the speech by Léon Brillouin (1889–1969), a faithful expression of this complex sentiment [Langevin; De Broglie (1912), 451–452. Emphasis in the original]: I would like to summarise the impression I got from reading the reports first, and even better from our discussions as a whole. Perhaps my conclusion will seem very timid to the younger among us; but, even so, it seems to me already very important. It now seems quite certain that it is necessary to introduce into our physical and chemical conceptions a discontinuity, an element varying by leaps, of which we had no idea a few years ago. How should it be introduced? That is what I see less clearly. Will it be the first form proposed by M. Planck, in spite of the difficulties it raises, or in the second form? Will it be in M. Sommerfeld’s form, or in some other form to be sought? I know nothing yet; each of these forms is well adapted to one group of phenomena, less well to others. Will we have to go

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much further, and upset the very foundations of classical electromagnetism and mechanics, instead of limiting ourselves to adapting the new discontinuity to the old mechanics? I doubt it a little, and however important the phenomena to which our attention has been directed, I cannot forget the enormous of physical phenomena in which the coordination of mechanics and electromagnetism has provided such good results, which I do not wish to compromise. The very uncertainty in which we remain as to the form and extent of the transformation to be effected, whether it be evolution or a complete overhaul, is a powerful stimulant; and it is certain that this concern will pursue us for many weeks, and that each of us will be passionately attached to the solution of the difficulties, of which our discussions have shown the inescapable character and importance in so many domains of Physics and Chemistry.

Finally, we present some of Einstein’s opinions about the development of the conference, which, let us not forget, represents his first participation in an international scientific meeting and, moreover, of highest level. This will bring us closer to a part of the human side of the character. His inexperience and a certain degree of arrogance are evident in some opinions he expresses about the meeting and about some of its famous participants. An opinion that, in most cases, would not change substantially throughout his life. As it could not be less, once back in Prague, formally and in a brief manner Einstein thanked Solvay for the invitation to the conference53 : Thank you again with all my heart for the wonderful week you provided for us in Brussels, and no less for your generous hospitality. The Solvey [sic] Congress will remain forever one of the most beautiful memories of my life.

His opinion about some of the people attending the event—including comments about an alleged secret romantic relationship between Langevin and Curie maliciously aired by some of the French press—can be read in the letter he sent, just arrived from Brussels to Heinrich Zangger (1875–1955), a Zurich physician friend of Einstein’s family54 : I returned last night from Brussels, where I spent much time with Perrin, Langevin, and Madame Curie, and became quite enchanted with these people. The latter even promised to come to visit us with her daughters. The horror story that was peddled in the newspapers is nonsense. It has been known for quite some time that Langevin wants to get divorced. If he loves Mme Curie & she loves him [sic], they do not have to run off, because they have plenty of opportunities to meet in Paris. But I did not at all get the impression that something special exists between the two of them; rather, I found all three of them bound by a pleasant and innocent relationship. Also, I do not believe that Mme Curie is power-hungry or hungry for whatever. She is an unpretentious, honest person with more than her fill of responsibilities and burdens. She has a sparkling intelligence, but despite her passionate nature she is

53

Letter from A. Einstein to E. Solvay, 22 November 1911. In Beck (1995), 227. Letter from A. Einstein to H. Zangger, 7 November 1911. In Beck (1995), 219–220. Zangger turned out to be first a friend of Einstein, then a confidant of Albert and Mileva to the point of becoming the couple’s legal intermediary when the marriage began to break up; after their divorce he took care of children’s interests.

54

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not attractive enough to represent a danger to anyone. Perrin’s determinations are much better than we thought. In particular, his determination of the particle diameter is impeccable. H. A. Lorentz chaired the conference with incomparable tact and unbelievable virtuosity. He speaks all three languages equally well, and his scientific acuity is unique. I largely succeeded in convincing Planck that my conception is correct, after he has struggled against it for so many years. He is a completely honest man who shows no consideration for himself.

These opinions on the development of the conference are following a week later, in another letter also addressed to Zangger in a more relaxed and, in a certain sense, harsher tone than in the previous one; not only against the meeting in general but also against Poincaré and Planck in particular55 : It was most interesting in Brussels. In addition to the French Curie, Langevin, Perrin, Brilloin [sic], Poinkaré [sic] and the Germans Nernst, Rubens, Warburg, Sommerfeld, also Rutherford and Jeans were there. And of course, also H. A. Lorentz and Kamerlingh Onnes. H. A. Lorentz is a marvel of intelligence and tact. He is a living work of art! In my opinion he was the most intelligent among the theoreticians present. Poinkaré was simply negative in general, and, all his acumen notwithstanding, he showed little grasp of the situation. Planck stuck stubbornly to some undoubtedly wrong preconceived opinions ... but as for knowing, nobody knows anything. The whole story would be a delight to diabolical Jesuit fathers.

However, perhaps Einstein’s most negative assessment of the Brussels meeting is the one he gives to his colleague Besso in a letter written almost two months after the first Solvay conference. While in his previous letter he referred to the development of the congress as “most interesting”, now, somewhat surprisingly, he replaces it with “nothing positive”56 : I have made no headway with the electron theory. In Brussels, too, they acknowledged the failure of the theory with much lamentation but without finding a remedy. In general, the congress in Brussels resembled the lamentations on the ruins of Jerusalem. Nothing positive has come out of it. My fluctuation arguments met with great interest and no serious objection. I did not find it very stimulating, because I heard nothing that I had not known before.

55

Letter from A. Einstein to H. Zangger, 15 November 1911. In Beck (1995), 221–222. Einstein and Poincaré had just met personally in Brussels and would not meet again, as the Frenchman died only a few months later. They never clearly manifested mutual admiration, especially as far as their respective formulations of the especial theory of relativity was concerned. For more details on the relationship between Poincaré and Einstein, see, for example, Pais (2005), 169–172. 56 Letter from A. Einstein to M. A. Besso, December 26, 1911. In Beck (1995), 241.

2.7 On the Necessity of Energy Quanta: Ehrenfest (1911) and Poincaré (1912)

2.7

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On the Necessity of Energy Quanta: Ehrenfest (1911) and Poincaré (1912)

On other occasions, we have shown the great interest that existed at the time of the first Solvay conference (1911) to discern with rigour whether some of the quantum discontinuities recently introduced were truly unavoidable. On the other hand, the various hypotheses introduced about the quantisation of energy were simply a convenient resource to obtain results in accordance with experimentation or, on the contrary, were they absolutely necessary premises to explain certain behaviours, such as, for example, Planck’s radiation law? [Navarro (2002–2003). In Spanish]. It was urgent to find the answer, given the different implications depending on the solution. If it turned out that quantum hypotheses were not necessary but a convenient and interesting methodological artifice, it was necessary to review the achievements made with them to try to obtain them with premises and methods more in accordance with end nineteenth century physics. If, on the contrary, those quantum hypotheses were absolutely necessary to account for recent reliable experimental results, it would be unavoidable to direct research to find new foundations for the edifice of physics or, at least, towards the rigorous revision of the existing ones. In addition, the very idea of quantisation of physical quantities forced researchers to reconsider the usual way of posing the laws of physics in terms of differential equations and, perhaps, to replace these by equations in finite differences, which in turn would imply a radical renovation of the mathematical methods usually employed in physics until then. It is generally accepted that the first rigorous demonstration of the necessity to introduce quanta into physics was presented by Poincaré in 1912. However, this does not conform to reality, as we will show. We shall see in particular that, in addition to Einstein’s contributions in this direction—already commented on in Sect. 2.4—there is a very remarkable contribution by Ehrenfest justifying that necessity, in 1911, published before the first Solvay conference. Poincare’s publication was subsequent to the Brussels meeting.

2.7.1

Ehrenfest (1911): “Weight Function” and the Necessity of Quanta

In October 1911, only a few days before the start of the first Solvay conference, a paper by Ehrenfest was published in which he analysed, with great rigour and originality, to what extent it is necessary to resort to quantum hypotheses to account for the known experiments on black body radiation. [Ehrenfest (1911)]. The work is not easy to read; here, we shall limit ourselves to summarizing its contents, with special emphasis on the method employed and on the scope of the results. In a brief introductory commentary, Ehrenfest suggests that although experimentation has not yet said its last word, certain features of the behaviour of black body radiation already offered sufficient clues to be able to determine to what

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extent the energy quantum hypothesis was or was not modifiable; that is, already existed reliable experimental data to elucidate the question of the necessity of such a hypothesis and he devotes the article to this topic. Ehrenfest begins by listing the results which, in his opinion and in the light of various widely tested experiments, any investigation into the nature of radiation must necessarily consider. In particular, Ehrenfest thinks that one must assume the validity of Wien’s displacement law, which he now writes in the following form: ( ν_ dν, ρ (ν, T )dν = αν 3 f β T

(2.26)

where ρ represents the spectral distribution energy of black body radiation, α and β are two irrelevant constants and f is a function to be determined. Furthermore, it must be admitted that for very large wavelengths, the Rayleigh-Jeans law is valid and for short wavelengths, one must avoid what Ehrenfest calls here the “RayleighJeans catastrophe in the ultraviolet”, as we have anticipated in Sect. 1.5.2. Ehrenfest states that instead of operating with Planckian resonators, he will follow Rayleigh and Jeans’ method by considering the normal modes of the oscillations in the cavity containing the radiation. However, he cites Planck to justify the expression that gives the number of independent eigenoscillations with frequencies in the interval with frequencies in the interval (ν, ν + dν): N (ν) dν =

8π L 3 ν 2 dν, c3

(2.27)

where L is the value of the side of a cubic cavity—with fully reflecting walls—containing the radiation. He also adopts a previous Rayleigh result: if the walls of the cubic cavity contract infinitely slowly, the energy associated with each eigenoscillation increases—at the cost of the work done in compression to overcome the radiation pressure—in direct proportion to the frequency and, therefore, in inverse proportion to the length L. In other words: Eν' Eν = and ν ' · L ' = ν · L ν' ν

(2.28)

To determine how the energy is distributed between the normal modes of oscillation, Ehrenfest anticipates that he will follow Boltzmann’s method for distributing energy among the molecules of an ideal gas: he will maximize the entropy of the system according to the corresponding constraints and thus obtain “the most probable distribution”. We must here recall that in Boltzmann’s approach there is no restriction for the energy values of each molecule and equiprobability is assumed: all the states of a molecule, characterized by their respective coordinates and generalized momenta, are equally probable a priori if they are compatible with the total energy of the system.

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However, the above assumption—a manifestation of the ergodic hypothesis—is not deduced from the laws of motion. Is a plausible and simple, but not absolutely necessary, assumption. Thus, its application to the modes of vibration can be subjected to criticism. Ehrenfest reminds us that the state of an eigenoscillation of frequency ν can be characterized by its energy and phase, the latter being independent of ν and completely random. He then introduces the so-called “weight function” (Gewichtsfunktion) γ : γ (ν, E ) dE represents the probability that, a priori, to an oscillation of frequency ν corresponds an energy in the range (E, E + dE). Ehrenfest demonstrates a result that will be key to his treatment: except for an irrelevant factor—which may depend on the frequency—the weight function is a function of a single variable: q ≡ E/ν. That is, it can be written in the following form [Ehrenfest (1911)]: ) γ (ν, E) = Q(ν) · G

E ν

) (2.29)

Ehrenfest and the Necessity of Quanta (1911)

Ehrenfest analyses the properties of the above function G(q) and its physical implications. To preserve the formalism from the possible existence of singularities in the weight function—associated with discrete values of the energy—, Ehrenfest admits a possible additional discrete domain (Punktbelegung); so to certain values q0 , q1 , q2 , … he associates to them the respective finite weights G0 , G1 , G2 , … Adding this possibility to the usual continuous domain (Streckenbelegung) represented by the function G(q), he obtains that the relation between this weight function and the radiation law is determined by: C · f (σ ) = −

] d [ log Q(σ ) , dσ

(2.30)

where C, σ, and Q are defined by the following relations: α c3 ν ; σ ≡β ; 8π T ∫ ∞ ⎲ exp (−σ qr ) · G r + Q(σ ) ≡ C≡

r=0

∞

dq · exp (−σ q) · G(q)

(2.31)

0

It is easy to verify that if we start from a purely continuous domain—that is, if all the Gr cancel each other out—we inevitably arrive to a law that leads to the catastrophe in the ultraviolet. To avoid it, one must necessarily

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assign to the point q0 = 0 a particular weight G0 /= 0.57 Taking into account (2.30) and (2.31) Ehrenfest deduces two important results: 1. The following numerical values of the weights: G(q) = 0; qr = r; G r =

1 r!

(2.32)

lead to the following relationship: C · f (σ ) = exp(−σ )

(2.33)

The latter expression, replaced in (2.26) and taking into account (2.31), allows us to obtain the radiation law (1.62) proposed by Wien and which was used by Einstein in his famous 1905 paper on energy quanta. 2. On the other hand, the following numerical values of the weights: G(q) = 0; qr = r; G r = A

(2.34)

lead, following the same procedure, to C · f (σ ) =

1 , exp(−σ ) − 1

(2.35)

expression which, substituted in (2.26) reproduces Planck’s law. It is true that, up to this point, Ehrenfest has obtained his main results from the behaviour required of ρ(ν, T ) for values of ν/T close to zero and to infinity and then this does not seem to be a solid argument to justify the need for energy quanta. However, the treatment allows to deduce rigorously the requirement of quantisation: it would be enough that, instead of starting from (2.34) to obtain (2.35), the process would be reversed in order to obtain—starting from Planck’s law now written in the form (2.35)—, to try to obtain the weights (2.34). For this purpose, it is convenient, and this is precisely what Ehrenfest does, to start from the following equality:

57

Ehrenfest also justifies the requirement of certain additional conditions—which we will not mention here—about the asymptotic behaviour of G(q). [Ehrenfest (1911), 197–198].

2.7 On the Necessity of Energy Quanta: Ehrenfest (1911) and Poincaré (1912) ∞ ⎲ r=0

G r · exp(−qr σ ) +

∫ ∞ 0

) ∫ ) dq· G(q) · exp ( − q σ ) = exp − C · f (σ ) · dσ ,

125

(2.36)

which follows trivially from (2.30) y (2.31). The functional equation (2.36) is the fundamental result of the paper: it implies that the knowledge of the function f (σ )—i.e., of the radiation law—leads to the univocal determination of the necessary weights, both discrete and continuous, to obtain this law. In connection with Eq. (2.36) without the summation term, Ehrenfest cites a paper by Bernhard Riemann (1826–1866) where it is solved by integration in the complex field. It is here that our spirits are lifted, for Ehrenfest states that, to illustrate the method, he will deduce the weights corresponding to Planck’s and Wien’s laws. It seems that, at last! we will start from (2.35) in order to obtain (2.34),—after solving the functional equation (2.36). In addition, similarly, to obtain the weights (2.32) from the radiation law (2.33). However, our disappointment is great when we see that Ehrenfest does not truly solve the Eq. (2.36), but directly proposes a solution which, he claims, has been provided by the comment of a certain unspecified friend [Ehrenfest (1911), 112]. In both situations—Ehrenfest writes—G(q) = 0 is verified, and the nonzero particular weights occur at the points qr = 0, 1, 2, . . .. If one accepts the clue provided by “Ehrenfest’s friend” certainly the solution of the Eq. (2.36)in both the Planck and Wien cases is very simple to obtain. Indeed, since in both cases the second member is of the following form: [ ] Q(σ ) = F exp( − σ ) ,

(2.37)

the functional equation (2.36) can be rewritten as follows: ∞ ⎲

[ ] G r · exp ( − σ )r =F exp ( − σ ) .

(2.38)

r=0

Thus, the particular weights Gr arise when developing F in series of powers of exp( − σ ), in each case. A simple calculation allows to obtain in this way both the particular Wien weights and the Planck ones. This is how the necessity of the radiation energy quantum has been demonstrated by Ehrenfest, in 1911.58

58

To solve the functional equation (2.36) there is no special difficulty at present. It is sufficient to derive the spectral decomposition of the function in the right-hand member, after substituting f (σ ) by (2.35) in the case of Planck, and by (2.33) in Wien’s case.

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Ehrenfest employs classical physics—thermodynamics, electromagnetism and statistical mechanics—in his rigorous treatment obtaining far-reaching conclusions. Although he assumes the validity of Planck’s formula to describe the behaviour of radiation, he also considers other possibilities such as, for example, Wien’s formula, which allows him to clarify certain aspects—hitherto unexplored—of the relationship between the laws of black body radiation and the quantisation of energy. Let us analyse one of them that seems to us as interesting as neglected. Among the conclusions obtained by Ehrenfest, it is worth mentioning that the weights (2.34) are necessary—and sufficient, of course—to obtain Planck’s law. This is because (2.34) is the only solution of (2.36) if in this equation f (σ ) is substituted by (2.35), which corresponds to the formula proposed by Planck in 1900. We have noted that if, instead of operating with Planck’s law, one starts from Wien’s law—in the form (2.33)—one obtains the weights (2.32), although with a notable difference in this case: while the probability associated with each value of qr in Planck’s case did not depend on r, to obtain the law proposed by Wien this probability must be proportional to 1/r!. This Ehrenfest’s result clarifies a problem related to the famous Einstein paper of 1905, in which, as we saw in Sect. 1.5.2, the existence of energy quanta is suggested on the basis of the validity of the black body radiation formula proposed by Wien. From a critical view, the following question may arise: if Planck’s law is ‘responsible’ for energy quantisation, how is it that Einstein could arrive at this property in 1905 without resorting to this law? It is worth mentioning that we have never seen such a puzzling situation analysed. Now, we have just seen how this work of Ehrenfest elucidates the question: strange as it may seem, the law proposed by Wien implies the same discrete values for the radiation energy as the law proposed by Planck. Thus, taking one or the other as the starting point—i.e., as experimental data—the same quantisation of the energy is obtained. According to Ehrenfest’s results, if Einstein had operated directly with Planck’s formula—instead of Wien’s—it is possible to think that his wit and sharpness of mind would have led him to the same result. However, he would have had to devise another procedure: it is easy to verify that by following his 1905 procedure, but with Planck’s law instead of Wien’s, the desired result is not obtained. Another interesting conclusion that can be drawn from Ehrenfest’s work is that the differences between the quantisation to which the respective Planck and Wien laws lead would possibly have been detected if one had operated, not only with values of the energy, but also with the corresponding probabilities of these values. However, this never happened. The early and resounding experimental confirmation of Planck’s formula—to the detriment of Wien’s—truncated the competition between the two candidates. It is one more case, among others that history frequently shows: certain problems were never solved, but were forgotten when they ceased to be problems or, at least, interesting problems.

2.7 On the Necessity of Energy Quanta: Ehrenfest (1911) and Poincaré (1912)

2.7.2

127

Poincaré (1912): Mechanisms of Equilibrium and the Necessity for Quanta

Poincaré’s first serious contact with quantum theory took place, as we have already pointed out, at the first Solvay conference, to which he was invited—without being commissioned to deliver a speech, given his lack of familiarity with the new quantum ideas—because of his high prestige as a mathematical physicist, on the assumption that his intervention in the discussions would help to raise the level, as it did. On his return from Brussels, now persuaded of the importance of the quantum subject, he devoted himself to it with such intensity that he had time to make a remarkable contribution to the necessity of quanta, although he died only nine months after the end of the meeting.59 In fact, only a month later he presented to the Académie des Sciences a short preview of his later famous article which appeared in January 1912, devoted to a rigorous and profound analysis of the issue of the sufficiency and necessity, in that order, of energy quanta. [Poincaré (1911) and Poincaré (1912), respectively]. Poincaré begins his article by ruling out the possibility suggested by Nernst in Brussels—already referred in Sect. 2.6.3—of avoiding the recourse to quantum discontinuities by the appropriate introduction in mechanics of a new concept of mass, now depending on both velocity and acceleration. The author affirms that his investigations have led him to rule out such a possibility, although without pointing out arguments. According to Planck, the energy radiated by solids is due to a great number of resonators, so that each one emits and absorbs only monochromatic radiation. To reach the equilibrium—represented by Planck’s law—it is necessary for some mechanism to make possible the exchange of energy between resonators with different frequencies. Poincaré cites two. One, based on the Döppler-Fizeau effect, in which the ether plays the role of intermediary: being resonators in motion, one might think that the respective frequencies are liable to undergo continuous changes. However, although he cites this possibility, he does not analyse it. Another mechanism is chosen: the exchange of energy between resonators is possible thanks to the collisions between the resonators—which are in continuous motion—and the atoms of matter. It is not that a resonator exchanges energy directly with another resonator because, considering the quantum hypothesis—and that, in general, their corresponding frequencies are incommensurable—the respective quanta are not interchangeable. However, on the other hand, a resonator can continuously alter its energy as a consequence of the incessant collisions with the present atoms; thus, the presence of matter guarantees the possibility of the energy exchange between resonators of arbitrary frequencies.60

59

Let us recall that the Brussels meeting took place between 30 October and 3 November. Poincaré died on 17 July 1912, in Paris. 60 The treatment through any possible mechanism has to lead to the same results, so that—as Poincaré indicates at the end of his article—he chooses which considers the most adequate one.

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Poincaré’s treatment is framed in the phase space of an isolated system. For the mechanical-statistical analysis, he introduces the probability density W, to which he assigns the usual properties.61 He notes how the validity of Hamilton’s equations can lead to equiprobability, that is, to W = 1—it is more appropriate to write W = const.—which, in turn, leads to the equipartition theorem of energy. Before dealing with the general case, Poincaré analyses the interaction between two resonators, one—which he will later assimilate to an atom—with a large period τ 1 and energy ζ. Another resonator—which will be responsible for the quantum effects—has a small period τ 2 and energy η. Both move along the same straight line, with different equilibrium positions, and can collide in certain instants, in which the integration of the equations of motion is not feasible because of the lack of knowledge of the interaction; for this reason, it is necessary to avoid any treatment requiring knowledge of the interaction. This is what Poincaré does, and he succeeds in justifying that the probability density of the system—the two oscillators—is a function only of the respective energies: W = W (ζ, η). If h is the total energy, X is the mean energy of the first oscillator and Y is the mean energy of the second oscillator, the following must be verified: ζ +η =h ;

X +Y =h

(2.39)

If there is no difference between the two oscillators—as in the classical treatment—we arrive at equiprobability and X = Y. However, one should be cautious in case this procedure is not adequate. It is now that Poincaré assimilates the large period oscillator to an atom that is supposed to follow the laws of classical physics. In contrast, it is conceivable that the small period oscillator—the one that truly plays the role of the Planckian resonator—may exhibit strange behaviour due to collisions with the first one. The probability density will now only be a function of the energy of the second oscillator: W = W (η). It is a matter to determine this dependence from experimental data. Since “X represents the absolute temperature (except for a constant factor which, with an appropriate choice of units, we can assume to be equal to unity)”, if we admit the experimental validity of Planck’s law, the relation between the mean energies of the two types of oscillators will be given by the following expression: Y =

ε , exp(ε/X ) − 1

(2.40)

where ε is a constant. This relation will be the basis for determining W = W (η).

61

Poincaré refers to W as “dernier multiplicateur” of the equations of motion, according to the usual terminology he used in his reports on integral invariants. Poincaré (1912), 6–7.

2.7 On the Necessity of Energy Quanta: Ehrenfest (1911) and Poincaré (1912)

129

Poincaré’s Oscillators and the Necessity of the Quantum Hypothesis (1912)

Poincaré begins by analysing the case of p resonators of the first type— those that will later be assimilated to atoms—and n of the second one. It is necessary to comment on some of the details of his cumbersome treatment. The Frenchman assumes that the oscillators of the same class must have the same mean value of energy “by reason of symmetry”, which makes it possible to generalize expressions (2.39) and now write the following ones: p ⎲ i=1

ζi +

n ⎲

ηk = h ; 〈ζi 〉 = X , 〈ηk 〉 = Y ; p X + n Y = h

(2.41)

k=1

Admitting that the collision between any two resonators must not affect the remaining ones, Poincaré justifies the factorization of the W function: W (ζi , ηk ) = ω' (ζ1 ) · ω' (ζ2 ) · · · ω' (ζp ) · ω(η1 ) · ω(η2 ) · · · ω(ηn ),

(2.42)

where ω' represents the probability density of a large period oscillator and that of one of the others. As in the case of only one resonator of each type, he assumes that ω' (ζi ) = 1 is true for the large period ones, which leads to: W = W (ηk ) = ω(η1 ) · ω(η2 ) · · · ω(ηn )

(2.43)

Therefore, the determination of W is subject to the knowledge of ω. A serious problem lies in the fact that, as Poincaré makes clear, the distribution of the total energy between the oscillators—actually the relationship between X and Y for each distribution ω—usually depends on the n and p values, so that even the equilibrium distribution would be meaningless. However, there is a fact that—as usual in statistical treatments—overcomes the difficulty: the realistic values for n and p turn out to be very high, which justifies introducing the following limits: n → ∞; p → ∞; pn → k(finite). Poincaré shows that, under such conditions and independently of the form of ω, the distribution of the energy between resonators in a state of equilibrium—the one of maximum probability—no longer depends on the relative number of oscillators. It is now that he properly deals with the question of the sufficiency of the quantum hypothesis to deduce Planck’s law; and he does it according to two different procedures. Although, logically, with both procedures he arrives at the same result, here we will only outline his second method, in which he uses the Fourier transform—, since it is the method that he will later follow to deduce the necessity of quanta.

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To this end, it begins by defining the integral transform of ω: ∫

∞

Φ(α) ≡

ω(η)exp(−α η) dη,

(2.44)

0

where α represents a complex variable with null real part and ω is the probability density of a Planckian resonator. Poincaré maintains the hypothesis that for large period resonators ω’ is constant. The inverse transform of (2.44) can be written as follows: ∫ 1 Φ(α) exp(α η) dα, (2.45) ω(η) = 2π i where the integration is now performed along the straight line α = iβ, parallel to the imaginary axis, so the latter expression can also be written in the following form: 1 ω(η) = 2π

∫

+∞

−∞

Φ(i β) exp(i β η) dβ

(2.46)

With the help of reasonings previously used, Poincaré demonstrates now the following two relations: Φ' (α) 1 = −Y ; X = Φ(α) α

(2.47)

The elimination of α between these two equalities provides a law that expresses the energy distribution between both types of resonators or, more properly, the average energy of a Planckian resonator as a function of temperature.62 By way of illustrative examples, Poincaré applies the above results to two different situations and deduces that: – if we start from ω = ηm , one obtains Y = (m + 1)X . On the other hand, – if we start from ω = exp(γ η), one obtains Y = 1−γX X . It is verified that, in both cases, the results agree with those obtained by Poincaré according to his first method, although now the treatment is notably simpler.

62

Remember here that X is proportional to the absolute temperature T.

2.7 On the Necessity of Energy Quanta: Ehrenfest (1911) and Poincaré (1912)

131

If one admits Planck’s hypothesis of the quantisation of the energy of a resonator, the integral of the expression (2.44) must be replaced by the corresponding summation, since only values of η that are integer multiples of a certain value ε are admissible: Φ(α) =

∞ ⎲

exp(−α m ε) =

m=0

1 1 − exp(−α ε)

(2.48)

The expressions (2.47) will now be rewritten in the following form: ε = −Y ; 1 − exp (α ε)

X=

1 α

(2.49)

By eliminating α between these last two expressions, we obtain without difficulty Planck’s law expressed in the form (2.40). It is now, finally, that the issue of the necessity of quanta is addressed. Thus far—as in the first part of Ehrenfest (1911)—a rigorous and original analytical method has been developed to prove that the quantum hypothesis inexorably leads to Planck’s law. Additionally, as in Ehrenfest’s case, once Poincaré underpins his own method, he reverses the procedure to deduce the necessity of the quantum hypothesis. In fact, if we take as a starting point the relation between X and Y —Planck’s law in our case—its substitution in (2.47) allows to deduce the function Φ(α), except for a multiplicative factor. The substitution of this function, in turn, in (2.45)—or in (2.46)—leads to ω(η), except for an irrelevant factor that can be fixed by normalising the function. This biunivocal correspondence between the expression of ω(η) —the energy distribution in a Planckian resonator—and the relation between X and Y —the radiation law—leaves the problem totally solved: the quantum hypothesis not only leads to the Planck radiation law, but that hypothesis is the necessary condition for obtaining this law. This allows Poincaré to draw the following conclusion: “Thus, the hypothesis of the quanta is the only one leading to the Planck’s law”. [Poincaré (1912): 27. Emphasis in the original]. Accepting the necessity of quanta, in any of its versions, had such undesirable implications that it was necessary to take extreme precautions. Poincaré affirms that an experimental law is always an approximate law. Is it then unthinkable that another law different from Planck’s, but experimentally indistinguishable from it, could lead to continuous ω(η) functions? With a reasoning—not excessively clear, by the way—based on Wien’s law and classical electrodynamics he reiterates his essential conclusion [Poincaré (1912): 30. Emphasis in the original]:

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Thus, whatever the law of radiation may be, if the total radiation is assumed to be finite, this leads to a function ω with discontinuities analogous to those given by the quanta hypothesis.

The conclusion seems to be exaggerated, because what Poincaré actually demonstrates in his paper is that if the total energy emitted by a black body is finite, the function ω(η) must have at least a discontinuity at η = 0. The last section of the article is essentially devoted to seating the assumptions that, in Poincaré’s opinion, has not been fully justified in the treatment. Thus, he provides additional arguments to justify that the probability density for a large period oscillator is a constant—i.e., to justify equiprobability—, which makes it possible to dispense with the variable ζ i in expressions such as (2.42). Poincaré also deals with systems with several degrees of freedom, being obviously interested in their energy distribution in equilibrium. Although now it would be necessary to start from probability densities that were functions of several variables, the necessary integration with respect to all the variables other than energy—because they are irrelevant—would solve the problem, which would again be a problem with the only variable η. Poincaré also addressed, albeit unsuccessfully, the issue of the necessity of quanta by analysing the energy exchange between resonators by means of the Döppler-Fizeau principle, outlined in his 1912 paper. However, this mechanism led him not to Planck’s formula, but to Rayleigh-Jeans’ law. The search for alternatives led him to consider “light atoms”, but he rejected them because they implied the equipartition of energy. He also groped for possible “time atoms”: undetectable time between states separated by a quantum jump. However, his early demise prevented him from developing these ideas. [McCormmach (1967), 49–50].

2.7.3

Two Very Different Impacts

It is difficult to understand why Ehrenfest’s 1911 article deserved so little attention. It was certainly the work of a young physicist in St. Petersburg since 1907, who had not made any outstanding contributions in the field and was not even invited to participate in the first Solvay conference. As we have already noted, his article appeared in the October issue, a few days before the Brussels conferences, in a journal as prestigious and widely circulated as Annalen der Physik. The paper rigorously answered certain questions that later arose at the meeting, without anyone mentioning it. Ehrenfest’s manuscript was known, at least by Sommerfeld [Klein (1985), 170] and Einstein.63 It was certainly not the work of a complete stranger. For instance, Felix Klein (1849–1925) had commissioned Paul Ehrenfest in 1906—and, if they so decided, to prepare it together with his wife Tatiana—the article that Boltzmann, because of his death, was unable to write for the prestigious Encyklopädie der mathematischen Wissenschaften on the foundations of statistical

63

See the letter of A. Einstein to M. Besso 21 October 1911. In Beck (1995), 215.

2.8 Law of the Photochemical Equivalent (Einstein, 1912)

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mechanics.64 Moreover, Ehrenfest must not have been so unknown when, in the spring of 1912, he received an initial proposal from Lorentz, near retirement, to sound him out as his possible successor in Leiden, in the light of “the profundity, clarity and penetration” of his research. [Klein (1985), 186]. The succession became a reality at the end of that same year. In a previous paper, we have analysed in greater depth various aspects related to Ehrenfest’s work. This has allowed us to highlight the relevant role that his geographical remoteness at that time may have played in the very limited repercussion of this article. During his days in St. Petersburg, he had hardly any contact with the main academic centres where quantum ideas were developed and debated.65 The impact achieved by Poincaré (1912), on the other hand, was remarkable. Not only because its treatment of physics was complemented by mathematical rigour; external factors must also have had an influence, such as, for example, the great international prestige of its author, not to mention the greater topicality of the quantum issue—from the beginning of 1912—due to the rapid publication and dissemination of the proceedings of the first Solvay conference [McCormmach (1967), 51–55]. It is worth mentioning here the important role played by Poincaré in the acceptance of the first quantum ideas outside of Germany, especially in Great Britain after the publication of Jeans’ famous Report in 1914. [Jeans (1914)]. It was not only because of the rigour employed, but also because it included, for the first time, a model for energy exchange between Planckian resonators, which was absent in Ehrenfest (1911). However, it should be noted that, on the contrary, Ehrenfest included aspects not dealt with by Poincaré, such as, for example, the comparison between the respective quantum hypotheses of Planck and Einstein.

2.8

Law of the Photochemical Equivalent (Einstein, 1912)

The next step in Einstein’s long way along a path that would lead to the birth of wave mechanics was in 1912, when he published a paper and a supplement very little known—or, at least, very scarcely cited by physicists and historians— on photochemical dissociation, that is, on the action of radiation on molecules. [Einstein (1912b)]. We are convinced of the great importance of this article in order to understand the evolution of Einstein’s thinking about the quantum of radiation; ideas that would lead him four years later to the establishment of the corpuscular structure of radiation. In the 1912 treatment we find a clear precedent of some reasonings that, in 1916, would lead Einstein to the photon. For this reason, we shall now dwell on that 1912 paper. Precisely because of the great relevance that

64

Ehrenfest (1959). Because of several incidents, the publication did not come out until 1912, with the Ehrenfests as authors. It soon became—and still is today—a classical text on the foundations of statistical mechanics and in particular for the comparison of Boltzmann’s and Gibbs’ views. On this last topic, one can also see Navarro (1998), 166–169. 65 On the limited impact of Ehrenfest (1911), see Navarro; Pérez (2004), 126–130.

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we assign to it, it has already been the object of special attention in a previous work by the Italian historian of science S. Bergia and me, to which we refer for further details.66 Einstein begins by stating the “law of photochemical equivalent”: the decomposition of a gram equivalent of any substance by means of a photochemical process by monochromatic radiation of frequency ν requires from this radiation the energy N h ν, where N is Avogadro’s number and h is Planck’s constant.67 Einstein’s concern for this topic is not accidental, since in his 1905 paper on energy quanta, he had already referred—in the third example—to the ionization of gases by ultraviolet light. And in the second of his 1909 papers he wondered why the occurrence or nonoccurrence of a particular photochemical reaction depends on the colour of the incident light and not on its intensity; and why the rays of small wavelength are, in general, more chemically effective than those of longer wavelengths. [Einstein (1909b). In Beck (1989), 386]. Today, the answer to such questions is trivial if we start from an interpretation of electromagnetic radiation in terms of aggregates of quanta. However, Einstein, at that time, insists on directing his ideas in exactly the opposite direction: he does not intend to apply the quantum hypothesis to obtain new results, but to deduce the existence of quanta from experimental data. It seemed to him that such a strange hypothesis, from the point of view of classical physics, required even greater experimental justification before it could be definitively accepted. A proper understanding of the aim of his 1912 paper requires this perspective: what Einstein truly intends is to demonstrate that the law of the photochemical equivalent—and hence another support for justifying the introduction of quanta— can be deduced, together with phenomenological Wien’s law (1.62), from simple and plausible hypotheses, spiced up with usual thermodynamic reasoning. Before entering fully into the contents of the article, we would like to make a slight reference to a not very mentioned polemic between Einstein and Stark because, in addition to its historical interest, it helps to clarify the meaning and the pretensions of Einstein’s paper. Of course, he was not the first to relate quantum ideas to photochemical processes. [See Mehra; Rechenberg (1982a, part A), 99–113]. Perhaps it was Stark the best known among those who for some time before—about five years—had been interested in finding phenomena that would allow certain aspects of quantum theory to be tested, understood rather along Planck’s lines as a useful instrument that, for the time being, did not imply in any way assuming a corpuscular aspect to radiation. It is therefore not surprising that, when Einstein published his 1912 paper on the law of the photochemical equivalent, Stark claims in a note the paternity of that law for himself, which may seem fair if, for example, one reads paragraphs of his earlier papers, such as this one from 1908 [quoted in Hermann (1971), 79]:

66

Bergia; Navarro (1988), especially 85–90. See also Editorial note: Einstein on the law of photochemical equivalence, in Klein et al. (1995), 109–113. 67 Einstein (1912 b). In Beck (1996), 89. Here Einstein still does not call N Avogadro’s number and, instead, continues referring to “the number of molecules in a gram-molecule”.

2.8 Law of the Photochemical Equivalent (Einstein, 1912)

135

The direct chemical action of light consists in the release of valence electrons from their binding as a result, according with the quantum law, of the absorption of more light by individual valence electrons than the amount of binding energy of these valence electrons.

He must have thought that the law enunciated by Einstein in 1912 did not go beyond Stark’s above statement. In addition, possibly Einstein himself participated in much of the same thinking, judging by his reply [Einstein (1912c). In Beck (1996), 125]. Emphasis in the original]: J. Stark has written a comment on a recently published paper of mine [Einstein (1912b)] for the purpose of defending his intellectual property. I will not go into the question of priority that he has raised, because this would hardly interest anyone, all the more so because the law of photochemical equivalence is a self-evident consequence of the quantum hypothesis. But I see from Stark’s remark that I did not bring out the purpose of my paper clearly enough. The paper was supposed to show that the derivation of the law of photochemical equivalence does not require the quantum hypothesis, but that it can be deduced from certain simple assumptions about the photochemical process by way of thermodynamics.

Some scholars, avoiding compromise, have chosen to refer to the law of the photochemical equivalent as the “Stark-Einstein law”. This is a good occasion to claim a higher regard—leaving aside his later Nazi affiliation and activism—for Stark’s role in the development and dissemination of early quantum ideas; especially for his search for, and conduct of, experiments whose results could be interpreted as consequences of the quantum hypothesis [See, for instance, Hermann (1971), 72]. It is worth asking: if Einstein affirms in 1912 that it is not necessary to resort to the quantum hypothesis to deduce the law of the photochemical equivalent, could this be interpreted as distancing, at least temporarily, from such a problematic notion as that of energy quanta? A reading of his works and of his correspondence with Besso, for example, suggests rather the opposite: he is so convinced of the necessity to definitively establish quanta in physics that his main interest lies, not in starting from their existence to study certain phenomena pending theoretical explanation but in deducing the reality of quanta directly from experimental facts. Shortly before Einstein submitted his article for publication, he sent a letter to Besso in which he gave him a full account of the content and significance of the paper68 : I proved in my last paper (in a thermodynamical way) that the radiant energy N h ν is always required for the absorption of the photochemical decomposition of one gram-molecule by Wien’s light. This can be deduced from the radiation formula and the law of mass action. So one does not need the quantum hypothesis for this. Yesterday I received a letter from Warburg, to whom I related this matter in Brussels, and he writes that he found the law exactly confirmed in one substance.

68

Letter from A. Einstein to M. Besso 4 February 1912. In Beck (1995), 105. Emphasis in the original. The substance referred to in this letter is possibly ammonia; see note 7 in Klein et al. (1993 b), 407.

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Let us now see an outline of Einstein’s paper on the law of the photochemical equivalent. He considers three “chemically different” gases, at thermodynamic equilibrium at absolute temperature T, enclosed in a volume V, where n1 , n2 and n3 are the respective numbers of moles. He admits that under the influence of thermal radiation of frequency ν0 , a molecule of type 1 can decompose into a molecule of type 2 and another of type 3 (decomposition reaction). The process is complemented by the corresponding recombination reaction and thermodynamic equilibrium requires that the respective reaction rates are equal. 1 → 2 + 3 (Decomposition) 2 + 3 → 1 (Recombination) Einstein starts from three assumptions about the mechanism of these reactions: I. The decomposition of a molecule of the first type takes place as if the other molecules were not present, which implies that the rate of decomposition only depends on the number n1 —keeping the other circumstances fixed—and not on the densities of the three gases. II. The probability that, in a unit of time, a molecule of type 1 decomposes under the action of the present radiation is proportional to the radiation energy density ρ . III. The recombination reaction, in which radiation has to be emitted, obeys the law of mass action: the number of molecules of the first type produced, per unit volume and time, is proportional to the product of the concentrations of the other two molecular species. The corresponding proportionality factor will depend on the temperature, but not on the energy density of the radiation. Considering hypotheses I and II, the number Z of molecules of type 1 that decompose per unit of time will be given by: Z = A ρ n1 ,

(2.50)

where the coefficient A depends on the temperature of the gas mixture. The number Z’ of molecules of type 1 originating per unit of time, due to the corresponding recombination, will be given, considering hypothesis III, by: Z ' = A' V

n2 n3 V V

(2.51)

To maintain the equilibrium gas-radiation, Z = Z’, which leads to: η2 · η3 A = ' ρ, η1 A where ηi represents the respective molecular concentrations.

(2.52)

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137

The thermodynamic part of the work consists of imposing the maintenance of thermodynamic equilibrium in the face of an isothermal change at a constant volume of the gas-radiation system. For a detailed calculation, as well as for the analysis of some delicate aspects, we refer to a previous work, in which we explain how it is concluded, without leaving the strictly thermodynamic framework, that ρ must be of the following form: ρ=

) ) A' α N" , exp − A RT

(2.53)

where α is a new constant, independent of the equilibrium temperature T, and Nε represents the energy transferred by the radiation to the gas mixture in the decomposition of a molecule gram of gas of type 1. Einstein then writes Wien’s phenomenological law (1.62) with the help of the constants h, k and c (Planck’s, Boltzmann’s and the speed of light in a vacuum, respectively), in the following form: ρ=

) ) hν 8π hν 3 exp − c3 kT

(2.54)

The comparison between the last two expressions shows that Wien’s law can be deduced by strictly thermodynamic reasoning, provided that the following two equalities are satisfied (remember that, as we saw in Sect. 1.3.4, the relation Nk = R is fulfilled): ε =hν ;

A' α 8π hν 3 = A c3

(2.55)

Einstein ends his paper by emphasizing the importance of the previous result [Einstein (1912c). In Beck (1996), 94. Emphasis in the original]: Thus, as the most important consequence we obtain [he refers to the relation ε = h ν], which states that a gas molecule that decomposes under the absorption of radiation of frequency ν0 absorbs (on average) the radiation energy h ν0 in the course of its decomposition. We assumed the simplest kind of reaction, but we could just as well have derived equation ε = h ν for other gas reactions occurring under light absorption in the same way. It is also obvious that this relation can be proved in a similar manner for dilute solutions. The relation may well be generally valid.

The scope of the above conclusion should be properly assessed, because it is not trivial that purely thermodynamic reasoning could lead to the deduction of the discontinuous behaviour of radiant energy in the elementary processes of absorption and emission; especially if one considers that hypotheses I, II and III do not make any reference to a possible discrete behaviour of radiant energy in those processes. Of course, Einstein’s deduction of the photochemical equivalence law and of the discontinuous character of absorption and emission of radiant energy—as well as

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his conclusions of 1905—limits its validity to the frequency range which conforms to the phenomenological Wien’s law. Finally, it should be pointed out that this 1912 work must be framed in Einstein’s attempts to obtain a proof of the physical reality of energy quanta from an experimental fact as was Wien’s law. Hence, also the great interest with which Einstein followed Warburg’s photochemical experiments at that time, which is reflected when he writes in 1912, referring to Warburg’s photochemical laboratory69 : [...] in which all the things I have been able to conceive vaguely in my dreams for years have become reality.

Einstein’s 1912 paper opens a new path that, four years later, would lead Einstein to the authentic deduction of the discrete nature, not of the energy of radiation, but of radiation itself. Or, in other words, it was to lead him to the birth of the photon, in 1916, as we shall see in Sect. 3.3.

2.9

Specific Heat of Gases and Quantum Theory (1911–1913)

Although Nernst’s theorem of 1906, to which we have referred in Sect. 2.3, applied to solids, he always hoped that it was more general and could be equally valid for gases. Since he also thought that incipient quantum theory could contribute to solving some of the different problems related to specific heats, he decided to turn to gases to test his ideas in this respect. In particular, Nernst saw molecular hydrogen as a suitable candidate to begin experimental investigations. At its very low liquefaction temperature—on the order of 20 K—it was necessary to add the fact that the smallness and the proximity of the two masses that make up the molecule in this case, made molecular hydrogen an ideal candidate for treating an ideal candidate for trying to measure small contributions to its specific heat. Arnold Eucken (1884–1950) worked in the laboratory run by Nernst in Berlin. There he had prepared his dissertation, presented in 1906 under the supervision of the director. He had participated intensively in the design and construction of an advanced model of calorimeter that was successfully used in the Berlin laboratory for the measurement of specific heats of solids. We emphasize that, as we have already pointed out, Nernst was convinced, especially after the first Solvay conference, that the new quantum ideas could shed light on the unexpected results obtained on the behaviour of specific heats at low temperatures. These ideas and his authority were enough to convince Eucken, approximately 1910, to

69

Letter from A. Einstein to E. Warburg, 25 April 1912. In Beck (1995), 289.

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turn his experience into a new and promising experimental field: obtaining precise measurements of the specific heat of molecular hydrogen.70

2.9.1

Planck’s “Second Theory” and “Zero-Point Energy” (1911)

At the beginning of 1910, Planck was already fully convinced of the necessity to introduce some discontinuity in the behaviour of the energy of the oscillators—Planck’s resonators—if the experimental behaviour of radiation was to be accounted for theoretically. His statements in a letter sent to Lorentz in early January confirm this71 : The discontinuity must enter somehow; otherwise one is irretrievably bound to the Hamiltonian equations and the Jeans theory [leading to the ultraviolet catastrophe]. Therefore, I have located the discontinuity at the place where it can do the least harm, at the excitation of the oscillator; its decay can then occur continuously with constant damping.

Planck’s subsequent research, in which he tried to incorporate such a discontinuity into classical physics with as little erosion of classical physics as possible, soon became known as “Planck’s second theory”. At the outset, Planck was prepared to admit the discontinuity in the absorption of energy by radiation, but he did not find it necessary to recognize the same property in the case of emission. In search of a mechanism, at least a provisional one, to explain that discontinuity Planck suggested the following additional property: the excitation of a monochromatic oscillator of frequency ν required the energy threshold h ν. It did not take Planck long to dispense with the energy threshold for oscillators and to find reasons—not entirely clear—to change the sense of the asymmetry between emission and absorption, assigning now to the latter a continuous character and making the emission responsible for the unavoidable discontinuity. Thus, in 1911, before the Deutsche Physikalische Gesellschaft [German Physical Society], he had to admit this hypothesis: “the emission of energy by an oscillator occurs by leaps, in accord with [the theory of] energy quanta and with the laws of chance”. [Quoted in in Kuhn (1978), 236]. These assumptions of Planck’s second theory—not only the discontinuity but also the new and surprising role of chance in emission—were a clear departure from classical physics. Despite the conservatism under which it is usual to view Planck’s attitude in the face of the development of quantum theory from his 1900 hypothesis onwards, he was obliged, for reasons of coherence, to revise and retouch certain topics of classical physics. This led him to obtain new results for the quanta and to point out precisions on phenomena that were still looking for

70

In this section we follow a part of our research on the subject, as shown in Navarro and Pérez (2006). 71 Letter from M. Planck to H. Lorentz, 7 January 1910. Excerpt quoted in Kuhn (1978), 274.

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satisfactory theoretical explanations. Among these, radioactivity, and even certain characteristics of the photoelectric effect itself.72 A consequence of the second theory was already stated by Planck himself in his intervention at the first Solvay conference in the autumn of 1911: although the new theory also led to his 1900 radiation law, the mean energy per oscillator of frequency ν, at absolute temperature T, was now given the following expression: Eν =

( exp

hν _

βν T

−1

+

hν , 2

(2.56)

while the usual expression—our (2.3)—was this: Eν =

hν ( _ βν exp T − 1

(2.57)

The last two expressions differ by h ν/2. At first sight, this summand seems to be an innocuous consequence, since with it included, we also arrive at Planck’s law and, on the other hand, it does not alter the theoretical predictions about the values of the specific heats because the derivative of this additional term with respect to the absolute temperature is zero because it is independent of T. Intuitively, the new summand appearing in (2.56) can be understood as follows. Let us admit, in accordance with Planck’s second theory, that a monochromatic oscillator of frequency ν absorbs energy continuously but cannot emit it until it reaches, at least, the value hν. To all effects, as before being able to emit its energy it adopts all the values between 0 and hν, it is as if the minimum energy of the oscillator—that of its fundamental state, we would say today—were hν/2. other words: the oscillator mean energy does not cancel for T = 0 but has a value of hν/2. In this sense, it is stated that expression (2.56) implies a “zero point energy”, contrary to the expression (2.57). This additional term, in spite of the little importance that even Planck himself had assigned to it, soon became an object of preferential attention. Especially since the publication by Eucken of his results on the measurement of the specific heat of hydrogen at low temperature. [Eucken (1912)]. Attempts at a theoretical explanation of these and subsequent results on the subject were to play an important role in the development of quantum physics.73 As did other little-known contributions of Einstein, to which we will now refer.

72

For a more detailed account of Planck’s second theory and its vicissitudes throughout the second decade of the twentieth century, see Kuhn (1978), Chap. 10. 73 For a precise analysis of the role played by the measurements of the specific heat of hydrogen, together with the corresponding theoretical interpretations, see Gearhart (2010).

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2.9.2

141

Einstein and Stern (1913): On the Specific Heat of Hydrogen

In 1912, Eucken published the first experimental results he had obtained on the rotational specific heat—the contribution of the rotational degrees of freedom to it—of molecular hydrogen. These results led to new difficulties in trying to find a theoretical explanation for them. In light of Eucken’s results, Einstein and Stern envisage a simple way of contrasting the existence of zero point energy. The first is to check with which of the last two expressions those results are compatible, since one and the other must lead to different results —when derived with respect to T to calculate the specific heat—if one thinks of a system in which the frequency depends on temperature. Eventually, such a system may be a gas of diatomic molecules with rotational motion. A first problem already arises: how can be related this rotation to the oscillations to which our last two expressions apply? It is well known that the rotational energy of a diatomic molecule is given by: Er =

J (2π ν)2 , 2

(2.58)

where J represents the moment of inertia of the molecule and ν is the frequency of its rotation. Einstein and Stern, essentially for reasons of simplicity, assume that at a given temperature T, this frequency is the same for all molecules. They solve the abovementioned problem by admitting—without much justification— that a rotating diatomic molecule emits, per unit of time, twice the mean energy that would be emitted by a one-dimensional oscillator of the same frequency. It is obvious that they have resorted to the energy equipartition principle, although without expressly mentioning, by associating two degrees of freedom to the rotation against only one to the oscillation. This allows them to equal the previous E r with twice the values given by (2.56) and (2.57). After clearing the temperature in both expressions and using the abbreviation p ≡ 2π 2 J , as in the original paper, one obtains: h −W ith zer o energy point, T = i.e., by using (2.56) : k

) log

ν h p ν− h2

)

(2.59)

+1

ν h −N o zer o energy point, _ ( T = i.e., by using (2.57) : k log h + 1 pν

(2.60)

dE r dν r The rotational specific heat can now be obtained as follows: cr = dE dT = dν · dT , dν r where dE dν is calculated from (2.58) and dT from (2.59) to (2.60), as appropriate.

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Fig. 2.3 Einstein and Stern show in their 1913 paper the results deduced for the rotational specific heat of molecular hydrogen at different temperatures, depending on whether or not the existence of the zero point energy is assumed. The experimental data of Eucken are represented by crosses. [In abscissae, absolute temperatures. In ordinates, rotational specific heats]

Figure 2.3 shows a comparison between different theoretical results obtained in this way and the experimental data published by Eucken. Curve I in Fig. 2.3 is obtained assuming that there exists the zero point of energy—i.e., using the expression (2.59)—and it shows a quite acceptable fit with Eucken’s experimental results. This is not the case for curve II, which is arrived at without assuming the existence of zero point energy—i.e., from (2.60)—and represents an option in clear disadvantage with the remaining ones. Curve III is obtained assuming that the frequency is independent of temperature, which makes it irrelevant whether or not a zero point of energy is admitted. In curve IV, as in curve I, the existence of zero point energy is assumed, but now with energy hν, instead of hν/2. Although curves III and IV also acceptably fit the experimental data for low temperatures, there is no doubt that curve I is globally favoured in the comparison. The first conclusion of Einstein and Stern is clear: the measurements by Eucken of the rotational specific heat of molecular hydrogen suggest the existence of a zero point energy of value hν/2, where the rotation frequency of the molecule depends on the temperature according to (2.59). They do not stop there, but in the second part of their article they present an original analysis of the equilibrium state between a radiation field and a system of resonators to arrive at a new deduction of Planck’s law, showing that the introduction of zero point energy makes it unnecessary to resort to the quantisation of the energy of resonators. Let us see.

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143

In Sect. 2.4.3, we have commented on the paper of Einstein and Hopf in 1910, in which they presented an original formalism based on the introduction of a stochastic field to study the influence of a radiation field on a resonator associated with each molecule of a gas, without assuming energy quantisation for the resonators. This led them inexorably to Rayleigh-Jeans’ law, which in turn implies the catastrophe in the ultraviolet. In 1913, Einstein and Stern took up that same formalism, also without introducing any quantisation, but assuming the existence of a zero point energy. The great novelty is that, as a final result, one now obtains the Planck’s radiation law, already definitively confirmed by experiments, although now they are forced to adopt the value hν, instead of hν/2, as zero point energy. In a footnote they point out that clarifying this discrepancy between the value of the zero point energy for hydrogen (hν/2) and for radiation (hν) requires future research with “more rigorous” calculations. And they end their paper by stating the following pair of conclusions [Einstein; Stern (1913). In Beck (1996), 145): 1. Eucken’s results on the specific heat of [molecular] hydrogen make probable the existence of a zero point of energy equal to hν/2. 2. The assumption of zero energy point opens a way for deriving Planck’s radiation formula without recourse to any kind of discontinuities. Nevertheless, it seems doubtful that the other difficulties can also be overcome without the assumption of quanta. This second conclusion clashed head-on with the ideas of Ehrenfest, among others, who considered to have demonstrated in 1911—as seen in Sect. 2.7.1—the necessity of admitting the quantum hypothesis, if the experimental behaviour of thermal radiation was taken into account. Hence, he immediately became involved in the subject of the specific heat of molecular hydrogen, trying to clarify the points of discrepancy between his ideas on quanta and this last conclusion of Einstein and Stern.

2.9.3

Ehrenfest’s Counterproposal (1913): Rotational Quantisation

In May, only two months after the appearance of Einstein and Stern’s article, Ehrenfest sends to publish a paper explaining his position on the matter, in which he reaffirms his belief in the necessity of quantisation. After referring to Eucken’s data and to the article of those with the introduction of the zero point of energy, Ehrenfest proposes an alternative based on a rigorous statistical treatment—which, in his opinion, was lacking in Einstein-Stern paper—and on a proper energy quantisation of the rotational motion. Ehrenfest specifies his starting point by means of two hypotheses concerning this quantisation [Ehrenfest (1913), 335]: 1. In the rotations around an axis, the only frequencies allowed are those that lead to a kinetic energy of rotation with value multiples of hν/2:

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εn =

L hν (2π ν)2 = n ; (n = 0, 1, 2, . . .), 2 2

(2.61)

where L—keeping the original notation—represents the moment of inertia. If q is the angle of rotation, the corresponding angular momentum will be the following: p = L q˙ = L(2π ν)

(2.62)

The last two expressions imply that, in the phasic plane (q, p) of a diatomic molecule—such as in the case of hydrogen—, considered a dipole with rotational motion around an axis, the only allowed regions are: (a) the point q = p = 0 and, in addition, h h h (b) the segment pairs p = ± 2π , ±2 2π , ±3 2π ,... 2. In relation to the statistical treatment, to all the points in each of the previous regions will be assigned the same probability; that is, they will be treated as equiprobable. In a footnote Ehrenfest states that, in writing equality (2.61), he has followed a suggestion of the young Danish physicist Niels H. D. Bohr (1885–1962) made at the first Solvay conference, about the quantisation of rotational energy.74 However, Ehrenfest has now used the quantum hν/2—instead of hν, which was the original proposal—because he considers it more adequate from a wide perspective, which he does not detail. However, the value of the quantum, Ehrenfest continues, is not relevant since it only influences the value of the moment of inertia of the molecule, which will always have to be determined from the experimental results. To calculate the kinetic energy of the rotation E R of a system of N molecules at thermal equilibrium at the absolute temperature T, Ehrenfest resorts, as usual in statistical mechanics, to “the most probable distribution”: ( εn ) n=0 εn exp − kT ( εn ) , ∑n=∞ n=0 exp − kT

∑n=∞ ER = N

(2.63)

where the rotational energy εn of a molecule is given, according to (2.61), by: εn = n2

74

h2 ; (n = 0, 1, 2, . . .) 8π 2 L

(2.64)

Niels Bohr would be awarded the Nobel Prize in Physics 1922 “for his services in the investigation of the structure of atoms and of the radiation emanating from them”.

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145

Substituting (2.64) in (2.63) and deriving E R with respect to the absolute temperature, Ehrenfest obtains the following expression for the rotational specific heat of a system composed of N diatomic molecules in thermal equilibrium: CR = 2Nkσ 2

d2 log Q(σ ) , dσ 2

(2.65)

where he has previously entered the following abbreviations: 2

σ ≡ 8π 2hLkT Q(σ ) ≡ 1 + exp(−σ ) + exp (−4σ ) + exp(−9σ ) + . . . + exp(−n2 σ ) + . . . (2.66) Ehrenfest determines the value of L from Eucken’s experiments and comes to the conclusion that his theoretical predictions obtained from (2.65) do not detract from those of Einstein and Stern—with the zero energy point hypothesis—when compared with experimental results. Therefore, he does not see sufficient reason to adopt Einstein and Stern’s point of view. What follows, according to Ehrenfest, is to quantize the rotational motion in a reasoned manner and rigorously apply statistical methods, which Einstein and Stern had not done. Perhaps it is worth insisting that Einstein and Stern assign a unique frequency varying with temperature according to (2.59) for all molecules of the gas. In contrast, Ehrenfest assumes quantisation of frequency according to the following rule: ν=n

h ; (n = 0, 1, 2, . . .), 4π 2 L

(2.67)

which is directly deduced from (2.61). This quantisation—something along the lines of Bohr’s atomic orbits, which Ehrenfest did not know—, added to the notion of the “most probable distribution” was the essence of his treatment.75 This work of Ehrenfest, as well as his correspondence with Einstein, must have contributed significantly to Einstein’s early abandonment of any future fickleness about the zero point energy. Einstein abandoned this idea completely very early, even in 1913, because it was not rigorously justified, nor was it excessively fertile. His formal renunciation was made at the second Solvay conference, also held in Brussels in the autumn of the same year, under the title “The structure of matter” [Goldschmidt et al. (1921), 108]: I should also point out in this respect that the arguments which I put forward, together with Mr. Stern, in favour of the existence of an energy at absolute zero, I do not consider anymore as valid. By developing further the considerations that we have made regarding the deduction of Planck’s radiation law, I have found indeed that this procedure, based on the zero point energy hypothesis leads to contradictions.

75

For more details about Ehrenfest’s treatment, as well as the comparison with that of Einstein and Stern, see, for example, Navarro; Pérez (2006), especially 215–223.

3

The First Harvest (1914–1924): In Search of the Photon

The quantum paper I sent out has led me back to the view of the spatially quantumlike nature of radiation energy. But I have the feeling that the actual crux of the problem posed to us by the eternal enigma-giver is not yet understood absolutely. Shall we live to see the redeeming idea? Albert Einstein, 1917

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 L. Navarro Veguillas, The Lesser-Known Albert Einstein, History of Physics, https://doi.org/10.1007/978-3-031-35568-4_3

147

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3 The First Harvest (1914–1924): In Search of the Photon

Diploma corresponding to the Nobel Prize in Physics 1921 —awarded in 1922—, that Einstein received “for his services to Theoretical Physics, and especially for his discovery of the law of the photoelectric effect” [Public Domain]

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3.1

149

Wartime and Peacetime in Berlin

At the end of Sect. 2.1, we had left Einstein in Zurich, but he was already preparing to join the University of Berlin. In April 1914, the family moved to the German capital, but in August of the same year, Mileva returned to Zurich with the children, never to share her life with Albert again. It is known that the decline of the marital relationship had begun earnestly in Zurich in 1909. From Prague, in April 1912, Einstein maintains epistolary relations with his divorced cousin Elsa Einstein (1876–1936)—who was living in Berlin with her two daughters Ilse and Margot— with whom he had been on good terms in his youth.1 ,2 Several of these letters from Albert to Elsa—her letters were destroyed at her express wish—show Einstein’s already resounding failure in his marriage, as well as his hope to find in his cousin a confidant and a lover to fill the gap left by Mileva. However, Einstein soon decides, still in Prague, to put an end to the correspondence with these vehement words.3 I am writing so late because I have misgivings about our affair. I have the feeling that it will not be good for the two of us as well as for the others if we form a closer attachment. So, I am writing to you today for the last time and am submitting again to the inevitable, and you must do the same. You know that it is not hardness of heart or lack of feeling that makes me talk like this, because you know that, like you, I bear my cross without hope. But preserve your kindly disposition toward me. I too will always be grateful for your warmth. If you ever have a hard time or otherwise feel the need to confide in somebody, then remember that you have a cousin who will feel for you no matter what the issue might be. Therefore, as soon as I am installed in Zurich, I will let you know the address at which you can write to me.

The epistolary relationship is re-established within a year, with the Einsteins already back in Zurich. When Albert accepts the offer from Berlin, he informs Elsa Einstein and is pleased to be able to live near her and thus be in a position to maintain a more personal relationship. However, Mari´c is not in favour of moving to Berlin. Not only because she is not attracted to the German character and way of life but also because Elsa lives there. Mileva, aware of the growing affection between Albert and her cousin, thinks that life in Berlin will increase her justified jealousy, which can only have a negative effect on family peace. Nevertheless, she travels with the family to the German capital, perhaps thinking of a last chance to save the marriage. Newly settled in Berlin, the intimacies between Einstein and his cousin reached such a point that they ended up triggering the already foreseeable break-up between Albert and Mileva. Since then, he has been pursuing a divorce and to make official his union with Elsa. Mileva did not seem willing to facilitate the

1

Letter from A. Einstein to M. Besso, 9 March 1917. In Hentschel (1998), 293. As single woman Elsa Einstein and when she married to her first husband Elsa Löwenthal. 3 Letter from A. Einstein to E. Löwenthal, from Prague, 21 May 1912. In Beck (1995), 300. Emphasis in the original. 2

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process. In these circumstances, on 18 July 1914, Albert wrote to her the following list of conditions that she had to commit herself in writing to fulfil to continue living under the same roof as him and to save social appearances.4 Conditions. A. You make sure (1) That my clothes and laundry are kept in good order and repair (2) That I receive my three meals regularly in my room. (3) That my bedroom and office are always kept neat, in particular, that the desk is available to me alone. B. You renounce all personal relations with me as far as maintaining them is not absolutely required for social reasons. Specifically, you do without (1) My sitting at home with you (2) My going out or traveling together with you. C. In your relations with me you commit yourself explicitly to adhering to the following points: (1) You are neither to expect intimacy from me nor to reproach me in any way. (2) You must desist immediately from addressing me if I request it. (3) You must leave my bedroom or office immediately without protest if I so request. D. You commit yourself not to disparage me either in word or in deed in front of my children. Mileva neither accepts nor signs these conditions. At the end of July 1914 —the First World War would begin on 1 August —she returned with the two children to Zurich, the city she considered ideal not only for living but also for educating her children. After long and very tense epistolary discussions, with the intervention of third parties —especially Dr. Zangger to whom we have referred in Sect. 2.6.3— the couple reached an agreement on the divorce, which was to take place in February 1919. Albert and Elsa, who had been living together since he was left alone in Berlin, get married in June of the same year. According to the divorce agreement, Einstein will not have to pay any monthly pension, but he will give Mari´c the full amount of the Nobel Prize, which both of them, like many scholars in those days, consider imminent. In fact, at the end of 1922 he was awarded the Nobel Prize in Physics 1921 “for his services to Theoretical Physics, and especially for his discovery of the law of the photoelectric effect”. Incidentally, it shows that relativity, despite the dissemination and fame that the theory soon acquired, had not yet achieved sufficient direct experimental confirmation to be considered a well-established theory.5

4

These conditions, made public in 1996, are recorded in the memorandum sent by Albert to Mileva in mid-July 1914. In Hentschel (1998), 32–33. Emphasis in the original. 5 For a detailed account of the vicissitudes through which Einstein’s candidacy had to pass until he received the Nobel Prize, see Pais (2005), Chap. 30.

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151

The beginning of the war is the occasion for Einstein to show in public, for the first time, some of his political ideas. The Germans soon invade Belgium in violation of the neutrality of this country. The beginning of the armed conflict has an extraordinary impact on all environments, including scientific environments. To mitigate the negative effects of the invasion and the excesses of German militarism, a group of German intellectuals, principally scientists and artists, wrote a manifesto addressed “to the civilized world”, in which they reject any German responsibility for the outbreak of war and the invasion of Belgium, both of which are presented as inevitable to safeguard Western culture; Germanic militarism is thus seen as inseparable from German culture.6 A manifesto in reply to the previous one was written by Georg Friedrich Nicolai (1874–1964), a German physician and professor of physiology at the University of Berlin, who would emigrate to South America in 1922.7 Einstein and the astronomer Wilhelm Foerster (1832–1921) collaborated with the author. After denouncing the danger of a war of annihilation such as the one foreseen, with the consequent irreparable damage to European cooperation and culture, the manifesto called for the union of “good Europeans” to avoid “perish from fratricidal war” because “the time has come where Europe must act as one in order to protect her soil, her inhabitants, and her culture”. The appeal ended thus [In Engel (1997), 29. Emphasis in the original]: But it is necessary that the Europeans first come together, and if-as we hope-enough Europeans in Europe can be found, that it is to say, people to whom Europe is not merely a geographical concept, but rather, a dear affair of the heart, then we shall try to call together such a union of Europeans. Thereupon, such a union shall speak and decide. To this end we only want to urge and appeal; and if you feel as we do, if you are likemindedly determined to provide the European will the farthest-reaching possible resonance, then we ask you to please send your (supporting) signature to us.

The authors of the countermanifesto were disappointed to receive only one explicit endorsement, that of Otto Buek (1873–1966), a philosopher at the University of Marburg, and so they did decide not to turn their appeal into a published manifesto, so its circulation was very limited. Three years later, however, the manifesto would appear in print, when Nicolai included it in the foreword of a book about certain relations between biology and war [Nicolai (1918)]. Einstein’s mental strength soon becomes evident, as he is able to give an enormous impulse to his research, in spite of the unfavourable conditions that surround him: marital breakdown and the First World War. During the first years of his stay

6

The manifesto “To the civilized world” —An die Kulturwelt— is dated 4 October 1914 and contains the signatures of 93 highly representative scientists of the time; hence it is sometimes referred to as the “Manifesto of the 93”. Among the signatories are M. Planck, W. Roentgen, F. Haber, W, Ostwald, W. Wien and W. Nernst. 7 The “Manifesto to Europeans” —Aufruf an die Europäer— appears in mid-October 1914. In Engel (1997), 28–29.

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in Berlin, he devoted his greatest efforts to the systematization of his relativistic and cosmological ideas. His interest in these topics had intensified in 1912, after his return to Zurich from Prague. His theory of gravitation —i.e., his general theory of relativity—was established in November 1915, when his derivation of the equations of the gravitational field was published. They provided a full explanation of the observed anomalies in the motion of the perihelion of Mercury. We will not address these titanic achievements of Einstein here precisely because they have already been widely publicized. Einstein himself contributed to their early and widespread dissemination with a book that soon became a classic of relativity.8 Einstein seems to have been convinced from the beginning of the disastrous results of the war, although he did not foresee the magnitude of its consequences. The German pretension of greatness and domination led to death, physical suffering, and the loss of dignity of millions of human beings. The conflict would cease four years later, when the Rethondes Armistice was signed on 11 November 1918 between the Allies, on the one hand, and the Austro-Hungarian Empire, Germany and Turkey on the other. Two days earlier, Emperor Wilhelm II had abdicated, which led to the birth of Germany’s first parliamentary democracy—the Weimar Republic—abolished by Nazism in 1933. It is highly surprising Einstein’s ability to abstract himself, to a large extent, from such unfavourable circumstances for scientific creation as those associated with general anguish and uneasiness in the face of such a cruel war. Perhaps it was his prejudice about the development of the last phase of the war that had a decisive influence on his apparent lack of concern for its final results. The fact is that the end of the war coincides with the beginning of the consecration of the “Einstein myth”, which is still firmly established among us today. The first steps towards its universal fame appear after being made public in a joint session of the Royal Society and the Royal Astronomical Society, on 6 November 1919, the results obtained by two expeditions proposed by Frank Watson Dyson (1868–1939)—to Sobral, in Brazil and to Principe Island, in Portuguese Africa, the latter led by Arthur Stanley Eddington (1882–1944)—to make certain measurements during the total solar eclipse of the previous May 29th. Those results are interpreted and disseminated, especially by Eddington, as the experimental confirmation of Einstein’s theoretical predictions, from his general theory of relativity, about the curvature of light rays coming from a distant star, due to the effect of solar gravity. Although today it is clear that the collection, selection and interpretation of the data obtained were far from the required rigor, its publication served to catapult Einstein to fame as a public figure. Indeed, the day after the joint session, The Times, the influential London newspaper carried these headlines:

8

Einstein (2001). For more details on the book see Editorial note: Einstein’s popular book on relativity, Kox et al. (1996), 417–419.

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Revolution in Science New Theory of the Universe Newtonian Ideas Overthrown In the same newspaper appears on November 28th an article, requested by the newspaper from Einstein himself, in which he presents to the readers a plain disclosure of the theory of relativity. At the end of the article the physicist includes an ironic—and prophetic—consideration about the image that The Times had offered of him, days before.9 Note: Some of the statements in your paper concerning my life and person owe their origin to the lively imagination of the writer. Here is yet another application of the principle of relativity for the delectation of the reader: today I am described in Germany as a “German savant”, and in England as a “Swiss Jew”. Should it ever be my fate to be represented as a bête noire, I should, on the contrary, become a “Swiss Jew” for the Germans and a “German savant” for the English.

The dissemination of the theory and its implications—the drastic modification of classical spatial–temporal notions, among the most striking—was to reach unsuspected limits. Some of these ideas did not limit their influence on the scientific field, but invaded other fields such as philosophy, literature, painting and even advertising!10 Einstein accepts invitations to visit many countries and travels to the United States, Europe and Japan. While travelling precisely to Japan, in November 1922, he receives the news of the awarding of the Nobel Prize in Physics 1921. From Japan, he travels to Palestine and, returning to Germany, he stops in Spain, being awarded honoris causa doctorate from the Universidad Central de Madrid.11 It was around this time that he began to broaden the scope of his social relations in a notable way, and at the same time he made contact—either personally or by mail—with a large number of intellectuals outside scientific practice and with artists of different tendencies, an attitude that would never cease.12 Einstein was soon overcome by a deep sense of anxiety, which would never leave him completely, in the face of such sudden and unexpected fame. This can be clearly perceived, for example, when he writes to his former collaborator Hopf in early 1920 in these terms.13 Saying no truly never was my strength. But in the predicament in which I find myself now, I am slowly learning. Since the influx of newspaper articles, I am being so terribly deluged

9

The full article is reproduced in Einstein (1954), 227–232. For a more detailed study of the scope of the process that led to the influence of Einstein’s ideas in the most varied fields, see, for instance, Friedmann; Donley (1985). 11 For details on Einstein’s visit to Spain in February–March 1923, see Glick (1998). 12 Just a sample: B. Russell, B. Shaw, H. Reichenbach, C. Weizmann —who was to become the first president of Israel in February 1949—, R. Tagore, M. Gandhi, C. Chaplin, S. Freud, among others. 13 Letter from A. Einstein to L. Hopf, 2 February 1920. In Hentschell (2004), 247. 10

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with inquiries, invitations, and requests that at night I dream I am burning in hell and the postman is the devil and is continually screaming at me, hurling a fresh bundle of letters at my head because I still haven’t answered the old ones. Added to that, I have my fatally ill mother here at home, as a result of the “momentous times” must attend countless meetings, etc. In short, I am nothing but a bunch of pitiful reflex motions. So, mercy and pity, that’s all I am asking. I am not lecturing at Zurich anymore, partly because I cannot leave here, partly because physics there is so superbly represented that my schoolteaching has become absolutely superfluous over there.

Einstein’s mother, who lived in Einstein’s home in Berlin, dies in 1920. Around this time, Einstein reacts publicly to the growing hostility towards him and his theory of relativity; an attitude instigated by anti-Semites, including the wellknown experimental physicist, Lenard. A statement by Einstein in the summer of 1921 shows his feelings on the subject, as well as his decision to join the Zionist movement [quoted in Seelig (1960), 285–286]. Until recently I lived in Switzerland, and while I was there, I didn’t realize that I was Jewish, nor was there anything in my country that would arouse my Jewish feelings or make them react. But things changed as soon as I moved to Berlin. It was here that I saw the misery of many young Jews. I saw how the anti-Semitic atmosphere made it impossible for them to pursue their studies in an orderly manner and to provide themselves with a secure existence. Above all, the Jews from the East, who were the most harassed and bullied. These and other experiences have awakened in me the Jewish national feeling. I am a national Jew in the sense that I claim the position of Jewish nationality, or any other nationality, as a fact in itself. I regard Jewish nationality as a fact, and my opinion is that every Jew must draw for himself the consequences of that fact. The elevation of Jewish selfconsciousness is, in my opinion, also convenient for the normal development of coexistence with non-Jews. This was the main reason that made me join the Zionist movement.

Regarding Einstein’s research in Berlin between 1914 and 1924, we will begin by analysing an episode related to experimental aspects. It is about his collaboration with the Dutch physicist Wander Johannes de Haas (1878–1960), in search of the experimental demonstration of the existence of the “Ampère’s molecular currents”. Nor has his unceasing concern for the nature and behaviour of electromagnetic radiation diminished over the years. One of his most decisive steps in this direction was taken in 1916, when he introduced the concepts—not the terminology—“photon” and “transition probabilities”. Moreover, in 1917, he will propose a new quantisation rule for mechanical systems, with certain advantages over the existing ones, which was to exert a notable influence on De Broglie when he introduced the “wave-particle duality” in 1924. We will end this chapter by exposing the impact, neither broad nor favourable, of Einsteinian ideas in relation to the nature and behaviour of electromagnetic radiation, particularly its corpuscular aspect —i.e., the existence of the photon—, which did not succeed in overcoming the prejudices of the time. In this connection we shall devote special attention to the “BKS proposal”, which appeared in 1924, as a striking attempt to eliminate any corpuscular vestige of radiation, its

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authors being ready to pay for it such a high price as the renunciation of the classical causality and the principles of conservation of energy and momentum. The experimental verdict would not be long in coming.

3.2

Einstein, De Haas and Ampère’s Molecular Currents (1915–1916)

Ampère was one of the most seriously affected when he learned about —and verified in his laboratory— the result of the experiment that the Danish physicist Hans Christian Oersted (1777–1851) had just carried out in Copenhagen in 1820. The experiment showed that electricity was capable of exerting mechanical action on a compass. This suggested that electricity and magnetism were related. Its clarification was the basic objective of a new discipline: “electromagnetism”. For a pro Newtonian—as was the case with Ampère—the electric fluid could only interact with the electric fluid and the magnetic fluid with the magnetic fluid. According to such a premise, the result of Oersted’s experiment could be trivially interpreted by admitting that electricity is only a form of magnetism or that magnetism is only a form of electricity. If the latter were the case, as Ampère intuited, two wires carried by two electric currents should interact, just as two magnets do. He proved this experimentally: two parallel conducting wires attract each other when they are crossed by electric currents in the same direction and repel each other when the currents have opposite directions. His conclusion was twofold. On the one hand, magnetism was nothing more than electricity in motion, so he was impelled to create the basis of another new discipline: “electrodynamics”, which was to study the interactions between electric currents. On the other hand, and as a corollary of the above, the French physicist thought that the effects of magnets could be reproduced with properly designed electrical currents. He himself confirmed this experimentally, thus taking the first step towards the introduction of “electromagnets” into experimental physics. Following his intuition, Ampère suggested, without rigorous justification, that the properties of a bar magnet could well be caused by hypothetical circular electric currents perpendicular to the axis of the bar. It was his young colleague Augustin Jean Fresnel (1788–1827)—one of the main promoters and systematisers of the wave theory of light—who made him see the inappropriateness of thinking in coaxial currents because there is not the slightest detectable trace of them (heat release, electrolysis produced by magnets, etc.). Instead, Fresnel suggested to his colleague that, given the scarce information then available on the nature of molecules, it might be more appropriate to think of molecular electric currents rotating randomly around the centre of each molecule. However, in a magnetized iron bar, for example, all its molecular currents would be oriented in the same direction. Soon began a race for the experimental detection of Ampère’s molecular currents, Maxwell himself being one of the first participants in the search. The experiments designed for that purpose obeyed the following scheme. An electric

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current was supposed to consist of electric particles in motion. Whatever the nature of these particles, this implied the existence of masses in rotational motion around the centre of the molecules; or what is the same, the existence of angular moments associated with the molecules because of the electric currents that surround them. Therefore, a variation in the magnetization—a rapid reversal of magnetization, for example—would imply a variation in the angular momentum of the bar. One could also think of the opposite effect: a demagnetized iron bar, if subjected to a rotation around its axis, would tend to orient the movement of the electric particles—which could be understood as elementary gyroscopes—in planes perpendicular to the axis, which would lead to bar magnetization. Several people soon tried to explain terrestrial magnetism in similar terms.

3.2.1

Einstein and De Haas Experiment

Einstein became interested in the subject of Ampère’s molecular currents as result of a report he had been asked to prepare as an expert in order to settle a dispute between two industrial companies concerning the design of a gyrocompass.14 He soon began a thorough research on the topic, as interesting as overlooked by physicists and historians. His inquiries in the field disprove the belief in Einstein’s exclusive interest in theoretical physics and, at the same time, they show the possible relationship between the above mentioned currents and other serious problems of the time, such as, for example, the zero point energy—which we have dealt with in Sect. 2.9.1—or the “spinning electrons”, which would become of interest later on, when the concept of “electron spin” emerged. Einstein’s ideas about molecular currents appeared, to a greater or lesser extent, in different publications. For us, the most interesting one—for its conciseness, clarity, and completeness—is the one signed in 1915, together with De Haas. It is entitled “Experimental proof of the existence of the molecular currents of Ampère” and it is the transcript of a report presented under the auspices of Lorentz at a session of the Amsterdam Academy of Sciences.15 The authors begin by supposing that because of the Lorentz electron theory of 1892, the Ampère currents had to originate from electrons in circular motion. Given the permanence of a magnetic state, the electrons, strange as it seemed at the time, had to conserve their energy, which should not be affected by friction or by the emission of radiation predicted by Maxwell’s electromagnetism. It thus seemed of the greatest interest to experimentally verify the existence of these currents. If they were confirmed, the next step would be to bring their existence into agreement with electromagnetic theory. We wish to emphasize our surprise that the

14

Recall that a gyrocompass maintains the geographic north–south direction, by virtue of the gyroscopic effect. The report was submitted on 5 February 1915 and is reproduced in Kox et al. (1996), 137–144. Its translation has not been included in the English version of that volume. 15 Einstein; De Haas (1915). The session took place on 23 April 1915. De Haas was married to Lorentz’s eldest daughter.

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article makes no reference to Bohr’s atomic model that appeared two years earlier, which, although for other reasons, postulates the conservation of the energy of the electrons revolving around the nucleus. On the other hand, write the authors, the Curie-Langevin law requires that the magnetic moment of a molecule be independent of temperature. Thus, this momentum would exist even at absolute zero, so that the energy of the revolving electrons could be interpreted as a very natural zero point energy. However, Einstein and De Haas explicitly state that “in the opinion of many physicists however, the existence of an energy of this kind is very improbable”.16 The experiment proposed by Einstein and De Haas to detect the Ampère currents consists of checking the required relationship between the angular momentum of the supposedly rotating electrons —remember that these are the ones that originate those currents—and their magnetic momentum. The theory predicts that these are two vector quantities with the same direction but in opposite senses (the electron has a negative electric charge) and proportional moduli. The corresponding theoretical explanation is very simple. Let us see the basis of their reasoning. Suppose an electron of mass m moves with uniform speed v along a circle of radius r, making n turns per second. The modulus of its angular momentum is given by: Mang = mvr = m(2πr n)r = 2π mr 2 n

(3.1)

On the other hand, Ampère’s electrodynamics allows to associate to this electron a magnetic moment whose modulus is given by: M mag = i πr 2 = π er 2 n,

(3.2)

where i represents the value of the intensity of the electric current originating from the movement of the electron and e designates its electric charge in absolute value. Since there are many rotating electrons in a body, the respective moments of the material are obtained by summing up the contributions of all electrons. If → μ its we denote by L the modulus of the angular momentum of the body and by − magnetization, the last two expressions lead to the abovementioned proportionality between the respective moduli: I ∑ −−→ I I I→I m II ∑ −−−→ II m II − I I → Mmag I = 2 μ I = 1 .13 · 10−7 I − μ I, (3.3) L≡I Mang I = 2 I e e where the numerical value of the proportionality factor has been deduced—as stated in the paper—from the then admitted value for the specific charge of the electron e/m, in the case of “negative electrons”.17

16

Ibid., 173. At the end of Sect. 2.9.3, we commented that Einstein was already of this opinion in 1913. 17 Ibid., 175. The currently accepted numerical, to two decimal digits, is 1.76·107 emu/g (electromagnetic units per gram).

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In the absence of external forces that give rise to rotations, the conservation of angular momentum requires that any change in the magnetization of the body is compensated by the corresponding transfer of angular momentum from the electrons to the body. In other words, any change in the magnetization of a body makes it rotate because it creates a torque with the following momentum: I I I − I I d ∑ −−→I I d→ μ II Θ = II Mang II = 1.13 · 10−7 II dt dt I

(3.4)

Thus, the experimental problem consisted of devising a setup that would make it possible to measure with rigour and precision the value of the torque that would cause the body to rotate due to a change in its magnetization. The body chosen was a magnetic rod suspended by a coaxial wire and susceptible to being introduced into a solenoid. When an electric current was passed through the solenoid, the rod rotated as a consequence of having changed its magnetization. A mirror conveniently attached to the wire allowed to evaluate the corresponding rotation measuring the angle of reflection of a ray of light incident on the mirror. The authors include in the article a detailed description of the experimental setup and the results obtained. After repeated measurements of the angle of rotation for different intensities of the current passing through the solenoid, Einstein and De Haas deduce the corresponding value for the proportionality coefficient given in our last expression: 1.11·10–7 , which “agrees very well with the theoretical one 1.13·10–7 ”. Although they “cannot assign to their measurements a greater precision than of 10%”, the authors consider that they have demonstrated the existence of Ampère’s currents. They end the article by thanking PTR —institution where the experiment was carried out and to which we have already referred in Sect. 1.5.1—“for the apparatus kindly placed at our disposition” [Einstein; De Haas (1915), 188]. Both Einstein and De Haas, aware of the limitations in the precision of an experiment such as the one they had devised, continued thinking about and publishing on possible improvements to the original design. In 1916, for example, Einstein alone signed an article entitled “A simple experiment for the demonstration of Ampère’s currents” in which he tried to improve the initial design, especially with a view to a better control of the disturbances originated by the magnetic field, and tried to achieve a more precise maintenance of the necessary verticality of the magnetic rod [Einstein (1916a)].

3.2.2 “Gyromagnetic Anomaly” and “Spinning Electrons” Einstein’s enthusiasm for the experimental search of the Ampère molecular currents is surprising, especially at a time when he was deeply involved in his crucial

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research on general relativity. His passion for the topic is clearly expressed in a letter addressed to Besso at the beginning of 191518 : As regards science, I have two fine matters to report to you: (1) Gravitation. Redshift of spectral lines. (Spectroscopic) binary stars have the same mean velocity within the radius of vision. The mass of the stars results from the Doppler periodic line fluctuation. The component of larger mass against the component of smaller mass ought to exhibit a mean redshift in the spectr. Lines [sic]. This is confirmed, since the stars’ radii can also be estimated (apparently from the (light intensity and) spectral type); thus there is even an approximate quantitative test of the theory, with satisfactory results. (2) Experimental confirmation of Ampère’s molecular current hypothesis. If paramagnetic molecules are spinning-top electrons [Elektronenkreisel], then each magnetic moment I corresponds to mechanical angular momentum M in the same orientation of the magnitude M=1.13·10-7 I. As I changes, a rotation moment appears (− ddtM ). When the magnetization of a small hanging rod is inverted, it experiences an axial rotation moment, the existence of which I, together with Mr. de Haas (Lorentz’s son-in-law), have proven with experiments performed at the Reich Institute. The experiment will be coming to an end soon. With it the existence of zero point energy has also been proven in a single instance. A wonderful experiment; what a pity that you can’t see it. And how traitorous nature is, when you want to deal with it experimentally! Experimenting is becoming a passion for me even in my old age [nearly thirty-six years old]. It is convenient to point out the radical difference between the points of view that Einstein expresses, regarding the possible existence of a zero point energy, depending on whether it is a matter of confidences between equals or of a formal publication. To the first type belongs the previous statement to Besso about the pretended demonstration of the existence of the zero point of energy when, almost a year and a half ago, at the second Solvay conference, he had already opted for the unfeasibility of such a concept, as we have emphasized at the end of Sect. 2.9.3. There were several physicists who continued to carry out experiments along the lines devised by Einstein and De Haas. However, far from confirming the experimental results obtained by them, the discrepancies became increasingly pronounced, to the point that, by 1919, it was admitted that the experimental value for the coefficient of proportionality between angular momentum and magnetization was 0.57·10–7 , with the units we have been using. This value is approximately half the result obtained by Einstein and De Haas which, in turn, seemed to fit acceptably well with the theoretical prediction. The question immediately arises: how could Einstein and De Haas have erred so seriously in their experimental investigations on the trail of Ampère’s molecular currents?

18

Letter from A. Einstein to M. Besso, 12 February 1915. In Hentschel (1998), 68–69.

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This discrepancy between theoretical predictions and experimental results was soon labelled “gyromagnetic anomaly”. To clarify it, two options had to be discerned: to find a possible flaw in the theoretical calculation or to reveal the source of the error in the experimental results or in its interpretation. Let us see, albeit briefly, how the dilemma was resolved after the introduction of spin in physics ten years later, in 1925. For the sake of convenience, we shall use current notation in writing expressions analogous to (3.3) for the case of an electron: e − e − → − → − → L; → μ S = −g S S μ L = −g L 2m 2m

(3.5)

− → − → → μ ( L represents the orbital angular momentum, S represents the spin, and − L − → and μS are the corresponding magnetic momenta, each with its respective “gfactor”). The minus sign indicates that the magnetic momenta have opposite sense to the angular momentum and spin, respectively. Today it is known that the values of the factor g for the electron are: gL =1 and gS =2 As a result of experiments that improved those of Einstein and De Haas, it was obtained for the specific charge e/m of the electron a value twice that predicted. Alternatively, in more modern terms, those experiments led to a value of 2 for the factor g-factor, twice the value obtained by Einstein and De Haas. The gyromagnetic anomaly can then be explained as follows: in the experiments carried out to detect the effect predicted by them, they were certainly working with materials whose atoms had a magnetic momentum determined by the spin, without a sensible contribution of the orbital angular momentum. It does not appear that Einstein and De Haas critically revised their results, in view of the abundant and rigorous experiments that refuted them. It was not until six years later that, at the third Solvay conference—“Atoms and electrons”, held in the spring of 1921—De Haas clarified, without Einstein being present, some important points regarding the original experiment [quoted in Pais (2005), 248]: The numbers which we found [for g) are 1.45 and 1.02. The second value is nearly equal to the classical value [g = 1) so that we thought that experimental errors had made the first value too large […] We did not measure the field of the solenoid; we calculated it […] We did not measure the magnetism of the cylinder, either; we calculated or estimated it. All this is stated in our original memoir. These preliminary results seemed satisfactory to us, and one can easily understand that we were led to consider the value 1.02 as the better one.

Einstein must have taken De Haas’s explanation for granted, for there is no record that he publicly alluded to later. In his scientific autobiography, he does not even refer to the incident, although it is the only important incursion of the myth into the experimental field. This episode illustrates the marked tendency of certain experimental physicists to find what they hope to measure on the basis of preconceived ideas. A tendency that, as we have just seen, completely dragged Einstein and De Haas in their quest for the experimental proof of Ampère’s currents.

3.3 Birth of the Photon (1916–1917)

3.3

161

Birth of the Photon (1916–1917)

However, let us return to Einstein’s concerns about the light quantum, which we had left at the end of Sect. 2.9.3, after dealing there with his article on the law of the photochemical equivalent of 1912, in which he had already dismissed the Planckian resonators as intermediary elements to explain the interaction between matter and radiation. The setting of the scene is now more sober and practical: it is limited to considering a cavity occupied simply by matter and radiation in thermal equilibrium. Inside its walls elementary processes occur, two at the time: decomposition in which radiation is absorbed by matter, and recombination, in which radiation is emitted by matter. He has dispensed, therefore, of the resonators and mirrors, so much used until then. Only four years later, Einstein will take the definitive step on his long road to the deduction of the discrete character of radiation. The Bohr atomic model, introduced in 1913 and already established in physics, clarified the nature of the emission and absorption of radiation by atoms: these could only exist in a discrete set of energetic states, and the transitions between these states gave rise to emission and absorption of radiation by matter. Einstein publishes in 1916—and republishes in 1917—his recent investigations about the nature of radiation in one of his most famous articles [Einstein (1916–1917)]. Now, the matter-radiation interaction is described in terms of three elementary processes, one spontaneous (emission) and two induced by the radiation present (one absorption and one emission).19 However, it seems to us appropriate, before going properly into the issue, to pay some attention to certain methodological questions, which will help us to better understand the evolution of Einstein’s ideas about the light quantum.

3.3.1

New Elementary Processes (1912–1916)

In a previous work, we justified that those decomposition and recombination processes considered by Einstein in 1912 when analysing the photochemical dissociation represent a clear precedent of those of absorption and (spontaneous) emission of radiation by molecules, which Einstein would consider four years later [Bergia; Navarro (1987)]. These two processes, together with a new form of emission—stimulated by the radiation itself—are the three elementary processes that would lead him to establish the corpuscular nature of radiation in 1916, as is schematically shown in Fig. 3.1. In 1916 Einstein begins by showing that if one assumes the existence of only two elementary processes—emission (spontaneous) and absorption (stimulated)— one inevitably arrives at Wien’s law, a serious setback because, in 1916, this law

19

Hereinafter, we will consider the terminology “induced emission” and “stimulated emission” as equivalent.

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1916

1912 3

2

1 DECOMPOSITION

2

ABSORTION

3

1 RECOMBINATION

EMISION (spotaneous)

EMISION (stimulated)

Fig. 3.1 Analogy between the two elementary processes considered by Einstein in 1912 to account for photodissociation and the three processes considered in 1916. Only by starting from these three processes was it possible for Einstein to deduce Planck’s law and, in addition, the existence of the light quantum, which would be baptized as “photon” in 1926 [Graphic: Tomás Navarro]

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had already been discarded in the face of the full experimental confirmation of Planck’s law. In addition, here appears the great novelty: Einstein shows now that the introduction of a third process —a stimulated emission—allows to redirect the situation. However, before jumping from 1912 to 1916, we must mention a publication from 1914, which, although with little later impact, provides an important clue regarding the evolution of Einstein’s thinking about quantum questions. This consideration appears at the beginning of the paper [Einstein (1914). In Engel (1997), 20]. Two considerations are presented here which—in some sense—belong together, as they show how far the most important newer results of the theory of heat, viz. Planck’s radiation formula and Nernst’s theorem, can be derived in a purely thermodynamical manner, utilizing basic ideas of quantum theory but not enlisting the help of the Boltzmann principle.

Einstein writes “using basic ideas of quantum theory”. It can be assumed that he is referring to the exclusive use of quantum concepts—that is, of a discrete nature—without resorting to Maxwell’s electromagnetism, as he will make more clear in his 1916 paper. The “basic ideas” consisted in considering that each molecule of gas was a carrier of a monochromatic oscillator of frequency ν, whose energy could only take values that were natural multiples of hν. Incidentally, the recourse to these monochromatic oscillators of undetermined nature proves that, in 1914, Einstein had not yet managed to completely get rid of the Planckian resonators. However, it is something different that we are interested in emphasizing, for we see that Einstein has here introduced a clear inversion of his usual procedure. It is the first time that he decides to start from a quantum hypothesis because, until then, he had always tried to obtain quantum properties as results; he had never used them as premises. Now, with the new hypothesis and additional thermodynamic reasoning, he obtains Planck’s expression for the mean energy of an aggregate of one-dimensional monochromatic oscillators, which leads directly to Planck’s radiation law. In the same work, the calculation is generalized without difficulty to oscillators with a larger number of degrees of freedom. It is quite possible that this inclusion of quantum hypotheses, replacing others of a more classical but obscure nature, was influenced by the recent appearance of Bohr’s atomic model. However, although this contribution by Einstein is recorded as having been read in an academic session on 24 July 1914, one year after the publication of Bohr’s model, the article makes no allusion to it, nor to his work of 1913. The references to thermodynamic deductions and to the nonuse of Boltzmann’s principle could be taken, at first sight, as an inconsistency with the mechanical-statistical approach under which Einstein had been treating the black body radiation problem, because those principle is the flagship of classical statistical mechanics. However, we shall see that such an inconsistency does not exist if one considers the corresponding context.

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Quantum hypotheses always limited the number of states under which a physical system (first oscillators, then atoms) can exist. In more colloquial language, quantum hypotheses force counting the number of possible states in a different way. This number will now be lower than the one obtained when operating classically, since with any of those hypotheses the number of accessible states is restricted. This being so, and given that Boltzmann principle—of which our expression (1.64) represents a usual statement—relates the number of microstates accessible to the system to its thermodynamic entropy, rigour obliged that, in those times, the compatibility between quantum ideas and Boltzmann’s principle should not be admitted a priori. It is in this context that Einstein’s cautions about Boltzmann’s principle should be framed and this is in spite of his proclaimed faith in its general validity.20 In 1914, precisely, he took a step forwards in the direction of justifying the compatibility between that principle and quantum ideas, since he not only presented his first deduction of Planck’s law—as we have already pointed out—and of Nernst’s theorem, but he obtains the following expression for the thermodynamic entropy of a macroscopic system: S=

R log Z , N

(3.6)

where now Z is “the number of elementary [microscopical] states possible under quantum theory”. Einstein states that the above equality “expresses Boltzmann’s principle in the formulation of Boltzmann-Planck” [Einstein (1914). In Engel (1997), 24–25]. Thus, in 1914, a clear reversal in Einstein’s procedure is perceived: instead of starting from Boltzmann’s principle and Planck’s law—already admitted as an experimental fact—to deduce the energy quantisation of a monochromatic oscillator, here starts from the quantum hypothesis and uses thermodynamics to obtain Planck’s formula, Nernst’s theorem, and Boltzmann’s principle.21 Finally, let us note that in this 1914 paper Einstein makes use of the “adiabatic Ehrenfest hypothesis”—which, by the way, he considers to be a generalization of Wien’s displacement law—that he enunciates as follows: according to quantum theory, in a reversible process every possible state evolves towards another possible state according to the same theory. Naturally, this implies that the number of accessible states does not change in an adiabatic reversible process. It is precisely this conclusion that allows him to justify the general validity of Boltzmann’s principle [Einstein (1914). In Engel (1997), 25]. For more details on the relationship

20

For example, in the discussion following his speech at the first Solvay conference, Einstein referred to the Boltzmann principle. After acknowledging reasonable doubts about its scope, he unequivocally said: “we must admit its validity without reservation”. Langevin; De Broglie (1912), 436. 21 For a more detailed analysis of Einstein’s attitude towards the compatibility between Boltzmann’s principle and the first quantum ideas, see Navarro; Pérez (2002). In Spanish.

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between Einstein and Ehrenfest about the “adiabatic hypothesis” —as it is coined by Einstein in this 1914 paper—we refer to the bibliography [See, for example, Navarro; Pérez (2006), specifically 253–258]. We are now approaching the very famous article of 1916 that certifies the reality of quanta, or equivalently, the birth of the photon. In the article we can distinguish two well-differentiated points: the deduction of Planck’s radiation law and the justification of the directional character of the elementary processes introduced for that purpose. The derivation of Planck’s formula was not truly a novelty: Einstein had published it before, also in 1916, in a short article that is rarely cited [Einstein (1916 b)]. However, we are obliged to digress here to comment on some aspects of this almost unknown paper, even if only briefly, since we believe that it provides essential clues for a better understanding of the evolution of Einstein’s ideas in relation to the radiation quantum. Einstein begins by recalling that, although Planck’s theory developed sixteen years earlier led to a formula for black body radiation, which was already fully confirmed by experiments, its deduction “was of unparalleled boldness” and, moreover, was based on two “incompatible” theories, such as electromagnetism and quantum theory. All of this caused great dissatisfaction in Planck himself and in the theorists who dealt with the subject. The introduction of Einstein’s paper ends as follows [Engel (1997), 212]: Since Bohr’s theory of spectra has achieved its great successes, it seems no longer doubtful that the basic idea of quantum theory must be maintained. It so appears that the uniformity of the theory must be established such that the electromagneto-mechanical considerations, which led Planck to equation (3.1) [our (1.59)], are to be replaced by quantum-theoretical contemplations on the interaction between matter and radiation. In this endeavor I feel galvanized by the following consideration, which is attractive both for its simplicity and generality.

What we want to emphasize is the double aspect contained in this Einstein’s reflection. On the one hand, the Planckian resonators are already definitively replaced by molecules that can undergo Bohrian transitions. On the other hand, three elementary processes are established (absorption, spontaneous emission and stimulated emission) that are ultimately responsible for the matter-radiation interaction. All that is incorporated in the following publication of Einstein, in which he will at last reveal the corpuscular behaviour of the radiation, as we shall now see.

3.3.2

Reality of Quanta (1916–1917)

Let us focus now on the very famous article of 1916 that certifies the birth of the photon. It reproduces the reasoning of his previous paper of 1916 publication that we have just commented on, but now in a more complete form. Therefore, from now on, we will refer only to this second publication, which includes important new features [Einstein (1916–1917)].

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It has arrived at the moment ‘chosen’ by Einstein to, by using Ockham’s razor, free quantum theory from mirrors and resonators. He now considers a cavity containing a material gas in thermal equilibrium with electromagnetic radiation, in which each molecule of gas can only occur in a discrete set of states Z1 , Z2 , …, Zn , … with energies ε1 , ε2 , …, εn , … respectively. The number of molecules in a state Zn is proportional to the canonical probability Wn , which is the one describing the thermal equilibrium: Wn = pn exp ( −

εn ), kT

(3.7)

where pn designates the statistical weight of the state Zn and does not depend on the temperature T. Einstein assumes that the molecules of gas can make transitions to higher energy states, absorbing the corresponding energy difference from radiation, and to lower energy states now emitting the difference in the form of radiant energy. The concrete laws that govern the elementary processes of absorption and emission are set out in the second section of this 1916–1917 paper, in which Einstein groups these processes into two classes: (a) Emergent radiation processes (Ausstrahlung). The molecule passes from a state Zm to another Zn of lower energy by means of spontaneous emission. The energy εm − εn is emitted in the form of radiation of frequency ν such that hv = εm − εn . Einstein assigns the following probability dW to this process occurring in the time interval dt: dW = Anm dt,

(3.8)

where Anm is a constant characteristic of the pair of states (Zm , Zn ). Einstein includes here an important consideration concerning the meaning of this probability: The assumed statistical law corresponds to a radioactive reaction, and the assumed elementary process to a reaction where only γ-radiation is emitted. One does not need to assume that this process requires no time; the time need only be negligible compared to the times during which the molecule is in states Z 1 , etc.

(b) Incident radiation processes (Einstrahlung). External monochromatic radiation, of frequency ν and energy density ρ , can originate two types of processes: – Absorption: the molecule passes from a state Zn to another Zm with higher energy, absorbing from the present radiation the energy εm − εn . The probability that Einstein assigns to this process is now given by: dW = B m n ρ dt,

(3.9)

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167

where B m n represents a characteristic constant of the state pair (Z n , Z m ). – Emission: under the stimulus of the radiation present, the molecule passes from a Zm state to another Zn state of lower energy, emitting radiation of the same frequency as the incident radiation and with energy εm − εn = hv. The probability of occurrence of this induced emission is now given by the following expression: n dW = Bm ρdt,

(3.10)

n is another characteristic constant of the state pair (Z , Z ). where Bm m n Considering the three elementary processes above, it is possible to arrive at an expression for the energy density ρ by imposing the condition of statistical equilibrium between gas and radiation:

pn exp ( −

) εm ( n εn ρ + Anm ) · Bnm ρ = pm · exp (− ) · Bm kT kT

(3.11)

Let us recall that in his 1912 work on the photochemical equivalent Einstein only considered two kinds of elementary processes, without any room for a process analogous to induced emission. Such a situation could be considered, on the basis of these 1916 reasonings, as a particular case in which the constant B nm is zero. In such a situation the last equality leads to the following relation: ρ=

) ( ε m − εn pm Anm , · exp − pn Bnm kT

(3.12)

which is nothing more than a concrete way of writing Wien’s law. Let us now verify that if, on the contrary, one considers the three elementary processes mentioned above—schematized in Fig. 3.2—one obtains Planck’s law.

Spontaneous emission

Absorption

Induced (or stimulated) emission

Fig. 3.2 In 1916–1917, Einstein showed that starting from the two usual elementary processes— absorption and spontaneous emission—one inevitably arrived at the then obsolete Wien’s law. In contrast, if one introduces a third process —induced emission—, with the trio as a starting hypothesis, he obtained the at the time fully confirmed Planck’s law… and more [Graphic: Tomás Navarro]

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Einstein explicitly admits that when the absolute temperature T tends to infinity, so must the energy density ρ. In view of (3.11), this forces the following condition n: to be imposed on the constants Bnm and Bm n pn Bnm = pm Bm

(3.13)

This relation, allows Einstein to rewrite the equilibrium condition (3.11) as follows: ρ=

n Anm /Bm , εm −εn exp (− kT ) − 1

(3.14)

which is a form of expressing Planck’s radiation law. But there is more: from the latter expression, taking into account the undisputed Wien’s displacement law as written in the form (2.26), the following two relationships can be deduced: Anm = α ν3 n Bm

(3.15)

εm − εn = hν

(3.16)

where α and h stand for universal constants to be determined. Einstein concludes his deduction of Planck’s law by stating that (3.16) not only expresses Bohr’s famous hypothesis about the emission and absorption of radiation, but that it also contains, although implicitly, the photochemical equivalence law [Einstein (1916– 1917), 225]. Let us recapitulate because the transcendence of the result deserves it. Einstein starts from quantum assumptions about the possible states of the molecules and assigns to three elementary processes the responsibility for the properties of the radiation-matter interaction. For each of these processes he associates a probability of occurrence: the latter known as “transition probabilities” between states. With these ingredients, after imposing the condition of equilibrium on the radiationmatter system, he not only deduces Planck’s radiation law but also comes to the conclusion that, in every elementary process, the energy exchanged between matter and monochromatic radiation of frequency ν has the same value: hν. However, these results must not have seemed to Einstein to be the most relevant part of his contribution. In fact, as we have already mentioned, all this had been published in his previous work of 1916 [Einstein (1916b)]. That he must not have given too much importance to the results he obtained in it is proven, for example, by the fact that in his correspondence with Besso, we have not found any other comment referring to that previous article than this brief paragraph22 :

22

Letter from A. Einstein to M. Besso, 14 May 1916. In Hentschel (1998), 213.

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169

I discovered a neat simplification of the thermodynamic derivation of the photochemical hν law [the photochemical equivalent law], somewhat in the manner of Van’t Hoff.

The results presented in his famous work of 1916–1917, on the other hand, earned a different assessment, as seen in fragments of three letters sent by Einstein to his colleague Besso in less than a month.23 A brilliant idea dawned on me about radiation absorption and emission; it will interest you. An astonishingly simple derivation, I should say, the derivation of Planck’s formula. A thoroughly quantized affair. I am writing the paper right now. The papers on gravitational waves and Planck’s formula have been lying around at your place for a long time now. You will enjoy the latter. The derivation is purely quantized and yields Planck’s formula. In connection with this, it can be demonstrated convincingly that the elementary processes of emission and absorption are directed processes. One just has to analyze the (Brownian) motion of a molecule (in the sense of that derivation) within a radiation field. In this analysis, which is being published in the Zurich Phys. Soc.’s issue in honor of Kleiner [the recently deceased supervisor of Einstein’s dissertation], there are no undulatory theory considerations either. But what is crucial is that the statistical considerations leading to Planck’s formula have become uniform and thus have become the most general imaginable, in that nothing more is assumed about the special properties of the molecules involved than the most general idea of the quantum. The result (not yet contained in the paper sent to you) thus obtained is that at each elementary transfer of energy between radiation and matter, the impulse quantity hν/c passed on to the molecule. It follows from this that any such elementary process is an entirely directed process. Thus light quanta are as good as established.

If the exchange of energy between matter and radiation in the processes of emission and absorption is always accompanied by a transfer of momentum, it is permissible to modify the terminology and state that in those elementary processes quanta of radiation are exchanged and not mere quanta of energy as hitherto. Thus, the quanta of monochromatic radiation of frequency ν become real particles that move at the speed of light, have zero mass—according to the relativistic prescriptions—, energy hν and momentum hν/c.

3.3.3

On Directionality of Elementary Processes

Although the issue has any formal complication, we will try to summarize Einstein’s ideas, as expressed in his 1916–1917 paper, on the directional character of the elementary processes and, with it, on the reality of radiation quanta. He does not limit himself to deriving Planck’s law but devotes half of the article to the analysis of the obtained results, a very understandable way of proceeding, in view of the strangeness of some of those results. Thus, the second part of the

23

Fragments of three letters from A. Einstein to M. Besso, dated August 11, August 24 and September 6, 1916. In Hentschel (1998), 243, 244 and 246, respectively. Emphasis in the original.

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3 The First Harvest (1914–1924): In Search of the Photon

paper contains calculations and considerations that give coherence to his previous results. His arguments are based on the skilful handling of his ideas about fluctuations, which is not surprising if one considers the high role that Einstein assigns in physics to statistical mechanics in general and to fluctuations in particular. Is in the fourth section of the paper where Einstein analyses the fluctuations of quantities associated with the motion of molecules in a radiation field, anticipating that he is going to employ methods already used in 1909 to analyse the Gedankenexperiment already commented in Sect. 2.4.1. Therefore, he recovers expressions (2.10)−(2.13), although with somewhat different notation: R is now the friction (formerly P) and k Planck’s constant (formerly R/N). To justify these expressions, he no longer resorts to resonators or mirrors, but to molecules slowed down by radiation because, although this “radiation is of the same nature in all directions, the molecule will—due to its own movement—suffer a force from the radiation that counteracts its movement”. About this force—which in the new notation is written as Rv, being R a constant to be determined—Einstein clarifies [Einstein (1916b). In Engel (1997), 226]: This force would bring the molecule to rest if the irregularities of the radiative actions would not force the molecule to receive a momentum Δ of changing sign and magnitude during the time τ.

In particular, the expression (2.13)—which, let us remember, was reached by imposing equilibrium condition on the gas-radiation system—is now written as follows: Δ2 = 2RkT , τ

(3.17)

where Δ represents the variation in the linear momentum of a molecule during a time interval τ as a consequence of the action of radiation. In the use of this expression Einstein introduces now a fruitful novelty: instead of applying it “from right to left”, as he had done in 1909 to deduce impulse fluctuation (remember Sect. 2.4.1), nor “from left to right”, as he had done in 1910 and 1913 in his attempts to deduce the radiation law (remember Sect. 2.4.3 Stochastic fields and quantum theory and 2.9.2 Einstein and Stern (1913): on the specific heat of hydrogen, he now uses it in a different way in order to test the solidity of his last conclusions. For this purpose, he separately calculates Δ2 and R on the basis of the new hypotheses on elementary processes and Planck’s law, previously demonstrated. If both values are subsequently substituted in (3.17), this will have to be satisfied identically. We emphasize that this is a check for the assumptions to derive Planck’s law. This test of compatibility is performed by analysing the statistical fluctuations of a system in thermal equilibrium. To that end, Einstein begins by calculating the value of the constant R associated with friction. Considering that Rτ v indicates the change experienced by the momentum of the molecule in the time interval τ, due to the effect of the radiation,

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171

it is a matter of evaluating the mean momentum transferred from the radiation to the molecule, taking into account the assumptions on the elementary processes. For simplicity, Einstein will consider that the molecule moves with constant velocity v along the X-axis of a certain reference system K. As usual, the hypotheses about emission and absorption had been initially formulated for molecules at rest. Thus, it was essential to introduce a new reference system K’, in which the molecule was certainly at rest, and to write in it the relations affecting the elementary processes. This, obviously, did not represent any difficulty for Einstein, and it is not necessary to reproduce his steps here. However, if someone follows them, he will soon find a difficulty which Einstein himself indicates how to overcome [Einstein (1916 b). In Engel (1997), 228–229]: One could object that Eqs. (3.14), (3.15), and (3.16) [standard relations for wave phenomena with moving focus] are based upon Maxwell’s theory of the electromagnetic field, a theory that is incompatible with quantum theory. But this objection touches the form more than the essence of the matter. Because, in whichever way the theory of electromagnetic processes may develop, it will certainly retain Doppler’s principle and the law of aberration.

The rest of the calculation is not properly based on Maxwellian electromagnetism but on the theory of special relativity. With this, he obtains an expression for the value of R that is not conditioned by the validity of Maxwell’s theory. It is in this part of the paper that Einstein arrives at the directionality of the elementary processes when he shows that the transfer of momentum must be carried out according to a certain direction. However, in regard to evaluating the mean impulse transferred to the molecule, only in induced processes appears a net contribution to the transference. Of course, there is no net transference in the case of spontaneous emission since those mean values are null because there is no privileged emission direction. In an induced process Zn → Zm , the transferred momentum is (εm − εn )/c, where c is the speed of light in vacuum. In the absorption of radiation, the change in the energy of the molecule is positive and the momentum is transferred to the molecule in the direction of beam propagation. In induced emission, the opposite occurs: the energy variation is negative, and the molecule now experiences a change in its momentum in the direction of propagation of the inducing beam, but in the opposite sense. Einstein clarifies that if the molecule is under the influence of several beams of radiation simultaneously, both energy and impulse are exchanged with only one of them, being precisely this that determines the direction and the sense of the impulse transferred to the molecule. The calculation of R leads to the following result: ( ) ( )] ( ε ) [ 1 ∂ρ hv hv n m pn Bn · exp − · 1 − exp − (3.18) R= 2 ρ− v c s 3 ∂v kT kT where S simply stands for the following abbreviation: S ≡ pn · exp(−

εm εn ) + pm · exp(− ) kT kT

(3.19)

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3 The First Harvest (1914–1924): In Search of the Photon

The penultimate section of the paper is devoted to the calculation of Δ 2 . It is assumed that the molecular speed is sufficiently small—in comparison with that of light, of course—for the molecule to be treated by the methods of classical mechanics. By admitting the independence among elementary processes, a short and simple calculation leads Einstein to the following expression for the mean square root of the momentum transferred to the molecule, per unit time: ( ) ( ε ) Δ2 2 hν 2 n pn Bnm ρ · exp − = τ 3S c kT

(3.20)

The paper ends with the full confirmation of the positive result of the proposed test: the substitution of (3.18) y (3.20) in (3.17), taking into account that ρ satisfies Planck’s formula, indeed lead to identity. Nothing could be more appropriate than to reproduce Einstein’s own opinion concerning the origin of his ideas and the meaning of the results obtained [Einstein (1916b). In Engel (1997), 231–232]. With this, our deliberations come to a close. They provide strong support for the hypotheses in §2 about the interaction between matter and radiation by means of processes of absorption and emission or, respectively, spontaneous and induced radiation processes. I was led to these hypotheses from a desire to postulate in the simplest manner the quantum-theoretical behavior of molecules in a manner analogous to the classical theory of Planck’s resonator. From the general assumption of quanta for matter, Bohr’s second rule (Eq. 3.9) and Planck’s radiation formula followed effortlessly.

3.4

Digression: A New Quantisation Rule (1917)

Between 1900 and 1917, several quantisation rules that limited the number of possible states of a physical system appeared. Perhaps the most famous of these was in connection with the atomic model that Bohr had established in 1913, in which a good part of the quantum seed sown at the first Solvay conference (1911) had borne fruit, specifically from the original ideas of Planck and Einstein during the first decade of the twentieth century. After the publication of Bohr’s atomic model various quantisation rules appeared which widened the field of validity of the prescriptions limiting the number of possible atomic states. However, the different rules did not arise deductively, from general principles, but as ingenious extensions of some previous rule. Only the adiabatic principle, developed by Ehrenfest between 1913 and 1916, was a remarkable attempt to provide a common foundation for the existing rules and to suggest a way to obtain new quantisation rules.24

24

For a detailed study of the genesis and formulation of Ehrenfest’s adiabatic principle, see Navarro; Pérez (2006) and Pérez (2009).

3.4 Digression: A New Quantisation Rule (1917)

173

Now, we will deal with a quantisation rule presented by Einstein with analogous pretensions to those of Ehrenfest just cited. It is, in our opinion, a work unjustly neglected, if not almost completely ignored, by physicists and even by a good number of historians of quantum physics, as it is easy to see.25

3.4.1

Quantisation Rules: Times of Splendour (1913–1916)

Bohr’s atomic model of 1913 starts from the basis of Rutherford’s planetary model of 1911 but freeing it from conceptual limitations arising from its mechanical and electromagnetic instability.26 The notion of “steady state” is key to the new model: the atom can only exist permanently in one of these. If the atom is in one stationary state, the rotation of any of its electrons is determined by the laws of mechanics. However, emission and absorption of electromagnetic radiation at transitions between atomic states of this kind do not conform to the classical laws but are phenomena regulated by the much-used expression ΔE = hν. Among other innovations, this law introduced by Bohr separated the frequency of the rotating electron from the frequency of the radiation emitted or absorbed, contrary to the common ideas of the time. To determine the energy of the stationary states, Bohr established a rule that considerably reduces the number of allowed atomic states compared to the predictions of classical physics: the angular momentum of the electron turning around the nucleus, has to be equal to an integer multiple of elementary units ℏ = h/2π , where h is the Planck’s constant. This rule applies, in principle, only to circular motions. Bohr himself, in 1916, tried to support and extend this rule to more general periodic motions, by using Ehrenfest’s adiabatic hypothesis, to which we have already referred in Sect. 3.3.1: in any reversible adiabatic modification of the parameters of a system, this remains in the same stationary state (i.e., its quantum number does not vary). Thus, in an adiabatic reversible transformation, the possible motions of the system are transformed into possible motions of the modified system. As a consequence, it was thought that the natural way to formulate a quantisation rule is to do so in terms of “adiabatic invariants”: physical quantities whose value is not altered in an adiabatic transformation. From Ehrenfest’s ideas on adiabatic transformations Bohr intuited a systematic way of obtaining quantisation rules. It was enough to imagine adiabatic modifications of the parameters of a given system (e.g., an electron rotating in an elliptical orbit) until it was reduced to the case of a simpler motion for which its quantisation rules were known (e.g., by reducing an elliptical orbit to a circular orbit). The rules for the second case must then also be valid for the original system. Such a

25

Following comments on Einstein’s rule content part of our research on this topic, essentially contained in Bergia; Navarro (2000). 26 Recall here that Rutherford had been awarded the Nobel Prize in Chemistry 1908.

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3 The First Harvest (1914–1924): In Search of the Photon

way of proceeding left the door open to establishing quantisation rules for systems with several degrees of freedom and for motions not strictly periodic, provided that these could be adiabatically related to periodic motions. The first generalizations of Bohr’s 1913 quantisation rules were obtained by Sommerfeld who, between 1915 and 1916, extended the preceding rules to apply them to elliptical orbits and to motions with relativistic velocities. This allowed him to obtain a theoretical explanation of the “fine structure” observed in the spectral lines of hydrogen [Sommerfeld (1915) and (1916a). The Sommerfeld rules required that for any stationary state of a system with a finite number of degrees of freedom g, and in which every canonical momentum pi was a periodic function of only the respective canonical coordinate qi , the following relation had to be verified: ∮ pi dqi = ni h, (i = 1, . . . , g), (3.21) where ni can be any integer and the integration extends over a complete period of the coordinate qi . Sommerfeld’s attitude, at least during the period we are dealing with, was that of refusing to ask for a rigorous demonstration of the validity of the quantisation rules. Although at the time they represented the most solid heuristic foundations of quantum theory, he expressly labelled them as “unproven and perhaps unprovable foundation of the quantum theory” [Sommerfeld (1916a), 6]. Ehrenfest, on the other hand, pointed out that his adiabatic hypothesis could support the validity of the conditions (3.21), given the character of adiabatic invariants of the integrals contained in these expressions. In any case, everything suggests that Sommerfeld never came to regard the adiabatic principle as a serious candidate for achieving the foundation of quantisation rules. It is appropriate here to note that William Wilson (1875–1965) and Jun Ishiwara (1881–1947) signed in 1915—some weeks before Sommerfeld made public his first generalizations—two papers that extended the range of validity of Planck’s quantisation rule (1900, for oscillators) and Bohr’s (1913, for circular motions). Both Wilson and Ishiwara formulated their generalizations in a manner analogous to the conditions expressed in (3.21), but since neither of them applied it to atomic spectra, the paternity of these rules has been exclusively assigned to Sommerfeld. Only rarely do some refer to them as “Sommerfeld, Wilson and Ishiwara quantum rules”.27 A problem soon arose concerning the appropriate coordinates for the application of the quantisation rules. For example, Kepler’s problem —motion due to a central force inversely proportional to the square of the distance to a fixed centre—can be approached with either polar or parabolic coordinates. However, in this example,

27

For details of Wilson’s and Ishiwara’s respective contributions, see, for example, Jammer (1966), 91–92. Jun Ishiwara translated into Japanese several of Einstein’s publications with which he was familiar after his stay in Europe between 1912 and 1914.

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175

conditions (3.21) lead to different results depending on which coordinates are used, which posed the dilemma of clarifying which kind of coordinates should be used for the quantisation rules to lead to valid results. Paul S. Epstein (1883–1966) and Karl Schwarzschild (1873–1916) approached the problem and soon—in 1916, independently—they came up with the solution: the correct application of the Sommerfeld quantisation rules requires that the system under consideration has a finite number of degrees of freedom and that the corresponding Hamilton–Jacobi equation is “separable”, since then each pi is a function of the respective qi . Moreover, from the previous century, it was known that systems with such a property present “multiperiodic” motions: in every bounded motion, each pair of conjugate canonical variables (qi , pi ) describes a closed orbit in its corresponding two-dimensional phase space.28 Schwarzschild also showed that the introduction of the action-angle variables—always possible when separability exists—allows for a tighter treatment of the quantisation rules.29 The concept of “degeneration” plays a relevant role in this kind of problems. A multiperiodic motion is said to be “degenerate” when, between the frequencies ν i of the g simple motions, there exists some relation of linear dependence: g ∑

si · νi = 0,

(3.22)

i=1

where si represents non-zero integers. If there are n relations of the above type, the degeneration is said “of order n”. It can be shown that the global motion is periodic if and only if there exist (g−1) relations of that kind that are independent of each other. In this case only one of the frequencies is truly independent; this occurs, for example, in Kepler’s problem if the relativistic correction is not considered. It can be shown that if there is no degeneracy, the coordinates allowing the separation of the Hamilton–Jacobi equation are unique, except for irrelevant details. In this case, the application of the conditions (3.21) to obtain the possible states of the system is free of difficulties. In contrast, when degeneracy is present, the separation of the Hamilton–Jacobi equation can be achieved with various coordinate systems, and therefore, the application of the rules leads to some ambiguity since the final result is not independent of the coordinates adopted. The actionangle variables play an essential role in the resolution of this problem, but all this is already too technical, so we prefer to refer here to an earlier and more detailed exposition of ours [Bergia; Navarro (2000), 326–332].

28

“Multiperiodic” motions are also called “quasi-periodic” or “conditionally periodic”. For a more precise exposition of Epstein’s and Schwarzschild’s respective contributions, as well as of the adiabatic invariance of the action-angle variables, see Jammer (1966), 101–109.

29

176

3.4.2

3 The First Harvest (1914–1924): In Search of the Photon

Einstein’s Quantisation Rule of 191730

It is a little-known contribution of Einstein’s, largely because it was published exclusively in the proceedings of the Deutsche Physikalische Gesellschaft —German Physical Society—of very scarce diffusion [Einstein (1917)]. In the paper, he tries to establish a formulation of the quantisation rules that does not depend on the coordinates chosen for their application, which does not happen if the rules are expressed in the form (3.21), as we have already pointed out. Such a dependence on the coordinate system, Einstein states, “probably has nothing to do with the quantum problem per se”, so he is going “to suggest a minor modification of the Sommerfeld-Epstein condition and thereby avoid this deficiency” [In Engel (1997), 436]. Einstein’s proposal is apparently simple since it consists of substituting the g conditions (3.21) by a single expression, which he writes as follows: ∮ ∑ pi dqi = ni h, (3.23) i

where the summation is done for the g degrees of freedom of the system, the integral is evaluated along any closed curve in the configuration space and ni designates an integer. But all this requires certain precisions. Einstein assumes the existence of a potential W for the pi . Then, the integrand of (3.23) represents the exact (or total) differential dW, which leads to a result that no longer depends on the adopted coordinate system: ∮ ∑ ∮ ∑ ∮ ∂W pi dqi = dqi = dW = ΔW = ni h (3.24) ∂qi i

i

(i) Of course, if W were a uni-valued function of the variables of the variables qi , since it is a variation throughout a complete cycle, the last expression would lead to ΔW = 0, with which the new rule would be meaningless. W is nothing more than the “characteristic function” or “action function” of the Hamilton– Jacobi theory.31 It is a multi-valued function of the coordinates qi , so ΔW needs not necessarily cancel out. Thus, the new Einstein rule stipulates that the variation of the action along a closed path in configuration space has to be an integer multiple of the Planck constant h. Always assuming the existence of the potential W for the field pi , Einstein recapitulates certain properties of the term appearing on the left in the last two

30

It should be noted that this section is practically a summary of a previous research work by S. Bergia and myself: Bergia; Navarro (2000). 31 Do not confuse this “action function” W with the “action variables” or with the “action integral” which is the left hand side term of (3.23).

3.4 Digression: A New Quantisation Rule (1917)

177

expressions. It is easy to see that such an integral takes the same value for all closed curves which, in the configuration space, can be transformed into each other by continuous deformations. Thus, for all the curves that, by means of a continuous transformation, can be reduced to a point, the integral will be zero. In the case of a usual bounded motion, the accessible region in that space is a multiply connected region, so that in this region, there are curves that cannot be reduced to a point by continuous transformations. Nevertheless, as Einstein clarifies, in the accessible region for the variables qi , there is only a finite number of closed curves to which all closed curves in that region can be reduced by means of continuous deformations. Einstein stops at “a quite essential point”: the analysis of the two possible types of trajectories that exist for bounded motions, over infinite time intervals, in a configuration space of g dimensions32 : (b) Movements in which there exist—always in the configuration space—a region of dimension g, such that, in the course of time, the trajectory approaches any point of this region as much as you wish to prefix. This is what happens, for example, for bounded motions in the classical Kepler problem, with the relativistic correction. (c) Movements in which the trajectory is always confined in a region of dimension less than g, as happens if the trajectory is completely closed. This is the case, for example, for bounded motions in the case of the classical Kepler problem, now without the relativistic correction. The first type, Einstein continues, corresponds to the most general case and, therefore, in the rest of the paper he basically refers to it. After justifying the need for the existence of a potential W in order for the rule (3.23) to be valid, he goes on to analyse in detail the relativistic Kepler motion; a useful example for him to show his conclusions. The article ends with a comment that, in our opinion, helps to determine the scope of the work. Suppose that one knows g independent constants of motion— among the (2 g−1) that can exist at most—of the following form: Rk (qi , pi ) = const.,

(3.25)

where Rk represent algebraic functions of the variables pi . Einstein states that, in this case, the integrand of (3.23) turns out to be an exact differential even if the system is not separable by the coordinates qi . This result is obvious, since, from the last expression, each pi can be expressed as a function of the set of variables qi , so that the new rule (3.23) is perfectly applicable and, moreover, coincides with Sommerfeld’s rule (3.21) when each pi turns out to be a function of the

32

Einstein (1917). In Engel (1997), 439. Einstein does not expressly write that he is referring to bounded motions, but it is clear from the context.

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3 The First Harvest (1914–1924): In Search of the Photon

corresponding qi . If, on the other hand, the number of known constants of motion of the form (3.25) is less than g, the variables pi cannot be expressed as a function of the corresponding qi and therefore the quantisation conditions do not apply, even in their general formulation (3.23). Thus far, the main content of Einstein’s article. It is important to emphasize that now it is a single rule, which is specified in a single relationship, in which all the degrees of freedom of the system are involved. For its invariance before changes in coordinates, the question is easy to solve. It is enough to write the first member of the equality (3.23) in the following form: ∫ ∫ ∑ g

dqi dpi

i=1

To recognize it as one of the “Poincaré’s canonical integral invariants”, which are invariants under canonical transformations. As a usual transformation of coordinates is a particular case of canonical transformation, the last expression is invariant under arbitrary changes in coordinatesf [See, e.g., Goldstein et al. (2013), 388-396]. It is highly surprising, at least for us, not to find any comment from Einstein about the adiabatic invariance of the action integral appearing in the first member of the condition (3.23), especially because it could lead to additional arguments to justify the new quantisation rule. It is even more surprising if one considers not only Einstein’s knowledge of the work of his friend Ehrenfest but also the visit that Einstein paid him in Leiden shortly before he published his new rule. Let us insist on the fact that, as we have already mentioned in Sect. 3.3.1, it was precisely Einstein who coined the terminology “Ehrenfest’s adiabatic hypothesis” in 1914, which would later be known as the “adiabatic principle”. Moreover, Einstein’s visit to Leiden, invited by Ehrenfest, took place in the first half of October 1916. At that time, the young Jan Martinus Burgers (1895– 1981) was preparing his dissertation under the supervision of Ehrenfest, who had assigned the study of the adiabatic invariance properties of the action integral as research topic.33 Einstein’s contribution received little attention from those involved in the development of quantum theory. The few analyses were, in general, quite critical of some theoretical aspects of the new proposal that, on the other hand, soon proved incapable of providing new results. Nevertheless, it is worth pausing, even if only minimally, to consider the most influential of these criticisms.

33

For more detail and bibliography on Burgers’ research, as well as on Einstein’s trip to Leiden, see Klein (1985), 290–291 and 302–305, respectively.

3.4 Digression: A New Quantisation Rule (1917)

3.4.3

179

First Impact: Epstein’s Criticism

It was Epstein —whose name appears together with Sommerfeld’s in the title of Einstein (1917), both as coauthors of the quantisation rule (3.21)—the first among the theoretical physicists of the time to criticize certain important aspects of the Einsteinian proposal; and he did so in the same journal in which, a few weeks earlier, the original publication had appeared.34 Epstein’s exposition is too technical to go into detail here. Nevertheless, we will point out his most important conclusions, since they contain the basis for other later criticisms, especially in relation to the problems that the new rule presents when there is degeneracy. Epstein admits the possibility that, with this rule, it is possible to reach the quantisation of more general movements than those solved by means of the preceding rules, but he believes that certain problems that the new rule presents if it is applied to degenerate movements must be eliminated beforehand. To highlight the ambiguity that can appear in the application of (3.23), Epstein deals explicitly with an example: the two-dimensional motion of a material point elastically bound to a fixed centre. His conclusion is rigorous and categorical: it is not true that, in all cases, the results arrived at after quantizing a system are independent of the coordinates chosen. Einstein emphasized that something as essential as quantisation rules for a physical system cannot be conditioned by something as accessory as the choice of the coordinate system. Then, Epstein’s criticism certainly affects the most striking aspect of Einstein’s new rule. Although Epstein shows with his example that the expression (3.23) is not as intrinsic—i.e., independent of the coordinates chosen— as its author claimed, he is not able to solve with generality the problem of the relationship between the validity of the new rule and the choice of the appropriate coordinate system. In fact, in his article he demands further research to determine whether the difficulties shown in his example are limited only to the case where degeneracy exists or whether the difficulty is deeper. Epstein’s criticism is not directed towards possible applications of the new rule but towards its theoretical foundation. If one considers the promptness of Epstein’s severe criticism and the fact that the new rule did not seem to extend the field of application of the previous ones, it is not surprising that it received so little attention. Even Epstein stopped referring to this quantisation rule in later works on the topic. Epstein’s judgment about Einstein’s rule must have been convincing, at least for a significant part of the German-speaking scientific community.35 We have found no mention of this rule in papers by physicists as committed to the quantum theory of the time as Bohr, Sommerfeld, Ehrenfest and even Einstein himself. Considering

34

Epstein (1917). The memoir Einstein (1917) is recorded as having been read at the May 11 session of the Deutsche Physikalische Gesellschaft and was published in the May 11 issue. Epstein (1917) was received for publication on July 2 of the same year. 35 For more details on the limited impact of Einstein’s quantisation rule among German-speaking physicists, see Bergia; Navarro (2000), 346–353.

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the prestige, influence, and attitude of such figures, it seems reasonable to think that the fundamental aspects of Epstein’s criticism must have been shared by most of the physicists involved in the development of quantum ideas.

3.4.4

Influence of Einstein’s Rule on Louis De Broglie

In contrast to Germany, the impact in France of Einstein’s new quantisation rule was remarkable. In particular, its influence on De Broglie’s quantum ideas would be relevant for his formulation of wave mechanics, as it is easy to detect in the Frenchman’s doctoral thesis which, under the title “Investigations on the theory of quanta”, was presented at the Sorbonne on 25 November 1924 [De Broglie (1924)]. Chapter 3 of this thesis where he refers to Einstein’s rule, indicating that it is “invariant with respect to changes of coordinates”, although without any mention of Epstein’s criticism. De Broglie notes that the integral appearing in (3.23) is the same as the one appearing in the Maupertuis principle, which he writes as follows:

δ

∫B ∑ A

pi dqi = 0

(3.26)

i

Since it was by then well known that this variational principle is equivalent to the principle of least action—which can be understood as the basic law of classical mechanics—De Broglie concludes that the action integral plays an essential role, not only in classical mechanics (through the Maupertuis principle) but also in quantum theory, now through Einstein’s quantisation rule. The above conclusion is extremely important in the scheme of the young French physicist, who was then trying to elaborate a new mechanics incorporating the peculiarities of quantum physics. He thought that a remarkable step in that direction could be taken by considering the analogies between classical mechanics and geometrical optics, considering, moreover, that the latter represents an approximation within the general (wave) theory of the electromagnetic field. Taking these analogies to its ultimate consequences, De Broglie tries to find a new general (wave) mechanics of which Newtonian (corpuscular) mechanics would be an approximation, just like geometrical optics turned out to be an approximation of electromagnetic optics. In view of such an approach, Einstein’s rule is of special interest, since in this rule appears the same integral action as in the Maupertuis principle, which is the image of Fermat’s principle within the mechanic-optical analogy mentioned above. To reveal the physical meaning of Einstein’s rule, De Broglie resorts to his notion of the “phase wave” associated with each material point in motion and whose rays could represent the possible trajectories of the mobile. In the case of an electron, the propagation of its phase wave is analogous—according to De Broglie—to the propagation of a liquid wave in a channel of variable depth, which

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closes in on itself. If the wavelength λ is constant, have a stable wave regime in a channel of length L, the relation L = nλ must be fulfilled, where n is a natural number. However, if, on the other hand, the wavelength is not constant, the relation to be verified is the following: ∮

ν dL = n, Vf

(3.27)

where ν is the (variable) frequency of the wave, V f is its propagation velocity and dL is the channel length element. In the second chapter of his thesis devoted to the analysis of the analogy between geometrical optics and Newtonian mechanics, De Broglie concluded that [De Broglie (1924), 56]: Fermat’s principle applied to the phase wave is identical to Maupertuis’ principle applied to the mobile; the dynamically possible trajectories of the mobile are identical to the possible rays of the wave. We think that this idea of [the existence of] a profound relation between the two great principles of Geometrical Optics and Dynamics could be a precise guide to perform the synthesis of waves and quanta.

For the above analogy to fit the theory of relativity, for the relation E = h ν to be feasible and for the analogy to relate physical quantities with the same dimensions—we have added the latter—De Broglie introduces an additional hypothesis, which he himself does not consider to be rigorously justified: the integrand of the Maupertuis principle (3.23) divided by h must be the one that corresponds to the integrand of Fermat’s principle expressed in the condition (3.27). Under such a perspective, Einstein’s rule (3.23) is nothing more than the expression that corresponds—considering the aforementioned optical-mechanical analogy—with the stability condition (3.27). De Broglie obtains in this way an original physical interpretation of Einstein’s rule (3.23). It is worth emphasizing the mechanical (corpuscular) character of the former against the optical (wave) character of the latter.

3.4.5

Einstein’s Rule and Wave Mechanics

In view of all the above, it seems clear that the essential reason for De Broglie to expressly show his inclination towards Einstein’s rule is due, not so much to its supposed invariance in the face of changes of coordinates, but because the enunciate of this new rule allowed its interpretation in physical terms that, in turn, seemed to give additional support to the Frenchman’s ideas: Einstein’s rule could be understood as the expression of a stability condition for the mechanical system, which corresponded to a resonance condition for the phase wave associated with

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it. Such a condition is a usual way of representing the stationary character of an undulatory motion. It is curious, to say the least, to note that the idea of interpreting a quantum rule as a resonance condition of a special wave emitted by an electron had already been suggested by Marcel Louis Brillouin (1854–1948) in 1919, as De Broglie himself would later recognize [see Kubli (1992), 129–132]. Moreover, this interpretation of Einstein’s rule represented strong support for De Broglie’s innovative ideas, as he expressly admits in his doctoral thesis [De Broglie (1924), 65]: This nice result [the interpretation of Einstein’s rule as a stability condition] whose demonstration is immediate once the ideas of the preceding chapter [devoted to the analysis of the analogy between mechanics and geometrical optics] are admitted, is the best justification we can give of our way of attacking the problem of quanta.

De Broglie considers that the Sommerfeld and Einstein rules—(3.21) and (3.23) respectively—are equivalent, so to the invariant character of Einstein’s rule must be added the success of its applications, especially to the hydrogen atom. However, it should not be thought that De Broglie considered the Einstein rule as a mere formal instrument, useful for the justification of his innovative and shocking ideas. We can see that the Frenchman, in his doctoral thesis, uses the condition (3.23) to obtain the quantisation of the general motion of two electric charges, a problem to which he devotes the whole of Chap. 4 of his thesis. That De Broglie held Einstein’s rule in high esteem—beyond the circumstance, already referred to, that it represented an additional support for his surprising new ideas—can be verified by his devotion to it even after the appearance of wave mechanics in 1926. That same year, for example, in a paper containing a brief exposition of the new theory, De Broglie emphasized the importance of Einstein’s rule “which constitutes the natural way of establishing quantum stability in the language of the old mechanics” [De Broglie (1926), 65–66]. In this same article of 1926, De Broglie includes an observation that, in our opinion, could have already been included in his dissertation of 1924. However, either out of caution or for lack of warning, it did not do so. We refer to the following consideration: in the same way that Fermat’s principle is valid only within a certain range of approximation (that of geometrical optics, within the general framework of electromagnetic theory), quantum conditions in general—which represent the mechanical image of that principle, according to the analogy between geometrical optics and mechanics—and Einstein’s rule in particular, must represent a certain approximation within a wave mechanical conceptual framework, something yet to be discovered in those days. Certainly, a reflection of this type was already within De Broglie’s reach in 1924, since he had all the necessary elements at his disposal. However, such a consideration had to wait until 1926, when, as we shall see later, the theoretical framework of wave mechanics was presented, in which those ideas found their natural place.

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Regarding the birth of quantum mechanics, it is worth recalling that the German physicist Werner Heisenberg (1901–1976) formulated the matrix version of quantum mechanics in 1925.36 However, is in the formulation of the wave version by Schrödinger in 1926 where Einstein’s rule comes into play. For example, in the first section of the second of Schrödinger’s four foundational papers—where the formal analogy between geometrical optics and classical mechanics is analysed—there is an express reference to it. Let us see. Schrödinger uses the following expression: pk =

∂W , ∂ qk

(3.28)

where W stands for the potential introduced in (3.24). This expression serves him to justify that the optical rays, which are orthogonal to the equipotential surfaces, must be identified with the trajectories of point masses, as De Broglie had anticipated. In addition, in a footnote, he comments on the physical meaning of this last expression, referring to 1917 Einstein’s paper. On the quantisation rule of Einstein in particular he writes [Schrödinger (1926b)]: The formulation of the quantum conditions contained here [in Einstein’s 1917 paper] is the closest, leaving aside all previous attempts, to the present [1926] formulation. De Broglie has returned to it.

Nevertheless, it cannot be affirmed that the influence of Einstein’s quantisation rule was determinant for the formulation of wave mechanics in 1926. However, what did have a major influence on Schrödinger was the contents of two Einstein’s papers in 1924 and 1925 on the quantum theory of ideal gases, a topic that we shall deal address in detail in the last two sections of Chap. 4.

3.5

First Reactions to the Photon Concept. Alternative BKS

At first sight, it may seem obvious that Einstein had been totally convinced of the reality of electromagnetic quanta, that is, of the validity of the hypotheses and conclusions in his 1916–1917 paper. With this in mind it may be surprising to read a paragraph addressed to Besso in mid-1918.37

36

Heisenberg was awarded the Nobel Prize in Physics 1932 “for the creation of quantum mechanics, the application of which has, inter alia, led to the discovery of the allotropic forms of hydrogen”. 37 Letter from A. Einstein to M. Besso, 29 July 1918. In Hentschel (1998), 613. Emphasis appears in the German original but not in the translation.

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Here I have been pondering for countless hours about the quantum problem again, naturally, without really making any headway. But I no longer have doubts about the reality of quanta in radiation, even though I’m still quite alone in this conviction. That’s how it will remain as long as no mathematical theory succeeds. I do want to present these arguments clearly sometime now, after all.

The pessimistic tone of this paragraph naturally leads to some questions: Why did the physicists of the time not share Einstein’s convictions? Why was Einstein still interested in the problem of quanta if he considered that he had proven their existence and established their properties? What does Einstein mean when he says that he has not made “any headway”? It is quite possible that the answers to such questions, if they exist at all, are neither clear, nor independent, nor unique. Nevertheless, let us hint at some possible approaches. From the abundant explicit references that exist, it is clear that the fundamental reason for the nonacceptance of Einsteinian ideas about the nature of radiation was the admitted and touted alleged incompatibility between electromagnetic field theory (wave) and, new quantum theory (corpuscular). Moreover, it is possible that the great weight assigned by Einstein to fluctuations in the foundations of quanta was a handicap when justifying them. For the vast majority of physicists, statistical fluctuations were little more than an academic matter of little relevance; thus, many could consider inconceivable a possible dismantling of the theory of the electromagnetic field—at that time fully contrasted and admitted—, on the basis of arguments substantially supported by statistical fluctuations. Thus, the generalised rejection of quanta since its birth, even maintained after the formulation of quantum mechanics, is understandable. It was decided to maintain at all costs the Maxwellian image of the free field and, as far as possible, to search hard for new models for matter-radiation interaction. All this with the permanent hope of removing any quantum residue as unnecessary. It is a good time to introduce an aside about terminology. In 1926, the American Gilbert Newton Lewis (1875–1946) first proposed the term “photon” to refer to the quantum of radiation. He did so in his article “The conservation of photons”, which appeared in the journal Nature, on 18 December 1926. His proposal is: “I therefore take the liberty of proposing for this hypothetical new atom, which is not light but plays an essential part in every process of radiation, the name photon”. The terminology was soon consecrated: the fifth Solvay conference held in October 1927 was already convened under the title “Electrons and photons”. From this point onwards, although somewhat improperly in view of the above, we will use throughout this book the term photon to refer to the electromagnetic radiation quantum and consequently to the light quantum. It is often written that the photon was definitively settled in physics after the discovery and the theoretical explanation of the “Compton effect”—scattering of X-rays by weakly bound atomic electrons—, in 1923. This explanation seemed to simultaneously confirm two theories, at that time still challenged: Einstein’s quantum theory of radiation, introduced when photons are used in the explanation of the effect, and the special theory of relativity, used to describe the photonelectron elastic collision. It would be naive to think that both independent theories

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could turn out to be false but that they compensated for their errors by achieving an accurate explanation of the precise experimental results about this effect. Thus, in particular, Einstein’s photon seemed to have been confirmed and accepted. However, this is not entirely true. To prove it, it is enough to take a look at a renowned article signed in 1924 by Bohr, together with his young colleagues Hendrik Anthony Kramers (1894–1952) and John Clarke Slater (1900–1976) [Bohr et al. (1924)]. It is the best known and most influential example of the prevailing philosophy on the issue. In the paper, known by the acronym BKS (initials of the authors’ surnames), unambiguously rejects Einstein’s photon, i.e., the quantum hypotheses introduced by Einstein in his 1916–1917 paper on the nature and behaviour of radiation in its interaction with matter. It is worthwhile to dwell, at least minimally, on the BKS proposal. The authors assume that the quantum discontinuities have come to physics to stay, since they are now indispensable to explain theoretically the observed properties of such interaction. However, this does not necessarily imply the validity of Einstein’s ideas in this respect. In other words, those discontinuities do not lead inexorably to the photon as a physical reality, or, therefore, to the corpuscular nature of radiation. In contrast, what is intended in the BKS article is to show an alternative way of making quantum jumps and the usual electromagnetic description (continuous) compatible. Not only are the arguments put forwards in the BKS paper surprising, but also its formal aspect. The seven pages it occupies in the journal contain only one mathematical formula—the well-known expression hν = E 1 − E 2 from of Bohr’s atom—although it proposes nothing less than a new theoretical model to analyse matter-radiation interaction. In the article, Bohr and Kramers assume previous ideas of Slater and jointly present the following textual proposal [Bohr et al (1924), 164–165]: We will assume that a given atom in a certain stationary state will communicate continually with other atoms through a time-spatial mechanism which is virtually equivalent with the field of radiation which on the classical theory would originate from the virtual harmonic oscillators corresponding with the various possible transitions to other stationary states. Further, we will assume that the occurrence of transition processes for the given atom itself, as well as for the other atoms with which it is in mutual communication, is connected with this mechanism by probability laws which are analogous to those which in Einstein’s theory hold for the induced transitions between stationary states when illuminated by radiation. On the one hand, the transitions which in this theory are designated as spontaneous are, on our view, considered as induced by the virtual field of radiation which is connected with the virtual harmonic oscillators conjugated with the motion of the atom itself. On the other hand, the induced transitions of Einstein’s theory occur in consequence of the virtual radiation in the surrounding space due to other atoms.

Thus, matter-radiation interaction is described in terms of interference between electromagnetic fields, whether they are real or virtual. These interferences give rise to phenomena of absorption or emission of radiation that—according to the authors—can only be described in probabilistic terms analogous to those used by Einstein in his 1916–1917 paper. In addition to recognizing that strict classical

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causality is incompatible with their model, the authors consider its implications and come to an extraordinarily striking conclusion, to say the least: in an individual process of emission or absorption of radiation, described on the basis of those virtual fields, the principles of conservation of energy and momentum are no longer universally valid. They are only statistically valid. Or, equivalently, they are fulfilled for mean values corresponding to a very large number of processes. In BKS paper, the Compton effect is described as a continuous process in which the scattering of radiation by electrons is considered “a continuous phenomenon to which each of the illuminated electrons contributes through the emission of coherent secondary wavelets”. Although the effect can be explained in terms of an elastic photon-electron collision, the authors—while admitting the conservation of energy and momentum in individual collisions—consider that “it seems at the present state of science hardly justifiable to reject a formal interpretation as that under consideration [the hypothesis of statistical conservation] as inadequate” [Bohr et. alt. (1924), 173]. The striking consequences of the bizarre assumptions of BKS served as a stimulus for certain physicists to try to submit some of those conclusions to a direct experimental test. The results appeared as early as 1925 and were convincing. Other experiments carried out later and with more sophisticated means would only confirm the first results. Let us see. Walther Bothe (1891–1857) and Hans Geiger (1882–1945) experimentally tested, in 1925, one of the implications of the BKS proposal: the violation of the causality principle [Bothe; Geiger (1925)]. They proved that the secondary photon originating in the Compton effect is created at the very instant of the photon-electron impact, as required by classical causality. There was no appreciable time interval between the impact and the appearance of that photon, contrary to what was required by BKS theory. This time interval appeared in BKS theory as an effect of the interaction between different radiation fields and would have to be detected with the experimental techniques of the time. On the other hand, Arthur Holly Compton (1892–1962) and his assistant Alfred W. Simon (1897-?) also carried out, in 1925, a very fine experiment—using the cloud chamber created by the Scottish-born physicist Charles Thomson Rees Wilson (1869–1959) in 1912—whose results fully confirmed the conservation of energy in individual processes.38 An aside on distinctions received by some of the physicists mentioned in the previous paragraph. Compton and Wilson shared the Nobel Prize in Physics 1927, “for his discovery of the effect named after him” and “for his method of making the paths of electrically charged particles visible by condensation of vapour”, respectively. Bothe and Born shared the Nobel Prize in Physics 1954 “for the coincidence method and his discoveries made therewith” and “for his fundamental

38

Compton; Simon (1925). For more details on the Bothe-Geiger and Compton-Simon experiments, see, for example, Pais (2005), 419–421, as well as the bibliography (original and secondary) cited there. Wilson.

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research in quantum mechanics, especially for his statistical interpretation of the wavefunction”, respectively. Let us return to the BKS alternative, whose death certificate was signed only one year after its birth. The experiments left no room for doubt. Being incompatible with the proposal, they provided, albeit indirectly, additional support for Einstein’s quantum theory. The opinion of Bothe and Geiger, in the paper where they give an account of their experimental results, is emphatic [Bothe; Geiger (1925), 662–663]: Concluding remarks. The described understandings [the results of their experiments] are not compatible with Bohr’s interpretation of the Compton effect […]. But not only with regard to the Compton effect, but also in general to Bohr’s new theory of radiation, since the result obtained here seems to pose very great difficulties for the interpretation of the Compton effect by Bohr, Kramers and Slater, which is very closely linked with the statistical conception of the energy and momentum theorem underlying this new theory. One must therefore assume that the concept of the light quantum [the photon] has a higher degree of reality than is assumed in this theory.

This did not mean, by any means, that Bohr and others stopped looking for arguments to close the door to the Einsteinian photon in physics, while at the same time trying to retain some aspects of the defeated proposal BKS. Bohr did not rule out having to resort to bolder assumptions in the near future. This is as true and documented as the undeniable influence that the BKS philosophy had on the gestation of matrix mechanics.39 Just as in Bohr’s mind there is no place for the photon for the time being, Einstein is unable to accept the philosophy contained in the BKS alternative. Lack of causality and nonconservation of energy and momentum in elementary processes are taxes that he is in no way willing to pay. On the other hand, with regard to the nature of radiation, from the very beginning, he had shown his conviction in the compatibility between the quantum description and the Maxwellian description. It is worth recalling here his opinion in Salzburg in 1909 [Einstein (1909b), 379]: All I wanted is briefly to indicate with its help that the two structural properties (the undulatory structure and the quantum structure) simultaneously displayed by radiation according to the Planck formula should not be considered as mutually incompatible.

However, Einstein was clearly unable, despite his strenuous efforts, to reconcile the two aspects of the behaviour of radiation: the quantum one, which is contained in his 1916–1917 theory, and the continuum of electromagnetic description. In 1924, already aware of the BKS proposal, he addressed Besso in these terms40 :

39

For Bohr’s reaction to the experimental rejection of the BKS proposal, as well as its influence on certain of Heisenberg’s ideas, see, for instance, Petruccioli (1993), 125–182. 40 Letter from A. Einstein to M. Besso, 24 May 1924. In Hentschel; Nollar-James (2015), 248.

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Scientifically, I’m dwelling almost without interruption on the quantum problem and do really believe I’m on the right track—provided it’s certain. The best thing that I have managed to do lately was the paper of 1917 in the Physikal. Zeitschrift [Einstein (1916−1917)]. My new efforts are in the direction of unifying quanta and the Maxwellian field. Among the experimental results of importance from the last few years there are actually only the experiments by Stern and Gerlach as well as the Compton exp. (scattering of Roentgen rays with a change in frequency), the former proving the unique existence of the quantum states; the latter, the reality of the momentum of light quanta.

There was an additional reason for Einstein’s concern—and later for his disenchantment—in the face of some aspects of his own contributions and also of further research by different physicists, such as the BKS proposal itself. The role that he had assigned to probability in his 1916–1917 paper was evolving in a direction that worried him greatly. That probability was beginning to reveal itself as a basic property of physical reality description and not as a mere tool of calculation, which is what happens in statistical mechanics. It is worth insisting on the reasons for Einstein disappointment. If the lack of information about the “when” and “how”, for example, of the spontaneous emission of radiation by an atom, is taken as a defect of the theory, it would be appropriate to wait for a new and more complete theory to appear. However, if indeed the most rigorous and precise theory possible on the nature of electromagnetic radiation had been obtained, then the starring role played by probability in the theory runs counter to absolute determinism and to the classical principle of causality. The dilemma is a strong one, and Einstein opts for the first option: his quantum theory, as well as later quantum mechanics, had only provisional validity, but never definitive; they are theories that he would later describe as incomplete. Einstein’s dissatisfaction with an eventual violation of the classical causality is clearly anticipated as early as 1920, four years before the appearance of the BKS proposal, in a letter to Born.41 That business about causality causes me a lot of trouble, too. Can the quantum absorption and emission of light ever be understood in the sense of the complete causality requirement, or would a statistical residue remain? I must admit that there I lack the courage of my convictions. But I would be very unhappy to renounce complete causality.

Obviously, the publication in 1924 of the BKS proposal only increased his fears. After learning of its contents, Einstein expresses his feelings in an expressive metaphor in a letter addressed to Born’s wife.42

41

Letter from A. Einstein to M. Born, 27 January 1920. In Born (1971), 23. Emphasis in the original. 42 Letter from A. Einstein to H. Born (M. Born’s wife), 29 April 1924. In Born (1971), 82. Emphasis in the original.

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Bohr’s opinion about radiation is of great interest. But I should not want to be forced into abandoning strict causality without defending it more strongly than I have so far. I find the idea quite intolerable that an electron exposed to radiation should choose of its own free will [aus freiem Entschluss], not only its moment to jump off, but also its direction. In that case, I would rather be a cobbler, or even an employee in a gaming-house, than a physicist. Certainly my attempts to give tangible form to the quanta have foundered again and again, but I am far from giving up hope. And even if it never works there is always that consolation that this lack of success is entirely mine.

The disagreement that Einstein expresses here, in relation to the essence of Bohr’s ideas, would be maintained for life. It would persist even after the formulation of the two theories that provide the current solution to the problems we have been explaining: quantum mechanics (matrix version by Heisenberg in 1925 and wave version by Schrödinger in 1926) and quantum electrodynamics (Dirac in 1927). The discrepancy between Einstein and Bohr about the interpretation of the quantum formalism never disappeared. Some people think that this initial confrontation, between their respective opinions about the BKS proposal, must be considered the real starting point of what would later become known as the “Bohr-Einstein debate”, which will be dealt with in the next chapter.43 The onset of such disagreements placed Ehrenfest between a rock and a hard place, for the dispute between the two colleagues, to whom he was linked by a close personal relationship, was becoming increasingly uncomfortable, especially as he had no solution to offer to end the bitter dispute. The early results of the experiments of Bothe-Geiger and Compton-Simon seemed to definitely invalidate the BKS proposal. The circumstance prompts Einstein to write to Ehrenfest, with relative enthusiasm, stating that “you know, of course, the result of Geiger-Bothe [experiment] certainly, neither of us doubted that”.44

43

For more detailed information on the beginnings of this debate, see Klein (1970), especially 23– 39. 44 Letter from A. Einstein to P. Ehrenfest, 18 August 1925. In Nollar-James; Hentschel (2018), 63.

4

The Last Collection (1924–1925): Formulation of The First Quantum Statistics

Einstein’s gas theory, and the short but infinitely far-seeing remarks that he suggested, form the link between De Broglie’s matter waves and Schrödinger’s wave mechanics. Martin J. Klein (1964), 46

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 L. Navarro Veguillas, The Lesser-Known Albert Einstein, History of Physics, https://doi.org/10.1007/978-3-031-35568-4_4

191

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4 The Last Collection (1924–1925): Formulation of The First Quantum Statistics

Paul Ehrenfest, Albert Einstein and Wassik, the youngest of Paul and Tatyana’s four children. The failure of the marriage and concern for the future of Wassik, who had Down’s syndrome, caused Paul to fall into a severe state of depression, which led him to shoot his son on one of his visits to the clinic where he was being treated. He then committed suicide by shooting himself on 25 September 1933. [Courtesy of the National Museum Boerhaave Leiden]

4.1 Fame and Unrest in Berlin

4.1

193

Fame and Unrest in Berlin

We had left Albert in Berlin somewhat overwhelmed by his growing fame as an extraordinary scientist and, moreover, as a person concerned with the social problems of his time. By the mid-twenties, however, his life with Elsa was generally peaceful, with hardly any epistolary contact with his former wife Mileva and their two children, who live with their mother in Zurich. Although anecdotally, to illustrate his ideas about women and marital happiness, we reproduce here some statements of Einstein, in Berlin, to a physics student—Esther Salaman, of Jewish origin—in the spring of 1925, when she confessed to him her difficulties in understanding theoretical physics.1 On the way to university, the student confessed to Einstein’s enquiry about how she was getting on in her studies that she felt she would never become a perfect theoretical physicist. She lacked the creative talent to do so. Einstein admitted: “Only a few minds are creatively talented. I would never let a daughter study physics. I am glad that my present wife [Elsa] doesn’t understand anything about scientific matters. My first wife [Mileva] was very different in this respect”. Esther Salam replied: “But Marie Curie was a creative woman!”. "Yes, we have spent some holiday days in Engadin with the Curie family. But Madame Curie never heard the birds singing”.

In 1924, he was definitively forced to accept German citizenship—due to a legal imperative corresponding to his academic appointments—without renouncing his Swiss citizenship. He was delighted to see the inauguration of Einstein Institute in the same year, was opened in the brand-new Einstein Tower (Einsteinturm) in Potsdam, near Berlin. The funding was private and came mainly from German industry and commerce. Although he would never work there, Einstein advised from the beginning in its construction and in the first steps of the research activities in the centre, directed to the analysis of the light that reaches us from the stars after passing through intense gravitational fields in their journey, which eventually could be used to test some of the predictions of the general theory of relativity. It is a time of full scientific and social success in which Einstein receives important distinctions. For example, on 30 November 1925, he was awarded the “Copley Medal” of the Royal Society of London. In February 1926, the “Gold Medal” of the Royal Astronomical Society, both of which were among the highest British scientific distinctions of the time. His prestige earned him various appointments and invitations from different international centres. In 1925, for example, he travelled to South America, where he visited Buenos Aires, Rio de Janeiro and Montevideo, giving lectures on relativity. His opinion on nonscientific topics—for example, on religion and politics—began to have an unexpected resonance. In those days, the rise of anti-Semitism and also of militaristic ideas in German society increased markedly. Einstein shows, in private and in public, his deep

1

In Seelig (2005), 316. Salaman would emigrate shortly afterwards to England, with a letter of recommendation from Einstein to introduce herself to Rutherford. Finally, she would end up devoting herself to the literature in that country.

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4 The Last Collection (1924–1925): Formulation of The First Quantum Statistics

concern for the near future. In 1925, he became a member of the board of trustees of the Hebrew University of Jerusalem, which was inaugurated in the same year. In remarks to Esther Salaman, the abovementioned student, when she confesses to him her intentions to leave Germany because of the growing anti-Semitism he confesses [in Seelig (2005), 316]: As far as I am concerned, I am very happy that I’m not German, but Swiss. And I am glad to have lived in a democracy [Switzerland] for so many years in my youth. The Germans say I am German when it suits them, and when it doesn’t suit them, they say I am Jewish. Anyway, I can’t get used to their way of being. Somehow, I can’t adapt to their style.

Around this time, Einstein joined—to the displeasure of some German political and scientific authorities—the cause of world peace, signing, together with other prominent pacifists, an international manifesto in favour of universal brotherhood and against compulsory military service.2 His scientific activity, somewhat muted in recent years, was spectacularly revived in 1924, stimulated by a letter from a young and unknown Bengali physicist. The latter’s new ideas on the statistical treatment of black body radiation inspired Einstein to extend them to ideal monoatomic gases—essentially developed in 1924–1925—in a form that constitutes the first formulation of a model for quantum statistical mechanics. Moreover, as we shall show, some of Einstein’s new ideas on ideal gases were to contribute to a large extent to the birth of wave mechanics. In 1925, Heisenberg developed the formalism of matrix mechanics and, in 1926, Schrödinger did the same with the formalism of wave mechanics. The rise of these two formulations represented a very hard blow against Einstein’s most intimate physical and philosophical conceptions, which were totally incompatible with the interpretation of the respective mathematical formalisms of both formulations; all of which would end up separating him from the later development of quantum mechanics, as we shall see in due course.

4.2

A Shooting Comet: Bose, 1924

In Sect. 3.5, we have shown some of the obstacles encountered by the photon before it was admitted as a fully-fledged entity in physics. There, we denounced the frequency with which it is asserted that, after the detection and theoretical explanation of the Compton effect, in 1922–1923, the photon was definitively incorporated into physics and, in the same section we justified the inaccuracy of this statement. In particular, we commented on the attempt that, in 1924, Bohr, Kramers and Slater made by suggesting a curious alternative that, keeping the wave description of the electromagnetic field, gave new bases for the study of matter-radiation interaction. This provided a theoretical explanation of the Compton effect, even if it was at the

2

Among others, the manifesto was also signed by Russell, Gandhi and Tagore.

4.2 A Shooting Comet: Bose, 1924

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cost of introducing possible violations of ordinary causality and the principles of conservation of energy and momentum, which would only have statistical validity. In the late spring of 1924, in the time between the appearance of the BKS proposal and its experimental rejection, Einstein receives a paper in English by the Bengali physicist Satyendra Nath Bose (1894–1974), who accompanied it with this handwritten note.3 Respected Sir: I have ventured to send you the accompanying article for your perusal and opinion. I am anxious to know what you think of it. You will see that I have tried to deduce the coefficient 8π v 2 /c3 in Planck’s Law independent of the classical electrodynamics, only assuming that the ultimate elementary regions in the Phase-space have [a volume with] the content h3 . I do not know sufficient German to translate the paper. If you think the paper worth publication I shall be grateful if you arrange for its publication in Zeitschrift für Physik. Though a complete stranger to you, I do not feel any hesitation in making such a request. Because we are all your pupils though profiting only by your teachings through your writings. I do not know whether you still remember that somebody from Calcutta asked your permission to translate your papers on Relativity in English. You acceded to the request. The book has since been published. I was the one who translated your paper on Generalised Relativity. Yours faithfully S. N. Bose

Bose was born in Calcutta on 1 January 1894, at whose university he graduated in physics and took the first steps of his research career. In 1924, he was assistant professor at Dhaka University, where he had been delivering courses on theoretical physics for three years: electromagnetism, statistical mechanics, relativity and atomic theory, among others. He was a critical reader of the works of great masters such as Planck, Einstein, Ehrenfest, Bohr and Sommerfeld. His publications up to that time—since 1918—were mainly on statistical mechanics and atomic theory. The paper that Bose sent to Einstein had been previously submitted to the prestigious British Philosophical Magazine, but it was not accepted for publication, in spite of the three articles signed by Bose himself that had already appeared in it; two of these in collaboration with Meghnad Saha (1893–1956), also a young Bengali physicist who would acquire international prestige in the field of astrophysics. Precisely in the first of these two papers, in which the influence of the finite volume of the molecules in the equation of state of gases is analysed, what later became known as the “Bose-Saha equation” is proposed.4 Bose’s paper was entitled “Planck’s law and light quantum hypothesis” and must have made a deep impression on Einstein, judging by his immediate attitude:

3

Letter from S. Bose to A. Einstein, 4 June 1924. Reprinted in Kormos-Buchwald et al. (2015), 399. 4 For more details about Bose’s academic and scientific life, as well as for the bibliography that justifies our statements in this section, see, for instance, Navarro (1996). In Spanish.

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in about a week, he studies the paper, translates it into German, recommends it for urgent publication in Zeitschrift für Physik and adds to the article a personal note as translator. This appears at the end and reads as follows5 : Translator’s comment. In my opinion, Bose’s derivation of Planck’s formula represents an important step forward. The method used here also provides a basis for the quantum theory of the ideal gas, as I will explain elsewhere.

The publication of the paper, with the translator comment, brought such recognition to Bose that he easily obtained a longed-for two-year leave to further his studies in Europe. In the early autumn of 1924, he arrived in Paris with the idea of collaborating in some experimental project. Through Bengali circles, he came into contact with Langevin who, having heard of his ambitions, recommended that he approach Madame Curie to work under her direction in experimental radioactivity. An anecdote. During the interview with the famous researcher, in English, the young Bose listened with rapt attention that she recommended, considering other precedents, that for the first six months he should forget physics and concentrate on the study of the French language and then join the team. Bose was unable to let his interlocutor know that he had previously studied French for fifteen years! It is a fact that the plans to collaborate with M. Curie never came to fruition. Nevertheless, during his stay in Europe, which was not very productive on the whole, Bose had the opportunity to meet leading physicists of the time.6 Two and a half months before leaving for Europe, Bose sends Einstein a second paper with some considerations of his own about equilibrium between radiation and matter. Einstein also translates this paper and, as he did with the first one, he sends it to Zeitschrift für Physik, where it would be published in September 1924, one month after the previous one. However, now the translator does not feel the same enthusiasm as before the first publication: in a commentary that he adds at the end of the new work, Einstein criticizes the considerations exposed there by Bose, at the same time that he expresses his deep disagreement with them. [Bose (1924 b)]. It was not until the end of 1925 that Bose came into personal contact with Einstein. The meeting takes place in Berlin and, in view of Bengali’s insistence working directly with him, Einstein proposes two possible topics. The first one consisted of determining if the new statistics, generated from Bose’s contribution, involved some interaction between photons. The second one, of much greater importance, was a proposal to analyse the relation, if any, between the new statistics and Heisenberg’s matrix mechanics, formulated a few months earlier. It is recorded that Bose did not make substantial progress in either case. At the end of 1926, he returned to the University of Dhaka, as a full professor. In 1945, he

5

Bose (1924 a), 181. This note is not included in the English translation cited in the bibliography. In France he was also in contact with the De Broglie brothers and in Germany, through Einstein, with Fritz Haber, Otto Hahn and Lise Meitner, among others.

6

4.2 A Shooting Comet: Bose, 1924

197

moved to the University of Calcutta, the city where he died in 1974 at the age of eighty. Since his return from Europe, Bose hardly engaged himself in research, devoting his time to teaching and management. He would still publish some research papers, but without resonance, on statistical mechanics and relativity. They were contributions of little interest, according to his own confession about his scientific work after his return from Europe7 : I never participated in science ever again. I was like a comet that appeared once and never came back.

Bose’s remarkable contribution is contained in the first of the articles sent to Einstein. It is a short paper that does not even include bibliographical citations. Although our interest lies in Einstein’s investigations, we shall dwell on this paper by Bose—devoted exclusively to obtaining a new deduction of Planck’s law— because of its importance in capturing the evolution of Einstein’s thinking about the light quantum. Bose disallows the pre-existing deductions of Planck’s radiation law because they incorporated, at one stage or another, the following relation: ) ( 8 π v2 dv E, (4.1) ρv dv = c3 that links the radiation spectral energy density ρv with the mean energy per oscillator E of a set of monochromatic oscillators of frequency v (c represents the speed of light in vacuum). The factor in the parentheses was always deduced by using Maxwell’s electromagnetism, which, according to Bose, is a serious incoherence if it is incorporated in derivations based on quantum hypotheses, as it is the case when introducing in (4.1) the mean value of quantized energy. For the Bengali, as for almost all the physicists of the time, electromagnetic field theory was incompatible with any quantum assumption [Bose (1924a). In Theimer; Ram (1976), 1056–1057]: In all cases it appears to me that the derivations [from Planck’s law] have insufficient logical foundation. In contrast, the combining of the light quanta hypothesis with statistical mechanics in the form adjusted by Planck to the needs of the quantum theory does appear to be sufficient for the derivation of the law, independent of any classical theory. In the following I wish to sketch briefly the new method.

Bose begins by considering thermal radiation of total energy E (not to be confused with E appearing in the last expression) enclosed in a volume V, containing Ns photons —in the paper they are still called quanta— of energy hv5 (from s = 0 to s = ∞). As we have been doing, we will try to keep the original notation as

7

Interview quoted in Mehra; Rechenberg (1982 a, part B), 571.

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4 The Last Collection (1924–1925): Formulation of The First Quantum Statistics

much as possible. In this rare notation, s always represents a superscript—never a power—designating concrete values of the frequency and the corresponding number of photons. Thus, the total energy can be expressed as follows: E=

Σ

Σ Ns hvs = V

ρv dv

(4.2)

s

According to Boltzmann’s statistical mechanics, the thermodynamic equilibrium state is the “most probable” state compatible with the constraints imposed by the macroscopic system. Bose, well versed in the discipline, has to begin by assigning a probability to each possible distribution of photons. Then, the equilibrium distribution will be determined by maximizing the corresponding probability, taking into account the constraint (4.2). Bose associates to each photon, as is usual in statistical mechanics, a point in a six-dimensional phase space of six dimensions, three spatial dimensions (X , Y , Z) and the three corresponding linear momenta (px , py , pz ) . As the momentum of a photon of frequency v has the value hv/c, another constraint appears: px2 + py2 + pz2 =

h2 v2 c2

(4.3)

To evaluate the volume that, in the phase space, corresponds to an interval of frequencies (v, v + dv), Bose integrates with respect to the other variables and obtains: Σ Ω=

(

hv dx dy dz dpx dpy dpz = V 4π c

)2 ( ) h3 v2 hv = 4π 3 V d v d c c

(4.4)

If the photon phase space is considered subdivided into cells of volume h3 , the number of cells corresponding to that interval of frequencies (v, v + dv) will be given by 4 π V v2 d v/c3 . Then, Bose states [in Theimer; Ram (1976), 1057]: Concerning the kind of subdivision of this type [into cells of volume h3 ], nothing definite can be said. However, the total number of cells must be interpreted as the number of the possible arrangements of one quantum in the given volume. In order to take into account polarization, it appears mandatory to multiply this number by the factor 2 so)that the number ( of cells belonging to an [elementary] interval dv becomes 8 π V v 2 d v/c3 .

Bose is aware of the somewhat heuristic character of the above assumptions, since it does not include any justification; a justification that, on the other hand, was not yet within the reach of physicists. In any case, the consideration in a monoparticular phase space of cells of volume h3 as receptacles of points corresponding to the possible states of a system was already common among the practitioners of quantum theory. Although today it could be justified on the basis of the indeterminacy principle, this is not a necessary invocation. Let us see, for

4.2 A Shooting Comet: Bose, 1924

199

example, a way of arriving at the same conclusion, using arguments from an incipient quantum theory, such as the one that existed even before the appearance of the Bohr model (1913) and Sommerfeld rules (1915). In 1900, Planck imposed the following restriction on the possible values of the energy of a monochromatic oscillator of frequency v: it could only have the values E = n hv, where n = 0,1,2, … The trajectory of the oscillator in its phase space (x, p) is determined by the following equation: p2 mω 2 x + 2m 2

2

= E,

(4.5)

where ω = 2π v. It determines an ellipse enclosing the following area: / √ 2E 2π E E S = π 2 mE = = 2 mω ω v

(4.6)

Thus, the Planckian hypothesis on the one-dimensional monochromatic oscillator leads to the fact that the area bounded by the ellipse corresponding to a certain energy —that is, to a certain value of n— has the value Sn = nh. Therefore, we will say that a state with n quanta ‘occupies’ an area of value nh, and the one corresponding to (n-1) quanta ‘occupies’ the area (n-1)h; all in its phase space, which is two-dimensional. Then, the state associated with a single quantum will occupy a cell of area h in that space. If itis extrapolated to the case of a three-dimensional harmonic oscillator, we can conclude that each cell now has a hypervolume h3 , in the phase space, which is now 6-dimensional. A reasoning such as the previous one can justify the reductive factor h3 used by Bose in 1924 to determine the number of cells, without having to resort to the indeterminacy principle —then still to be discovered— or to other anachronisms. The next step -—according to Boltzmann’s method— -consists in the deters mination of the thermodynamic probability W of the macroscopic state. ( 8 8 ) If N 8 denotes the number of photons in the frequency interval v , v + d v , in how many different ways can they be distributed among the As cells corresponding to that frequency range? To answer such a question, numbers ( )s Bose enters in his treatment the occupancy ( )s of the respective cells: p0 is the number of cells vacant cells, p1 the number ( )s of cells containing one photon, p2 the number of cells containing two photons, etc. Then, the respective expressions for the number of cells As and the number of photons Ns will be: Σ ( )s (vs )2 s ( )s ( )s ( )s pr dv = p + p + p + . . . = 0 1 2 c3 r Σ ( )s ( )s ( )s ( )s Ns = 0 · p0 + 1 · p1 + 2 · p2 + . . . = r · pr

As = 8π V

r

(4.7) (4.8)

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4 The Last Collection (1924–1925): Formulation of The First Quantum Statistics

Be careful with this original notation! It should be emphasized that s does not represent a power but a superscript that characterizes the range of frequencies considered. Continuing within this frequency range, the number of possible distributions Ws —of the Ns photons between the As cells and in the indicated conditions—will be given by this expression: Ws =

As ! (p0 )s !(p1 )s !(p2 )s ! . . .

(4.9)

Now, considering the above results, Bose obtains the following expression for the (relative) probability (of )a macroscopic state, which he assumes to be s characterized by the set of all pr : As ! W = Π Ws = Π ( )s ( )s ( )s s s p 0 ! p1 ! p2 ! . . .

(4.10)

Starting from such a background, the Bengali physicist obtains for the first time —and always in Boltzmann’s line, he says— a quantum derivation of Planck’s law. The first quantum derivation of Planck’s law (Bose, 1924)

To determine the equilibrium state of the system —electromagnetic radiation in a cavity— Bose calculates the maximum of W, with a prefixed value for the total energy E, which is given by (4.2). As it is usual in statistical mechanics, for reasons of convenience, log W —instead of W— is maximized an Stirling approximation is applied, given the large value of the numbers involved in (4.10). Thus, Bose calculates the maximum conditionate by the constraint (4.2) of the following expression: log W =

Σ

As log As −

s

Σ Σ ( )s ( )s pr log pr s

(4.11)

r

A standard calculation —which we are not going to reproduce here— leads to the determination of the equilibrium state, i.e., to the respective values of all (pr )s : ) ( r h vs , (pr )s = Bs exp − β

(4.12)

where β is the corresponding multiplier, which is to be determined by means of the constraint (4.2), and the coefficient Bs is given by the following expression: [ B =A s

s

)] ( h vs −1 1 − exp − β

(4.13)

4.2 A Shooting Comet: Bose, 1924

201

From the last two equalities we obtain the law of distribution of photons among the considered frequency ranges:

Ns =

Σ r

) ( s As exp − hvβ ( )s ) ( r · pr = s 1 − exp − hvβ

(4.14)

Bose presents the above result in a slightly different way, by substituting this value of Ns in the expression (4.2) and then taking into account the value of As , given by the first equality in (4.7). This allows him to write the following expression for the energy of the system:

E=

Σ

Ns hvs =

s

Σ 8π h(vs )3 c3

s

) ( s exp − hvβ ) ( dvs V s 1 − exp − hvβ

(4.15)

On the other hand, substituting in the relation (4.11) the values obtained for the set of all the (pr )s , Bose obtains the following equality for the entropy of the radiation in the cavity: {

)]} [ ( E Σ s hvs S = k log W = k A log 1 − exp − − β β s

(4.16)

The thermodynamic relation ∂∂ ES = T1 —where T represents the absolute temperature—, along the last expression allows to identify the multiplier β with kT, which, on the other hand, represents the usual way of introducing the temperature when starting from the microcanonical formalism in statistical mechanics. Substituting now this value for β in the expression (4.15) Bose arrives at the following result: E=

Σ 8π h(vs )3 s

c3

V

1 ( s) dvs , hv exp kT − 1

(4.17)

“which is equivalent to Planck’s formula”, according to the last sentence of the paper. In order to recognize in this expression the usual form of Planck’s law for the spectral energy density ρv , it is sufficient to compare (4.17) with (4.2) to obtain the well-known formula: ρv (T ) d v =

1 8 π h v3 ( ) dv · hv c3 exp k T − 1

(4.18)

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It is on the basis provided Bose’s paper that one can understand the next step in the evolution of Einstein’s thinking about the photon. Therefore, it is appropriate to outline some relevant aspects of Bose’s contribution, which could be obscured by his somewhat schematic presentation, sometimes imprecise. However, first an aside, if only as a curiosity, to show some examples of this not very careful presentation. The superscript s, which labels the frequencies, begins by being introduced as a subscript. There are typographical errors—for example, in the original of (4.7) the superscript is missing in the frequency and the volume V —which have not been corrected in Theimer and Ram’s translation cited in the bibliography, either; we don’t know whether this is due to an excess of zeal or a lack of warning. Moreover, and although their identification is simple, we miss an explicit clarification of the meaning of the different symbols that appear in the paper: c, V, S, T, k… However, the inaccuracy is not only in the formal aspect of the article. It seems that the essential aim of the paper—the derivation of Planck’s law without resorting to Maxwellian electromagnetism—is achieved, but to obtain it Bose had to include certain characteristics of photons which, without explicitly mentioning them, are essential to reach the proposed goal, at the same time that they represent substantial innovations with respect to the status in force at that time. Bose considers an aggregate of massless particles, without expressly stating so, but which can be deduced from his use of (4.3). On the other hand, he associates to each photon something like an inner degree of freedom —the polarisation— with two possible values, without the slightest clarification. The treatment, we would say today, seems to be based on the microcanonical formalism of statistical mechanics. This is what must be assumed when the entropy is obtained from the distribution that maximizes the probability of the macroscopic state and then introducing the absolute temperature as the inverse of the derivative of entropy with respect to energy. However, the Bose formalism that he actually uses could not be considered orthodox today. Let us see. In fact, by considering from the very beginning the total energy as a constraint, it indicates that he is going to use the microcanonical ensemble. However, in the microcanonical formalism, the number of particles that integrate the system must be fixed and, in the paper, this condition is not imposed. Only one multiplier appears in the derivation, precisely the one associated with the total energy. On the other hand, one might think that Bose has opted for the canonical formalism, in which the temperature is prefixed. However, this is not entirely convincing either, since in this case, the energy admits fluctuations, while in his article, only a fixed energy appears. Nor does the treatment fit adequately into the grancanonical formalism because, although it is true that in practice no condition has been imposed on the total number of photons, the constancy of the total energy of the radiation has been emphasized, which is not compatible with the rigorous use of this formalism. In short, Bose’s treatment does not coherently conform to any formalism based on classical ensemble theory. How is it possible to arrive at a correct result—Planck’s formula—without conforming to a concrete and precise formalism? A hint could go in the following direction. Without going into further detail, one can be stated that the results

4.2 A Shooting Comet: Bose, 1924

203

obtained through any of the usual statistical ensembles leads to the same result when large values are admitted for the number of particles (which is assumed when applying the Stirling approximation) and large volume (implicit in admitting a continuous spectrum for the photon momentum). Hence, the inadequate mix of ensembles contained in the presentation of the Bengali physicist can eventually lead to Planck’s law. Bose’s deduction, already refined to conform rigorously to the grancanonical formalism, is found today in classical texts.8 Apart from the previous comments on the poor presentation of Bose and on the almost nonexistent justification of the treatment used —criticisms that, by the way, are not usually seen in writing— there are other aspects of greater importance that should be highlighted. Perhaps it is the expression (4.10) for the (relative)  s probability of a state —defined by the set of all pr — that has a greater number of transcendent implications, in spite of being qualified as “evident” (offenbar) in the publication. Let us see. Considering Bose’s undoubted knowledge of Boltzmann’s ideas and statistical methods, it is reasonable to think that he was familiar with Boltzmann’s expression for thermodynamic probability. Consider the thermodynamic state of a gas of N molecules —all of them identical and independent, according to the feeling of the time— such that they are distributed in the elementary subdivisions (cells) of the molecular phase space as follows: N0 in cell c0 , N1 in cell c1 , N2 in cell c2 , etc. The (relative) probability W assigned by Boltzmann to such a macroscopic state is the following: W=

N! N0 ! N1 ! N2 ! . . .

(4.19)

The then admitted distinguishability of the molecules, although without being explicitly recognized, is manifested in the factorials of the denominator: the exchange of molecules ‘housed’ in different cells implies new configurations, the thermodynamic probability being proportional to the number of different configurations compatible with the quantities defining the macroscopic state of the system: total energy, number of molecules and volume, in the case of gas. Comparison between (4.9 and 4.19), for example, shows that Bose’s reasoning involves a subtle but crucial deviation from Boltzmann’s ideas: the role that in this case—i.e., in (4.19)—is played by molecules, in Bose’s case—i.e., in (4.9)—is played by cells. Thus, the number of these must be prefixed, but not the number of photons, which never appears as a constraint. The distinguishability among Boltzmann’s molecules is now translated —not because Bose specifies it but as a consequence of his treatment— into the distinguishability among cells; hence the factorials in the denominator of (4.9).

8 See, for example, Huang (1987), 278–283. Recall that in the grancanonical ensemble the variables characterizing the thermodynamic system are volume, temperature and chemical potential, which is zero in the case of a photon gas.

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4 The Last Collection (1924–1925): Formulation of The First Quantum Statistics

Since, in Bose’s derivation, it is only the number of photons in each cell that matters, photon indistinguishability is a fact, although in the paper he does not emphasize this new property, which breaks with the traditional ideas about classical distinguishability among the gas molecules. Thus, it turns out that the photons, considered ‘Bose-like’, follow a new statistics, different from the classical one of Maxwell–Boltzmann for molecules. Let us recall, moreover, that in the classical treatment of ideal gases the statistical independence of its molecules is assumed. In a different treatment, such as that of Bose, such independence does not seem to be guaranteed a priori, even though no interaction between photons has been made explicit. We will have to go deeper into this matter and its consequences later on, since all this turned out to be of great importance. In Bose’s article there is no clue to discover his awareness of the fact that he has introduced, with his original treatment, a new statistic, at least for a photon gas. We shall see that Einstein, shortly afterwards, went much further than Bose as far as the ultimate implications of the new method are concerned. Let us close this section with Bose’s honest acknowledgement of the limitations with which he conceived his own work [quoted in Pais (2005), 424]: I had no idea that what I had done was really novel [...] I was not a statistician to the extent of really knowing that I was doing something which was really different from what Boltzmann would have done, from Boltzmann statistics. Instead of thinking of the light-quantum just as a particle, I talked about these states. Somehow this was the same question which Einstein asked when I met him [in October or November 1925]: how had I arrived at this method of deriving Planck’s formula?

Before concluding this section, we would like to highlight an aspect that, at first sight, may seem surprising. We have just explained how the statistical method unconsciously introduced by Bose, clearly different from that of Boltzmann, arrives at Planck’s formula, which expresses the equilibrium energy distribution of radiation, understood now as a photon gas. However, Einstein took a first step towards the photon in 1905 using Boltzmann’s statistics. How can both situations be fitted? The apparent paradox has a clear explanation if we turn to current knowledge. In his 1905 paper Einstein was moving in the zone of validity of Wien’s experimental law for black body radiation, which corresponds to a physical situation in which the inequality h v/k T >> 1 holds. Today, we know that, in this case, the mean value of the occupancy numbers of the different cells is so small that Bose–Einstein statistics yields the same result as the Boltzmann statistics, in spite of the radically different assumptions of one and the other. Hence, the equivalence between the two statistics is within the region of validity of Wien’s formula. In any case, it is important not to confuse two different aspects, although both are involved in the Bose–Einstein statistics: the quantum character of the treatment that is introduced by considering cells of volume h3 in the one particle phase space and the attribute of “indistinguishability” implicitly assigned to photons. These are two independent aspects since the quantum treatment in certain cases —for example, when there is a certain degree of localization of the particles, as in the

4.3 From Photons to Molecules: Quantum Theory of Ideal Gases, 1924

205

case of the atoms of a crystalline lattice— is compatible with the distinguishability of the particles. Returning to Einstein’s 1905 treatment, tacitly considering the approximation h v/k T >> 1 is what allowed him to ‘ignore’ the indistinguishability of quanta, but not the discreteness of the energy of radiation, which was actually the property then detected by Einstein. Thus, the assumption implied by the above inequality made possible the surprising result that, operating ‘Boltzmann-like’, Einstein took the first steps in pursuit of the photon. It is perhaps even more surprising to realize that if Einstein had used Planck’s law instead of Wien’s law, he would not have arrived at the energy quantum in 1905: it is not difficult to verify the incompatibility between Planck’s law, Boltzmann’s statistics, and quantisation of the energy of radiation. In contrast, such compatibility exists if, instead of Planck’s formula, Wien’s formula is used, as Ehrenfest demonstrated in 1911. [See Navarro; Pérez (2004), 110–126].

4.3

From Photons to Molecules: Quantum Theory of Ideal Gases, 1924

In spite of Einstein’s promptness and success in extending Bose’s ideas on radiation to ideal gases, it must be noted that Einstein had little admiration for the Bengali physicist. For example, in Einstein’s correspondence with his friend Besso there is no reference to Bose’s name or work, although he commented sometimes on his new dedication to obtain the quantum theory of ideal gases.9 Nor does he mention Bose’s contribution in his scientific autobiography. [Einstein (1949a)]. However, it is recorded that, only one week after submitting Bose’s translated article for publication in Zeitschrift für Physik, Einstein presents a paper at the Prussian Academy of Sciences —session 10 July 1924— in which he uses Bose’s method to obtain new and interesting results on the behaviour of ideal gases. [Einstein (1924)]. As we shall see below, this paper forms a unit with another one, to be published in 1925, presented at the session of January 8 of this year at the same institution. [Einstein (1925a)]. What was the true appeal of Bose’s treatment in Einstein’s eyes? In other words, what did he see in Bose’s paper that he described —in the enclosed note as translator— as an “important step forwards” that provided “a basis for the quantum theory of ideal gases”, which he himself undertook to develop promptly? We have no doubt that a strong motive for the positive appraisal of Bose’s work must have been that, for the first time, a derivation of Planck’s law free from any dependence on Maxwell’s electromagnetism was obtained. Photons were treated as if they were ordinary particles, with the only exception, apart from the two possible

9

See, for example, the letter from A. Einstein to M. Besso, 5 June 1925. In Nollar-James et al. (2018), 26–27.

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4 The Last Collection (1924–1925): Formulation of The First Quantum Statistics

values for their polarisation, of having zero mass. Basically, all this could be interpreted as additional support for Einstein’s hypothesis in 1916 on the directionality of elementary processes of elementary processes and the reality of quanta, which we have already dealt with in some detail in Sect. 3.3. However, we believe that there were even more important reasons why Bose’s contribution attracted Einstein’s attention. It is certain that he immediately realized that Bengali’s treatment could be transferred without major difficulties to the ideal gas by virtue of the existing analogy between black body radiation and the ideal gas of molecules. An analogy that was already exploited by Einstein in 1905, although in reverse, in his first paper on the energy quantum of radiation. If Bose’s method leads to the quantum theory of radiation represented by Planck’s law, its analogue applied to the ideal gas should provide a quantum theory of ideal gases, yet to be formulated. To confirm Einstein’s aims in 1924, as well as his view of the method proposed by Bose, nothing better than the introduction of his article [Einstein (1924). In Hentschel; Nollar-James (2015), 276]: A quantum theory for the monatomic ideal gas free from arbitrary assumptions still does not exist today. This lacuna ought to be filled in the following on the grounds of a new approach derived by Mr. D. Bose [sic], upon which this author has based a highly noteworthy derivation of Planck’s radiation formula. The path to be taken below, following Bose, is to be described thus: The phase space of an elementary structure (here of a monatomic molecule) is divided, with reference to a given (three-dimensional) volume, into “cells” of extension h3 . If many elementary structures are present, then their (microscopic) distribution as regards thermodynamics is characterized by the ways and means by which the elementary entities are distributed across these cells. The “probability” of a macroscopically defined state (in Planck’s sense) is equal to the number of different microscopic states by which the macroscopic state can be thought to be realized. The entropy of the macroscopic state, and therefore the statistical and thermodynamical behavior of the system, is then determined by Boltzmann’s principle.

Let us now analyse now the theory proposed by Einstein, some of whose aspects, as we will have the opportunity to verify, would have had a certain influence in the appearance of wave mechanics. His first paper, in 1924, is structured in five sections. The first two sections are devoted to the study of cells in phase space and of the probability and entropy of a macroscopic state, while in the third he characterizes the state of thermodynamic equilibrium as the state of maximum entropy. Finally, in the last two sections, the connection between the new treatment and the classical treatment is analysed. Einstein’s paper can be understood as a transcription of Bose’s, but substituting the characteristics of photons for those of molecules and developing new ideas with his own touch. Thus, he begins by writing the corresponding expression for the volume in phase space of the monoatomic molecule: Σ Φ=

Σ ...

dx dy dz dpx dpy dpz

(4.20)

4.3 From Photons to Molecules: Quantum Theory of Ideal Gases, 1924

207

After reminding us of the expression for the energy of a (free) molecule of mass m, E=

) 1 ( 2 px + py2 + pz2 , 2m

(4.21)

and taking into account (4.20), Einstein writes the expression of the accessible hypervolume, in the molecular phase space, for the state of a molecule with energy equal to or less than E, enclosed in a given volume V: 3 4 Φ = V · π (2 m E) 2 3

(4.22)

Extrapolating Bose’s reasoning, the number of cells corresponding to the narrow energy band in the interval (E, E + ΔE) (assuming ΔE E 1 h3 n

(4.34)

Einstein numerically evaluates the first member—which of course is a dimensionless quantity—of the last inequality for the case of hydrogen, under normal conditions of pressure and temperature, obtaining the approximate value of 60,000, which is “very large compared to 1. Classical theory thus still delivers a quite good approximation here”. The approach entailed by the classical treatment is all the more valid the more strongly the inequality (4.34) holds. Moreover, the use of the classical approximation is less justified the higher the density of molecules—represented by the ratio n/V—and the lower the absolute temperature. This situation of lack of validity of the classical treatment, Einstein justifies, occurs, for example, in the case of helium in the neighbourhood of its critical state, where one can expect a behaviour far from that predicted by the classical theory. Admitting that the right conditions are given for inequality (4.34) to be satisfied, a simple calculation leads to the expression (4.28) for the entropy of the gas, which can then be written as follows: ] [ 1 V (4.35) S = ν R log e5/2 3 (2 π m k T )3/2 , h n where ν now represents the number of moles contained in V, and R is the universal constant of gases. This expression for the entropy of the ideal gas had already been obtained by other researchers, as Einstein recognizes [Einstein (1924). In Hentschel; James (2015), 281]: This result about the absolute value of the entropy is in agreement with well-known findings of quantum statistics.

4.3 From Photons to Molecules: Quantum Theory of Ideal Gases, 1924

211

The above expression for the entropy can also be arrived at by introducing some kind of quantisation by means of usual rules and maintaining the habitual distinguishability of molecules, but introducing a factor of reduction of the number of states ad hoc —the well-known N!—to avoid the lack of extensivity of the entropy to which the classical statistical treatment leads. We will return to this topic in the next section.11 At the epoch under consideration, the validity of Nernst’s theorem in its ‘strong’ statement—entropy cancels at zero absolute temperature—was beyond doubt, so its theoretical deduction could be taken as a crucial test. Einstein’s fine instinct does not miss the opportunity to refer sharply to that question in light of his own results. When reasoning with low temperatures he could not simply use the expression (4.35), which had been deduced with assumptions that, at first sight, seemed plausible only in the case of high temperature. Nothing could be more explicit in this respect than the paragraph with which he closes the penultimate section of his 1924 paper [Einstein (1924). In Hentschel; James (2015), part B, 281]: According to the theory given here, Nernst’s theorem for ideal gases is satisfied. Our formulas [for the entropy in particular] however do not permit direct application to extremely low temperatures, because we presumed in their derivation that the (pr )s change only relatively infinitely little when s changes by 1. Yet one immediately perceives that the entropy must vanish at absolute zero. For, then all the molecules are located in the first cell; however, for this state there is just a single distribution of the molecules in the sense of our counting. From this the correctness of the assumption directly follows.

Einstein devotes the final section to a comparison between the new expressions and those deduced according to the classical formalism. For this purpose, he introduces the following dimensionless parameter: λ≡

1 , exp (A)

(4.36)

which, under the conditions of validity of the classical treatment, coincides with the inverse of the first member of the inequality (4.34). This allows him to justify that the classical approximation is more justified the smaller the value of λ is in relation to unity. Therefore, Einstein presents his new results concerning the ideal gas as developments in power series of the parameter λ, so that the successive terms provide additional corrections to the classical results.

11

Among others, they had previously justified expressions similar to (4.35) for the entropy of the ideal monoatomic gas by the following: O. Sackur (1911), H. Tetrode (1912), W. H. Keesom (1914), W. Lenz (1915), A. Sommerfeld (1915), P. Scherrer (1916) and M. Planck (1916). See on this subject Mehra; Rechenberg (1982a, part B), 573–574.

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4 The Last Collection (1924–1925): Formulation of The First Quantum Statistics

Einstein’s Model for Monoatomic Ideal Gases (1924)

The main expressions then obtained by Einstein are the following: 3

n = (2π mkT ) 2

∞ V Σ −3 τ τ 2λ h3 τ =1

E=

∞ 3 V Σ 5 3 τ − 2 λτ k T (2 π m kT ) 2 3 2 h

(4.37)

τ =1

∑∞ − 5 τ τ 2λ E 3 = kT ∑ τ =1 3 ∞ n 2 τ−2 λ τ τ= 1

In view of this last equality Einstein writes the following comment12 : The mean energy of the gas molecule at this temperature (as well as the pressure) therefore is always less than the classical value, namely, the factor expressing the reduction the smaller, the larger the degeneracy parameter λ.

If λ is small enough for λ2