From c-Numbers to q-Numbers: The Classical Analogy in the History of Quantum Theory (California Studies in the History of Science)

375 Pages·1993·1.42 MB·English

by  Olivier Darrigol

From c-Numbers to q-Numbers
The Classical Analogy in the History of Quantum
Theory
Olivier Darrigol
UNIVERSITY OF CALIFORNIA PRESS
Berkeley · Los Angeles · Oxford

À mes Parents

Preferred Citation:
Darrigol, Olivier.
From c-Numbers to q-Numbers: The Classical Analogy in the History of Quantum Theory.
Berkeley: University of California Press, 1992.
Contents
ACKNOWLEDGMENTS
CONVENTIONS AND NOTATIONS

INTRODUCTION
PART A PLANCK'S RADIATION THEORY
Introduction
Chapter I Concepts of Gas Theory
Chapter II Planck's Absolute Irreversibility
Chapter III On Irreversible Radiation Processes
Chapter IV The Infrared Challenge
PART B THE CORRESPONDENCE PRINCIPLE
Introduction
Chapter V The Bohr Atom (1913-1916)
Chapter VI Postulates and Principles
Chapter VII Harmonic Interplay
Chapter VIII A Crisis
Chapter IX The Virtual Orchestra
Chapter X Matrix Mechanics
PART C DIRAC'S QUANTUM MECHANICS
Introduction
Chapter XI Classical Beauty
Chapter XII Queer Numbers
Chapter XIII Quantum Beauty

BIBLIOGRAPHICAL GUIDE
ABBREVIATIONS USED IN CITATIONS AND IN THE BIBLIOGRAPHY
BIBLIOGRAPHY OF SECONDARY LITERATURE
BIBLIOGRAPHY OF PRIMARY LITERATURE
INDEX
ACKNOWLEDGMENTS
I first conceived the project of this book three years ago, during a conversation with John
Heilbron in the Berkeley hills. Since then, and before, John has encouraged me in many
ways: commenting on my work, giving me professional advice, and inviting me for long
stays at an ideal working place, the Office for the History of Science and Technology at
Berkeley.
There I made friends with Edward Jurkowicz, without whom the project of writing a
book m English would have been beyond my reach. Not only did Ed correct and clarify
my English, but his highly detailed comments helped to structure and strengthen many of
my arguments.
In France I received the highest intellectual stimulation from Catherine Chevalley, who
discussed with me several issues of the history and philosophy of quantum theory, let me
read her work on Bohr (Chevalley 1991a) prior to publication, and commented on large
portions of my manuscript.
Since history of quantum theory is not a new field of research, I have sought advice from
competent scholars. Norton Wise, a perceptive analyst of Bohr's early work and also an
expert on analogical thinking, offered several suggestions for improving my manuscript.
The convergence of our views on Bohr's role made our exchange particularly pleasant.
The part of my manuscript on Dirac's quantum mechanics benefited from comments by
the Danish historian Helge Kragh, who has recently published a major biography of
Dirac, and by the philosopher Edward MacKinnon, who has long been interested in
Dirac's methodology.

― xii ―
Some physicist friends have been kind enough to check the more technical content of my
manuscript. Jean-Michel Raimond, an authority on atomic physics, read my
considerations on the old spectroscopy of fine structure and anomalous Zeeman effects.
Bruno Jech, a professor of physics and historian, went through the arcana of my
presentation of Planck's radiation theory.
I have also tested the reaction of historians of science who have no specialized
knowledge of the history of quantum theory. Mario Biagioli suggested improvements in
the general introduction of this book; Mathias Dörries helped prune numerous obscurities
in the nontechnical summaries.
All my research has been done under the auspices of the Centre National de la Recherche
Scientifique, both in Paris and during my stays abroad. My research director, the
philosopher-physicist Bernard d'Espagnat, has been, from the beginning, supportive of
the type of work in which I was engaged. I have much benefited from his exceptionally
deep understanding of the foundations of quantum theory. I owe special thanks to the
authorities of CNRS who gave me the freedom to do my work in the best environments.
In Paris I have profited from the intellectual ambience at REHSEIS (équipe du CNRS
pour les Recherches en Epistémologie et Histoire des Sciences et des Institutions
Scientifiques). In this group, I have especially appreciated the guidance of Michel Paty
and Roshdi Rashed.
At the final stage of this project, the collaboration of my sponsoring editor, Elizabeth
Knoll, of the University of California Press, has been pleasant and constructive. Some
anonymous reviewers of her choosing made valuable suggestions for improving my
manuscript.
Aage Bohr has kindly granted me permission to quote from his father's unpublished
manuscripts and letters (which are deposited in the Bohr Archive in Copenhagen). For
Heisenberg's unpublished letters (which belong to the Heisenberg Archive in Munich) I
owe a similar favor to Helmut Rechenberg.
However small my project and modest the results, my debts appear to be extensive and
numerous. I am very thankful to the colleagues and friends just mentioned, and to all
those who, by their friendship or their research, helped me in an indirect, but invaluable,
manner.

― xiii ―

CONVENTIONS AND NOTATIONS
Vector notation is used throughout the book, even when it is anachronistic, for instance in
Maxwell's and Planck's cases. Knowing that Maxwell used points in a geometric space
(in the context of his dynamic theory of gases) and that Planck used Cartesian
coordinates, my reader will easily imagine a more authentic form of their equations.
In conformity with the convention found in most works described in this book, Gaussian
units are employed, which gives Maxwell's equations (in vacuo) the form

Then the Poynting vector is given by (c /4p )E × B and the energy density by (1/8p )(E2 +
B 2 ).
For Hamiltonian systems collective coordinates q and p are introduced according to
and the "dot product" of two such coordinates q ' and q " is defined by

― xiv ―
Frequencies in the strict sense (cycles per unit time) are denoted by the letter v , whereas
angular frequencies (radians per unit time) are denoted by the letter w (therefore, w = 2pv
).
Infinitesimal solid angles are denoted by dW and the corresponding direction by x .
In order to distinguish them from radiation frequencies, orbital frequencies will be
written with a bar: v instead of v .
As for physical constants, c denotes the velocity of light, k Boltzmann's constant, h
Planck's constant, Planck's constant divided by 2p , e the arithmetic value of the electron
charge (e > 0), m the electron mass.
For quantum numbers I have mainly used Bohr's notation, n for the principal quantum
number, k for the azimuthal one, j for the inner one, and m for the magnetic one.
However, I have used other letters when the normalization was essentially different (as in
Landé and Sommerfeld). The correspondence between Bohr's quantum numbers, those of
his colleagues, and the modern ones will be given in footnotes.
Significant simplifications have been introduced in some original proofs, for instance in
Boltzmann's and Planck's proofs of their H -theorems. Unless otherwise indicated, these
simplifications are of a purely mathematical nature and do not alter the main logical
arguments.
Citations of sources are in the author-date format and refer to works listed in one of the
two bibliographies (primary or secondary literature). Square brackets ([ ]) enclosing a
date indicate that the work in question is an unpublished manuscript and is listed in the
bibliography of primary literature. Abbreviations used in citations and in the
bibliographies are listed and explained on pp. 354-355 below.
Translations are generally mine, unless I am quoting from sources which are, or already
contain, a translation (which is always the case for manuscripts included in Bohr's
Collected works .)

― xv ―
INTRODUCTION
In the radiation theory just as in the gas theory, one could determine a state of maximal
probability.
Boltzmann, 1897

As we go from the kinetic theory of gases to the theory of thermal radiation ... we come
across relations which are very similar in a certain sense.
Planck, 1901

Using a metaphor, we may say that we are dealing with a translation of the
electromagnetic theory into a language alien to the usual description of nature, a language
in which continuities are replaced by discontinuities and gradual changes by
immutability, except for sudden jumps, but a translation in which nevertheless every
feature of the electromagnetic theory, however small, is duly recognized and receives its
counterpart in the new conceptions.
Bohr, 1924

The quantum theory has now reached a form ... in which it is as beautiful and in certain
respects more beautiful than the classical theory. This has been brought about by the fact
that the new quantum theory requires very few changes from the

― xvi ―
classical theory, these changes being of a fundamental nature, so that many of the
features of the classical theory to which it owes its attractiveness can be taken over
unchanged into the quantum theory.
Dirac, 1927

The genesis, maturation, and final formulation of quantum theory owed much to
analogies with classical theories.[1] Even modern quantum mechanics is still an art of
"quantization." Any application of it starts with formally defining a classical system, and
the quantum-theoretical level is then reached by applying a precise mathematical
procedure followed by interpretative rules. In the early history of quantum theory,
analogies with classical theory were not so sharply formulated, but they already were
fairly detailed and articulate. Not just a vague illustrative resemblance, they concerned
entire pieces of logical and mathematical structures and were able to produce new laws
and formalisms. The aim of this book is to analyze the structure and development of such
analogies in three cases: Planck's radiation theory, Bohr's atomic theory, and Dirac's
quantum mechanics.
In 1926 Dirac introduced a parallel between "c -numbers" and "q -numbers," capturing in
symbols the correspondence between classical and quantum (or "queer," as he
humorously said) mechanics. The formal expression of ordinary dynamic laws was
maintained in the new theory, while the related quantities no longer behaved like ordinary
numbers and no longer received a space-time interpretation. The c and q in my title
alludes to this ultimate perception of the analogies under discussion. Before quantum
mechanics, analogies with classical theories were not usually expressed in terms of the
exact transference of mathematical formulae. However, one can still speak of the
analogies as being formal in Bohr's sense of the word: "We are ... obliged to be modest in
our de-
[1]

― xvii ―
mands and content ourselves with concepts that are formal in the sense that they do not
provide a visual picture of the sort one is accustomed to in the explanations with which
natural philosophy deals."[2] Bohr understood that the analogy involved in the description
of the interaction between atoms and radiation could not be of a visual nature. Moreover,
this analogy, originally combined with the orbital picture of atoms, ended up, at the dawn
of quantum mechanics, being independent of any visual model of atomic motion.
Accordingly, this book is not directly about classical models as visualization tools in the
quantum theory, it is about formal classical analogies.
What is to be meant by classical theories in the historical episodes evoked in this book?
The answer is immediate in the case of Bohr's quantum theory and Dirac's quantum
mechanics, which are the main objects of parts B and C. A little before the beginning of
Bohr's atomic theory, classical (or "ordinary" in Bohr's words) theory already had the
meaning that is now familiar to us: it covered Newtonian mechanics, Lorentz's
electrodynamics, and, if necessary (for instance in Dirac's case), Einstein's relativity.
Obviously, these theories could not be called classical before a consensus had been
reached, in the early 1910s, about the need for a radically new physics in the realm of
atoms.[3]
At the turn of the century not only was such hindsight impossible but there was no
uniform conception of mechanics, electrodynamics, and their relations to
thermodynamics. At that time, opinions varied about the role to be played by
microphysical entities in the organization of macroscopic physics. In this context the
word "classical" is therefore misleading, unless it is used in a limited conventional way,
referring to mechanical and electrodynamic laws commonly accepted at the macroscopic
level. This remark has to be kept in mind in the first part of this book, which is dedicated
to Planck's radiation theory.
The analogy that guided Planck's work around 1900 really was an analogy between
Boltzmann's anterior gas theory and a new thermal radiation theory. In a sense
Boltzmann's theory can be called classical, meaning that it subjected gas molecules to the
well-established (in the macroscopic realm) laws of Newton's mechanics. But this should
not hide the fact that the kinetic molecular theory was not universally accepted
[2][3]

― xviii ―
at the end of the nineteenth century. Planck himself converted to this theory only in 1896-
1897, while developing his radiation theory.
Further classification of the analogies studied in this book is obtained by looking at the
nature of the "target" theory, that is, the theory which the analogy helps to construct. In
the case of Planck's celebrated work of 1900, the target theory was intended to consist of
a simple extension or transposition of Boltzmann's methods—originally designed for
gases—to a system made of electromagnetic radiation and sources. Until 1907-1908
Planck actually believed the sources of thermal radiation to comply with ordinary
electrodynamic laws. In this approach the target theory was as classical as Boltzmann's
gas theory could be, and we may call the relevant analogy "horizontal." Again, the
present use of the word "classical" should not hide the fact that there was no universally
accepted formulation of electrodynamics around 1900.[4]
The product of Planck's horizontal analogy was not meant to break with accepted
theories. It was not a "quantum discontinuity."[5] This does not mean that "the father of
the quantum theory" did not introduce anything substantial in 1900. He isolated the
fundamental constant h , and he gave the formal skeleton of what could later be regarded
as a quantum-theoretical proof of the blackbody law. This is just a first example of a
recurrent characteristic of the history of quantum theory: the "correct" interpretation of
new mathematical schemes generally came after their invention.
In the years 1905-1907 Einstein introduced an intrinsic discontinuity of the energy of
microscopic entities, the so-called quantum discontinuity, and used it most successfully
in a new theory of specific heats. In 1913 Bohr exploited the same discontinuity in his
first atomic theory. By that time the rudiments of a radically new "quantum theory" were
known, and the possibility was open for genuine "vertical" analogies, connecting the
nascent theory to the now classical theory. However, such analogies could not flourish
before the new "language of atoms" was sufficiently known, that is to say, after
Sommerfeld generalized Bohr's original quantum condition in 1916.
Bohr then recognized that some laws of classical electrodynamics had a formal
counterpart within the quantum theory. This analogy, which led
[4][5]
― xix ―
to what Bohr named the "correspondence principle" in 1920, was at least heuristically
important, because the resulting quantum-theoretical laws could not be deduced from the
general assumptions of the Bohr-Sommerfeld theory, which was clearly incomplete. In
the absence of a rigorous deductive scheme, the qualitative or semiquantitative validity of
these laws was essentially controlled by two conditions: one empirical, the compatibility
with observed atomic spectra; and one intertheoretical, the asymptotic agreement with the
corresponding classical laws in the case of relatively small quantum jumps.
An analogy between two theories, one of which is essentially incomplete and provisional,
should not be expected to be unambiguous.[6] Accordingly, Bohr wished to formulate his
correspondence principle in a not too sharp form. In part B we will observe the
multiplicity of uses of this principle. A crucial ambiguity lay in the extent to which the
analogy maintained the space-time description of the classical theory. In the spring of
1925, at the end of a crisis that started in 1922, Bohr and some of his disciples cut the by
then dead branches of the correspondence principles, namely, its visual elements, and
retained only the idea of a symbolic translation of classical laws. Within this stream of
thought Heisenberg devised quantum mechanics.
What Heisenberg proposed in the summer of 1925 was a complete mathematical scheme
interpreted in terms of the original postulates of Bohr's theory, that is, in terms of
stationary states and atomic transitions. Formally, the analogy between this theory and
classical mechanics could hardly be closer, since the formal expression of dynamic laws
was integrally maintained (though transcribed in bold and gothic types). All the same, the
distance between classical and quantum concepts was larger than ever, for the dynamic
variables were now represented by infinite matrices instead of ordinary numbers.
At that stage, reference to the correspondence principle became unnecessary, because all
properties of atomic spectra could be deduced from the new scheme. Yet an early
follower of Heisenberg's ideas, Paul Dirac, found deeper connections between new and
old mechanics, ones involving algebraic structures. He brilliantly exploited these
structural connections in consolidating and developing Heisenberg's ideas. In part C the
originality and power of Dirac's approach is shown to have also depended on a rather
different kind of classical analogy. The latter was not so much
[6]

― xx ―
between the formal contents of two theories as between characteristic strategies for
theory-building. The model here was Einstein's general relativity, as perceived by Dirac's
philosophy teacher, C.D. Broad, and England's foremost relativist, Arthur Eddington.
To summarize, four types of classical analogies will be described in this book: Planck's
abusively conservative but formally suggestive "horizontal" analogy, Bohr's tentative
"vertical" analogies between classical electrodynamics and an incomplete quantum
theory, Heisenberg's and Dirac's analogies between the mathematical schemes of classical
and quantum mechanics, and Dirac's reference to the relativistic strategy of theory-
building.
The originators of each of these analogies all made general comments about the function
they served. Planck declared he had reached, despite remaining obscurities in the proof of
his blackbody law, a fundamental unification of gas theory and radiation theory; for he
had managed to apply the same formula, S = k In W, in both theories, with the same
fundamental constant k . Moreover, comparing the resulting expression of the blackbody
law with empirical measurements provided the best available access to Avogadro's
number, through the constant k . Who would not agree with Planck that horizontal
analogies, if successful, bring unity in the architecture of physics?
When Bohr introduced the correspondence principle, he first emphasized its heuristic
power: it was a means to compensate for the incompleteness of the quantum theory. Even
the enemies of this principle came to agree with this. However, Bohr soon attributed
more fundamental functions to his "principle": bring more structure into the quantum
theory, and show the overall harmony of its various assumptions. Characteristically, he
regarded the formal connections obtained by analogy as part of the quantum theory, even
before these connections could be expressed in a precise quantitative way. In his opinion
a reasonable degree of conceptual clarity and consistency could be achieved even before
the advent of a more complete and definitive theory.
Sommerfeld and many other quantum theorists of lesser importance were out of
sympathy with Bohr's strategy of rational guessing. Sommerfeld's concept of rationality
demanded a sound complete mathematical framework or, as long as nothing better was
available, a set of clear mathematical models. Yet, be it a historical contingency, a
necessary outcome of a rational attitude, or something in between, the correspondence
principle did play the most important part in the construction of

― xxi ―
the first version of quantum mechanics. Moreover, the main source of the early
confidence in Heisenberg's strange kinematics of infinite matrices was its close formal.
analogy with classical mechanics. The precise expression of this analogy suggested the
mathematical completeness and consistency of the new scheme even before a rigorous
proof could be given; it automatically warranted the necessary asymptotic agreement
with classical mechanics; and it integrated in a more quantitative form earlier-verified
predictions of the correspondence principle. Last but not least, this analogy preserved
what Dirac found to be the beauty of classical mechanics.
That the construction of quantum theory heavily relied on previous leading theories
should not be a matter of surprise.[7] The modern theoretical physicist is almost more
concerned with relations between different theories than with empirical data. Many