The correspondence principle of Niels Bohr
Charlotte Fraza (3829693)
Department of History and Philosophy of Science
Utrecht University
“There was rarely in the history of physics a comprehensive theory which owed so much to
one principle as quantum mechanics owed to Bohr's correspondence principle”1
Introduction
In modern literature the Correspondence principle is often defined as the requirement for
the quantum theory to go over to the classical theories in the limit of large quantum
numbers, or in other words as ℎ → 02. Already in the time of Bohr it was the trend to
perceive the principle in this matter, and to use it accordingly. Nobelprize winner Max Born
also cites Bohr’s correspondence principle as such. In his lectures of 1933 Born states that;
“die neue Mechanik für den Grenzfall großer Massen oder großer Bahndimensionen in die
klassische Mechanik übergeht”3. In 1925, in the paper The Fundamental Equations of
Quantum Mechanics, P. A. M. Dirac named the correspondence principle as the requirement
that: “the classical theory gives the right results in the limiting case when the action per
cycle of the system is large compared to Planck’s constant ℎ, and in certain other special
cases.”4 According to these formulations, the correspondence principle functions as a
constraint on theorizing; this means that it only acts in a limit. However this correspondence
limit sometimes called ‘Planck’s correspondence principle’ does not capture the full body of
the correspondence principle as defined by Niels Bohr. Léon Rosenfeld recalled that he once
suggested to Bohr that Planck’s found correspondence between classical theory and
quantum theory was the first germ of the correspondence principle, but that Bohr disagreed
fervently: ‘He said emphatically that; “it is not the correspondence argument. The
requirement that the quantum theory should go over to the classical description for low
modes of frequency, is not at all a principle. It is an obvious requirement of the theory”.5 But
if it is not as Bohr himself states a limiting principle, what is the full definition of the
correspondence principle?
1
Jammer, Max. The conceptual development of quantum mechanics. Tomash, 1989., p. 118.
As was the case with Planck’s radiation law in relation to the Rayleigh-jeans law.
3
Born, Max. "Moderne Physik." The Journal of Physical Chemistry 37.6 (1933): 830-831.
4
Dirac, Paul AM. "The fundamental equations of quantum mechanics."Proceedings of the Royal Society of
London. Series A, Containing Papers of a Mathematical and Physical Character (1925): 642-653.
5
Rosenfeld, L., 1979 [1973], “The Wave-Particle Dilemma”, in R. Cohen and J. Stachel (eds.) Selected Papers of
Léon Rosenfeld (Boston Studies in the Philosophy of Science, Volume 21), Dordrecht: D. Reidel Publishing Co.,
688–703. Originally published in J. Mehra (ed.)The Physicist's Conception of Nature, Dordrecht: D. Reidel
Publishing Co., pp. 251–263
2
1
While doing a survey on the work of Bohr we will find that there is quite some
ambiguity surrounding the correspondence principle and that it may not be fully clear what
this principle exactly covers. We will see that for this reason the correspondence principle is
often ill understood by fellow scientists. It is therefore important to differentiate between
Bohr’s own understanding of the correspondence principle and how his peers eventually
used it. In the first chapter we shall mainly focus on Bohr’s own writing on the
correspondence principle and in the second chapter elaborate on what status it held in the
larger physics community. Also I want to give some special attention to the philosophical
considerations of Bohr in chapter 3, as I feel that this philosophical side shaped him as a
physicist. I shall refer to earlier papers on the correspondence principle and elaborate on
what these papers conclude. It is worth mentioning that Bohr’s statements concerning the
correspondence principle changed considerably in his later works and therefore it does not
reflect well the way he thought about the correspondence principle in the beginning of his
career. For this reason we will leave out his later statements concerning the correspondence
principle in the first two chapters and focus primarily on the time period between 19131925.
A. Bokulich divided the formulations made by Bohr on the correspondence principle
into three categories6; ‘first we have the frequency theorem interpretation, according to
which the correspondence principle is an agreement between the Fourier decomposition of
the classically defined frequency and the quantum frequency. This interpretation holds in
the limit of large quantum numbers and is therefore only an asymptotic agreement.
Secondly, there is the intensity interpretation, which signifies the thought that there is a
statistical relation between the classical intensity and the quantum intensity. The quantum
intensity is then depending on the number of quantum transitions and therefore the
probability for this transition to occur. The classical intensity is equal to the square of one
component of the classical described motion (Fourier decomposition). Finally, there is also
the selection rule interpretation, which is to be understood as the principle that each
allowed quantum transition corresponds to one harmonic component in the classical
motion. We will see that in the works of Bohr and his peers we will encounter all three
interpretations of which we will try to define ‘the’ correspondence principle. I make use of
these three interpretations when analyzing Bohr’s work, as they are generally used in
literature surrounding Bohr’s correspondence principle. Although not all Bohr’s remarks on
the correspondence principle can be subdivided into these categories, it is good as a basic to
think about whilst trying to define the correspondence principle. However the use of these
three interpretations will actually also underline the point I will make; that the
correspondence principle is very difficult to place under one coherent interpretation. I will
also show in the paper that the correspondence principle was in its first formulations more a
mathematical principle, although not clearly stated in one version by Bohr, and that it slowly
changed. At the end of 1925 we will see that it had become more of a guiding principle,
which helped Bohr and his peers to look in the right direction for answers surrounding
6
Bokulich, Alisa. "Three Puzzles about Bohr's Correspondence Principle." (2009).
2
atomic questions. So the role and the formulation of the correspondence principle
underwent changes that can be brought back to the changes in the atomic model on which
it was based.
1 The changing correspondence principle in the work of Bohr (1913-1925)
1.1 The foundation of the correspondence principle
Niels Bohr was a Danish physicist who lived from 1885 until 1962. He was born and died in
Copenhagen. In 1922 he won the Nobel Prize in physics and is best known for his work on
quantum physics. Besides his scientific work, Bohr spent a great deal of his time on the
philosophical interpretation of it; in his later years he even became more known for these
philosophical contributions than for his contributions to physics. He was in the beginning of
his career first and foremost a physicist. His lasting contributions to the quantum theory
included the Copenhagen interpretation, the Bohr atom, the thesis of complementarity, and
the correspondence principle. The correspondence principle can be called one of the most
fascinating and interesting parts of Bohr’s philosophy. It reflects perfectly his personal way
of thinking and it was a prime example of what Einstein called Bohr’s ‘musicality’ and
‘unique instinct and tact’.7 To understand the basics correspondence principle one first
needs to understand the context in which it was formed: the old quantum theory.
The old quantum theory was an “interim” theory, developed on the basis that classical
theories could not fully describe atomic systems. Max Jammer, an Israeli physicist and
philosopher of physics, aptly describes the pre-1925 'theory' as follows:
“Every single quantum-theoretic problem had to be solved first in terms of classical
physics; its classical solution had then to pass through the mysterious sieve of
quantum conditions or, as it happened in the majority of cases, the classical solution
had to be translated into the language of quanta in conformance with the
correspondence principle. Usually, the process of finding "the correct translation"
was a matter of skilful guessing and intuition rather than of deductive and systematic
reasoning.”8
Especially this ‘mysterious sieve of quantum conditions’ was a part of the old quantum
theory that was more based on the intuition of a scientist, than on hard mathematical rules.
This was also the reason that many scientists first rejected the quantum theory.
7
Kragh, Helge. Niels Bohr and the Quantum Atom: the Bohr model of atomic structure 1913-1925. Oxford
University Press, 2012, p. 271.
8
Max, Jammer. "The conceptual development of quantum mechanics.", (1966).
3
The first theory that seemed to solve the problem that classical theories could not
describe atomic systems was proposed by Niels Bohr in the Philosophical Magazine of July
1913 in a three-part paper titled ‘On the Constitution of Atoms and Molecules’. It is called
an ‘epoch-defining paper’9 and caused the first definite split with the classical theories. In
the paper Bohr articulated the postulates of the old quantum theory. In a long letter to
Ernest Rutherford, a British physicist known for his great experimental contributions to the
atomic theory, on March 6, 1913 he sent his paper and wrote:
"Enclosed I send the first chapter of my paper on the constitution of atoms. [...] I
hope that you will find that I have taken a reasonable point of view as to the delicate
question of the simultaneous use of the old mechanics and of the new assumptions
introduced by Planck's theory of radiation. I am very anxious to know what you may
think of it all...".10
We see that Bohr knew that his blending of quantum and classical ideas was highly
‘delicate’, but he defined the rules and combined the two theories in an ingenious manner.
Bohr used in his paper Ernest Rutherford's model of the atom, according to which a
relatively high central charge is concentrated into a very small volume in comparison to the
rest of the atom (the nucleus) and with this central volume also containing the bulk of the
atomic mass of the atom. The electrons orbit the nucleus in planetary trajectories. However,
in an attempt to explain some of the properties of matter with this atom-model Bohr
encountered a problem; it was unstable according to classical electrodynamics. According to
the model, the electron, an accelerated charged body, radiates energy in such a manner
that it would eventually collapse into the nucleus. Bohr's solution to this problem was to
incorporate Max Planck's theory:
"The result of the discussion of these questions seems to be a general
acknowledgment of the inadequacy of the classical electrodynamics in describing the
behavior of systems of atomic size. Whatever the alteration in the laws of motion of
the electrons may be, it seems necessary to introduce in the laws in question a
quantity foreign to the classical electrodynamics, i. e. Planck's constant".11
Bohr summarized his quantum theory by means of two assumptions or postulates:
9
Kragh, Helge. Niels Bohr and the Quantum Atom: the Bohr model of atomic structure 1913-1925. Oxford
University Press, 2012, p.1.
10
Applied Physics. Physics 50 years later: Papers as presented to the XIV General Assembly of the International
Union of Pure and Applied Physics on the occasion of the union's fiftieth anniversary, September 1972. Ed.
Sanborn Conner Brown. Vol. 14. National Academies, 1973.
11
Bohr, Niels. "I. On the constitution of atoms and molecules." The London, Edinburgh, and Dublin
Philosophical Magazine and Journal of Science 26.151 (1913): 1-25.
4
“(1) That the dynamical equilibrium of the systems in the stationary states can be discussed
by help of the ordinary mechanics, while the passing of the systems between different
stationary states cannot be treated on that basis.
(2) That the latter process is followed by the emission of a homogeneous radiation, for
which the relation between the frequency and the amount of energy emitted is the one
given by Planck's theory."12
So the atomic systems can only exist in a series of individual states, which can be thought of
as stable periodic orbits around the nucleus. Bohr also often compared these orbits to the
orbits of our planetary system, from which he got his inspiration. These periodic orbits are
labeled by the principal quantum number 𝑛, with the lowest orbit labeled 𝑛 = 1, the next
𝑛 = 2 etc. When the electron is in one of these states, its motion can be correctly described
by classical mechanics. However, when an electron makes a jump from one state to the
next, the theory no longer applies. So we see a mix of quantum and classical theories.
Whereas the first postulate still relies on the classical mechanics as defined by Isaac
Newton, the second postulate is the official break with the classical theories. We will see
that, in later papers, this second postulate will develop in to the frequency interpretation.
Caution is needed when we speak of the quantum jumps made from the stationary
states, because it is now generally accepted that a photon is released, however Bohr himself
did not embrace this theory up until the mid-1920s. Bohr preferred to think of the emitted
radiation as a wave, rather than a particle. He stated his distrust of light quanta clearly in his
lecture in 1922 where he said that: “in spite of its heuristic value, however, the hypothesis
of light-quanta, which is quite irreconcilable with so-called interference phenomena, is not
able to throw light on the nature of radiation.”13 Bohr’s resistance for accepting the particle
theory was also tied to the importance of being able to analyze radiation into its harmonic
components, which was in essence tied to the basis of the correspondence principle. As
John Stachel notes: “It was indeed, his reliance on the correspondence principle that seems
to have been a principal motive for Bohr's distrust of the photon concept”14. In the end Bohr
would be led to embrace the concept of the photon and recognize its heuristic value,
although not before the collapse of his atomic model.
1.2 “On the quantum theory of line spectra” (1918)
Bohr continued to elaborate his work from 1913 and finally in 1917 sent his first part of the
paper called “On the quantum theory of line spectra” to the proceedings of the Danish
Academy. Bohr had recently been selected a member of the prestigious society and worked
12
Idem
Bohr, Niels. "The structure of the atom." Nobel lecture 11 (1922): 14.
14
Stachel, John. "Bohr and the Photon." Quantum Reality, Relativistic Causality, and Closing the Epistemic
Circle. Springer Netherlands, 2009. 69-83.
13
5
hard to prove the value of his atomic theory. He chose first to only publish the paper in the
academy proceedings rather than submitting it to a regular physics journal. This decision
meant a more limited circulation of his paper, but Bohr had his reason, as he wrote to
Rutherford in 1917:
“This last term I have used all my spare time to complete a long paper on the general
principles of the quantum theory. On account, however, of the great difficulties of
postal communication I have thought it advisable first to publish it here in the
transactions of the Royal Danish Academy, where I could publish it in English, and
where I had the special advantage to be able to start the printing before the end of
the long paper was quite finished… As soon as I have got a corrected proof of it all I
look forward to send it to you.”15
We can see his hesitation to make his paper public, as Bohr felt himself that it was not yet
finished. He felt that certain assumptions made in the paper would not be received well,
especially concerning the mixing of quantum and classical concepts, and therefore chose to
wait before sending it to the larger scientific journals. In 1918 Bohr wanted to publish a
series of three papers, which would give a clear overview of the quantum theory thus far.
The trilogy of 1918 was “somewhat unmanageable”16 as Bohr himself stated. The sentences
are long and often repeated. Bohr did not break up his chapters and often made no clear
conclusions. Altogether it was quite hard for the reader to get a grip of Bohr’s thoughts.
Pauli later called it the ‘subtlety of Bohr’s style’: “Bohr knew well what he wished not to say
when he strove in long sentences to express himself in his scientific papers”17, but it was
probably this ‘subtlety’ that got him misunderstood a lot of times.
In the introduction of the paper we will find the leading motive in the coming work
of Bohr; the search for connections between the classical theory and the quantum theory,
that isn’t to say that Bohr believed that there would be such a connection, but he found it a
good starting point for further scientific reasoning:
“On this state of the theory it might therefore be of interest to make an attempt to
discuss the different applications [of the quantum theory] from a uniform point of
view, and especially to consider the underlying assumptions in their relations to
ordinary mechanics and electrodynamics. Such an attempt has been made in the
present paper, and it will be shown that it seems possible to throw some light on the
outstanding difficulties by trying to trace the analogy between the quantum theory
and the ordinary theory of radiation as closely as possible.”18
15
Nielsen, J. Rud, ed. The correspondence principle (1918-1923). Elsevier, 2013.
Kragh, 2012
17
Idem
18
Bohr, Niels. On the quantum theory of line-spectra. Courier Corporation, 2005.
16
6
By “the relation to ordinary mechanics” Bohr meant the adiabatic principle, as described by
Paul Ehrenfest, a Dutch and Australian theoretical physicist. This principle plays a role not
only in fixing or determining the allowed stationary states, but also in justifying the use of
the classical electro-dynamical laws to characterize motion in those states. The Adiabatic
Principle “demands that the conditions for the stationary states must be of such a kind that
they define certain properties of the motion of the system, which will not change during an
adiabatic transformation, if the motion is described ... by help of the usual electrodynamic
laws”19. In modern quantum theory, this is to say that a system remains in its eigenvalue if
perturbation acting on it is changing slow enough20. With “the relation to electrodynamics”
Bohr meant what he would later call the correspondence principle and what we defined as
the frequency correspondence principle.
One of the interesting changes in the paper compared to his paper in 1913 is that he
now called his postulates “fundamental assumptions” in which we can see a growing
confidence in the vital nature of his postulates. Moreover, he altered his second assumption
to the form of the now better-known frequency condition: “II. That the radiation absorbed
or emitted during a transition between two stationary states is ‘unifrequentic’ and
possesses a frequency 𝜈, given by the relation 𝐸’ − 𝐸’’ = ℎ𝜈, Where ℎ is Planck’s constant
and where 𝐸’ and 𝐸’’ are the values of the energy in the two states under consideration.”21
The second assumption is obviously in contrast with the classical ideas of electrodynamics,
as the radiation given off in a transition between stationary states could not be described by
classical mechanics. Bohr stated however that it was a necessary assumption to account for
experimental facts. The renewed belief in his own ideas can be traced back to Debye and
Sommerfeld who had been able to deduce the Zeeman effect in the hydrogen atom from
the from the general quantization of multi periodic systems. And Einstein’s theory of
radiation had given independent support to the relation 𝐸’ − 𝐸’’ = ℎ𝜈.
In the outline of the old quantum theory, Bohr’s two postulates did not yet suffice in
giving the correspondence between classically allowed orbits and orbits corresponding to
stationary states. First the stationary states and their orbits needed to be determined. In
order to determine these stationary states a “quantum condition” needed to be used:
∮ 𝑝𝑑𝜃 = 𝑛h
Here the integral is taken over one period of the orbit of the electron, and 𝑝 is the angular
momentum, 𝜃 is the angle in the plane of the electron orbit, and n is the quantum number.
Max Jammer summarizes: “applying the old quantum theory consists of essentially three
steps: first, the application of classical mechanics for the determination of the possible
motions of the system; second, the imposition of certain quantum conditions for the
19
Bohr, Niels, Hendrik Anton Kramers, and John Clarke Slater. "LXXVI. The quantum theory of radiation." The
London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 47.281 (1924): 785-802.
20
M. Born and V. A. Fock (1928). "Beweis des Adiabatensatzes". Zeitschrift für Physik A 51 (3–4): 165-180.
21
Bohr 1918a
7
selection of the actual or allowed motions; and third, the treatment of the radiative
processes as transitions between allowed motions subject to the Bohr frequency formula”22.
To understand how exactly the correspondence principle was used in this context, it
is useful to first look at how a one-dimensional atomic-system undergoing a simple periodic
motion is classically described. The trajectory is than given by 𝑥(𝑡), which would be the
solution of the equations of motion from Newton and will be periodic with a frequency 𝜔,
also known as the fundamental frequency. A Fourier series can then describe the position of
the motion as follows:
𝑥(𝑡) = 𝐶1 cos(𝜔𝑡) + 𝐶2 cos(2𝜔𝑡) + 𝐶3 cos(3𝜔𝑡) + ⋯
According to the classical theory the frequency of the atom as described by this series
should be given by the frequencies in the harmonics23 of the motion: 𝜔, 2𝜔, 3𝜔, etc.; and
therefore the spectrum of such a motion is by classical interpretation described by a series
of discrete evenly spaced lines. If we follow the reasoning of Bohr in 1918 the radiation of
the electron in its orbit is not a result of an accelerated motion (classically thought), but
rather an electron jumping from one of the stationary states to the other and by this jump a
single frequency, 𝜈, is emitted. The value of this frequency 𝜈 is then given by the Bohr
frequency condition24. The spectral lines as seen from experiments are built up out of atoms
undergoing transitions between different stationary states, and these lines will be discreet
and evenly spaced in the limit of large quantum numbers. 1918 was the first time Bohr
mentioned the full description of what could be interpreted as the frequency interpretation
of the correspondence principle:
“Although, of course, we cannot without a detailed theory of the mechanism of
transition obtain an exact calculation of the latter probabilities, unless 𝑛 is large, we
may expect that also for small values of 𝑛 the amplitude of the harmonic vibrations
corresponding to a given value of τ will in some way give a measure for the
probability of a transition between two states for which 𝑛’ – 𝑛’’ is equal to 𝜏.”25
He also speaks of an expectation for the small values of 𝑛 to have an agreement with the
classical harmonic system, but he does not further specify how this relation should be
interpreted and why it would also hold for small quantum numbers. We can see that
although Bohr considered extending his principle to small quantum numbers, he did not
formulate his reasoning for this extension, making it quite obscure.
Bohr tested this correspondence hypothesis by using it as a premiss in a theoretical
calculation of the Rydberg constant, which is a physical constant relating to the atomic
spectra. The calculated value of the constant appeared to agree well with the
experimentally determined value. Before Bohr’s atomic model this constant could not be
22
Jammer 1966, p. 90.
A integer multiply of the fundamental frequency (usually defined for a wave).
24
Bokulich, Alisa. "Bohr's correspondence principle." (2010).
25
Bohr 1918a
23
8
calculated by any of the then existing theories, so this was a strong indicator that Bohr’s
atomic model was indeed correct. We should notice that Bohr always spoke of 'agreement
of calculations’ concerning this type of correspondence by which he meant that there was a
statistical asymptotic agreement between the quantum frequency, 𝑣𝑛’→𝑛’’ of radiation
emitted in a quantum jump of difference 𝜏 from state 𝑛’ to 𝑛’’ and the classical frequency
𝜔𝜏 in the 𝜏 𝑡ℎ harmonic of the classical motion in the 𝑛’ stationary state:
𝑣𝑛’→𝑛’’ = 𝜔𝜏 = 𝜏𝜔, for large 𝑛,
where 𝑛′ − 𝑛″ = 𝜏.
We note here once more that this equation only holds for large quantum numbers, and not
for low quantum number transitions, that is to say the equation is only true in the
appropriate limit. Bohr described this also aptly during the Solvay conference: “In the limit
where the action involved is sufficiently large to permit the neglect of individual quanta, the
quantum phenomena can be described as a rational generalization of the classical physical
description.”26 So we see here that Bohr now states that the relation only holds when the
action is ‘sufficiently’ large, but he does not state when this sufficiency is to be expected and
also his earlier considerations to extend the principle to small quantum numbers does not
align with this statement.
While classically all frequencies will be sent out at the same time, in the quantum
mechanical interpretation only a single wave will be emitted with a single frequency
between the quantum jumps, therefore one must consider an ensemble of these atoms to
compare with the classical spectrum. Einstein introduced this statistical agreement in his
theory of heat radiation, on which Bohr commented: “this raises the serious question of
whether we must rest content with statements of probabilities for individual processes. As
matters stand at present, we are so far from being able to give a real description of these
processes that we may well assume that Einstein’s mode of treatment may actually be the
most appropriate”27. There seems to be a hint of skepticism in Bohr’s writing on this use of
ensembles of atoms to describe quantum processes. Although he embraces the results this
method brings he does not quite accept its origins, which rested mostly on the belief that
light can be thought of as particles (light quanta).
He also declared explicitly that in this context the ties with the classical physics
should be mostly understood as a mere heuristic device: “You understand, of course, that I
am by no means trying to give what might ordinarily be described as an explanation;
nothing has been said here about how and why the radiation is emitted. On one point,
however, we may expect a connection with the ordinary conceptions, namely, that it will be
possible to calculate the emission of slow electromagnetic oscillations on the basis of
electrodynamics”.28 Bohr refrained from giving his thoughts on the origin of radiation and
26
Bohr, N. "The Solvay meetings and the development of quantum mechanics." (1958).
Bohr 1922 unpublished lecture; BCW 4, p. 348.
28
N. Bohr ‘Om Brintspektret’ Fysisk Tidsskrift 1914
27
9
instead wanted to give attention to the relation with classical theories that the atomic
system shows. He did not want to give some vague explanation without experimental
groundings. He merely noticed the problems surrounding the origin of radiation and defined
the use of classical concepts in these problems as heuristically interesting, but nothing more
than that. One of the main reasons for this disinclination to give an interpretation of the
relations between quantum mechanics and classical mechanics was that they were easily
misunderstood as trying to gloss over the differences between both theories, which would
not be well received by other physicists.
The intensity interpretation of the correspondence principle is also found in his paper
of 1918, according to which in the limit of large quantum numbers there is a
correspondence between the probability of the quantum jumps and the amplitudes of the
classical motion: “We must further claim that a relation, as that just proved for the
frequencies, will, in the limit of large 𝑛, hold also for the intensities of the different lines in
the spectrum. Since now on ordinary electrodynamics the intensities of the radiations
corresponding to different values of 𝜏 are directly determined from the coefficients 𝐶𝜏 , we
must therefore expect that for large values of 𝑛 these coefficients will on the quantum
theory determine the probability of spontaneous transition from a given stationary state for
which 𝑛 = 𝑛’ to a neighboring state for which 𝑛 = 𝑛’’ = 𝑛’ − 𝜏””.29
𝑃𝑛’→𝑛” ~𝐶𝜏 (𝑛)2 for large 𝑛
Thus in the limit when 𝑛 gets large the amplitudes of the harmonic components of the
electron’s classical orbit can be used to calculate the intensities of the spectral lines. In the
classical case the intensity is directly related to the amplitude of the electron, however in
the quantum mechanical case the intensity of the spectral line will be determined by how
many photons will be admitted and therefore the probability of a particular quantum jump
to occur. This result was partially supported by observations on the absorption spectra,
where only certain lines with corresponding intensities where viewed.
He also elaborates in his paper on conditionally period systems, to prove that the
found correspondences hold even outside of periodic systems. He comments: “As far as the
frequencies are concerned, we thus see that for conditionally periodic systems there exists a
connection between the quantum theory and the ordinary theory of radiation of exactly the
same character as that shown to exist in the simple case of periodic systems of one degree
of freedom.”30 This was a big step forward in the quantum theory, as before no one could
fully explain systems that where not fully periodic.
So the second postulate together with the quantum condition gave predictions
about the frequencies of the radiation emitted or absorbed in the transitions between the
stationary states and gave a connection between the classical amplitudes of the motion and
quantum intensities. Despite the empirical successes of the correspondence principle, some
physicists remained skeptical about the renowned predictions it was supposed to give.
29
30
Bohr 1918a
Idem P.34
10
Rynasiewicz, professor in history of science at the Johns Hopkins University, also
noticed that the formulizations of 1918 do not yet yield any predictions about the state of
polarization of emitted radiation and also they do not make any predictions about the
emission and absorption probabilities. Although later the correspondence principle will get
the reputation to do just that: “The above formulation (second postulate of 1918) of the
correspondence principle as a constraint on quantum theorizing provides for empirical
content at best constraining the determination of stationary states in the limit (of slow
frequencies or large quantum numbers). It cannot milk out of the two fundamental
postulates any information about transition probabilities or polarization.”31 So the
correspondence principle must hold something more than these two relationships or at
least it is accounted to hold something more.
In 1918 the third use of the correspondence principle can also be found, the selection
rule interpretation, which says that; a transition from a stationary state 𝑛’ to another
stationary state 𝑛’’ whose separation is 𝜏 is allowed if and only if the exists a 𝜏 𝑡ℎ harmonic in
the classical motion of the electron in the stationary state. A visual description of this
interpretation is depicted in figure 1.
Figure 1: The selection rule interpretation of the correspondence principle where a
transition between stationary states corresponds to one harmonic component of the
classical motion. If this harmonic component is not found in the classical motion the
transition will not occur.32
Bohr concluded that if there was no corresponding harmonic in the Fourier series for large 𝑛
that quantum jump would not occur: “Thus in general there will be a certain probability of
an atomic system in a stationary state to pass spontaneously to any other state of smaller
energy, but if for all motions of a given system the coefficients 𝐶 [in the Fourier series] are
zero for certain values of 𝜏 , we are led to expect that no transition will be possible, for
which 𝑛’ – 𝑛’’ is equal to one of these values.”33
We can illustrate this with an simple example34; suppose the solution to Newton’s
second equation 𝐹 =
𝑚𝑑2 𝑥
𝑑𝑡 2
and the quantum condition ∮ 𝑝𝑑𝜃 = 𝑛h is;
31
Rynasiewicz, Robert. "The Correspondence Principle." (2013).
Based on Fig. 3 of Fedak and Prentis 2002, [http://plato.stanford.edu/entries/bohr-correspondence]
33
Bohr 1918a P.16
34
Bokulich, Alisa. "Bohr's correspondence principle." (2010).
32
11
1
1
𝑥(𝑡, 𝑛) = 𝑛 cos (𝑛2 𝑡) + 𝑛1/2 cos (3𝑛2 𝑡)
Which is the Fourier series of a classical periodic motion of an electron in an allowed
1
stationary state 𝑛. We see here that the fundamental frequency is given by 𝜔 = 𝑛2 . There
are only first and third harmonics present in the classical motions. Now according to Bohr’s
rule of selection, this means that there can only exist quantum jumps between the
stationary states that are one or three stationary states apart from each other. We see here
that unlike the other interpretations of the correspondence principle this one holds for all
quantum numbers and is not only applicable for the limit of large ones. He writes:
“This peculiar relation suggests a general law for the occurrence of transitions
between stationary states. Thus we shall assume that even when the quantum
numbers are small the possibility of transition between two stationary states is
connected with the presence of a certain harmonic component in the motion of the
system.”35 As this interpretation of the correspondence principle holds for all
quantum numbers it has let some of the contemporary history of science researchers
to point this relationship as ‘the’ correspondence principle. One of the firmest
among them is A. Bokulich who points out that: “Bohr emphasizes in his work that
the correspondence principle holds even for low quantum number transitions. For
those who see the correspondence principle as only a asymptotic agreement in the
region of large quantum numbers between the classical theory and the quantum
theory, this will prove a difficult point.”36
Bohr used all three corresponding relationships in his work of 1918, but he did not yet name
any one of the ‘the’ correspondence principle, he merely noticed the existence of these
correspondences, and instead named them ‘formal analogies’. Therefore, we must look at
some of his later work concerning the correspondence principle to get a better background
on it.
1.3 “Uber die Serienspektra der Elemente” (eleboration of the CP between 1920-1923)
In 1920 Bohr gave in Berlin a lecture called “Uber die Serienspektra der Elemente” to the
Deutsche Physikalische Gesellschaft. It was during this occasion that for the first time the
term “Korrespondenzprinzip” was used, instead of the term ‘formal analogy’. With this
formal naming of his analogies Bohr made clear that he then considered the
correspondence principle a fully-grown concept. He reiterated the frequency interpretation
of the correspondence principle and also mentioned the intensity interpretation. He explains
his expectations as to how the frequency theorem will lead to an expectation for the
intensities. He states: “Through this agreement, we obviously obtain a connection between
the spectrum and the atomic models of ~ hydrogen that is as intimate as one could hope, if
one considers the fundamental difference between the ideas of quantum theory and the
35
1920, “On the series spectra of the elements”, Lecture before the German Physical Society in Berlin (27 April
1920), translated by A. D. Udden, in Bohr (1976), 241–282.
36
Bokulich, Alisa. "Three Puzzles about Bohr's Correspondence Principle." (2009).
12
conventional radiation theory in mind”. 3738 Bohr keeps on emphasizing this ‘fundamental
difference’ between quantum theories and classical ones, where the correspondence
principle would serve as a bridge to connect the two.
He restates this consideration in 1920 in Berlin: “If we now inquire into a deeper
meaning of the correspondence established, we are naturally led to expect first that the
correspondence arises not only in an agreement of the frequencies of spectral lines
determined by the two methods, but will remain valid also for their intensities; an
expectation that is equivalent to the proposition that the relative probability of a given
transition between two stationary states is connected in an easily stated way with the
amplitude of the corresponding harmonic component of the motion.”39 He doesn’t explicitly
explain how the connection to the intensities was discovered, but he says that he was
‘naturally led’ to this conclusion, where we see the famous ‘physics intuition’ Bohr was
renowned for having.
In the years after 1920 Bohr continued to elaborate and rethink his atomic theory
and the correspondence principle. He admitted that “the correspondence principle, like all
other notions of quantum theory, was of a somewhat formal character”, by which he meant
that the validity is independent of any empirical procedure. He also emphasized the
successes experimentally obtained with the principle, to soften this notion of a formal
character. He said “a reality of the assumptions of spectral theory” of a kind that allowed
other physical and chemical properties of atoms to be explained on the same basis. 40 In
other words the electron orbits were maybe not as real as our solar system, but to Bohr
they provided a universal explanation, which included chemical as well as electoral
properties of different atoms.
There were, however, a few withstanding problems with Bohr’s atomic theory, as
Sommerfeld expressed in a few letters to Bohr: “It seems to me that the goal of determining
the number of electrons in each ring still lies far in the future”; “there is too much that is
hypothetical about the positions and populations of the rings”41. Sommerfeld noted here
one of the biggest difficulties with the model, although it could explain quite some
experimental facts, it was not able to determine the position and number of electrons in the
rings. This was crucial to know in order for a lot of the chemical properties of the atoms to
be explained. Bohr also noted this problem with the atomic theory himself, but he could not
seem to find a solution to it.
37
Own translation off: “Durch diese Übereinstimmung haben wir offenbar eine Verbindung zwischen dem
Spektrum und dem Atommodelle des ~Wasserstoffs erhalten, die so innig ist, wie man es nur hoffen konnte,
wenn man sich den grundsätzlichen Unterschied zwischen den Vorstellungen der Quantentheorie und der
üblichen Strahlentheorie vor Augen hält.“
38
Bohr, Niels. "Über die Serienspektra der Elemente." Zeitschrift für Physik A Hadrons and Nuclei 2.5 (1920):
423-469.
39
Niels Bohr. Uber die Serienspektra der Elemente. Zeitschrift für Physik, 2:423–469, 1920. Translated into
English by A. D. Udden as, 431.
40
Niels Bohr. Atomic structure. Nature, 107(2682):104–107, March 1921.
41
Sommerfeld to Bohr, 18 May 1918, quoted from J. L. Heilbron, "The Kossel Sommerfeld Theory and the Ring
Atom"
13
Bohr’s thoughts on the atomic theory after 1920 appeared in different formulations
and versions of paper that were not necessarily mutually consistent. Although not
consistent in some manner, his thoughts where fruitful as he came to insight that there was
a problem with the logical point of view behind the correspondence principle. He once again
noticed that the use of classical conception in the formulation of a quantum law may not
seem consistent. He therefore altered his original expression for the correspondence
principle “a formal analogy between the quantum theory and the classical theory”42 and
remarked on this insight: “Such an expression might cause misunderstanding, since, in fact
… The correspondence Principle must be regarded purely as a law of quantum theory.
Which can in no way diminish the contrast between the postulates and electrodynamic
theory.”43 The thought; ‘that the essence of the quantum theory and the classical theory
was not overbridged, or trying to be overbridged, by the correspondence principle’, was so
fundamental to Bohr that he repeated it later in his paper: “In the limiting region of large
quantum numbers there is in no wise a question of a gradual diminution of the difference
between the description by the quantum theory of the phenomena of radiation and the
ideas of classical electrodynamics, but only of an asymptotic agreement of the statistical
results.”44 He noted here that there was indeed only a asymptotic agreement of the existing
correspondences, which is primarily the essence of the frequency and intensity
interpretation of the correspondence principle, but as we see Bohr also states that the
correspondence principle is a ‘law of quantum theory’, which implies that it should hold for
all quantum numbers. This is only accounted for by the selection rule correspondence. By
these statements it not uniformly clear which is the correspondence principle. Although we
do see that Bohr used the correspondence principle every time a connection was made
between the classical theory and the quantum theory.
So Bohr became to call the correspondence principle a law of quantum mechanics
and dropped his original inclination to perceive it only as a heuristic tool. He started to
consider it as a principle that holds universally true, irrespective of the value of the quantum
number (so not only in the region of large quantum numbers). Which would therefore
imply, as earlier stated, that the correspondence embodies more than just the frequency or
intensity interpretation. He concluded his Nobel lecture in 1922 with the statement: “In
spite of the fundamental differences between the quantum points of view and the ordinary
conceptions of the phenomena of radiation, it still appears possible on the basis of the
general correspondence between the spectrum and the motion in the atom to employ these
conceptions in a certain sense as guides in the investigation of spectra”.45 We get here the
feeling that instead of a well-defined principle, the correspondence principle would become
42
1918, “The quantum theory of line-spectra”, Det kongelige Danske videnskabernes Selskab, Matematiskefysike Meddelser, 4(1): 1–36; reprinted in Bohr (1976), pp. 67–102.
43
[1923] 1924, “On the application of the quantum theory to atomic structure,” Proceedings of the Cambridge
Philosophical Society (supplement), Cambridge: Cambridge University Press, pp. 1–42. First published
in Zeitschrift für Physik, 13 (1923): 117. Reprinted in Bohr (1976), pp. 457–499
44
Idem
45
Nielsen, J. Rud, ed. The correspondence principle (1918-1923). Elsevier, 2013.
14
more of a guiding tool, in the end, that would eventually lead to modern quantum
mechanics. This was also noted by Jan Faye who said: “The guiding principle behind Bohr's
and later Heisenberg's work in the development of a consistent theory of atoms was the
correspondence rule.”46 Although the correspondence principle started as a set of semiformal rules, after the fall of Bohr’s atom model it lost most of its grounds on which it was
founded on. We will see that especially after this decline of the old atomic models, the
correspondence principle developed towards a guiding principle.
1.4 The second atomic theory
The optimism of the previous year’s surrounding Bohr’s atomic model was slightly
reduced as Bohr’s theory encountered more and more empirical and conceptual problems.
One of these problems was that the energies and spectra of many elements and the
interactions between radiation and matter such as dispersion could not be explained fully
according to Bohr’s atomic theory47. Eventually these faults in Bohr’s theory would bring it
to be replaced by Heisenberg’s matrix mechanics and Schrodinger’s wave mechanics and
Bohr’s theory would from henceforth be called the old quantum theory. Of course, the role
of the correspondence theory was altered in the face of the loss of Bohr’s atomic theory on
which it was originally based.
In June 1922 Bohr gave a lecture in Göttingen about the so-called ‘second atomic
theory’. The need for a renewal of his old atomic theory was no doubt stimulated by the
vexing case of the helium atom. It became apparent after several scientists48 tried to solve
the problem with help of Bohr’s atomic model of 1918 that it was inadequate to explain it.
None of them could find an electron configuration that agreed with the experimental data
on the spectrum of helium. As early as 1921 Bohr already stated that the earlier physical
models contradicted the correspondence principle and where in fact wrong. He said certain
symmetries couldn’t be upheld, from which “we are led... directly to the conclusion that we
cannot expect in actual atoms configurations of the type in which the electrons within each
group are arranged in rings or configurations of polyhedral symmetry, because the
formation of such configurations would claim that all the electrons within each group should
be originally bound by the atom at the same time.”49 Why the correspondence principle was
the key to understanding the atomic structure was not explained explicitly, as Bohr spoke
only in general terms about the principle50. He probably noted that the atomic models did
not explain the found experimental data correctly, but the correspondence principle did still
46
Faye, Jan, "Copenhagen Interpretation of Quantum Mechanics", The Stanford Encyclopedia of
Philosophy (Fall 2014 Edition), Edward N. Zalta (ed.),
47
Max, Jammer. "The conceptual development of quantum mechanics.", chapter 3 and 4, (1966).
48
Among the physicists who tried to solve the helium problem were J. Franck, F. Reiche, Land, E. C. Kemble, J.
H. Van Vleck, Langmuir, Born, Werner Heisenberg, Pauli, Bohr, and H. A. Kramers.
49
Bohr, "Atomic Structure" (February 1921), p. 106.
50
Niels Bohr's Second Atomic Theory, Helge Kragh, Historical Studies in the Physical Sciences, Vol. 10 (1979),
pp. 123-186
15
have some heuristic value, as the ties between quantum and classical theory still remained
in a certain sense.
From the manuscript of the lecture in Göttingen, we can construct the general
concept of the revised atomic theory that was primarily built on the correspondence
principle. The model that Bohr proposed was, in short, the following; electrons make up
various groups and subgroups that correspond to certain quantum numbers and orbits,
which are in line with the classic Keplerian ellipses. During the orbiting the electrons
penetrate the region of internal electrons, causing a coupling of the revolving electrons,
whose motions and numbers where specified by the correspondence principle.
Figure 1; Bohr's illustrations for his atomic theory of 1921 extracted from the paper "On the
Ground States of the Atoms of the Elements", (Bohr Archive, Copenhagen.) The figures and
formulas in the lower left square refer to the 1913 model, whereas those in the right square
refer to the 1921 model, which introduced elliptical orbits and the penetration effect; the
formulas refer to the mechanical treatment of the new model as Bohr explained in
Göttingen in 1922.
Although the results of the second atomic theory where quite astonishing; as for the first
time the constitution of several noble gases could be explained. The origins of his theory
remained quite obscure. No one except for maybe Bohr and his wonder student Kramers
could quite follow how the correspondence principle was to be applied and how it restricted
the motions that the electrons where allowed to make. The reactions on the theory were
mainly positive though. For example Ernest Rutherford expressed his high value of Bohr’s
new theoretical ideas, but also announces his bafflement over its origins, he stated: “it is
very difficult for me to form an idea of how you arrive at your conclusions. Everybody is
eager to know whether you can fix the 'rings of electrons' by the correspondence principle
or whether you have recourse to the chemical facts to do so.”51 This was the general
response of most of the physicists; that Bohr had with his second atomic theory found a way
to formally deduce the atomic movements, but that it was just very ‘difficult’ to understand.
51
Rutherford to Bohr, 26 September 1921 (BSC).
16
Bohr seemed to have made simultaneous and intricate use of two types of
considerations while formulating his second atomic theory52; the first one was mostly
inductive and relied on the empirical stream of information provided by series spectra, X-ray
spectra, chemical properties etc. So although the evidence provided by the spectra was
valid, the conclusion drawn on the basis of this evidence was probable. The second type
pointed to the deductive use of the general assumptions of the quantum theory, for which
Bohr had the hope that they would be self-sufficient. The correspondence principle played a
major part in the two types of considerations. Firstly, as a means of connecting empirical
spectra and electronic orbits and as an a priori procedure for deriving “exclusion rules” that
forbid certain electronic configurations from occurring. The second atomic theory is mostly
based on an analogy to the Bohr-Kramers theory of the helium atom, which assumed that
the element was stable, and would therefore prove eventually to fail when it appeared that
the helium atom is not stable.
In 1922 Bohr was awarded the Nobel Prize “for his services in the investigation of the
structure of atoms and of the radiation emanating from them”53. In Nobel lectures on the
atomic theory Bohr did not overestimate the ground on which his reasoning’s where based
and knew them to be shaky. He therefore emphasized the empirical power of the
correspondence principle where he paid particular mind to the calculations done by Kramer
as proof that: “It is seen that the theory reproduces completely the main feature of the
experimental results, and in the light of the correspondence principle we can say that the
Stark effect reflects down to the smallest details the action of the electric field on the orbit
of the electron in the hydrogen atom, even though in this case the reflection is so distorted
that, in contrast with the case of the Zeeman effect, it would scarcely be possible directly to
recognize the motion on the basis of the classical ideas of the origin of electromagnetic
radiation.”54 Bohr always considered empirical evidence the most important factor of a
strong theory and therefore placed strong faith in the correspondence principle, as it was
align with the experiments. From here on Bohr continues to emphasize that the allowed
transitions between stationary states are connected to the harmonic components in the
classical motion, and if these do not occur that the transition will not happen. In the 7th
Guthrie lecture, given on March 24, 1922, he says:
“This law, which has been called the correspondence principle, states that the
occurrence of each transition between two stationary states accompanied by
emission of radiation is correlated to one of the constituent harmonic oscillations
into which the electric moment of the atom considered as a function of time can be
resolved, to the extent that the probability of the occurrence of a transition shall
depend on the amplitude of the corresponding harmonic oscillation of the atom, in
52
Darrigol, Olivier. From c-numbers to q-numbers: the classical analogy in the history of quantum theory. Univ
of California Press, 1992.
53
The Nobel Prize in Physics 1922". Nobelprize.org. Nobel Media AB 2014. Web. 2 Apr 2015.
<http://www.nobelprize.org/nobel_prizes/physics/laureates/1922/>
54
Bohr, 1922, Nobel speech, reprinted in; Bohr, Niels. The Periodic System (1920-1923). Vol. 4. Elsevier, 1977.
17
such a way that in the limit when the quantum-number is large, the intensity of the
emitted radiation in unit time in the mean shall be the same as that which would
follow from the classical laws of electrodynamics. A similar connection with the
classical theory will be exhibited by the polarization of the emitted radiation. If, for
instance, the corresponding harmonic oscillation in all states of the atom is a linear
vibration or a circular rotation, the radiation will have the same constitution as that
which on the classical theory would be emitted by an electron executing harmonic
motion of that type.”55
Bohr thus presents all three correspondence relations: the selection rule correspondence,
the intensity correspondence and the frequency correspondence. Would the correspondence
principle then be a combination of all of these? He does not explicitly states what the
correspondence principle entails, but it seems clear that almost every found experimental
connection between the classical theories and quantum theory could theoretically be a form
of the correspondence principle. As the atomic model begins to crumble, however, we will
see that the correspondence principle will undergo a new metamorphosis to be included in
the new quantum mechanics.
1.5 The collapse of the old quantum theory
At the beginning of 1924 problems began to accumulate surrounding Bohr’s atomic theory.
Some of them where old, others new, but in the end they all caused the same effect: the
replacement of the old atomic theory. One of the more severe problems belonged to the
anomalous Zeeman effect, which showed op for atoms with an odd atomic number (mainly
hydrogen). This could not be explained by the rules following Bohr’s model. According to
Kragh ‘the atomic models after 1922 became increasingly abstract, symbolic, and formal,
which soon would lead to the proposal of a total abandonment of the model concept’56. We
can notice with this abstraction of the atomic theory a firm resistance of Bohr, and other
physicists, to fully let go of the atomic models, which until then worked quite well. One of
the first that would wholly refuse to use the atomic models any further would be Pauli,
followed by Heisenberg.
In 1924 Bohr, Kramers and Slater proposed a new atomic theory: The Bohr-KramersSlater (BKS) theory. This was perhaps the last attempt to understand the interaction of
matter and electromagnetic radiation on the basis of the so-called old quantum theory. It
was an attempt to understand radiation from atoms without making use of the concept of
light quanta. One aspect, the idea of modelling atomic behaviour under incident
electromagnetic radiation using "virtual oscillators" at the absorption and emission
55
Niels Bohr. The effect of electric and magnetic fields on spectral lines. Proceedings of the Physical Society
(London), 35:275–302, 1922-3. The Seventh Guthrie Lecture
56
Niels Bohr's Second Atomic Theory, Helge Kragh, Historical Studies in the Physical Sciences, Vol. 10 (1979), p.
313.
18
frequencies, rather than the (different) apparent frequencies of the Bohr orbits. Following
Slater, the basic description of the BKS theory was to assign to each atom with an
unspecified number of virtual oscillators that produced a virtual radiation field through
which the atoms entered in a mutual communication. It remained unclear, however, what
these virtual oscillators were, besides that they were to be taken to be abstract and
unphysical quantities. This abstract use of the virtual oscillators made the theory vague and
unmanageable for many other theorists in the same field. This was also one of the main
reasons for the short lifespan of the theory.
As mentioned earlier one of the disadvantages of light quanta was that the picture of
correspondence was not aligned with it. This could have been one of the reasons why Bohr
tried to deny its existence until the very end. In the BKS-theory the correspondence
principle was included in the following manner: “It must be remembered that the analogy
between the classical theory and the quantum theory as formulated through the
correspondence principle is of an essentially formal character, which is especially illustrated
by the fact that on the quantum theory the absorption and emission of radiation are
coupled to different processes of transition, and thereby to different virtual oscillators” 57.
We can see here that the correspondence principle has fully developed in its form as a
guiding principle. The principle changed from a set of rules for the connections between
classic and quantum theory to a general relation between quantum and classical theory. The
new BKS theory was the most appropriate way to include the correspondence idea in terms
of space-time pictures. There was a price to be paid though and Bohr made it in form of a
relaxation of the energy principle and the “virtualization” of the electromagnetic field. He
said: “we abandon on the other hand any attempt at a casual connection between the
transitions in distant atoms, and especially a direct application of the principles of
conservation of energy and momentum so characteristic for the classical theories.”58 The
BKS theory maintained only statistical energy conservation. Which was one of the biggest
reasons that the majority of physicists rejected the theory. For example, Pauli wrote as a
response to the draft version of the BKS theory: “I laughed a little… about your warm
recommendations of the words “communicate” and “virtual” …On the basis of my
knowledge of these two words… I have tried to guess what your paper is all about. But I
have not succeeded”.59 It was nonetheless influential and would eventually lead to the
development of what we now call modern quantum mechanics.
Bohr himself was not actively involved in the creation of the new quantum
mechanics, but he did talk about the recent developments in the quantum theory. After
having received a copy of the work of Heisenberg he commented on it as “a brilliant
realization of the efforts which hitherto have served as a guideline in the development of
57
Kragh, Helge. "Bohr—Kramers—Slater Theory." Compendium of Quantum Physics. Springer Berlin
Heidelberg, 2009. 62-64.
58
Bohr, Niels, Hendrik Anton Kramers, and John Clarke Slater. "LXXVI. The quantum theory of radiation." The
London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 47.281 (1924): 785-802.
59
Pauli to Bohr, 21 February 1924, reprinted in Hermann et al. 1979, p.147.
19
the atomic theory”.60 He further noted that: “In the nature of matter, there is no longer a
question of an independent correspondence principle in the above-mentioned scheme, but
rather the whole formulation of quantum mechanics may well be regarded as a sharpening
of the content of this principle”61. We see that in Bohr’s opinion the new quantum theory
should be regarded as a revolution that found its origin in the correspondence principle.
Which is quite interesting, because in the new quantum theory the correspondence
principle would not be specifically used. The only form in which it was used was as a guiding
principle in translating the classical concepts to the quantum concepts.
Although Heisenberg used the correspondence principle in the formulation of his
new quantum theory, he denied the fundamental nature of the correspondence principle
and did not call it, unlike Bohr, a law of quantum mechanics: “it must be emphasized that
this [correspondence] is a purely formal result; it does not follow from any of the physical
principles of quantum theory”62. So it was only a mathematical result that did not have any
further physical implications. This also signifies a general difference in the attitude of Bohr
and Heisenberg in physics. While Heisenberg remained decidedly mathematical and formal
in his theories, Bohr took a more philosophical approach and always searched for higher
serving relationships between theory and reality.
In the end Bohr restricted the principle to the case of the transition from classical
theory to quantum theory. He stated: “The correspondence principle expresses the
tendency to utilise in the systematic development of the quantum theory every feature of
the classical theories in a rational transcription appropriate to the fundamental contrast
between the postulates and the classical theories”.63 So the original formulations of the
correspondence principle as the frequency theorem interpretation, intensity interpretation
and selection rule interpretation were mainly based on the old atomic model, as defined by
Bohr, and after this model turned out to be insufficient the correspondence principle
developed into a possibility to transcribe the classical components to the quantum ones,
which therefore gave a bridge between classical and quantum theoretical theories.
1.6 The Correspondence Principle Defined
While reading the works of Bohr and trying to define “the” correspondence principle one
comes across a few hardships. First of all, there are two main analogies between the
classical and quantum mechanics that Bohr uses throughout his work; the frequency
analogy and the intensity analogy. Secondly, there is the use of the selection rule principle
that is also named as a correspondence rule; Thirdly, the scope in which the correspondence
principle applies is unclear: is it a law of quantum mechanics that is valid for all quantum
numbers or is it only valid in an approximation to large quantum numbers? Finally, if the
60
Stolzenburg, Klaus, ed. The Emergence of Quantum Mechanics (Mainly 1924-1926). Elsevier, 2013.
Bohr 1925a. Manuscript on ‘Atomic Theory and Mechanics, in Kragh 2012 p. 355.
62
Heisenberg, W. "The Physical Principles of the Quantum Theory (1930)."Scientific Review Papers, Talks, and
Books Wissenschaftliche Übersichtsartikel, Vorträge und Bücher (2013): 118., p. 83
63
Bohr, Niels. "Atomic theory and mechanics." Nature 116.2927 (1925).
61
20
correspondence principle is one of these or a combination of these analogies, why does it,
according to Bohr, still hold in modern matrix quantum mechanics, as defined by
Heisenberg, while the new quantum mechanics is in general seen as the break with all
classical relations?
One of the important things to remember while looking at the correspondence
principle is that it was primarily based on Bohr’s atomic model, which underwent quite a
few changes and with it also the correspondence principle. ‘He first considered strictly
periodic systems obeying ordinary mechanics (1913-1916), motions of multi periodic
systems subjected to ordinary mechanics (1917), multi periodic motions of not necessarily
multi periodic systems still subjected to ordinary mechanics (1918-1922), multi periodic
motions eluding ordinary mechanics (1922-1925); and finally after 1925 he gave up on the
notion of definite electronic orbits.’64 We saw that in the beginning the correspondence
principle was primarily the frequency interpretation. Later also the intensities of the system
were considered and the selection rules came about, but as soon as Bohr gave up the notion
of the definite electronic orbits, these rules also lost their grounds. After that the
correspondence principle, became a more ‘obscure’ guiding rule that gave the last
connections to the classical theories.
One of the most influential scientific elaborations on the correspondence principle
appeared in the work of Max Jammer ‘The conceptual Development of Quantum
Mechanics’. Jammer writes: “In the limit, therefore, the quantum-theoretic frequency
coincides with the classical mechanical frequency. By demanding that this correspondence
remain approximately valid also for moderate and small quantum numbers, Bohr
generalized and modified into a principle what in the limit may be regarded formally as a
theorem.”65 Jammer focuses in his interpretation of the Correspondence principle mainly on
the frequency relation. He thereby accounts for Bohr’s writing; that the correspondence
principle should hold also for small quantum numbers, by stating that this it approximately
valid for small quantum numbers. Jammer is thereby rather dismissive of Bohr's claim that
the correspondence principle should be thought of as a law of quantum theory. He writes:
“For taking resort to classical physics in order to establish quantum-theoretic predictions, or
in other words, constructing a theory whose corroboration depends on premises which
conflict with the substance of the theory, is of course a serious inconsistency from the
logical point of view. Being fully aware of this difficulty, Bohr attempted repeatedly to show
that the correspondence principle must be regarded purely as a law of the quantum
theory.”66 He states Bohr wanted to cover up the inconsistencies of his theory by declaring
the fundamental nature of the correspondence principle.
Oliver Darrigol gives a slightly different interpretation in his overview of the quantum
history in his book ‘From c-numbers to q-numbers’. He sees the correspondence principle as
primarily the intensity interpretation: “Bohr assumed that, even for moderately excited
states, the probability of a given quantum jump was approximately given by the intensity of
the corresponding harmonic component of the motion in the initial stationary state. This is
what Bohr called the correspondence principle.”67 Strictly speaking this intensity
correspondence is exact only in the limit of large quantum numbers, and cannot be
64
Darrigol, Olivier. From c-numbers to q-numbers: the classical analogy in the history of quantum theory. Univ
of California Press, 1992.
65
Jammer 1966, p. 111
66
Jammer 1966, p. 116
67
Darrigol 1992, 126
21
extended to small quantum numbers. Darrigol noticed this inconsistency in Bohr’s writing,
but he states about the correspondence principle that: “the correspondence principle even
in the hands of its creator, failed to provide unambiguous guidance in the building of
atoms.”68 Because of the many, not always in alignment, statements of the correspondence
principle both Jammer and Darrigol agree that there could also be a possibility that there is
no such thing as ‘the’ correspondence principle. Jammer, for example, writes, “Bohr’s
numerous and often somewhat conflicting statements, on the essence of the
correspondence principle make it difficult, if not impossible, to ascribe to Bohr a clear-cut
unvarying conception of the principle”.69 Similarly Darrigol writes: “Confronted with this
paradoxical appearance of Bohr’s work, many physicists and historians have renounced the
project of giving a rational account of it. In their opinion, Bohr’s success owed much to an
unusual tolerance for contradiction …”70. One would except in face of these many
contradictions, that the correspondence principle was not well received, however, it is
noticeable, that the reactions on the work of Bohr, where mainly positive. There was a
general belief in the physics community that there was a concrete basis for the use of the
correspondence principle. This belief can be traced back to his original successes with his
first atomic theory and therefore could have allowed the later theories of Bohr to hold ‘an
unusual tolerance for contradiction’; we will see this also in the second chapter, which
elaborates more on the reactions of the physics community on the correspondence
principle.
Alisa Bokulich states in her paper Three Puzzles about Bohr’s Correspondence
principle that the correspondence principle is defined as the selection rule interpretation.
Stating that it is the formulation that each allowed quantum transition has a harmonic
component in the electron’s classically described motion and if this harmonic is missing then
the quantum transition is not allowed. She recognizes in her paper the other two
correspondences as formulated by Bohr, but dismisses them as evidence for the selection
rule correspondence principle, she writes: “It is important to recognize, however, that none
of these particular correspondences, which, in the limit of large 𝑛, allow for a direct
calculation of various quantum quantities from the classical harmonic components, are
themselves the correspondence principle. Rather they can be used alternatively as inductive
evidence for the correspondence principle, or, once the correspondence principle is
established, as applications or consequences of the correspondence principle. The
correspondence principle is a more general relation underlying these various particular
correspondences.”71 So Bokulich found that Bohr intended the correspondence principle to
be primarily the selection principle, however this leads to a counterpoint that the selection
principle is just a special case of zero probability, but Bohr connects his principle explicitly
with both probabilities and polarizations as noted by Rynasiewicz.
In general all three correspondence relationships are found in the works of Bohr and
all three of them can be accounted as the correspondence principle. We can find parts in
the work of Bohr where he names only one of them: “The possibility of the occurrence of a
transition, accompanied by radiation, between two states of a multiply periodic system, of
quantum numbers for example 𝑛1 , … , 𝑛𝑢 ′ and 𝑛1 ′′, … , 𝑛𝑢 ′′, is conditioned by the presence
of certain harmonic components in the expression given by…[the Fourier series expansion of
68
Idem
Jammer 1966, 117
70
Darrigol 1997, p. 546
71
Bokulich, Alisa. "Three Puzzles about Bohr's Correspondence Principle." (2009).
69
22
the classical electron motion]. The frequencies 𝜏1 𝜔1 + ⋯ + 𝜏𝑢 𝜔𝑢 of these harmonic
components are given by the following equation 𝜏1 = 𝑛1′ − 𝑛1′′ , … , 𝑛𝑢′ − 𝑛𝑢′′ . We, therefore,
call these the “corresponding” harmonic components in the motion, and the substance of
the above statement we designate as the “Correspondence Principle.”72 This is a prime
example of the correspondence principle as defined by the selection rule interpretation. We
can however also find writings where two or all three of the interpretations are
simultaneously used together: “The demonstration of the asymptotic agreement between
spectrum and motion gave rise to the formulation of the “correspondence principle,”
according to which the possibility of every transition process connected with emission of
radiation is conditioned by the presence of a corresponding harmonic component in the
motion of the atom. Not only do the frequencies of the corresponding harmonic
components agree asymptotically with the values obtained from the frequency condition in
the limit where the energies of the stationary states converge, but also the amplitudes of
the mechanical oscillatory components give in this limit an asymptotic measure for the
probabilities of the transition processes on which the intensities of the observable spectral
lines depend.”73 Here we see that Bohr uses all three of the interpretations, but he again
doesn’t explicitly define the correspondence principle.
As mentioned before, the atomic theory of Bohr underwent quite some changes in
the years between 1913 and 1925 and therefore the correspondence principle also
changed. For this reason there is a last opinion in explaining what the correspondence
principle holds, which I also account for, that there is no such thing as the correspondence
principle. It got altered, changed in basis and formulation and later even used on a more
philosophical ground, and did not remain unambiguous in definition. Rynasiewicz also takes
this somewhat more extreme vision on the correspondence principle. He says “Bohr was not
always clear nor necessarily consistent”74 and states that the correspondence principle
evolved as the quantum mechanics evolved; “One comes to suspect that, to the extent that
the content of “the” correspondence principle can be articulated, that content must be seen
to evolve hand in hand with the evolution of the quantum theory.”75 Thus the
correspondence principle changed, got altered and evolved together with its basis: the
atomic theory. One can therefore not define ‘the’ correspondence principle. One can only
state what the main correspondence relationship was during a certain period in the atomic
theory. A general feature of the correspondence principle does remain in all Bohr’s work: a
bridge between classical and quantum theory. Also Darrigol writes; “The precise expression
and scope of the CP depended on the assumptions made about the electric motion.
Whenever this motion was a priori determined the ‘correspondence’ aided in deducing
properties of emitted radiation. In the opposite case, characteristics of the electronic
motion could be induced from the observed atomic spectra. This ambiguity made the CP a
very flexible tool that was able to draw the most from the permanent inflow of empirical
data”76. So the correspondence principle was not set in its formulation and changed
according to the context it was used in and as this context, the atomic theory, changed so
did the correspondence principle.
72
Bohr 1924, p. 22
Bohr 1925a. Manuscript on ‘Atomic Theory and Mechanics
74
Rynasiewicz, Robert. "The Correspondence Principle." (2013).
75
Idem
76
Darrigol 1992, p.83
73
23
2 The reception of the correspondence principle among Bohr’s peers
In chapter one we already saw that there was quite some flexibility in the definition
of the Correspondence principle and that there was not one coherent interpretation of it. If
the principle was however not even definably clear in the work of its founder Bohr, how did
his peers perceive it? It is important to understand the distinction between the
correspondence principle in Bohr’s own understanding and what it became to mean for the
physics community of his time. We can divide the responses to Bohr’s correspondence
principle into roughly three categories77; ‘those who misunderstood the principle (e.g., Born
and Rosenfeld), those who embraced and developed it (e.g., Kramers and van Vleck), and
those who seemed to understand it, but nevertheless mistrusted it (e.g., Sommerfeld and
Pauli)’. We cannot say who understood the correspondence principle completely and if it
was even possible to do so, as it was so flexible in its definition. Even when we look at the
supporters of the correspondence principle we can see that there were quite a few whom,
according to Bohr, didn’t fully grasp the correspondence principle. For example Léon
Rosenfeld, a Belgian physicist and collaborator of Bohr, wrote about Bohr’s annoyance ‘over
his failure to have correctly understood the substance of this principle’.78
Bohr himself was quite aware of the misinterpretations that his often-difficult
written essays on the topic could induce. He realized a lot of physicists regarded his work as
being ‘philosophical’ and ambiguous. For example he wrote to Sommerfeld: “I should not
like you to get the impression that my inclination to pursue obscure, and undoubtedly often
false analogies makes me blind to that part of the formation of our conceptions that lies in
unveiling of the systematic of the facts. Even if I were blind, I would only need to glance at
your book to be healed”79. The book Bohr talks about was Sommerfeld’s famous Atombau
that at that time gave one of the most complete overviews of the quantum theory.
There is indeed quite an obscurity surrounding the correspondence principle and its
use, which persists even in today’s literature. A possible explanation could be that Bohr’s
general writing style was complex and sometimes difficult to read, seeing how Bohr’s
process of writing was quite different than most scientists. A common writing process is first
thinking a thought through before writing it down, for Bohr writing was an important part of
his cognitive process. He tried to explain his thoughts in his papers with as much clarity as
possible, which sometimes led to a painful process vividly described by Heisenberg: “Bohr
would always change the sentences again and again. He could have filled half a page with a
few sentences and then everything was crossed out and changed again. And even when the
whole paper was almost finished- say, ten pages or so – the next day everything would be
77
Bokulich, Alisa, "Bohr's Correspondence Principle", The Stanford Encyclopedia of Philosophy (Spring 2014
Edition), Edward N. Zalta (ed.).
78
Rosenfeld, L., 1979 [1973], “The Wave-Particle Dilemma”, in R. Cohen and J. Stachel (eds.) Selected Papers of
Léon Rosenfeld (Boston Studies in the Philosophy of Science, Volume 21), Dordrecht: D. Reidel Publishing Co.,
688–703. Originally published in J. Mehra (ed.) The Physicist's Conception of Nature, Dordrecht: D. Reidel
Publishing Co., pp. 251–263.
79
Bohr to Sommerfeld, 21 November 1924, in Stolzenburg 1984 p.38.
24
changed over again. So it was a continuous process of improvement, change and discussions
with others… The final text of Bohr’s paper was so subtle and he would think about half an
hour whether in a certain case he would use indikativ or konjunktiv and so” 80. We can see
that Bohr considered it important to clarify his thoughts as much as possible, however, this
sometimes had the opposite effect, making it even more ambiguous. Could it have
happened that, because of this ambiguity and the certain flexibility in the interpretation of
Bohr’s texts on the correspondence principle, Bohr’s original principle underwent
supplementation, modification and metamorphosis? Did it come to have a different
function in the physics community than originally intended by Bohr? Multiple physicists, for
example Hendrik Kramers and John van Vleck, used and extended the term
“correspondence principle” to cover a wide range of correspondence-type arguments that
were important in the development of quantum mechanics, but which were not necessarily
implied in the formulations of Bohr.
Among the physicists whom ‘correctly’ understood Bohr’s principle there were also a
lot of critics. Their main argument against Bohr’s atomic theory, not entirely unjustified, was
that the basis of it was a messy, patchy combination of classical and quantum elements. Or
as Henry Mergenau later phrased it: “Bohr’s atom sat like a baroque tower upon the Gothic
base of classical electrodynamics”.81 Among skeptics the most noticeable were Arnold
Sommerfeld and Wolfgang Pauli.
In this chapter I will try to give some insight into the question “how was the
correspondence principle, as formulated in the old quantum theory, received by the physics
community?” I will pay attention specifically to the thoughts of the physicists who were
closely involved in the development of the quantum theory such as; Sommerfeld,
Heisenberg, Pauli, Kramers and others. I believe by looking at the general responses on the
correspondence principle of Bohr, we can obtain more insight in the correspondence
principle and how it was used and altered.
2.1 Sommerfeld; a magic wand; Die Zauberkraft des, Korrespondenzprinzips.
Even though the correspondence principle successfully explained various
phenomena, most notable the derivation of selection rules for quantum transitions, it didn’t
convince one of Bohr’s biggest competitors at the time, Arnold Sommerfeld. As Darrigol has
recounted in detail: “Sommerfeld was never comfortable with Bohr's correspondence
principle, and only begrudgingly admitted its fertility”.82 As early as 1919 Sommerfeld wrote
to Bohr as a reaction on his paper On the quantum theory of line spectra: “Your formal
theory is very interesting and fruitful. However, the hypothesis of Rubinowicz, although not
nearly as far-reaching, seems for the present more satisfactory to me.”83 At that time
Rubinowicz, a polish physicist, had indeed succeeded in deriving some of the quantum
80
Heisenberg, Werner. "Interview conducted by TS Kuhn (Munich, February 19, 1963)." Archive for the History
of Quantum Physics (1963): 16-17.
81
Margenau, Henry. "The nature of physical reality: A philosophy of modern physics." (1950).
82
Darrigol 1992, pp. 138–145.
83
Sommerfeld to Bohr, 5 feb 1919, reprinted in Darrigol 1992.
25
selection rules without the help of the Correspondence principle. He had used an extension
of the conservation of angular momentum to radiation processes. Although Bohr found
Rubinowicz’s deduction interesting and added his findings in a footnote in the second part
of his 1918 memoir, he paid little attention to Sommerfeld’s comment; that it was
‘physically more instructive’ than his correspondence principle. In 1920 in his Berlin lecture
Bohr made the following comment: “The conclusions which we can draw by means of the
correspondence principle are of a more detailed character than can be obtained solely from
a consideration of the conservation of angular momentum. For example that 𝑘 cannot
change by more than unity, while the correspondence principle requires that 𝑘 shall vary by
unity for every possible transition and that its value cannot remain unchanged. Further, this
principle… also enables us to draw conclusions about the relative probabilities of the various
possible types of transitions from the values of the amplitudes of the harmonic
components”84. From the comment we can see that Bohr had a strong faith that the
correspondence principle would prove to be the better theory. One of the strongpoints of
the correspondence principle was that it was highly supported by the experimental data.
This experimental data accounted for the fact that azimuthal quantum number should be
𝑘 = ±1, which was in alignment with Bohr’s principle, but not that of Rubinowicz’s, which
was not able to exclude 𝑘 = 0. This was where the correspondence principle thrived and
the theory of Rubinowicz would end up lacking.
One of the most important contributions to the quantum physics of Sommerfeld was
his famous book Atombau und Spectrallienen, first published in 1919. It was perceived at the
time as a good indicator of the current knowledge on quantum physics. Sommerfeld also
mentioned the correspondence principle in his book. Although not in the most positive
manner “In this way, by remarkably rigorous manner of deduction, reminiscent of the
incontrovertible logic of numerical calculations, we have arrived from the principle of
conservation of angular momentum at a principle of selection and a rule of polarization […]
On the other hand, Bohr has discovered in his principle of correspondence a magic wand,
which allows us immediately to make use of the results of the classical wave theory in the
quantum theory”.85 The term “magic wand” was clearly ironically intended, as no doubt
Sommerfeld preferred a rational claim to the use of magic. Sommerfeld could not get clear
insight in the exact use of the correspondence principle, only that it provided the same
solutions as the experimental data. The basis of the principle was vague, according to
Sommerfeld, and therefore not the right approach to the solution of the quantum problems.
In a letter to Bohr on 11 November in 1920 Sommerfeld expressed again his
uneasiness with the Correspondence principle, but he also softens his criticism by noting its
heuristic value; “In the appendixes to my book you can see that I have gone to some pains
to treat your correspondence principle better than in the first edition… Nonetheless I must
84
Bohr, Niels. "Über die Serienspektra der Elemente." Zeitschrift für Physik A Hadrons and Nuclei 2.5 (1920):
423-469.
85
Sommerfeld, A. "Atombau und Spektrallinien. Friedrich Vieweg und Sohn, Braunschweig (1919)." English see
Atomic Structure and Spectral Lines, translated from the third German edition by HL Brose, Methuen (1923).
26
confess that your principle, the origin of which is foreign to the quantum theory, is still a
source of distress to me, however much I recognize that through it a most important
connection between quantum theory and classical electrodynamics is revealed”.86 This
distress was caused by Bohr’s use of classical concepts in quantum theoretical context.
According to Sommerfeld, Bohr used these concepts without proper explanation and,
although they gave the right answers, Sommerfeld found that this was not an appropriate
way to conceive physics.
In the second (1921) and third (1922) editions of Sommerfeld’s Atombau there was
however an honorable place for the correspondence principle: “Bohr’s method is not only of
greater consequence [than Rubinowicz’s] in the question of intensity, but also leads to
sharper and more definite results as regards the question of polarization… In the matter of
method the principle of correspondence has the great advantage that it postulates that
Maxwell’s theory be generally valid for long waves, and that it does not throw overboard
the many useful results, which the classical theory gives for optical waves and Rontgen rays.
In the matter of atomic models we must now recognize the complete superiority of the
correspondence principle. For here it seems that Bohr, by applying classical mechanics and
electrodynamics to obtain definite statements about the periodic system and the atomic
shells, has succeeded to advance where other roads have been inaccessible”.87 This change
in attitude came out of the fact that the correspondence principle was able to successfully
explain many new empirical results. Sommerfeld had adopted Kramers’ correspondencebased analysis, concluding that now the fine structure of one-electron atoms was perfectly
understood: “The exemplary agreement with experiment, which Kramers has obtained in
particular in the case of Paschen’s He+ observations and also in the case of the stark effect…
confirms convincingly the fertility of Bohr’s correspondence principle.”88 Kramers’
calculation explained the lines observed by Paschen in the hydrogen spectrum. The
hydrogen spectrum is an emission spectrum that is divided into a number of spectral lines,
where the lines represent the electron making a transition between two energy levels. The
Paschen series are then the series of transitions of electrons going from a layer n to the
layer n = 3. Kramers’ calculations explained perfectly the transition from the (4,1) → (3,1),
however, according to the same reasoning as used in the theory there should have been a
fairly noticeable component from the (4,3) → (3,1) transition which was missing in the
experiments, but this was strangely not noted for a long time.
You could state that Sommerfeld never fully came to terms with the correspondence
principle. Although he learned to appreciate its heuristic value he said: “Nonetheless I
cannot view it as ultimately satisfying on account of its mixing of quantum-theoretical and
classical viewpoints.”89 We once again see here Sommerfeld’s critical attitude towards the
mixing of classical and quantum concepts by Bohr. Sommerfeld was not alone in his aversion
86
Sommerfeld to Bohr, 11 November 1920 in Rud Nielsen 1976, p. 690.
Sommerfeld, A. "Atombau und Spektrallinien.
88
Idem.
89
Idem.
87
27
towards the ambiguous nature of the correspondence principle. In France, Louis de Broglie
commented on the correspondence principle with a hint of criticism. He wrote in 1924 in his
thesis: “This idea of correspondence, which seems still imprecise and rather elastic, will
have to guide the bold researchers who wish to set up a new electromagnetic theory that
agrees better with quantum phenomena than the present theory does”.90 This elasticity was
the point that most physicists found offensive and could not accept.
In contrast to this ‘elastic’ principle, Sommerfeld and his followers tended to strive
toward a more axiomatic deductive approach, where you start with a few self-evident
axioms and from there, through the rules of logic, derive new theories. According to him,
Bohr’s approach didn’t have the desirable approach in this direction. We can see from
Sommerfeld’s opposing attitude that there was definitely some reluctance towards the
correspondence principle. This could have been, because of its ambiguous nature or
because Bohr never fully explained what it entailed. As we saw the main point of the
opposition was the mixing of the quantum and classical rules, as the two theories were
fundamentally different. Sommerfeld's critical attitude towards the correspondence
principle would prove influential on Wolfgang Pauli and Werner Heisenberg, both of whom
were his doctoral students. Especially Pauli would later follow his teacher and fully oppose
the correspondence principle.
2.2 Heisenberg; ‘The correspondence principle, as important as the entire quantum theory’
In the beginning of his career Heisenberg, then a young student in München, did not
quite agree with his esteemed professor Sommerfeld on the ambiguous nature of the
correspondence principle. In his earlier development of the core model of the atom, he
placed strong faith in the correspondence principle. In 1921 he wrote to Landé: “The fact
that the entire model interpretation of a process can be calculated from purely empirical
material is another brilliant achievement of the Bohr correspondence principle, which I am
beginning to want to consider as important as the entire quantum theory”91. We see here
an example of this faith in the theory of Bohr, which can be traced back to the glamour
surrounding Bohr’s name at the time.
In June 1922 Bohr gave a series of lectures sponsored by the Wolfkehl Foundation in
Gottingen, which became known as the Bohr Festspiele (Bohr festival). He explained his
ideas to a large group of physicist, among them also young practitioners Heisenberg, Pauli
and Jordan. Heisenberg, then 21 years old, was very impressed with Bohr’s core model and
his philosophical views on the atomic theory. He stated: “Bohr’s long stream of thoughts, of
which only the beginning was clear, and whose end disappeared in the darkness of a
philosophical attitude that was very stimulating to me”.92 Hund, a German physicist whom
90
De Broglie, Louis. Recherches sur la théorie des quanta. Diss. Migration-université en cours d'affectation,
1924.
91
Heisenberg to landé, 29 October 921, quoted in Cassidy 1979 p. 211.
92
Heisenberg 1969, p 45, quoted in Kragh 2012 p. 213.
28
also attended the festival, said about it: “The glamour that surrounded this even cannot be
communicated in words today; for us it was as brilliant as the Händel-Festival93 of those
days in Göttingen”.94 We can see that Bohr had made quite a name for himself among
young physicists, which also gave him great authority in the matters of atomic theory. For
example we see this authority in a letter from Heisenberg to Pauli: “Bohr can only explain
the diamagnetism by assuming that in a magnetic field He is always oriented in such a way
that its moment about the axis of the field is 0. This doesn’t please me very much, but of
course one cannot doubt that Bohr is right”95. Heisenberg mentions here one of the
problems with the Bohr model for par helium; that the total angular momentum was in
apparent contradiction with the experimentally established fact that helium is diamagnetic,
but of course no one could doubt that Bohr was right.
This reputation provided a large audience to which Bohr could display his new ideas
and it also attracted a lot of bright students to his lab, including Heisenberg, who would
arrive in 1924 in Copenhagen to work together with Bohr. Here he would come to
appreciate the correspondence principle even more. He wrote to Pauli about the role of the
correspondence principle in the multiplet problem: “Together with Bohr I have again
examined the problem carefully, and we arrived at the conclusion that it is not as
Sommerfeld says that the sum-rules cannot be understood with the help of the
correspondence principle; on the contrary they are a necessary consequence of the
correspondence principle. We are very happy about this interpretation for now the attacks
against the correspondence principle are completely refuted. Since recently the
correspondence principle has been blamed so much, it would be good to publish it [your
results confirming the correspondence principle] [for the greater glory of the
correspondence principle].”96 The sentences ‘we are very happy with this interpretation’
referred to the findings that he had made with Bohr to explain the intensities of the
multiples lines. It was however not completely clear if the sum-rules could be derived from
the correspondence principle or if they were only in consensus with it, but as we will see,
Bohr must have been quite convincing in his belief in the correspondence principle, for
almost every young student whom visited his lab became convinced of the superiority of the
correspondence principle.
Someone who was not easy to convince was Pauli, who tended to side with
Sommerfeld, believing that the correspondence principle did not reach as far as people
wished to believe. Sommerfeld had replied to all the uproar about the correspondence
principle by admitting ‘that the correspondence principle is a (highly valuable) limiting
theorem of quantum theory, but not its foundation (fundamental principle)’97. Pauli felt in
particular that the claim that the correspondence principle could derive the Ornstein sum
93
International music festival that started in 1922 and concentrated on the music of George Friedrich Handel.
Hund 1922, quoted in Kragh 2012, p. 224.
95
Heisenberg to Pauli, 6 march 1922, in, Hermann et al. 1979, p.57.
96
Heisenberg to Pauli, September 30th, 1924; quoted in Mehra and Rechenberg pp. 156-157
97
Sommerfeld to Kramers, 5 July 1924, Mehra, Jagdish, and Helmut Rechenberg. The discovery of quantum
mechanics, 1925. Vol. 2. Springer Science & Business Media, 2001.
94
29
rules was false. These rules stated that the sum of the intensities of all the lines of a
multiplet, which belong to the same initial or final state, is proportional to the statistical
weight of the initial or final state98. Concerning this Heisenberg commented: “If by
correspondence principle one means, as you do, the false statement that one can come
from averaging the classical intensities to those of quantum theory, then you are right that
one cannot come to Ornstein’s rule by means of the correspondence principle; however, if
one understands a meaningful logical connection to the classical theory, then I am right.”99
This comment illustrates that there was indeed some dispute on what the correspondence
principle meant and that it could probably be understood in different ways. Heisenberg said
that the correspondence principle is ‘a meaningful logical connexion’, but how this
connection is translated into mathematical terms and formulas, is not specified. We can
consider this ambiguity surrounding the correspondence principle also one of its strengths.
As almost no one was quite sure how the correspondence principle came exactly to play in
the quantum theory, it could be bendable to the situation and user.
In 1923 Heisenberg would for the first time start with the symbolic translation of the
correspondence principle to the new theory of quantum mechanics. He wrote: “The model
representations have only a symbolic meaning; they are the classical analogue of the
‘discrete’ quantum theory”.100 Hereby Heisenberg denied any direct physical meaning to the
previous made atomic models (which came to a failure in the helium model), but he did
integrate a symbolic meaning, which came to play a role in the new discrete theory. In other
words the correspondence principle gave a formal analogy between classical mechanics and
new discontinuous mechanics.
Bohr wrote a letter, which was published in Nature in December 1925 about the
importance of Heisenberg’s attempt to use the correspondence between quantum
theoretical spectrum and the harmonics of classical motion to define a new kind of quantum
theory. He wrote: “In contrast to ordinary mechanics, the new quantum mechanics does not
deal with a space-time description of the motion of atomic particles. It operates with
manifolds of quantities, which replace the harmonic oscillating components of the motion
and symbolize the possibilities of transitions between stationary states in conformity with
the correspondence principle. The quantities satisfy certain relations which take the place of
the mechanical equations of motion and the quantization rules … In brief, the whole
apparatus of the quantum mechanics can be regarded as a precise formulation of the
tendencies embodied in the correspondence principle”101. So Bohr stated that the new
quantum theory was in full conformity with the correspondence principle and was actually
fully based on the correspondence principle. It is true that, as earlier said, Heisenberg made
use of a symbolic translation, but to state that this was based on the correspondence is
98
Herzberg, Gerhard, and John William Tranter Spinks. Atomic spectra and atomic structure. Courier
Corporation, 1944
99
Heisenberg to Pauli, 8 October 1924, in Hermann et al. 1979, p. 167.
100
Darrigol, Olivier. From c-numbers to q-numbers: the classical analogy in the history of quantum. theory. Univ
of California Press, 1992, p.197.
101
Bohr, Niels. "Atomic theory and mechanics." Nature 116.2927 (1925).
30
quite far fetched. It is better formulated that Bohr and the correspondence principle
inspired Heisenberg’s translation. Heisenberg, however, also stated that his work was “an
attempt to establish a theoretical quantum mechanics analogous to classical mechanics
(with use of the correspondence principle)”, but Darrigol has refuted this. He stated this
acknowledgment of Heisenberg ‘that the correspondence principle was the basis of his
work’ to be a matter of endless debate; was he really trying to describe his own strategy? Or
was he trying to win over Bohr, Pauli, and Born, who, at the time, all advocated a radical
elimination of unobservable quantities? We can conclude that Heisenberg did a bit of both.
He did inscribe in his program a translation of the classical quantities to the quantum ones,
but in the end we cannot know how much he was thereby inspired by the correspondence
principle and if this use of the CP is in accordance with the original meaning as defined by
Bohr. The roots are found in the principle; namely the idea that a formal analogy exists
between the laws of quantum theory and those of classical theory.
It is interesting to note the change in Heisenberg’s attitude towards the
correspondence principle after it became apparent that it failed in describing the helium
atom and its properties. He increasingly began to distance his work from the
correspondence principle. Mara Beller noted “Heisenberg does not cite Bohr´s work at all (in
the 1925 matrix paper), despite the fact that the paper was built on Bohr's correspondence
principle in a fundamental way… Instead Heisenberg presented his work as flowing from the
positivist principle of elimination of unobservables”.102 Heisenberg expresses his increasing
distrust towards the correspondence principle in his book The Physical Principles of the
Quantum Theory: “It is true that an ingenious combination of arguments based on the
correspondence principle can make the quantum theory of matter together with a classical
theory of radiation furnish quantitative values for the transition probabilities… Such a
formulation of the radiation problem is far from satisfactory, however, and easily leads to
false conclusions.”103 We see that although Heisenberg did use the correspondence
principle to guide his work, but he did not give it the original role as a law of quantum
mechanics. Heisenberg would continue his belief that the quantum theory does not depend
on laws from outside, such as the original classic theories. So the correspondence with the
classical theory is a purely mathematical result, not revealing any deeper physical
connection. He calls the quantum mechanics a ´Abgeschlossene Theorie´ by which he meant
that it was a closed theory, that is to say an axiomatic system complete in and on itself.
2.3 Pauli; The model concepts are in a severe, fundamental crisis, which, I believe, will finally
end with a further radical sharpening of the contrast between classical and quantum
theory…104
In 1945 Pauli received the Nobel Prize for his exclusion principle and him general
contribution to Quantum mechanics. In his speech he described the two main players, Niels
102
Beller, Mara. "Jocular Commemorations: The Copenhagen Spirit." Osiris(1999), p.140.
Heisenberg, Werner. "The physical principles of quantum mechanics." U. Chicago Press, Chicago (1930): 21.
104
Pauli to Sommerfeld, 6 december 1924, in Eckert and Märker 2004, p.177.
103
31
Bohr and Arnold Sommerfeld, in the quantum area and their course to dealing with the
quantum theory: “At that time there were two approaches to the difficult problems
connected with the quantum of action. One was an effort to bring abstract order to the new
ideas by looking for a key to translate classical mechanics and electrodynamics into
quantum language, which would form a logical generalization of there. This was the
direction, which was taken by Bohr’s correspondence principle. Sommerfeld, however,
preferred, in view of the difficulties which blocked the use of the concepts of kinematical
models, a direct interpretation, as independent of models as possible, of the laws of spectra
in terms of integral numbers…”105. This overview of Pauli actually gave the perfect
description of the two main ‘sides’ that were picked in quantum theory. One side preferred
to maintain the successes of the classical theories and incorporate them in the new
quantum theory. The other side, however, wanted an independent theory that did not really
on concepts from outside of the theory. Pauli stated to have been influenced by both the
sommerfeldian approach and the borisch correspondence approach, but found himself
more and more inclined to the second one. He had two major issues with Bohr’s atom
theory. First, Pauli objected to the use of a ‘formal’ notion of an electron orbit, on which the
correspondence principle is primarily based. Second, Pauli opposed to the thought that the
correspondence principle would be able to explain the closing of the electron shells as Bohr
had hoped. Pauli believed that the orbital meaning of the azimuthal quantum number
(through the correspondence principle) would not hold in the context of multi periodic
models. He wrote in his letter to Bohr of 1924: “Against the point of view which you were
still holding last fall, I now believe that even for the quantum number k (not only j) essential
features of the true laws cannot be reproduced by the theory of multi periodic systems.”106
In Pauli’s opinion, Bohr was wrong to retain classical concepts while he gave up classical
laws. In a letter of 1923 to Eddington he alleged that this was the root of all the paradoxes
surrounding quantum theory and for that reason should be disposed of. If quantum theory
was ever to establish itself as a proper theory it should be formed on an entirely new set of
concepts.
Pauli understood the correspondence principle mostly as a kind of selection rule. In
other words that transitions between stationary states are connected with harmonics in the
classical motions. He did not object to this primarily, but he disregarded the extension of the
principle; to be able to explain the closing of electron groups in atoms. He wrote to Bohr in
1924: “I have already often said to you that I am of the opinion that the Correspondence
Principle has in reality nothing to do with the problem of the closing of the groups in the
atom. The exclusion of certain stationary states (not transitions), which is what is in
question here, has more similarity in principle with the exclusion of the state 𝑚 = 0 or
𝑘 = 0 in the H-atom than, for example, with the selection rule 𝑘 = ±1. Do you still cling
to your application of the Correspondence Principle in this case? There is moreover no need
whatever to talk of harmonic interplay.”107 The closing of the electron shells would in the
end be explained by the exclusion rule of Pauli, which said that ‘the numbers of the
electrons in a closed shell of an atom can be properly reduced to only one electron per
state, by which the state is defined as the limited number of quantum states available’.
The only problem with the exclusion principle of Pauli was that it couldn’t be
properly justified. When Bohr suggested that the correspondence principle was the
105
Pauli, Wolfgang. "Remarks on the History of the Exclusion Principle." Science103.2669 (1946), p.27.
Darrigol, Olivier. From c-numbers to q-numbers.
107
Pauli to Bohr, December 12th, 1924; quoted in Serwer 1977, p. 235.
106
32
foundation needed for this justification, Pauli replied: “I personally do not believe, however,
that the correspondence principle will lead to a foundation of the rule For weak men, who
need the crutch of the idea of unambiguously defined electron orbits and mechanical
models, the rule can be grounded as follows: If more than one electron have the same
quantum numbers in strong fields, they would have the same orbits and would therefore
collide. The justification of the exclusion of the above-mentioned cases in the H-atom by
pointing to the collision with the nucleus has never pleased me much. It would be much
more satisfying if we could understand directly on the grounds of a more general quantum
mechanics (one that deviates from classical mechanics).”108 Pauli was actually quite
offended by the idea that the classically based correspondence principle would be the basis
of his exclusion principle. We see in his writing the growing influence of Sommerfeld’s
‘direct’ interpretation and his distrust of the model-based reasoning. Pauli would become
one of the firmest anti-model voters and would together with Sommerfeld continue for a
more direct approach (without the need of models) of the quantum mechanics. They would
continue to look for “a radial sharpening of the opposition between classical and quantum
theory”, whilst Bohr and Heisenberg opted for a “sharpening of the correspondence
principle”.
2.4 Kramers; “In dieser Nacht von Schwierigkeiten und Unwissenheit ist nun das von Bohr
1917 aufgestellte Korrespondenzprinzip ein Leuchtpunkt“109
In 1919 Kramers had already reached, in his dissertation on the “Intensities of
spectral lines, some quantitative expressions for the intensities of the hydrogen lines with
the help of the correspondence principle: he took the Fourier components of the electron
that was orbiting around its nucleus, to give an estimation of the probability of the
corresponding quantum transitions. Kramers summarized his conclusions as follows: “… the
results obtained as regards the applications to the Stark effect and to the fine structure of
the hydrogen lines must be considered as affording a general support for Bohr’s
fundamental hypothesis of the connection between the intensity of spectral lines and the
amplitudes of the harmonic vibrations into which the motion of the electron in the atom
may be resolved”.110 With help of the very extensive calculations, Kramers had arrived at
theoretical values of the relative intensities of the Stark lines, which he modestly described
as ‘convincing’. The theoretical values and the experimental data matched so well that it
gave a strong support for the correspondence principle of which they were derived. Kramers
was in everyway a strong supporter of almost all of Bohr’s theories. He could, with the help
of these and his strong mathematical knowledge, derive certain properties of atoms that
other physicists couldn’t even dream of. It was Kramers who helped the correspondence
principle win in conviction with all his different applications of it.
Bohr’s correspondence principle was for Kramers and Bohr a lot of times like a
guiding principle, which pointed them in the right direction when looking for answers. This
108
Pauli to Bohr December 31st, 1924; quoted in Heilbron 1983, p. 306 and Serwer 1977, p. 236.
‘In this night of difficulties and ignorance it is only the correspondence principle of Bohr from 1917 that
established a bright spot’ [kramers 1923]
110
Kramers, Hendrik Anthony. Intensities of spectral lines on the application of the quantum theory to the
problem of the relative intensities of the components of the fine structure & of the Stark effect of the lines of
the hydrogen spectrum. AF Host & son, 1919.
109
33
direction usually seemed to be the strong connection between quantum physics and
classical theories. Kramers could even prove that the principle was useful even for systems
without the original limitation of periodicity. In 1920 Kramers published the first concrete
application of the correspondence principle outside of the context of multi-periodic
systems. The determination of the effect of a small electric field on the fine structure of the
hydrogen spectrum: “that one could expect to be able to determine theoretically, according
to the principle of correspondence, a relationship between the motion of the atom and the
emitted radiation and the polarization state of these components and that it was possible to
draw a conclusion about their intensity”111112. We see that Kramers also expected to be able
to calculate the intensity components by using the correspondence principle, although nor
he nor Bohr gave a direct description on how to do that. It appeared that there was an
inside connection between Kramers and Bohr in understanding the quantum theory that
other physicists couldn’t understand. In his book of 1923, coauthored by Helge Holst,
Kramers mentioned the correspondence principle and that “It is difficult to explain in what it
consists, because it cannot be expressed in exact quantitative laws, and it is, on this
account, also difficult to apply.”113 So we see that even the favorite student of Bohr had
difficulty grasping the full body of the correspondence theory, although he was himself an
acknowledged master of making mathematical sense of the correspondence philosophy. If
even Kramers couldn’t fully explain the correspondence principle, then perhaps explaining it
should be deemed impossible.
2.5 The reception of the second atomic theory; “Wouldn’t that be grand! But the tragedy is
that we understand not a word of how he does it.”114
When the news surfaced of a second atomic theory developed by Bohr, the German
physicists responded decidedly positive and with a noticeable lack of a critical attitude.
Landé, a German physicist, was among the many enthusiasts, he said: “[the atomic theory]
will be of the very greatest significance for the future of the atomic theory […] on the whole
it seems to me that until your full and detailed exposition appears there is no sense in doing
any more theoretical work in the atomic theory”115. Another follower of Bohr’s work Born
wrote to Frank, who at that time was staying with Bohr in Copenhagen: “Wouldn’t that be
grand! But the tragedy is that we understand not a word of how he does it. He only says
that it is based on the correspondence principle … Dear, good Franck, be a nice chap and
write to us about it as well as you understand it, or ask Bohr or Kramers to write something
clear. Otherwise, we will explode from curiosity.”116 We see here many positive reactions on
a theory from which its basis was actually not fully understood. Many physicists assumed
111
das man auf Grund des nach dem Korrespondenzprinzip zu erwartenden Zusammenhanges zwischen der
Bewegung des Atoms und der emittierten Strahlung auch den Polarisationszustand dieser Komponenten
theoretisch bestimmen konnte und das es möglich war, gewisse Schlusse über ihre Intensität zu ziehen.
112
Kramers, Hendrik A. "Über den Einfluß eines elektrischen Feldes auf die Feinstruktur der
Wasserstofflinien." Zeitschrift für Physik A Hadrons and Nuclei3.4 (1920): 199-223.
113
Kramers to holst 1955, reprinted in Kragh 2012, p.204.
114
Born to Frank, reprinted in Kragh 2012, p.298.
115
Landé to Bohr, 3 march 1921, in Rud Nielsen 1977, p. 722
116
Born to Franck, 22 February 1921, as quoted in Greenspan 2005, pp. 104-5, reprinted in Kragh 2012, p.204.
34
Bohr’s theory to be based on formal calculations and therefore a major step towards the
mathematization of the physics. Even Sommerfeld’s response was initially largely positive as
he wrote: “I am certainly convinced that your way is the right one; if, as it seems, you can
mathematically reconstruct the numbers of the periods 2, 8, 18, … then that is indeed the
fulfillment of the most daring hopes in atomic physics.”117 In his third edition of Atombau,
he stated how excited he was over ‘Bohr’s consideration based on calculations’. Sommerfeld
was not the only physicist that read Bohr’s work as being based on rigorous calculations,
although an explicit formulation of these calculations was missing in the article send to
Nature. The Finnish mathematical physicist Gunnar Nordström, professor at the University
of Helsinki, after having studied Bohr’s latest work wrote: “The great and simple lines that
you -in spite of the complex relationships- have found to govern the structure pf atomic
systems provide your new theory with a suggestive power which makes it appear convincing
even in the absence of compelling proof from quantum theory. You have taken a giant step
in understanding the world of atoms, and I heartily congratulate you with this new
progress”118. Another positive comment came from Rubinowicz, to whom Bohr had sent a
reprint of his first report in Nature, he wrote as follows: “Even if your letter to Nature does
not let us see the finer details of the wonder-world which you now again have opened up to
us, but only foreshadows them, I believe nevertheless to have gathered so much from it that
by your work you once more lead us to a summit from which we now again can view a wide
field as a clear and beautiful harmonic unity”119. It is interesting that although Bohr’s theory
obviously lacked hard proof and some physicists noted this, it didn’t seem to cause any
distress and the theory was received as an important step to the complete mathematization
of the sciences of matter.
Although it was largely received positively, there were also some physicists that
expressed their puzzlement over the foundations of Bohr’s work. For example Ehrenfest,
whom had also received a copy of the manuscript before the publication in Nature, wrote: “I
have read your Nature letter with great eager. The picture you sketch there is really so
much more beautiful than that of the quantum numbers decreasing again toward the
outside. Of course it now interests me even more how you saw it all in considerations of the
correspondence. It interests me especially: what were the difficulties like that taught you to
find the way from the conception of the first Nature letter to that of the second. I am VERY
curious to know that.”120 Most of the leading physicist thought Bohr’s work to be open a
world for a new kind of quantum mechanics, based on formal mathematics and everyone
was very curious to know how this derivation looked like. Very few considered the thought
that if there was not an explicit derivation in the original paper of Bohr, that this derivation
may not exist. It could have been Bohr’s fame that caused the blind acceptance of his new
theory while the full derivation was missing. Also Bohr’s writing style always had some
117
Sommerfeld to Bohr, 25 April 1921, in Rud Nielsen 1977, p. 741.
Nordström to Bohr, 2 February 1922, reprinted in Kragh 2012, p. 299.
119
Rubinowicz to Bohr, 1922, reprinted in Kragh, p. 299.
120
Ehrenfest to Bohr, 27 September 1921, reprinted in Kragh 1979, p. 137
118
35
ambiguity in it that usually dissolved in further papers, but this time the ambiguity
remained.
Kramers recounted in 1935 how Bohr had, contrary to popular belief arrived at his
theory, not on calculations, but rather on qualitative arguments based on the
correspondence principle and empirical data. He wrote: “It is interesting to recollect how
many physicists abroad thought at the time of the appearance of Bohr’s theory of the
periodic system that it was extensively supported by unpublished calculations which dealt in
detail with the structure of the individual atoms, whereas the truth was, in fact, that Bohr
had created and elaborated with a divine glance a synthesis between results of a
spectroscopic nature and of a chemical nature.”121 So we see that Bohr had arrived at his
theory ‘with a divine glance’ which came from his renowned physics intuition. Rather than a
formal deduction through mathematics the theory was the based on matching empirical
data with chemical properties. Physicist how lacked this ‘divine glance’ could not, but fail to
understand how this produced the final theory.
As soon as it became known that his new atomic theory was indeed not based on
formal calculations, the positive response quickly cooled. Sommerfeld stated in a detailed
treatment of the theory: “Bohr’s theory is, of course, not based on mathematics but is
conceived intuitively”122. The ‘of course’ shows Sommerfeld’s sceptic attitude towards most
of Bohr’s, model based, quantum theories. We see that the enthusiasm that had surfaced at
the first release of the second atomic theory had mostly evaporated, although eventually
the reception of Bohr’s second atomic theory lost its positivism, the original acceptance of it
was blind trust and a firm belief in the ability of Bohr to help the quantum theory to a new
level. We can see that the name of a physicist has a lot of impact on the original acceptance
of his theory. This can also be the reason that even though the correspondence principle
was quite ambiguous in nature and most physicists didn’t fully comprehend it, it held
ground for a very long time as one of the pillars of the old quantum theory.
2.6 The general reception of the correspondence principle
So we have seen three general responses to the correspondence principle; full
acceptance, distrust and misunderstanding. In this chapter we saw that even in the time of
Bohr there where only a handful of physicists that could ‘successfully’ apply the
correspondence principle. Which makes it quite clear that the writings of Bohr where not
unambiguous. The correspondence principle had a wide set of applications and therefore
received much admiration. These various contributions were reasons that Bohr was
awarded the Nobel Prize in 1922. For the Nobel Prize the speaker Svante Arrhenius,
renowned physicist himself, emphasized the worth of the correspondence principle of Bohr.
Arrhenius illustrated the use of the correspondence principle by explaining its first use as
121
122
Kramers 1935, reprinted in Kragh 2012, p. 300.
Sommerfeld 1924b, reprinted in Kragh 2012, p. 300.
36
defining the structure of the helium atom: “It occurs in two different modifications: one is
called parhelium, and the other is called ortho-helium – these were supposed at first to be
different substances. The principle of correspondence states that the two electrons in
parhelium in the track of rest [stationary orbits] run along two circles, which form an angle
of 60 degrees to one another. In ortho-helium, on the other hand, the tracks of the two
electrons lie in the same plane, the one being circular, while the other is elliptical.”123 We
see here that Arrhenius does not try to explain how these successes where derived with
help of the correspondence, seeing how this was a very difficult task. The most that the
physics community got from the correspondence principle was that it was widely applicable,
it solved many, in that time, experimental problems and gave a direct link between
quantum physics and classical theories, but how all this came to play in the formal theory
was a mystery to most physicists.
There was certain flexibility in how the correspondence principle was to be
understood and used. Ehrenfest emphasized the need to maintain this adaptability of the
correspondence principle: “It is not desirable that, with the most automatic application in
view, one already casts in a rigid form the condition of correspondence, which up to now
has been variable and groping”124. So the quantum theory had well-established quantum
rules, but the correspondence principle was the connecting tool that could be different in
each hand. For this reason there were also a lot of extensions on the basics of the
correspondence principle. For example van Vleck wrote two articles called; ‘the most
sophisticated application of the correspondence principle’ and ‘Some extensions of the
correspondence principle’, which included a correspondence theory of absorption. In these
articles he looked for new areas in which the correspondence principle could also be used.
So the correspondence principle was not only used in its original form, but also extended
and formed to be applicable in situations, even outside of the scope of physics. Later the
term correspondence principle was remolded to serve as a general principle of the
reduction of a new scientific theory to the old one in appropriate limits, although this was
not how Bohr had defined it. There were also complete misinterpretations of the
correspondence principle in the physics community. Campbell had claimed that the
electrons in the Bohr-Sommerfeld theory do not really move on the orbits and, in
justification, had cited the correspondence principle. Bohr was completely against such an
interpretation; “On the contrary of we admit the soundness of the quantum theory of
spectra, the principle of correspondence would seem to afford perhaps the strongest
inducement to seek an interpretation of the other physical and chemical properties of the
elements on the same line as the interpretation of their series spectra.”125 Where these
miscomprehensions caused by the ‘unilluminating’ papers of Bohr or was it the term
‘correspondence principle’ that was misused in the physics community? It was probably a
123
Nobel Prize in Physics 1922 - Presentation Speech. Nobelprize.org. Nobel Media AB 2014. Web. 21 May
2015. <http://www.nobelprize.org/nobel_prizes/physics/laureates/1922/press.html>
124
Ehrenfest 1923a, reprinted in Kragh 2012, p. 142.
125
Bohr, Niels. Atomic structure. Printed in Great Britain by the Cornwall Press, 1921.
37
bit of both, as Bohr was not very clear, physicists could fill in their own understanding on the
use of the correspondence principle.
In general we can state the correspondence principle was well received. In the
beginning of the old quantum theory the correspondence principle was more a formal
mathematical application, but as the atomic theory changed and with it the correspondence
principle, this mathematical application made room for a more philosophical interpretation,
which we shall elaborate in the third chapter of this paper. During the second atomic model
we saw how much influence the name Bohr had, because of earlier success there was room
for some ambiguity in Bohr´s newer theories. In every hand the correspondence principle
could take a new form and in the end it seemed that all connections from former classical
theories to quantum theories could be deemed a part of the correspondence principle. A
large part of this was fueled by the explanations the correspondence principle provided for
experiments, also that it was quite easy to visualize and fitted in the general belief of the
existence of one universal theory.
38
3. Bohr as a Philosopher
“The significance of science […] is concerned with learning the structures of the world that lie
beyond the expressions and concepts of any particular language”126
Niels Bohr was renowned for his contributions to the quantum theory, but also for
his philosophical views. Heisenberg even stated that ‘Bohr was primarily a philosopher not a
physicist’127. However if one reads Bohr’s work, it is quite clear that Bohr does not consider
himself a full-fledged philosopher: “It is not my intention to enter into an academic
philosophical discourse for which I would hardly possess the required scholarship”128. Even
though Bohr did not consider himself a philosopher, his work in physics was primarily
spurred by his philosophical considerations. Rosenfeld also noted this: “epistemological
considerations played a decisive part in ... investigations, and that, inversely, the results of
the latter led … to a deeper understanding of the theory of knowledge”129. Bohr was
convinced that a new mathematical formulation had to come after a physical understanding
of the problem. According to Heisenberg: “Bohr would always say, ‘we’ll, first we have to
understand how physics works. Only when we have completely understood what it is all
about can we then hope to represent it by mathematical schemes’”130. Bohr believed
mathematics to be the tool to discover the ‘real’ physics. He did not believe a description of
nature based solely on mathematics to be a good one, he wanted a further reaching
interpretation linked to reality. This was also the big difference between Bohr and
Heisenberg whom “wanted to start entirely from the mathematical scheme of quantum
theory…”131, however this was something Bohr would never quite accept, as he also wanted
a further reaching interpretation linked to physical world.
Bohr’s philosophy can be said to be sui generis, or one of a kind, although when we
take a closer look at the works of Bohr we will see hints of Kant’s, Kierkegaard’s, Høffding’s
and other philosopher’s thoughts. The relation of the works from these philosophers to
Bohr’s theories is, however, far from obvious, as Bohr does not in his work explicitly quote
or mentions them. Even though the similarities between Bohr and other philosophers are
not quite so obvious, it is interesting to study these similarities, primarily Høffding and Kant,
because by comparison and differentiating we will be able to understand Bohr’s work
better. The Kantian part of the philosophy of Bohr has been analyzed by a number of
historians, among them Hooker 1972, Honner 1982 and recently Brock 2003. I will also
address some of the Kantian parts of Bohr’s philosophy in this chapter. Although Bohr
himself never explicitly mentions having been influenced by the thoughts of Kant, we can
126
Hesse, Mary. "Comment on Von Weizsäcker." Transcendental Arguments and Science. Springer
Netherlands, 1979. 159-170.
127
Heisenberg on Bohr, reprinted in Plotnitsky, Arkady. Reading Bohr: Physics and Philosophy: Physics and
Philosophy. Vol. 152. Springer Science & Business Media, 2006. P.146.
128
Bohr, Niels. "Essays 1958-1962 on atomic physics and human knowledge." (1963).
129
L. Rosenfeld ‘Biographical sketch’, reprinted in Honner, John. "The transcendental philosophy of Niels
Bohr." Studies in History and Philosophy of Science Part A 13.1 (1982): 1-29.
130
Interview with Heisenberg by T. S. Kuhn on 19 February 1963. American Institute of Physics, Niels Bohr
Library and Archives.
131
Interview with Heisenberg, 25 February 1963, reprinted in Kalckar, Jørgen, ed. Foundations of Quantum
Physics I (1926-1932). Vol. 6. Elsevier, 2013.
39
find some of the same thought patterns in both works. Honner stated that Bohr’s
philosophical views can be seen as transcendental, which means that there is a certain
allowance for paradoxes in the conditions to obtain knowledge of the system. We will later
see how this transcendental part of Bohr’s philosophy relates to Kant.
In this chapter I will not try to label the philosophy of Bohr as being either positive or
pragmatic, for I think Bohr’s philosophic thoughts are too subtle for that. Moreover, his
thoughts and theories changed quite a bit over the course of time, making it more difficult
to label them. Seeing how Bohr is sometimes called primarily a philosopher, I intend to take
a closer look at how these philosophies correlate to his work in physics, and perhaps
stimulated the first expressions of his atomic model and with it the correspondence
principle. I think it is necessary to thoroughly analyze the philosophical considerations of
Bohr in order to understand the origin of the correspondence principle. During this analysis I
will make extensive use of his later work, as Bohr articulates his philosophical thoughts
more clearly then. While using these works I will keep in mind that Bohr’s ideas could have
altered significantly compared to the time of the first formulation of the correspondence
principle. There are however a few leading considerations made by Bohr that can be found
starting from his earlier works until his final papers.
3.1 Inconsistencies in the quantum theory and empirical efficiency
Until 1925 when Heisenberg came up with the new formulation of the quantum theory, the
quantum theory contained some significant inconsistencies. The Bohr model, which was
then primarily used to describe the quantum system, was in its founding not fully consistent,
some of its concepts came from classical theory, while they applied in a quantum
environment. Even though Bohr was aware of these inconsistencies it seemed he allowed
them. With regard to these inconsistencies, Heisenberg and Thomas Kuhn expressed their
opinions in 1963. Kuhn was of the opinion that Bohr was usually eager to get rid of the
inconsistencies although there was a sort of ‘higher inconsistency’, which he was willing to
except. Heisenberg, however, pointed out that whilst reading Bohr’s work one should
distinguish between ‘paradox’ and ‘inconsistency’, and that Kuhn’s ‘higher inconsistency’
might belong to the first class.132 Bohr, according to Heisenberg, strongly believed that
certain paradoxes would remain even in the final version of quantum theory: “Because he
was so much impressed by these paradoxes which were apparently unavoidable he counted
always on the possibility, ‘Well, these paradoxes may even, in the long run, mean some kind
of inconsistency which cannot be avoided’. On the other hand, in the course of the
development, Bohr realized that an inconsistency is something still much worse than a
paradox because inconsistency means that you talk just nonsense- that you do not know
what you are talking about. A paradox may be very disagreeable but still you can make it
work. An inconsistency can never be made to work”133. This comment shows us that Bohr
expected the quantum theory, or theories in physics in general, to always keep some form
of inconsistency, or in his words paradoxes. According to Bohr these inconsistencies did not
come from nature, but from our own observations of nature. He claims that as our means to
make observations will never be perfect and always create a superficial context, a certain
extent of inconsistency is to be expected. That is not to say that Bohr did not believe that
132
133
Kragh 2012, p. 366.
Interview by T. S. Kuhn of 25 February 1963. American Institute of physics.
40
nature could not be fully explained by physical laws, he only stated that he believed
everything was context related. We cannot create a frame in which our observation
methods do not alter the object that we are observing, so it is only possible to a certain
extent to give a full representation of nature in the form of physical laws, based on
measurements. The unfinished manuscript in 1927 ‘Philosophical Foundations of Quantum
theory’ summarizes one of the main points Bohr was trying to make within his philosophical
considerations: “Due to the contrast between the principles underlying the ordinary
description of natural phenomena and the element of discontinuity characteristic for
quantum theory, we must be prepared that every concept used in accounting for the
experimental evidence will have only restricted validity when dealing with atomic
phenomena.”134 He stresses once again that we must keep in mind that we use classical
descriptive words for the observations made on a quantum level. This cannot be
unambiguously done and must therefore always be taken into account when trying to
define a new theory on atomic level. These reflections between words and reality, classical
and quantum will remain a reoccurring problem in the works of Bohr and this Bohrisch
sense of objectivity will be more extensively explained later in this chapter.
Although sometimes a theory turns out to be wrong, as was the case with Bohr’s
theory at the end of 1924, Bohr believed that a future theory needed to contain the same
strong values as the previous one. In the case of Bohr’s atom model, its strong empirical
record should still agree with the future theory. He wrote to Born: “It is necessary for us to
regard the usual formulation of the quantum theory, as it is elaborated for atomic structure,
as a limiting case of a more general theory.”135 According to Bohr the connection to the
classical theory was one of the important features of the old theory that the new theory
should also contain. According to Darrigol, Bohr “did not regard logical completeness as
prior to empirical efficiency. He believed instead that progress was possible within a
manifestly incomplete theory, under the guidance of organizing principles like the
correspondence principle”136. From this we can conclude that if, before describing the
physical reality, a theory can explain the experimental data, Bohr would consider it
sufficiently correct. According to him the theory does not have to be a reflection of the ‘real’
world. The Borhish approach to physics was largely empirically driven. He used well-defined
phenomena as a good starting point and as new quantum effects were introduced, he
altered the theory to satisfy these effects. He attempted to determine the atomic structure
from the experimentally found quantum properties. He used the spectral evidence without
having a real understanding of the physics behind the emission of this radiation. Bohr
sometimes referred to conclusions reached through this method as “empirical deductions
from the spectral evidence”.137 Tanona said this process explicitly required a tool to allow
one to bridge the gap between phenomena and theory. The correspondence principle was
134
N. Bohr, ‘Philosophical foundations of quantum theory’, "Essays 1958-1962 on atomic physics and human
knowledge." (1963).
135
Bohr to Born, 9 December 1924, in Stolzenburg 1984, p.73.
136
Darrigol, Olivier. From c-numbers to q-numbers: the classical analogy in the history of quantum theory.
University of California Press, 1992.
137
Bohr, Niels, Hendrik Anton Kramers, and John Clarke Slater. "LXXVI. The quantum theory of radiation." The
London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 47.281 (1924): 785-802.
41
this tool, which together with Bohr’s frequency condition acted as the bridge that allowed
him to make connections between classical and quantum theory based on empirical
phenomena.
3.2 Formal through the eyes of Bohr
It is sometimes difficult to understand Bohr’s subtle physics and philosophical theories as he
had his own set of definitions. One of them was the word formal; a formal principle
according to Bohr: ‘has significance because of the role it plays in a theory, rather than
because of any direct physical reality’138. The definition of the word formal actually gives
quite a good summary of Bohr’s general striving. Instead of giving physics the role of
describing the physical world as it is, he wants it to give a descriptive approximation that is
as close to reality as possible. In this manner he does not consider mathematical laws to
give a representation of the real physical world, but an idealization in formal laws. Darrigol
said that ‘Analogical thinking usually works with a touch of blindness: formal relations of a
given theory are tentatively applied to new objects, and if the operation is empirically
successful, the concepts originally underlying these relations are assumed to extend to
these objects.’139 So if the theory is confirmed by the experiment, it is as close as we can get
to a good description of the physical world.
The correspondence principle was not, as many think, a way for Bohr to secretly
include the classical theories in the new quantum ones. It was primarily formal and was not
meant to give a reflection of reality, but a relation to it. In quantum mechanics Bohr still
used the same concepts as momentum, energy and frequency as formulated in the classical
theory, related through the correspondence principle. This led to quite some discussion and
confusion, as it seemed that the quantum theory was self-contradictory and, although for its
heuristics value, its foundations were unclear, as these were largely based on classical
concepts. Bohr realized that certain fundamental concepts from the classical theory could
still be used in the quantum theory, within a range of certain quantum numbers (although
Bohr made some contradicting statements about this). He used new empirical data to
support the use of classical concepts in quantum physics and his theories evolved around
this process. Here we must again remember that although Bohr was stated to be a
philosopher he placed strong emphasis on the empirical data and was always ready to
reconsider his theories if they did not match the experiments. This is a classical
characteristic of a physicist, but sometimes overlooked in the case of Bohr.
3.3 The relation to the classical theories and the unity of science
138
MacKinnon, Edward. "Niels Bohr on the Unity of Science." PSA: Proceedings of the Biennial Meeting of the
Philosophy of Science Association. Philosophy of Science Association, 1980.
139
Darrigol, Olivier. From c-numbers to q-numbers: the classical analogy in the history of quantum theory. Univ
of California Press, 1992.
42
Although Bohr called the relation between quantum theories and classical theories primarily
formal, that is not to say that he considered the relation to classical theories dispensable.
Everything in our daily lives is lived through the laws of classical physics. Honner stated that
Bohr saw classical physics and quantum physics as ‘mutually complementing each other,
despite their inherent contradiction.’140 It is possible to predict with classical physics the
state of a system at present and future time, if the necessary parameters are provided,
however for a quantum system this is not as simple, as the system is altered by the
measurement. Bohr repeatedly pointed out that through the measuring apparatus there
would exist an indispensable link between quantum physics and classical physics.
Heisenberg commented on this view of Bohr: “Bohr has rightly pointed out on many
occasions that the connection with the external world is one of the necessary conditions for
the measuring apparatus to perform its function, since the behavior of the measuring
apparatus must be capable of being … described in terms of simple (classical) concepts, if
the apparatus is to be used as a measuring instrument at all. And the connection with the
external world is therefore necessary”141. Heisenberg rightfully summarised Bohr’s view in
the following manner: that quantum systems are, in the end, part of a bigger system, and
through the interaction with this bigger system, classical concepts come into play. The
bigger system is the unknown factor in the quantum equations and gives a ‘new statistical
element into the description’142. It was, according to Bohr, therefore not possible in the
description of quantum systems to completely rid oneself of classical concepts.
A. Bokulich also noticed the strong emphasis Bohr placed on classical concepts and
therefore states that Niels Bohr’s general approach to physics is in many ways more that of
a continuity theorist than a revolutionary. He is a continuity theorist in the sense that he
tries to maintain and emphasize those features of the predecessor theory that are
preserved in the transition to the successor theory. I agree that Bohr does try to maintain a
link to the classical theories, which he deems necessary, but not because he thinks that a
theory cannot evolve. He states that our perception of reality cannot be broken from the
classical view of the world. It must be emphasized here that Bohr thought the use of the
classical concepts indispensable to the quantum theory. So it was not, as for example
Heisenberg said, just a case of we do use the classical concepts, but a case of we must us the
classical concepts. This emphasis on the use of the classical concepts was because Bohr
thought these were necessary to provide an unambiguous communication in physics. He
stated: “the unambiguous interpretation of any measurement must be essentially framed in
terms of classical physical theories, and we may say that in the sense the language of
Newton and Maxwell will remain the language of physics for all time”143. When reading this
passage of Bohr it does bring one question to mind; why can we describe observations only
in a classical matter? One time Schrodinger posed the question, but Bohr’s reply simply
restated “the seemingly obvious fact that the functioning of the measuring apparatus must
140
Honner, John. "The transcendental philosophy of Niels Bohr." Studies in History and Philosophy of Science
Part A 13.1 (1982), p 21.
141
Heisenberg, Werner. "The development of the interpretation of the quantum theory." Niels Bohr and the
development of physics 1 (1955): 12, pp. 26-7.
142
Heisenberg, 1989, pp. 121-2
143
Bohr 1931, reprinted in Greenberger, Daniel, Wolfgang L. Reiter, and Anton Zeilinger, eds. Epistemological
and experimental perspectives on quantum physics. Vol. 7. Springer, 1999.
43
be described in space and time”144, which obviously does not suffice as an answer to the
question; why this is to be done so? Aage Petersen, a Danish professor and assistant of
Bohr, argued that “Bohr’s remarks on the indispensability of classical concepts are based on
his general attitude to the epistemological status of the language and to the meaning of
unambiguous conceptual communication, and that they should be interpreted in that
background”145. In order for a measurement to be able to be done in the quantum domain
and conveyed, we must consider that at least part of the instrument must be described
according to the classical rules.
Honner said that objectivity according to Bohr “lay in the refinement of
unambiguous communication based on a well-defined use of pictures and ideas referring to
the events of daily lives”146. This also sums up the relation that Bohr assumed necessary
between classical and quantum theory, because all our descriptive methods were based on
objects described in a classical nature, it was not possible according to Bohr to, even in the
realms of quantum theory, completely let loose of these terms and definitions, because of
the need for unambiguous communication. Bohr did not think full objectivity in physics was
possible, because people were already pre-instructed on the situation and properties of the
experiment just by daily life. It is therefore impossible to achieve objectivity such as ‘the eye
of God’, or, as Bohr calls it: “… a radical revision of our attitude toward the problem of
physical reality. In fact, as we shall see, a criterion of reality like that proposed by the named
authors contains – however cautious its formulation may appear—an essential ambiguity
when it is applied to the actual problems with which we are here concerned.”147 Bohr did
not view this lack of objectivity as a limit in physics alone, but a general limit in all sciences.
The framework we live in will always present a certain context that will limit our objectivity.
He stated this key idea of conceptual frameworks in 1955, where he wrote: “The main point
to realize is that all knowledge presents itself within a conceptual framework adapted to
account for previous experiences and that any such frame may prove too narrow to
comprehend new experiences … . When speaking of a conceptual framework, we refer
merely to unambiguous logical representation of relations between experience.”148
Especially the last sentence gives a good overview of Bohr’s general concern when involved
with physics. One should not strive to have the finest objectivity, but strive to give the best
unambiguous relationships between experience and theory. Physics is a science of
relationships, according to Bohr, and physicists should aspire to find these relationships and
present them in comprehensible theories. He explicitly stated this task for physicists in
1934: “… our task must be to account for such experience in a manner independent of
individual subjective judgment and therefore objective in the sense that it can be
144
Bohr 1996, pp. 511-2, reprinted in Schlosshauer, Maximilian, and Kristian Camilleri. "The quantum-toclassical transition: Bohr's doctrine of classical concepts, emergent classicality, and decoherence." arXiv
preprint arXiv:0804.1609 (2008).
145
A. Peterson, Quantum Physics and the Philosophical Tradition, MIT Press, Cambridge, 1968, p. 179.
146
Honner, John. "The transcendental philosophy of Niels Bohr." Studies in History and Philosophy of Science
Part A 13.1 (1982).
147
Bohr, Niels. "Can quantum-mechanical description of physical reality be considered complete?." Physical
review 48.8 (1935): 696.
148
Bohr, Niels 1958b, pp. 6-7, "Essays 1958-1962 on atomic physics and human knowledge." (1963).
44
unambiguously communicated in common human language”149. So a physicist should be an
objective observer who tries to make an unambiguous account of that which he observes.
The analysis that came after the observation was something that Bohr had made an art out
of. Or as said by his brother Harald “His intuitive grasp of the situation was overwhelmingly
convincing notwithstanding the necessary subsequent farther-reaching quantummechanical analysis”150. Bohr was equally infamous for this intuitive grasp, as it allowed him
to penetrate the situation, but not to explain it, in a well-rounded manner, to his maybe
‘less’ talented peers.
As already stated before, Bohr was not a reality searcher in the sense that he
believed that physics should reflect reality or objects as they actually exist in nature. As
Heisenberg explains: “Bohr has expressed himself in discussions somewhat as follows;
classical physics and the quantum theory, taken as descriptions of nature, are both
caricatures; they allow us, so to speak, to asymptotically represent actual events in two
extreme regions of nature” 151 We see here that Bohr in general perceived theories to be
more asymptotically caricatures of reality. Bohr describes the quantum theory as ‘a rational
generalization’ of the classical theories. He sees the new matrix mechanics as defined by
Heisenberg as a continuous development of the correspondence principle, which laid the
link between the classical theories and the quantum theories with the help of the Fourier
development. He emphasizes this point in 1929 when he writes, “We are dealing here with
an unbroken development . . . which, beginning with the fundamental works of Planck on
black body radiation, has reached a temporary climax, in recent years, in the formulation of
a symbolic quantum mechanics”152. He states here that there is a continuity in the sense
that the old theory, the classical theory, is incorporated in the new one, the quantum
theory, where the former is a special case only allowed in a limited context of the preceding
theory. We see here once again the search for relationships between nature and reality, but
also between classical theory and quantum theory. In his quest to generalize classical
theories, Bohr primarily used the correspondence principle as his tool. For example he
writes: “The correspondence principle expresses the tendency to utilize in the systematic
development of the quantum theory every feature of the classical theories in a rational
transcription appropriate to the fundamental contrast between the postulates and the
classical theories”153. Bohr tries to bring quantum theory and classical theory together into
one coherent whole. It is interesting to note the change in his philosophical views
concerning the correspondence principle. In the first formulations of it Bohr emphasizes
multiple times that it was mainly a heuristic tool, but we can see in his later work that Bohr
takes the generalization to classical mechanics to be rational. This insistence on the
149
Bohr, Niels, Philosophy of Science Vol. 37 (1934), p. 157, and in The Truth of Science : Physical Theories and
Reality (1997) by Roger Gerhard Newton, p. 176
150
R. Courant, ‘Fifty years of Friendship, in Rozental, Niels Bohr, p. 304’
151
Heisenberg, Physics and Beyond, p. 210
152
Dingle, Herbert. "Atomic Theory and the, Description of Nature 1: Four Essays, with an Introductory
Survey." Nature 133 (1934): 962-964. P. 92.
153
Bohr 1925, reprinted in Kragh 2012, p.218.
45
rationality of the correspondence principle was also quite different from how it was
perceived by most physicists, who took it at best as a “clever bricolge”154. In his later
writings however we continue to see this pursuit to a consistent whole theory that
incorporates both classical and quantum concepts.
Bohr begins to see that classical theories and quantum theory are complementary in
the sense that they fill the holes in the foundations of both theories, despite the inherent
contradiction of both theories. There is certain ‘circularity’ in the connection between the
two theories, as the classical theory provides the fundamentals of the quantum theory, with
help of the correspondence principle, and the quantum theory will then, within certain
limits, give the same solution as the classical theory. Bohr sensed wholeness in the two
theories that could not exist independent of each other. One was the generalization of the
other: “At the same time, it affords some indication of how we may perceive in the
quantum theory, in spite of the fundamental character of this departure, a natural
generalization of the fundamental concepts of the classical electrodynamic theory.”155 Bohr
was listed as the first member of the advisory committee of the unity of science movement.
A movement that tried to promote the idea that every science is in a way linked to the
other, however it must be said that Bohr only contributed a one-page paper to the
movement. It is certain though that Bohr did believe, to some extent, in the unity of nature
and always emphasized the relationships to be found in different theories.
3.4 Bohr’s correlation to Høffding
Before beginning on the similarities between Bohr’s and Høffding’s philosophy, I first want
to clarify their relationship, as some state that Bohr was, throughout his carrier, extensively
influenced by Høffding’s reasonings and through him by other philosophers such as Soren
Kierkegaard and William James. Høffding was one of Bohr’s professors during his time at the
Copenhagen University and also an old friend of the Bohr family. He wrote two, at that time,
renowned books called ‘Outlines of Psychology’ and ‘A History of modern Philosophy’. In his
expression of his philosophies Kant and Kierkegaard primarily influenced him, among others.
Although Bohr is assumed to have attended Høffding lectures in his early scholar carrier,
after this initial relationship almost all contact ceased. Only a few letters from the
correspondence between him and Høffding were found, most being about matters outside
of philosophy or physics. We can state that Høffding was important for the forming of
Bohr’s general stand in philosophy, but he did not aid to the specific formulations of Bohr’s
theories on philosophy.156
One of the main ideas of Høffding was that one cannot regard an individual on its
own. There will always remain or exist a context that directly influences the individual and
should therefore be taken into account when regarding the individual or in Høffding’s own
words: “Research may and must admit that, as regards the individual, it does not succeed in
giving every detail, that there is always something which escapes it, -that the individuality
appears in consequence an irrational whole, which admits of only approximate
determination”. This emphasis on context is also found in the works of Bohr. The idea that
there is no such thing as a free mind and unspoiled context is one of the main debate
154
Darrigol uses this phrase to describe the physics community’s perception of Bohr’s old quantum theory.
Bohr, Niels. "The structure of the atom." Nobel lecture 11 (1922): 14.
156
Favrholdt, D., ed. Complementarity Beyond Physics (1928-1962). Vol. 10. Elsevier, 2013.
155
46
questions in the more philosophical papers of Bohr. However a difference must be pointed
out: Høffding only applied this context based reasoning to problems within psychology,
biology and philosophy, but he never extended it to chemistry or physics. So although the
reasonings are similar the, area in which they are applied are different.
Bohr believed that all definitions pre-assumed continuity, for both ordinary language
and quantum theory. In the case of ordinary language words in isolation naturally contain a
form of ambiguity. They need context to acquire their precise meaning. Various meanings of
the same word can only be compared if there is a continuous connection between the
various contexts in which the word plays a part. For Bohr’s theories the atomic model also
contains a certain form of ambiguity as it can be found in different stationary states; the
various states can only be compared if there is a continuous deformation of the system
connecting the two states. Bohr saw this condition as an essential part of the foundations of
his theories, but the condition could not be fulfilled in the later stages of his atomic model.
On this problem he wrote to Høffding; “In my personal opinion these difficulties
[surrounding the atom model] are of such nature that they hardly allow us to hope that we
shall be able, in the world of the atom, to carry through a description in space and time of a
kind which corresponds to our ordinary sensory images. Under these circumstances one
must naturally constantly bear in mind that one is operating with analogies, and this step, in
which the application of these analogies is delimited in each case, is of decisive significance
for progress ”157. Bohr claims that we cannot expect the concepts used to describe physics in
the observable physical world to also apply in the world of the atom, therefore all
measurements and theories done on atoms are idealizations of nature. We cannot expect
terms such as momentum and time, which are defined through our sensory experiences, to
apply in the same manner in the quantum theory. Bohr did discuss some of his overthinking
with Høffding, but it is not quite clear how much Høffding’s work influenced him, but Bohr’s
context related ideas did show quite some similarities to Høffding. These reasonings also
have quite a Kantian feeling, as we shall see below.
3.5 The transcendental part of Bohr’s philosophy
I get to think about my own thoughts of the situation in which I find myself. I even think that
I think of it, and divide myself into an infinite regressive sequence of ‘I’s who consider each
other. I do not know at which ‘I’ to stop as the actual, and in the moment I stop at one, there
is indeed again an ‘I’ which stops at it. I become confused and feel dizziness as if I were
looking down into a bottomless abyss, and my ponderings result finally in a terrible
headache.158
One of the first philosophical problems that Bohr countered was the problem of context and
how it affected objectivity within experiments; solutions are not universal, but only
accountable in a specific frame. He already mentions his overthinking in a letter from 1910
written to his brother: “sensations, like cognitions, must be arranged in planes that cannot
157
Bohr to Høffding 22 September 1922. Kragh 2012 p. 353.
Adventures of a Danish student by Poul Martin Møller, one of the favorite books of Bohr and often quoted
by him during lectures and in papers.
158
47
be compared”159. Later this slight remark on the planes of objectivity would be worked out
in full-fledged papers. Bohr assumed that full objectivity, understood as not being
influenced by prejudice, independent of thought and part of reality, was impossible within
preforming and describing experiments. He did not think it was a problem of the
measurement apparatus, as can be observed clearly in the following statement: “The very
fact that quantum phenomena cannot be analyzed on classical lines thus implies the
impossibility of separating a behavior of atomic objects from the interaction of these objects
with the measuring instruments which serve to specify the conditions under which the
phenomena appear.”160 It was not the experiment that was the problem. The problem is
that we cannot find an independent reality that is free of any context. In the quantum
theoretical case, quantum concepts and theory will always remain, to a certain degree,
dependent on the classically described context, which places limitations on our objectivity.
Both Kant and Bohr were interested in exploring these limits of objectivity. Kant was
one of the first philosophers that explicitly stated the dependency of knowledge on the
knowledge seeker: “Thus far it has been assumed that all our cognition must conform to
objects. Let us try to find out by experiment whether we shall not make better progress, if
we assume that objects must conform to our cognition.”161 According to Kant we can state
that the form of objects, as seen by the observer, is predetermined by a set of cognitive
conditions from said observer. Objectivity according to Kant is therefore an intersubjective
agreement between multiple observers and not a direct reflection of a ‘real’ truth. We
recognize some of Bohr’s thinking in these statements, as Bohr said: “The very fact that
quantum phenomena cannot be analyzed on classical lines thus implies the impossibility of
separating a behavior of atomic objects from the interaction of these objects with the
measuring instruments which serve to specify the conditions under which the phenomena
appear.”162 According to Bohr there is a limit to our objectivity in theorizing atomic physical
processes, because we cannot measure them in a definite descriptive quantum context.
Only with the help of our experimental tools can we make certain assumptions about
quantum processes and these experimental tools have to be described in classical terms.
This is not to say that they cannot be described according to the quantum rules, but, in the
end, to describe the phenomena classical terms have to be used. Bohr stated; “… In spite of
their limitations, we can by no means dispense with those forms of perception which color
our whole language and in terms of which all experience must ultimately be expressed”163.
So it was not an ontological problem that gave rise to the need of classical terms, but an
epistemological problem, because our whole framework of thought is colored by the
classical context.
In classical physics we can at the end of our experiment subtract the influence of the
measuring instruments, but this cannot be done in quantum theory, as said measuring
159
Bohr, N. "Collected Works V. 1. Early Work (1905-1911)/Ed. J. R Nielsen." (1972) P. 96.
Niels Bohr, ‘On the notions of causality and complementarity’, dialectica 2 (1948): 312-319.
161
Pluhar, Werner, and I. Kant. "Critique of pure reason." (1996): A820B848, p.21.
162
Bohr, Niels. ‘On the notions of causality and complementarity’, dialectica 2 (1948): 312-319.
163
Bohr, Niels. "The quantum of action and the description of nature." Bohr (1934)(1929): 92-101.
160
48
instruments are described in classical terms. Even though describing the measuring
instrument in classical terms forms an ambiguity in the description of quantum processes,
Bohr stubbornly held on to the use of classical concepts, even within the new quantum
theory. He states that: “... it would be misconception to believe that the difficulties of the
atomic theory may be evaded by eventually replacing the concepts of classical physics by
new conceptual forms... It continues to be the application of these concepts alone that
makes it possible to relate the symbolism of quantum theory to the data of experience.”164
Bohr stated that even if we do measurements on non-classical objects ‘the account of all
evidence must be expressed in classical terms’165. As noted earlier, Bohr considered this an
epistemological problem. In a recent essay on Bohr’s philosophy, Henry Folse noticed this
distinct trait of Bohr’s epistemology: “his constant concern with the problem of the
applicability of individual concepts to the description of phenomena and his deep conviction
that one of the principal tasks of science was to develop as it went along ‘a conceptual
framework adequate for such a description”166. For Bohr, in the physical world, this
conceptual framework has to be described using classical concepts, as our ordinary
language is full of the classical notions. He deems it impossible to be completely objective;
in the sense that the context is already predetermined by our use of language.
According to Kant it is impossible to subtract the contribution of the knower from
what is known; ‘pure’ objectivity does not exist. He stated: “Whatever [characteristics] we
are acquainted with in matter are nothing but relations (what we call its intrinsic
determinations is intrinsic only comparatively); among these relations there are
independent and permanent ones, through which a determinate object is given to us”167.
Bohr also noticed this “Impossibility of a strict separation of phenomena and means of
observation”168. We can call this consideration the ‘relational status of attributes’. The
knower is not known in the act of knowing is Kant’s reasoning, which tells us that there are
certain preconditions of experience that shape the experience without being part of it. Bohr
used the same kind of reasoning in stating that the instrumental preconditions of a quantum
description cannot be described quantum-mechanically. According to Bohr when doing an
experiment the measuring object is part of the knower and not of the known. This is not to
say that the measuring object has in sense different intrinsic properties, but because of the
way we epistemologically use the measuring objects classically, we have to describe them
classically. According to Bohr there has to be intersubjective agreement about the result,
which means that we must use the same language to describe the properties. That way
there is only a practical boundary, not an absolute boundary, between items described
classically and objects described quantum mechanically. Bohr stated this as the
“impossibility of any sharp separation between the behavior of atomic objects and the
interaction with the measuring instruments which serve to define the very conditions under
164
Idem, p.16
Bohr, Niels. Quantum physics and philosophy: causality and complementarity, p.39.
166
Folse, Henry J. "The philosophy of Niels Bohr: The framework of complementarity." (1985).
167
Kant, Immanuel. "Critique of Pure Reason, 1781." Translated by Norman Kemp Smith,. 9291nallimcaM:
nodnoL (1908).
168
Bohr, Niels. "The quantum of action and the description of nature." Bohr (1934)(1929): 92-101.
165
49
which the phenomena appear”169. This condition of separability was quite vague, as it was
not fully clear how one should choose where to place the separation. Bohr explained that
the separation was bendable depending on the context.
Bohr stated in his paper Atomic Theory and the Description of Nature: “Ultimately
every observation can, of course, be reduced to our sense perceptions. The circumstance,
however, that in interpreting observations use has always to be made of theoretical notions
entails that for every particular case it is a question of convenience at which point the
concept of observation involving the quantum postulate with its inherent irrationality is
brought in.” In order to make an observation to a quantum mechanical system one has to
permit a certain interaction with the surroundings, which are unfortunately described by
classical theories. If a description of the quantum system is made with these difficulties in
mind, one allows for a certain ‘idealization of observation and definition’ according to Bohr.
“The relation between subject and object”, as said by Bohr, “forms the core of the problem
of knowledge”170 and the reason for the irrationality. Strawson described this problem as:
“The investigation of that limiting framework of ideas and principles, the use and application
of which are essential to empirical knowledge, and which are implicit in any coherent
conception of experience which we can form”171. According to Bohr there was no such thing
as an independent reality, as it would, in the physical world, always be dependent on the
agencies of observation. The problem with quantum mechanics was that these agencies
were classically described while the measured objects contained quantum elements. With
this separation of measured objects and measuring devices, one has to keep in mind that
this vaguely defined separation was able to shift.
In order to do any experiments on a quantum system we must allow for an
interaction between the two separated systems. The systems being the object and
measuring instruments, where one is described in quantum terms and the other in classical
terms. This concern with the necessary conditions in accumulating knowledge and giving an
unambiguous report about the foundings can be described as transcendental. Honner
described the term transcendental as ‘signifying a fundamental concern with the necessary
conditions for the possibility of (experiential) knowledge’ According to Honner it is a word
that can “describe the character of Bohr’s philosophy […] The term ‘transcendental’ is a key:
and understanding of Bohr lies behind the door that unlocks it”172. This transcendentalthought was one of the doctrines founded by Kant. If we consider the transcendental part of
Bohr’s philosophy, we can understand his stress on the need for classical concepts in the
quantum theory better. Someone who also made this observation was Weizsäcker, who
said: “Niels Bohr is the only physicist in our time who-as far as I know, without having been
influenced by Kant-proceeded from fundamental insights similar to Kant’s … I join in the
conjecture that these principles, to put it in Kant’s terminology, will be neither transcendent
nor empirical, but transcendental in nature. In other words, they will formulate neither
metaphysical hypotheses nor particular experiences, but merely the preconditions of the
possibility of experience as such. Only in this framework will physicists be able to do justice
169
Bohr, Niels. "Discussion with Einstein on epistemological problems in atomic physics." Quantum theory and
measurement (1949): 1, p.210.
170
Niels, Bohr, Atomic Theory and the Description of Nature (Cambridge: Cambridge University Press, 1934),
p.117.
171
Strawson, Peter Frederick. The Bounds of Sense: An Essays on Kant's Critique of Pure Reason. Methuen,
1966.
172
Honner, John. "The transcendental philosophy of Niels Bohr." Studies in History and Philosophy of Science
Part A 13.1 (1982): 1-29.
50
to Bohr’s doctrine of the indispensability of classical concepts.”173. If one chooses to do a
measurement on a quantum system there will always be preconditions of the context in
which it is applied and both Bohr and Kant have made the important point on unambiguous
communication in the language.
There are, however, also differences to be spotted in the work of Bohr and Kant and
I would not like to claim that Bohr is considerably Kantian in nature. For example an
important difference noted by Henry Folse: “this (Bohr’s philosophy) … has a certain Kantlike appearance, for it issues in statements about the logical requirements for the proper
use of concepts in experience. But such an appearance is revealed to be mistaken when we
recall that these claims have nothing to do with how experience is obtained, as the Kantian
statements concerning the proper use of concepts imply, but only has to do with the
communication of an observation or an experience once that experience has already come
to pass. In this sense, Bohr’s reference to the use of concepts is tellingly non Kantian.”174 I
fully agree with the point being made here and I do not have the intention to push Bohr
over to Kant’s side, but I am of the opinion that there is a similar style to their philosophies.
Bohr is not explicitly Kantian, but there are several features of Bohr’s philosophy of physics
that are very isomorphic to Kant’s philosophy of knowledge. Their general framework of
thought shows quite some similarities and I hope that with the comparison of these two
great minds, we came to understand the work of Bohr better.
3.5 The position of Bohr within philosophy
There are two general themes that are repeated over and over again in the philosophical
considerations of Bohr, which we have covered in this paper. First there is the search of
relationships between theories. A new theory does not replace the other, but is a new
revised version of the old one, where the old theory is true in a limited context. Secondly
there is the focus on concepts, and then especially the use of unambiguous language. There
must be a complete awareness of the presumptions that are made when doing
experimental research and when describing the discovered data from the researches. This
awareness is crucial for a clear communication in physics. Bohr strongly emphasized this
need for unambiguous communication within physics, even at the boundaries of human
experience. The quantum realm was one of these boundaries, as we cannot directly interact
with the system. Only with the help of measuring mechanisms can we become more
knowledgeable on the quantum systems, but these can only be described in classical terms.
Even quantum systems are in the end ‘tied to the concepts-or, much more, the wordswhich are used in our ordinary description of nature’175. He believed that classical concepts
were indispensable for this description of nature, because these concepts have adapted to
the way humans describe the world. If we want to provide logical rules in common language
173
Weizsäcker in E. W. Bastin (ed.), Quantum Theory and Beyond (Cambridge, Cambridge University Press,
1971), p. 372.
174
Folse, H. J. (1985). The philosophy of Niels Bohr: The framework of complementarity.
175
Letter to Einstein, 13 April 1927, BSC:10, reprinted in Honner 1982.
51
to all men, we must use these concepts. We need these conceptual representations, for else
we would not be able to understand how we should perceive what happens on quantum
level. From his elaborate philosophical reasoning within quantum theory, Bohr realized that
we could never be only the spectators, seeing how context always has an effect on the view
of the experiment or as Bohr said: 'We are both onlookers and actors in the great drama of
existence: this is 'the old truth' of which 'the new situation in physics... has so forcibly
reminded us'176.
Pauli once wrote to Bohr to congratulate him on having ‘omitted all physics’ and that
he could now fully focus on philosophy. I do not think, however, that Bohr viewed it as his
task to solve general philosophical problems. In most of his work he focused primarily on
quantum and atomic problems and from there extended them to wider subjects, such as
biology and psychology. I consider, although not everyone credits him in this name, Bohr
one of the major philosophers of science of the twentieth century. He has influenced his
peers in physics and also other sciences to rethink their conduct in science and look beyond
the context presented by the framework of instruments. If we want to describe Bohr as a
philosopher it is best to do so in his own words: “The task of philosophy may be
characterized as the development of conceptual means appropriate for communication of
human experience”177. In this quote Bohr gives an overview of what is to be the relationship
between philosophy and science, but it is also a good self-assessment and description of the
position Bohr holds within philosophy.
Conclusion
While doing the research on the correspondence principle of Bohr it can be noticed that it is
called one of the most important theories in the old quantum theory. It was named the
‘cornerstone of Bohr's philosophical interpretation of quantum mechanics’ by A. Bokulich.
The fact that this cp is so well-known is quite interesting, as it is largely an ambiguous
principle. How is it that such an indefinable principle became one of the ‘cornerstones’ of
the quantum theory? The quantum theory is said to be primarily formal and set on clear
quantum rules, even though the correspondence principle represents a grey area in the
theory, were also some flexibility is allowed. In chapter three, we saw that Bohr usually
considered quantum problems to be connected with the physical world. He not only desired
to describe the quantum world according to mathematical formulas, but also wanted to
describe a further reaching connection to the physical world. From his aspirations towards
physics it is clear why he deemed the correspondence principle indispensable to the
quantum theory, as it was the only remaining link with the classical theories, which color
our language and perspective of the world. So it is epistemologically needed, in the sense
that we cannot describe quantum effects without the use of classical concepts, because
176
177
Bohr, Niels. "The quantum of action and the description of nature." Bohr (1934)(1929): 92-101.
Note, 4 January 1958, reprinted in Honner 1982.
52
classical concepts are the only ones we can really comprehend in a world classically
described.
The correspondence principle was also deemed indispensable and used extensively
by other physicists, even though not all physicists comprehended it entirely. Why did the
physics community accept and aspire to use a principle that was considered ambiguous? It
could have been that other physicists were impressed by the explanations Bohr could give
with help of the correspondence principle for the found experimental data done on atomic
systems. They therefore also tried to apply and understand it. A fault in this argument was,
however, as we saw, that the correspondence principle also extended to the new quantum
theory, which actually explained these experimental data without direct need for the
correspondence principle. In the final stages of the correspondence principle, it could be
argued that it was actually dispensable. The correspondence principle gave a bridge
between the new quantum theory and the classical theory and was applied according to the
old quantum rules, but in the new theory correspondence was already included in the
theory itself. Or as Born, Heisenberg and Jordan said in their famous Dreimännerarbeit:
“This similarity of the new theory with the classical theory also precludes any question of an
independent correspondence principle in addition to the theory; rather the theory itself can
be regarded as an exact formulation of Bohr’s correspondence ideas
[Korrespondenzgedanken].”178 So as the correspondence principle lost direct application
with the change of the quantum theory, only the concept remained. The term
correspondence principle is still used though, in today’s literature for example, Joseph Ford
and Giorgio Mantica in their article, “Does Quantum Mechanics Obey the Correspondence
Principle? Is It Complete?” quote the correspondence principle and state it to be
understood as; “any two valid physical theories which have an overlap in their domains of
validity must, to relevant accuracy, yield the same predictions for physical observations”179.
It could be deemed that only the general concept of correspondence between theories is
the one that is debated in today’s literature. The correspondence principle as first
articulated in 1913 did not survive the transition from the old quantum theory to the new
one, but the general concept of correspondence between classical mechanics and the new
quantum theory did survive. So although the correspondence principle is still a subject of
today’s literature, we can state that it is different from the original version.
The correspondence principle and its formulation changed according to the context
it is used in and it is therefore a good reflection of the general consensus, in a certain time
period, on how the atomic theory should be perceived. We saw that with the fall of the old
atomic model the correspondence principle remained as a bridge between the classical and
178
Max Born, Werner Heisenberg, and Pascual Jordan. Zur Quantenmechanik II. Zeitschrift für Physik, 35(89):557–615, 1926. English translation in [46], pp. 321–385.
179
Ford, Joseph, and Giorgio Mantica. "Does quantum mechanics obey the correspondence principle? Is it
complete?." American journal of physics 60.12 (1992): 1086-1098.
53
quantum theory, but the direct application became lost. If we follow the correspondence
principle, we get a good notion of the changes the quantum theory went through in the
course of time. The correspondence principle gives a good overview of Bohr’s general
thinking and shows beautifully the changes to his concept of quantum physics and the atom
theories throughout the years. I do believe that if anyone wants to understand Bohr theory
of physics, one must try to begin with understanding the correspondence principle, as it was
the link that Bohr saw between the classical mechanics, the old quantum theory and new
quantum mechanics.
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