## 1. Background and Scientific Context



Niels Bohr was a Danish physicist who lived from 1885 until 1962; he
was born and died in Copenhagen. He is best known as one of the
founders of quantum theory and for his work on the structure of atoms,
which earned him the Nobel Prize in Physics in 1922. In addition to
his scientific work, he devoted much of his career to the
philosophical interpretation of quantum mechanics. He was involved in
a famous long-standing debate with Albert Einstein over whether
quantum mechanics was a complete description of atomic phenomena (the
"Einstein-Bohr Debate"), and his lasting contributions to
the philosophical interpretation of quantum theory include the
Copenhagen interpretation, the thesis of complementarity, and the
correspondence principle.



The correspondence principle was first introduced in the context of
the old quantum theory, which was developed in the period between 1900
and 1925. The old quantum theory was an "interim" theory
developed in response to the recognition that Newtonian mechanics and
classical electrodynamics are inadequate for the description of atomic
systems. Although the old quantum theory had remarkable empirical
successes, it was criticized for having what many thought were
inconsistent foundations. The old quantum theory was ultimately
replaced by modern quantum mechanics with Werner Heisenberg's
development of matrix mechanics in 1925 and Erwin
Schrodinger's development of wave mechanics in 1926.



In order to understand the substance of the correspondence principle,
it is necessary to first understand the basic framework of the old
quantum theory. Niels Bohr first articulated the postulates of the old
quantum theory in 1913, in a three-part paper titled "On the
Constitution of Atoms and Molecules" (Bohr 1913). Bohr had
adopted Ernest Rutherford's model of the atom, according to
which most of the mass of the atom is concentrated in a small central
nucleus, while the electrons orbit the nucleus in planetary
trajectories. The key challenge facing Rutherford's model was
that it was unstable: according to classical electrodynamics, the
electron, which is an accelerated charged body, should radiate energy
and so rapidly collapse into the nucleus. Bohr's solution was to
incorporate Max Planck's theory of radiation, which postulates
that "the energy radiation from an atomic system does not take
place in the continuous way assumed in ordinary electrodynamics, but
that it, on the contrary, takes place in distinctly separated
emissions" (Bohr 1913, p. 4; BCW 2, p.
 164).[1]



Bohr summarized his quantum theory by means of two assumptions or
postulates. According to the first postulate, electrons cannot travel
in any path around the nucleus; rather, atomic systems can only exist
in one of a series of discrete "stationary states," in
which the electron is in a particular allowed stable periodic orbit
and is not emitting radiation. Intuitively, these stationary states
can be thought of as a series of concentric circular orbits around the
nucleus, along which the electron travels; these stationary states are
labeled by means of the principal quantum number \(n\), with the
lowest allowed orbit (the "normal" or ground state)
labeled \(n = 1\), the next stationary state of higher energy
\(n = 2\), and so on. Bohr postulates that when the electron is
in one of these stationary states, its motions can be adequately
described by means of classical mechanics; however, when the electron
makes a transition from one stationary state to another, the classical
theory no longer applies. The second postulate of Bohr's old
quantum theory is that, when there is a transition between different
stationary states, \(n'\) and \(n''\), the
emitted radiation is of a single frequency, \(\nu\), that is given by the
difference in the energy of the two states,
\((E\_{n'} - E\_{n''})\), divided by Planck's
constant.



\[\tag{1}
\nu = \frac{E\_{n'} - E\_{n''}}{h}
\]


This formula is typically referred to as the "Bohr-Einstein
frequency condition." The second postulate constituted a
significant break from classical electrodynamics, according to which a
variety of radiation frequencies would be emitted and those
frequencies would be determined solely by the motion of the
source.



It is important to note that while it is now accepted that this
emitted radiation is a photon, Bohr himself was surprisingly skeptical
of the photon concept up until as late as the mid-1920s (Stachel
2009). Bohr preferred to think of the emitted or absorbed radiation as
a wave, rather than as a particle. Bohr's hesitation in
accepting the existence of photons during this period seems to arise
from his concern that such a particle conception of light would be
irreconcilable with familiar interference phenomena. For example, Bohr
writes in 1921,



> 
> Thus according to this theory of light quanta, electromagnetic
> radiation from an atom should not spread as a system of spherical
> waves, but should be propagated in a definite direction as a
> concentrated entity, containing within a very small volume the energy
> \(h\nu\). On one hand such a conception seems to offer the only
> simple possibility of accounting for the phenomena of photoelectric
> action, if we adhere to an unrestricted application of the notions of
> conservation of energy and momentum. On the other hand, it does not
> appear reconcilable with the phenomena of interference of light, which
> constitute our only means of analysing radiation in its harmonic
> constituents and determining the frequency and state of polarisation
> of each of these constituents. (Bohr 1921a unpublished; BCW 3, pp.
> 412-413)
> 



It is interesting that Bohr's initial rejection of the photon
concept was also tied to the importance of being able to analyze
radiation into its harmonic components, which was essential for the
application of his correspondence principle (discussed below). 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 and his related
willingness to give up energy-momentum conservation to save the
classical electrodynamic picture of radiation" (Stachel 2009,
p.72). Stachel argues that it was, in part, the results of the
Bothe-Geiger and Compton-Simon experiments that finally led Bohr to
embrace the concept of the photon. For these historical reasons, care
needs to be exercised in invoking the photon concept when discussing
Bohr's early views; nonetheless the concept of photons can be
very helpful in understanding the physics behind the correspondence
principle (as well as Bohr's views after the early-1920s), and
hence, will be invoked in the following exposition.



Returning to the outline of the old quantum theory, Bohr's two
postulates were not yet enough to pick out, from all of the
classically allowed orbits, those orbits corresponding to stationary
states. In order to determine the stationary states, the following
"quantum condition" also needs to be introduced:



\[\tag{2}
\oint p\_{\theta} d\theta = nh \,.
\]


where the integral is taken over one period of the electron orbit, and
\(p\_{\theta}\) is the angular momentum, \(\theta\) is the angle in the plane of the electron orbit,
and \(n\) is the quantum number.[2] As 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 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"
(Jammer 1966, p. 90). The old quantum theory was a blending of
classical and quantum ideas: while it is assumed that the motion of an
electron *within* a particular stationary state can still be
described on the basis of the classical theory, the radiation given
off in a transition *between* stationary states (the
"quantum jumps") cannot.



In order to understand the substance of the correspondence principle
it is helpful to consider a simplified model of the atom as a
one-dimensional system, where the electron is undergoing simply
periodic motion. Classically the trajectory of the electron is given
by \(x(t)\), which is the solution to Newton's
equation of motion, and is periodic, which means it simply retraces
its steps over and over again with a frequency, \(\omega\), known as the
fundamental frequency. Because the motion is periodic, the position of
the electron can be represented by a Fourier series as
 follows:[3]



\[\tag{3}
x(t) = C\_1\cos(\omega t) + C\_2\cos(2\omega t) + C\_3\cos(3\omega t) + \cdots\,.
\] 


Each of these terms in the sum is known as a harmonic, and the
\(\tau\)th harmonic is given in terms of the time, \(t\),
an amplitude, \(C\_{\tau}\), and a frequency,
\(\omega\_{\tau}\), which is an integer multiple of the
fundamental frequency, \(\omega\_{\tau} = \tau \omega\) (these
multiples of the fundamental frequency are referred to as the
"overtones"). According to *classical*
electrodynamics, the frequencies of the radiation emitted by this atom
should just be given by the frequencies in the harmonics of the
motion: \(\omega,\) \(2\omega,\) \(3\omega,\) etc.; hence the spectrum of this
classical system should be a series of discrete evenly spaced
lines.



According to Bohr's old quantum theory, by contrast, the
radiation is not a result of the accelerated motion of the electron in
its orbit, but rather of the electron jumping from one stationary
state to another; and rather than giving off all of the harmonic
"overtones" together, only a single frequency, \(\nu\), is
emitted, and the value of that frequency is given by the Bohr-Einstein
frequency condition (Equation 1). The spectral lines are built up by a
whole ensemble of atoms undergoing transitions between different
stationary states, and these spectral lines, though they exhibit a
pattern of regularity, are not evenly spaced--except in the limit
of large quantum numbers.



Following Bohr, one can label the difference between the
\(n'\) stationary state and the \(n''\)
stationary state by \(\tau\); that is, if the electron jumps to the
nearest stationary state, \(\tau = 1\); if it jumps two stationary states
away, \(\tau = 2\); and so on. Although according to the classical
definition \(\tau\) specifies a particular harmonic component of the
classical motion, according to the quantum mechanical definition \(\tau\)
specifies the change in the quantum number in a particular jump.


## 2. The Correspondence Principle Defined



Among current Bohr scholars there is a consensus that Bohr did not
intend his correspondence principle to designate some sort of general
requirement that quantum mechanics recover the predictions of
classical mechanics in the classical limit, despite the prevalence of
this interpretation in the physics literature (see Section 6). There
is far less agreement, however, concerning precisely which relation
between classical and quantum mechanics Bohr intended to designate by
the correspondence principle. There are three prominent contenders in
the literature, all of which find support in Bohr's writings.
These three interpretations can be labeled: the frequency
interpretation, the intensity interpretation, and the selection rule
interpretation.



According to the *frequency interpretation*, the correspondence
principle is defined as a statistical asymptotic agreement between the
(quantum) frequency, \(\nu\_{n' \rightarrow n''}\), of radiation
emitted in a quantum jump of difference \(\tau\) from state \(n'\) to
\(n''\) and the (classical) frequency \((\omega\_{\tau})\) in the
\(\tau\)th harmonic of the classical motion in the \(n'\)
stationary state, namely



\[\tag{4}
\nu\_{n' \rightarrow n''} = \omega\_{\tau} = \tau \omega, \text{ for
large } n,
\]


where, \(n' - n'' = \tau\).



Note that this equality between the quantum frequency and one
component of the classical frequency only holds in the limit of
*large* quantum numbers--not for low quantum-number
transitions. Moreover there is only a statistical agreement since
classically all the frequencies will be given off together, while
quantum mechanically only a single photon is emitted with a single
frequency in any given transition between stationary states; hence,
one must consider a statistical ensemble of atoms to compare with the
classical spectrum.



According to the *intensity interpretation*, the correspondence
principle is defined as the agreement, in the limit of large quantum
numbers, between the probability, \(P\_{n' \rightarrow n''}\), of a
transition between two stationary states separated by \(\tau\) and the
square of the amplitude, \(C\_{\tau}\), of the \(\tau\)th
harmonic component of the classical motion:



\[\tag{5}
P\_{n' \rightarrow n''} \propto |C\_{\tau}(n)|^2 \text{ for large } n.
\]


Thus in the limit of large \(n\) the amplitudes of the harmonic
components of the electron's classical orbit can be used to
calculate the intensities of the spectral lines. While classically the
intensity of radiation is determined by its amplitude, quantum
mechanically the intensity of a spectral line is determined by how
many photons are emitted at that particular frequency. Hence, the more
probable a particular quantum transition is, the more photons will be
given off, and the greater the intensity. Note that, as with the
frequency interpretation, this is a correspondence that holds only
statistically and only in the limit of large quantum
 numbers.[4]



According to the *selection rule interpretation*, Bohr's
correspondence principle is best understood as the statement that
*each allowed quantum transition between stationary states
corresponds to one harmonic component of the classical motion*.
More precisely, Bohr's selection rule states that the transition
from a stationary state \(n'\) to another stationary state
\(n''\) is allowed *if and only if* there exists a
\(\tau\)th harmonic in the classical motion of the electron in
the initial stationary state; if there is no \(\tau\)th
harmonic in the classical motion, then transitions between stationary
states whose separation is \(\tau\) are not allowed quantum mechanically.
The essence of Bohr's correspondence principle is depicted in
Figure 1.



![sum of harmonics](Figure1.png)
Figure 1.
A classical periodic orbit, \(x(t)\), can be represented as a Fourier
sum of "harmonics" which are integer multiples of the
fundamental frequency, \(\omega\), representing the periodicity of the
motion. According to the selection rule interpretation, the
correspondence principle is Bohr's insight that each allowed
transition between stationary states corresponds to one harmonic
component of the classical motion. (Based on Fig. 3 of Fedak and
Prentis 2002)




Bohr's selection rule can be illustrated by the following
simplified
 example.[5]
 Suppose that the solution to Newton's equation,
\(F = m d^2 x/dt^2\),
and the quantum condition \(\oint pdx = nh\)
 is[6]



\[\tag{6}
x(t,n) = n\cos(n^{\frac{1}{2}}t) + n^{\frac{1}{2}}\cos(3n^{\frac{1}{2}}t),
\]


which is the Fourier decomposition of the classical periodic motion of
the electron in an allowed stationary state \(n\). For this
stationary state \(n\), the fundamental frequency (i.e.,
periodicity of the electron motion) is \(\omega = n^{\frac{1}{2}}\). Note that there are only the first
\((\tau = 1)\) and third \((\tau = 3)\) harmonics present in the classical
motion (Equation 6). According to Bohr's selection rule, this
means that there can only be quantum jumps between stationary states
that are one or three stationary states apart. So, for example, there
can be transitions from the \(n = 100\) stationary state to the
\(n = 99\) or \(n = 97\) stationary states; but there cannot
be transitions from the \(n = 100\) stationary state to the
\(n = 98\) stationary state, because there is no second harmonic
in the classical electron orbit.



Unlike the frequency and intensity interpretations of the
correspondence principle, the selection rule interpretation is not an
asymptotic relation--it applies to *all* quantum number
transitions, including small \(n\).



It is important to recognize that Bohr discovered and wrote about all
these correspondence relations; the disagreement among Bohr scholars
is simply *which* (if any) of these correspondence relations
Bohr meant to designate as the correspondence principle. Some scholars
have even gone so far as to doubt whether Bohr ever had a well-defined
and unvarying definition of the correspondence principle at all. These
and other interpretations of the correspondence principle are
elaborated in more detail in Section 5 below.


## 3. Bohr's Writings on the Correspondence Principle (1918-1928)



Most of Bohr's writings on the correspondence principle are
collected in Volume 3 of *Niels Bohr Collected Works*
(abbreviated here BCW), which is titled *The Correspondence
Principle (1918-1923)*. As Bohr himself reports (Bohr 1922),
the first germ of the correspondence principle can be found in his
1913 lecture "On the constitution of molecules and atoms,"
although the term does not appear in his writings until 1920. In the
years before Bohr adopted the expression "correspondence
principle," he described the relation as an analogy between
classical and quantum mechanics. For example, in his 1918 paper
"On the Quantum Theory of Line Spectra." (which he refers
to as "Q.o.L.") Bohr writes, "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" (Bohr 1918, p. 4; BCW 3, p.
70). In his later writings, however, Bohr explicitly rejects this view
that the correspondence principle can be thought of as an analogy. He
writes,



> 
> In Q.o.L [Bohr 1918] this designation has not yet been used, but the
> substance of the principle is referred to there as a formal analogy
> between the quantum theory and the classical theory. Such expressions
> might cause misunderstanding, since in fact--as we shall see
> later on--this Correspondence Principle must be regarded purely
> as a law of the quantum theory, which can in no way diminish the
> contrast between the postulates and electrodynamic theory. (Bohr
> [1923] 1924, fn. p. 22)
> 



Not only does Bohr reject the view that the correspondence principle
is an analogy, but the fact that he refers to it as a *law* of
quantum theory suggests, first, that he takes it to be a
*universal* principle (not just applicable in a limited
domain), and, second, that it is an essential part of quantum theory
itself, not some sort of general methodological constraint coming from
outside of quantum theory.



In his 1920 paper where the terms "correspondence" and
"correspondence principle" first appear (Nielsen 1976, p.
21), Bohr writes,



> 
> [A]lthough the process of radiation can not be described on the basis
> of the ordinary theory of electrodynamics, according to which the
> nature of the radiation emitted by an atom is directly related to the
> harmonic components occurring in the motion of the system, there is
> found, nevertheless, to exist a far-reaching *correspondence*
> between the various types of possible transitions between the
> stationary states on the one hand and the various harmonic components
> of the motion on the other hand. This correspondence is of such a
> nature, that the present theory of spectra is in a certain sense to be
> regarded as a rational generalization of the ordinary theory of
> radiation. (Bohr 1920, pp. 23-24; BCW 3, pp. 245-246)
> 



The correspondence Bohr describes here is between allowed transitions
from one stationary state to another and the harmonic components of
the classical motion. Moreover, he takes this correspondence to be the
ground or justification for the view that quantum theory is a rational
generalization of classical mechanics (for a discussion of this latter
view see Bokulich and Bokulich 2005). A few pages later, in a section
titled "The Correspondence Principle," Bohr goes on to
describe both the frequency correspondence and the intensity
correspondence:



> 
> This correspondence between frequencies determined by the two methods
> must have a deeper significance and we are led to anticipate that it
> will also apply to the intensities. This is equivalent to the
> statement that, when the quantum numbers are large, the relative
> probability of a particular transition is connected in a simple manner
> with the amplitude of the corresponding harmonic component in the
> motion. 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. (Bohr 1920, pp. 27-28; BCW 3, pp. 249-250)
> 



In typical Bohr style, he fails to come out and clearly say whether
one, all, or none of these correspondences is what he means to
designate by "the correspondence principle."



Bohr also argues that the correspondence principle is something which
explains why only certain spectral lines are observed in experiments.
He writes,



> 
> The above view, which may be termed the correspondence principle
> ... has offered an immediate interpretation of the apparent
> capriciousness, involved in the application of the principle of
> combination of spectral lines, which consists in the circumstance,
> that only a small part of the spectral lines, which might be
> anticipated from an unrestricted application of this [Rydberg-Ritz
> combination] principle, are actually observed in the experiments.
> (Bohr 1921b unpublished; BCW 4, p. 150)
> 



Bokulich (2008, and 2009 [Other Internet Resources]) has argued that
interpreting the correspondence principle in terms of the selection
rule not only illuminates Bohr's claim that it explains the
capriciousness of the spectral lines, but also his claim that the
correspondence principle is a law of quantum theory. Indeed, the
correspondence that he refers to as a "law" is the
selection rule correspondence, which holds also for small quantum
numbers, not just in the classical limit. It is a *law* because
it is a universal (i.e., applying to all \(n)\) restriction on
the allowed quantum transitions. To understand why it is a law *of
quantum theory* (as opposed to a law of classical electrodynamics)
it is helpful to consider Bohr's following remarks:



> 
> [T]he occurrence of radiative transitions is conditioned by the
> presence of the corresponding vibrations in the motion of the atom. As
> to our right to regard the asymptotic relation obtained as the
> intimation of a general law of the quantum theory for the occurrence
> of radiation, as it is assumed to be in the Correspondence Principle
> mentioned above, let it be once more recalled that in the limiting
> region of large quantum numbers there is 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 an asymptotic agreement of the
> statistical results. (Bohr [1923] 1924, p. 23; BCW 3, p. 480)
> 



In this passage it is clear that Bohr takes quantum mechanics to be a
universal theory. Despite the statistical agreement of results in this
limit, the physics of the frequencies and intensities, for example,
remains different, and Bohr is insistent that it is the quantum
account that is the strictly correct one--even in this high
\(n\) or "classical" limit. Hence, when Bohr
discovered that the allowable quantum transitions are those for which
there is a corresponding harmonic in the classical motion, what he had
discovered was something about quantum theory.



In his 1922 lectures on atomic theory in Gottingen, Bohr again
emphasizes that the correspondence principle holds even for low
quantum number transitions. This can particularly be seen in his
discussion of the well-known red and green spectral lines of the
Balmer series in the visible part of the hydrogen spectrum. The red
spectral line (which really is red at a wavelength of around 656 nm)
is typically labeled \(\rH\_{\alpha}\), and is the result of
radiation emitted in the jump from the \(n = 3\) to \(n =
2\) stationary state. The green line (labeled \(\rH\_{\beta}\) with a
wavelength of around 486 nm) is a result of the electron in a hydrogen
atom jumping from the \(n = 4\) to \(n = 2\) stationary
state.



![missing text, please inform](Figure2a.png)
  

![missing text, please inform](Figure2b.png)
Figure 2.
Red and green spectral lines of the Balmer series in the visible part
of the hydrogen atom




Regarding these low-quantum-number transitions Bohr writes,



> 
> We may regard \(\rH\_{\beta}\) as the octave of \(\rH\_{\alpha}\),
> since \(\rH\_{\beta}\) corresponds to a jump of 2 and
> \(\rH\_{\alpha}\) to a quantum jump of 1. It is true that
> \(\rH\_{\beta}\) does not have twice the frequency of
> \(\rH\_{\alpha}\), but it corresponds to the octave. This
> relationship we call the 'correspondence principle.' To
> each transition there corresponds a harmonic component of the
> mechanical motion. (Bohr 1922 unpublished lecture; BCW4, p. 348)
> 



In other words, although the "frequency correspondence"
does not hold for these low quantum numbers (nor can the intensities
of these lines be calculated directly from the classical amplitudes
via the "intensity correspondence"), what Bohr is here
calling the more general *correspondence principle* does hold;
specifically, these \(\tau = 1\) and \(\tau = 2\) transitions are allowed
because there is, in the Fourier decomposition of the electron's
classical orbit in the initial stationary state, a first and second
harmonic component. Such passages are a challenge for those who want
to interpret the correspondence principle as some sort of asymptotic
relation holding in the classical limit.



There is a remarkable continuity in Bohr's statements on the
correspondence principle. In 1923 one again finds Bohr emphasizing
that the allowed or possible transitions between stationary states are
connected to the harmonic components present in the classical
motion.



> 
> 
> [T]he possibility of the occurrence of a transition, accompanied by
> radiation, between two states of a multiply periodic system, of
> quantum numbers for example \(n\_1', \ldots, n\_u'\) and \(n\_1'',
> \ldots, n\_u''\), is conditioned by the presence
> of certain harmonic components in the expression given by [the Fourier
> series expansion of the classical electron motion]. The frequencies
> \(\tau\_1\omega\_1 + \ldots + \tau\_u\omega\_u\) of these
> harmonic components are given by the following equation
> 
> 
> 
> \[
> \tau\_1 = n\_1' - n\_1'' \quad\ldots\quad \tau\_u = n\_u' - n\_u''.
> \]
> 
> 
> We, therefore, call these the "corresponding" harmonic
> components in the motion, and the substance of the above statement we
> designate as the "Correspondence Principle." (Bohr [1923]
> 1924, p. 22; BCW 3, p. 479)
> 
> 
> 



In 1925 Werner Heisenberg published his famous "Umdeutung"
paper introducing matrix mechanics (Heisenberg 1925). This would be
the beginning of modern quantum mechanics, which replaced the old
quantum theory. In a supplement to *Nature* published in
December of 1925, Bohr describes what he takes to be the relationship
of the new quantum mechanics to his correspondence principle. He
begins with a statement of the correspondence principle.



> 
> 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. (Bohr 1925, pp. 848-849;
> BCW 5, pp. 276-277)
> 



In this quotation Bohr notes all three correspondence relations: the
selection rule correspondence, the asymptotic frequency
correspondence, and the asymptotic intensity correspondence. Bohr then
turns to a discussion of Heisenberg's matrix mechanics paper,
arguing that



> 
> It [the new matrix mechanics] operates with manifolds of quantities,
> which replace the harmonic oscillating components of the motion and
> symbolise the possibilities of transitions between stationary states
> in conformity with the correspondence principle... In brief, the
> whole apparatus of the quantum mechanics can be regarded as a precise
> formulation of the tendencies embodied in the correspondence
> principle. (Bohr 1925, p. 852; BCW 5, p. 280)
> 



What Bohr seems to be saying in this passage is that the new quantum
mechanics symbolizes the allowed transitions in conformity with his
selection rule.



Bohr continued to discuss the correspondence principle even after the
advent of modern quantum mechanics. In his 1928 Como lecture, for
example, he writes,



> 
> The aim of regarding the quantum theory as a rational generalisation
> of the classical theories led to the formulation of the so-called
> correspondence principle. The utilisation of this principle for the
> interpretation of spectroscopic results was based on a symbolical
> application of classical electrodynamics, in which the individual
> transition processes were each associated with a harmonic in the
> motion of the atomic particles to be expected according to ordinary
> mechanics. (Bohr 1928, p. 584; BCW 6, p. 152)
> 



Note that Bohr here speaks of a "symbolic" application of
classical electrodynamics--the electron trajectories in the
stationary states, for example, are not to be interpreted literally or
even "formally." This shift in Bohr's understanding
of the status of the electron trajectories being employed in the
correspondence principle has been emphasized in particular by Olivier
Darrigol (Darrigol 1997, p. 558).



Nonetheless, as in 1920 when he first introduced the term
correspondence principle, Bohr is here in 1928 (after the replacement
of the old quantum theory by modern quantum mechanics) interpreting
the correspondence principle as a connection between transition
processes and harmonics in the classical motion of the particles.
Moreover, just as in 1920, he is explicitly tying the correspondence
principle to his view that quantum mechanics is a rational
generalization of classical mechanics. The correspondence principle
thus provided a conceptual link between classical mechanics (and
electrodynamics), the old quantum theory, and modern quantum
mechanics. This rational generalization thesis in turn was the
foundation of Bohr's views on complementarity, and more broadly,
his Copenhagen interpretation of quantum mechanics. Thus the
correspondence principle was not only technically important in the
discovery of modern quantum mechanics, as Jammer (1966, p. 118) has
emphasized, but it was also was the very foundation of Bohr's
philosophy.


## 4. Early Responses



Early responders to Bohr's correspondence principle can be
roughly divided into three categories: 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, Pauli, and
Heisenberg).



It is unclear how well many of Bohr's contemporaries understood
what he meant by the correspondence principle. For example, even
Leon Rosenfeld, one of Bohr's closest students and
collaborators, recalls Bohr's annoyance over his failure to have
correctly understood the substance of this principle (Rosenfeld [1973]
1979, p. 690). This is perhaps not entirely surprising given the
turbidity of Bohr's writings. Bohr himself was quite conscious
of the fact that his dense writing style often led others to
misunderstand him. In a letter to Arnold Sommerfeld, Bohr laments,



> 
> In the last few years I have often felt myself scientifically
> very lonesome, under the impression that my efforts to develop the
> principles of the quantum theory systematically to the best of my
> ability have been received with very little understanding... I
> understand quite well how little the matters are clarified as yet, and
> how helpless I am at expressing my thoughts in easily accessible
> form. (Bohr to Sommerfeld, April 30, 1922; BCW 3, p. 39)
> 



Indeed one quickly sees a wide variety of different interpretations of
the correspondence principle emerging in the physics
literature--a variety of interpretations that persists to this
day. Hence, it is important to distinguish Bohr's own views on
the correspondence principle from what the correspondence principle
came to mean to the broader physics community (for a detailed account
of how the correspondence principle drove research and discussions in
this period, see Jahnert 2019). Both Hendrik Kramers and John van
Vleck, for example, 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
(see, for example, Duncan and Janssen 2007; 2019).



Among those who seemed to correctly understand the core of
Bohr's correspondence principle, even fewer embraced it. Indeed
many of Bohr's contemporaries were distrustful of the principle,
despite their willingness to opportunistically make use of it to
obtain concrete results when other methods failed. Into this third
category fall three of the most important contemporaneous critics of
the correspondence principle: Arnold Sommerfeld, Wolfgang Pauli, and
Werner
 Heisenberg.[7]


### 4.1 Sommerfeld



One of the earliest and most forceful critics of Bohr's
correspondence principle was Arnold Sommerfeld (1868-1951). As
Darrigol has recounted in detail, Sommerfeld was never comfortable
with Bohr's correspondence principle, and only begrudgingly
admitted its fertility (Darrigol 1992, pp. 138-145). Sommerfeld
preferred to view quantum theory as a self-contained set of formal
rules, and wanted to derive the selection rules "by a remarkably
rigorous manner of deduction, reminiscent of the incontrovertible
logic of numerical calculations" (Sommerfeld [1919] 1923, pp.
265-266). In the first edition of his book *Atombau und
Spektrallinen* (*Atomic Structure and Spectral Lines*)
Sommerfeld disparagingly refers to Bohr's correspondence
principle as a "magic wand":



> 
> Bohr has discovered in his *principle of correspondence* a
> magic wand (which he himself calls a formal principle), which allows
> us immediately to make use of the results of the classical wave theory
> in the quantum theory. (Sommereld [1919] 1923, p.
>  275)[8]
> 



In the second edition of *Atombau* Sommerfeld softens his
criticism of the correspondence principle, noting its great fertility.
In a letter written to Bohr in November of 1920 Sommerfeld writes,



> 
> In the appendices of my book, you can see that I have taken some pains
> to show the value of your correspondence principle better than in the
> 1. edition... Nonetheless, I must confess that the origin of your
> principle, which is foreign to quantum theory, is still distressing to
> me, however much I recognize that through it a most important
> connection between quantum th. and classical electrodynamics is
> revealed. (Sommerfeld to Bohr, 11 November 1920; BCW 3, p. 690)
> 



Suman Seth (2008) has argued that Sommerfeld's dissatisfaction
with the correspondence principle stemmed not so much from a
commitment to formal rules as Darrigol suggests, but more from his
general distrust of analogical thinking and what he took to be an
illegitimate mixing of quantum and classical concepts. We see this,
for example, in Sommerfeld's 1924 article on Bohr's atomic
models, where he writes,



> 
> The magic of the correspondence principle has proved itself generally
> through the selection rules of the quantum numbers, in the series and
> band spectra... Nonetheless I cannot view it as ultimately
> satisfying on account of its mixing of quantum-theoretical and
> classical viewpoints. (Sommerfeld 1924, p. 1048; quoted also in Seth
> 2008, p. 345).
> 



Sommerfeld's critical attitude toward the correspondence
principle would prove influential on Wolfgang Pauli and Werner
Heisenberg, both of whom were his doctoral students.


### 4.2 Pauli



In his lecture accepting the Nobel Prize (which was awarded in 1945
for his discovery of the exclusion principle) Wolfgang Pauli describes
the fundamental methodological differences between Bohr and
Sommerfeld:



> 
> 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 these. 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... (Pauli
> 1946, p. 27)
> 



Although in hindsight, Pauli claims that both methodologies influenced
him, in his correspondence dating from 1924 and 1925, one finds Pauli
almost exclusively defending a Sommerfeldian approach, dismissing both
the correspondence principle and model-based reasoning more generally.
Pauli's critical remarks about Bohr's correspondence
principle center on two issues. First, Pauli objected to the use of
even a "formal" notion of an electron orbit, which is
central to the application of the correspondence principle (e.g.,
Pauli to Bohr, February 21st 1924). Second, Pauli doubted that the
correspondence principle would be able to explain the closing of the
electron shells as Bohr had hoped. In 1924 Pauli wrote to Bohr:



> 
> 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 \(\Delta k = \pm 1\). 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. (Pauli to Bohr, December 12th, 1924;
> quoted in Serwer 1977, p. 235)
> 



Reading between the lines, Pauli seems to understand Bohr's
correspondence principle as a selection rule about transitions between
stationary states that are connected with harmonics in the classical
motion. His objection is not to this correspondence principle, but
rather with attempts at extending this principle to explain the
closing of electron groups in the atom--the sort of extension of
the correspondence principle that Pauli elsewhere referred to as
"correspondence principle imperialism" (*Imperialismus
des Korrespondenzprinzips*) (Pauli to Heisenberg, February 28th,
1925; quoted in Serwer 1977, p.233). The closing of electron shells
would ultimately be explained by Pauli's exclusion principle,
though initially Pauli was dissatisfied with such an explanation
insofar as his exclusion rule could not be given a more general
grounding and justification. When Bohr suggested that the
correspondence principle might provide such a grounding for the
exclusion principle, 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). (Pauli to Bohr December 31st, 1924; quoted in Heilbron
> 1983, p. 306 and Serwer 1977, p. 236)
> 



In the above quotation we see Pauli siding with Sommerfeld, in his
distrust of the correspondence principle and model-based reasoning in
general, and in his preference for a "direct"
interpretation in terms of a new quantum mechanics.


### 4.3 Heisenberg



As Jagdish Mehra and Helmut Rechenberg (1982) have recounted, when
Werner Heisenberg first arrived in Copenhagen to work with Bohr in
1924, he enthusiastically embraced the correspondence principle and
took up the task of defending this principle to the broader physics
community. This is especially clear in a letter Heisenberg wrote to
Pauli on September 30th, 1924:



> 
> 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... [S]ince
> recently the correspondence principle has been blamed so much, it
> would be good to publish it [your results confirming the
> correspondence principle] *ad majorem correspondentiae principii
> gloriam* [for the greater glory of the correspondence principle].
> (Heisenberg to Pauli, September 30th, 1924; quoted in Mehra
> and Rechenberg pp. 156-157)
> 



By 1925, however, Heisenberg increasingly began to distance himself
from the correspondence principle. Daniel Serwer attributes this shift
in Heisenberg's views to Pauli's influence:



> 
> [I]n early March 1925, Pauli came to Copenhagen. This visit was
> crucial to Heisenberg. Under pressure from Pauli, he began to change
> his views on many issues...[including] the Correspondence
> Principle. ...But until Pauli's visit, Heisenberg had
> remained convinced that ... the Correspondence Principle would
> bring results. (Serwer 1977, p. 222)
> 



Despite the fact that Heisenberg relied heavily on the correspondence
principle in developing matrix mechanics, he no longer publicly
defended
 it.[9]
 Indeed as Mara Beller has noted, "Heisenberg does not cite
Bohr's work at all [in the 1925 matrix mechanics paper], despite
the fact that the paper was built on Bohr's correspondence
principle in a fundamental way... Instead Heisenberg
[inaccurately] presented his work as flowing from the positivist
principle of elimination of unobservables" (Beller 1999, p.
 140).[10]
 Heisenberg expresses his increasing ambivalence 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. (Heisenberg 1930, p. 82)
> 



While Heisenberg is willing to admit the limited utility of the
principle, he does not view it as a fundamental principle of quantum
theory the way that Bohr does. Instead Heisenberg argues that
"[i]t must be emphasized that this [correspondence] is a purely
formal result; it does not follow from any of the physical principles
of quantum theory" (Heisenberg 1930, p. 83). In other words, the
correspondence is to be interpreted as a purely mathematical result,
not as revealing any deep connection between the quantum and classical
theories, and certainly not as a principle of quantum theory itself.
For Heisenberg, quantum mechanics is a closed theory
[*Abgeschlossene Theorie*], an axiomatic system complete in
itself, not one that in any way depends on classical
 mechanics.[11]


## 5. Interpretations in the History and Philosophy Literature



Arguably any interpretation of the correspondence principle faces the
following four challenges: First, determining which (or which
combination) of the various analogies or relations between classical
and quantum mechanics Bohr intended to designate by the correspondence
principle; second, determining the scope of the correspondence
principle (i.e., does it apply only to large quantum numbers or all
quantum numbers); third, offering an explanation for why Bohr thought
that the correspondence principle should be thought of as a
*law* of quantum theory; and finally, offering an explanation
for why Bohr thought that the correspondence principle had been
formalized, and hence preserved, in modern (matrix) quantum
mechanics.



One of the most influential discussions of Bohr's correspondence
principle appears in Max Jammer's 1966 book *The Conceptual
Development of Quantum Mechanics*. Jammer takes the correspondence
principle to be a relation between the kinematics of the electron and
the properties of the emitted radiation. Like many interpreters of the
correspondence principle, he focuses primarily on the frequency
relation. He notes that Bohr also found an asymptotic correspondence
between the intensities of the spectral lines and the amplitudes in
the classical harmonic components, as well as a correspondence between
the polarization of the emitted radiation and the character of the
classical motion. Although Jammer notes these other correspondences,
he seems to interpret the frequency correspondence as the primary
correspondence principle. Regarding the status of this correspondence
as a principle, Jammer writes,



> 
> In the limit, therefore, the quantum-theoretic frequency
> \(\nu\_{qu}\) coincides with the classical
> mechanical frequency \(\nu\_{cl}\). 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.
> (Jammer 1966, p. 111)
> 



According to Jammer, the correspondence principle, interpreted as the
frequency relation, applies by fiat to all quantum numbers and hence
obtains the status of a "principle," even though it is an
"approximate" relation that is only exact for large
quantum numbers.



Jammer is 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'. (Jammer
> 1966, p. 116)
> 



On Jammer's view, Bohr's claim that the correspondence
principle is a law is simply an attempt to cover up the inconsistent
foundations of the old quantum theory. In opposition to Bohr's
claim that quantum theory is a rational generalization of classical
mechanics, Jammer interprets Bohr as viewing quantum and classical
mechanics as irreconcilable, and hence interprets the correspondence
principle as only a "formal analogy of heuristic value."
At the end of his discussion of the correspondence principle, Jammer
concludes "his [Bohr's] numerous and often somewhat
conflicting statements, made from 1920 to 1961, 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" (Jammer 1966, p. 117).



It is perhaps for this reason that Jagdish Mehra and Helmut Rechenberg
(1982), in their comprehensive history of the development of quantum
theory, make no concrete commitment as to which of the several
correspondence relations discussed by Bohr should be designated as the
correspondence principle. Like Jammer they struggle to understand why
Bohr thought the correspondence principle applied to small quantum
numbers and why it should be considered a law of quantum theory. They
write, "[T]he extension of the analogy 'in some way'
to small quantum numbers represented a most daring assumption...
[Bohr] felt confident that an extension of his relations for
frequencies and intensities, which were valid in the high quantum
number limit, might eventually be justified also when low quantum
numbers were involved" (Mehra and Rechenberg 1982, p. 250).
Although they offer one of the more technically detailed discussions
of the correspondence principle, they do not discuss either
Bohr's claim that it should be considered a law nor Bohr's
claim that the correspondence principle is formalized into the new
quantum mechanics.



An alternative interpretation of Bohr's correspondence principle
has been defended by Olivier Darrigol in his excellent and highly
readable history of quantum theory *From \(c\)-Numbers to
\(q\)-Numbers*. Instead of viewing the correspondence
principle as a statement about quantum and classical frequencies, he
interprets it as what was earlier called the *intensity
correspondence*: The quantum transition probability between two
stationary states separated by \(\tau\) is proportional to the squared
modulus of the classical amplitude of the \(\tau\)th harmonic
vibration, which classically is a measure of the intensity (see
Equation 5).



Darrigol writes,



> 
> 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'. (Darrigol 1997, p. 550; see also
> Darrigol 1992, p. 126)
> 



Strictly speaking this intensity correspondence is exact only in the
limit of large quantum numbers, and cannot be extended to small
quantum numbers. The one exception to this is when the classical
amplitude is zero, the quantum transition probability will also be
zero (a special case of what was earlier called *Bohr's
selection rule*), which does hold for all quantum numbers.



Unlike some interpreters of the correspondence principle Darrigol does
believe that there is a coherent and rational account of Bohr's
correspondence principle in terms of the intensity correspondence. He
argues that Bohr is justified in calling it a *law of quantum
theory* because "[i]t was indeed a relation between two
quantum theoretical concepts: the probability of a quantum jump, and
atomic motion" (Darrigol 1997, p. 553). Moreover, Darrigol
cogently argues that this correspondence was incorporated directly
into matrix mechanics in Heisenberg's 1925 paper, thus
vindicating Bohr's claim that the new quantum mechanics can be
regarded as "a precise formulation of the tendencies embodied in
the correspondence principle" (Bohr 1925, p. 852; BCW 5, p.
280).



Despite this continuity, Darrigol argues that there was a fundamental,
though implicit, shift in Bohr's understanding of the
correspondence principle:



> 
> Originally, the motion to which the 'correspondence' was
> applied was the motion in stationary states, formally described by the
> orbiting electrons. In the end, this motion had nothing to do with the
> description of stationary states, it was purely
> 'symbolic.' (Darrigol 1997, p. 558)
> 



By 'formally' Darrigol means that the space-time relations
and dynamical relations were retained in the description of the
electron's motion of the atom, although it was recognized that
there was no possibility of actually observing this motion. Once it
became a 'symbolic' relation, the space-time description
was lost completely.



Hans Radder has also argued that Bohr's understanding of the
correspondence principle evolved over his career. Radder (1991)
identifies three different phases. In the first phase (from about 1913
to 1915) the correspondence principle was merely a "numerical
correspondence"; that is, it concerned a numerical agreement of
the values of certain quantities in classical mechanics (and
electrodynamics) and the old quantum theory. In the second phase,
lasting between 1916 and 1922, the correspondence principle was not
just a numerical agreement but involved a conceptual continuity as
well: "The same fundamental concepts \(\omega\_{n,\tau}\) and
\(C\_{n,\tau}\), which govern the motion of the electrons in their
orbits, and the same function \(f\) determining the transition
probabilities are claimed to underlie both kinds of theory"
(Radder 1991, p. 206). In the third phase from 1923 until 1925 when
the new matrix (quantum) mechanics was introduced, the conceptual
continuity was abandoned in favor of a "formal"
correspondence. Radder means something different than Darrigol when he
describes the correspondence principle as "formal." He
explains,



> 
> Now conceptual correspondence, linked as it was to the mechanical
> orbital model was rejected as a basis for atomic theory and the
> orbital model was provisionally replaced by the so-called
> 'virtual field model'... [T]hese relations between
> terms are *formal* (not conceptual) *correspondences*,
> expressing the existence of certain relations of mathematical identity
> or substitution. (Radder 1991, p. 207)
> 



Radder's chief concern is relating the various incarnations of
Bohr's correspondence principle to the more general
correspondence and heuristics arguments in the philosophy of science
literature (see Section 7 below).



Other recent interpretations of Bohr's correspondence principle
include Robert Batterman's, according to which "the CP is
a statement to the effect that the radiative processes to which the
second postulate \([h\nu = E' - E'']\) applies are
'correlated' with, or 'correspond' to,
mechanical vibrations or periodic motions of the charged particles,
the electrons, in the atom" (Batterman 1991, p. 203). Batterman
rightly notes that the correspondence principle is not the claim that
quantum mechanics must contain classical mechanics as a limiting case,
and argues that the asymptotic agreement of classical and quantum
frequencies (the frequency interpretation) is not the correspondence
principle, but rather something that is *justified and
explained* by the correspondence principle.



Similarly, Scott Tanona (2004) rejects the view that Bohr's
correspondence principle is about an asymptotic agreement of quantum
and classical theories. He argues instead that Bohr's
correspondence principle should be understood primarily as a
connection between spectra (radiation) and orbital motion. More
broadly, Tanona argues that the correspondence principle should not be
understood as a heuristic tool for theory construction, but rather as
an epistemological tool, whose "main purpose within Bohr's
empirical approach was to bridge the epistemological gap between
empirical phenomena and the unknown atomic structure ... by
associating classical properties of spectra with atomic
properties" (Tanona 2004, p. 683).



More recently, Alisa Bokulich (2008, and 2009 [Other Internet
Resources]) has argued that Bohr's correspondence principle
should be understood in terms of what she calls *Bohr's
selection rule*. This selection rule states that "the
transition from a stationary state \(n'\) to another
stationary state \(n''\) is allowed *if and only if*
there exists a \(\tau\)th harmonic in the classical motion of
the electron in the stationary state; if there is no
\(\tau\)th harmonic in the classical motion, then transitions
between stationary states whose separation is \(\tau\) are not allowed
quantum mechanically" (Bokulich 2008, p. 85). Thus, the
correspondence principle, on this reading of Bohr, is defined as the
statement that each allowed quantum transition between stationary
states corresponds to one harmonic component of the classical motion.
She argues that the asymptotic agreements of frequencies and
intensities emphasized by Jammer and Darrigol respectively are
applications or *consequences* of the correspondence principle,
but not the correspondence principle itself. Moreover, she argues that
the selection rule interpretation of Bohr's correspondence
principle best makes sense of Bohr's claims that the
correspondence principle applies to small \(n\), that it is a law
of quantum theory, and that it is formalized and preserved in matrix
mechanics.


## 6. Interpretations in the Current Physics Literature



One typically finds a very different understanding of Bohr's
correspondence principle in the current physics literature. Even as
early as Max Born's classic text on the new quantum
 theory,[12]
 first published in 1933, one finds the correspondence principle
defined as follows:



> 
> The leading idea (Bohr's *correspondence principle*,
> 1923) may be stated broadly as follows. Judged by the test of
> experience, the laws of classical physics have brilliantly justified
> themselves in all processes of motion... It must therefore be
> laid down as an unconditionally necessary postulate, that the new
> mechanics ... must in all these problems reach the same results
> as the classical mechanics. In other words, it must be demanded that,
> for the limiting cases of large masses and of orbits of large
> dimensions, the new mechanics passes over into classical mechanics.
> (Born [1933] 1957, p. 103).
> 



There are two strands to Born's rereading of Bohr's
correspondence principle that have become quite common in the physics
literature: first, the correspondence principle as a very general
requirement that quantum mechanics ought to be able to recover the
empirical successes of classical mechanics; and, second, the more
specific requirement that it recover the predictions of classical
mechanics in the limit of large masses and orbits of large
dimension--this latter limit is typically expressed as \(n
\rightarrow \infty\).



In David Bohm's classic textbook on standard quantum mechanics,
the correspondence principle is similarly interpreted as a general
relation between the two theories: "[T]he correspondence
principle, which was first given by Bohr ... states that the laws
of quantum physics must be so chosen that in the classical limit,
where many quanta are involved, the quantum laws lead to the classical
equations as an average" (Bohm 1951, p. 31).



In his 1984 *Physics Today* article "The Correspondence
Principle Revisited," Richard Liboff argues that Bohr's
correspondence principle is not a general claim about recovering
classical mechanics from quantum mechanics, but the following more
restricted claim: "the quantum frequency spectrum of a periodic
system approaches the classical spectrum in the limit of large
\(n\)" (Liboff 1984, p. 52). Liboff goes on to note that even
this restricted form of the correspondence principle is violated for
systems such as a particle in a box and a rigid rotor. He notes that
"[i]n both of these cases, the limit \(h \rightarrow 0\) leads
to frequency correspondence, but the limit \(n \rightarrow \infty\)
does not" (Liboff 1984, p. 52).



Following on Liboff, Ghazi Hassoun and Donald Kobe (1989) have argued
that in order to obtain a meaningful classical limit of quantum
mechanical eigenvalues it is necessary that "both formulations
[of the correspondence principle] are used concurrently in the sense
that the Planck constant goes to zero and the appropriate quantum
number goes to infinity ... subject to the constraint that \(nh =
J\), where \(J\) is the appropriate classical action" (Hassoun
and Kobe 1989, p. 658). On this reading, Bohr's correspondence
principle is construed simply as the limit \(n \rightarrow
\infty\). Although they call this limit "Bohr's
correspondence principle", they note that Bohr was able to
obtain the correct classical orbital frequency precisely
"because he did not take the mathematical limit as the quantum
number goes to infinity. Instead, Bohr considered that for large
quantum numbers the quantum result was very nearly the classical
result. Thus the quantum number \(n\), say, was \(n \gg 1\), but was
not infinite" (Hassoun and Kobe 1989, p. 661). They conclude
that the correspondence principle is "often thought to be
imprecise because of Bohr's use of large, but not infinite,
quantum numbers. However, when the Bohr and Planck formulations are
synthesized, as shown in this article, the correspondence principle
has a precise mathematical form" (Hassoun and Kobe 1989,
p. 661). The correspondence principle on this reading is understood
broadly as the attempt to get a mathematically precise formulation of
the classical limit.



Much of the contemporary physics community's interest in
Bohr's correspondence principle (variously understood) is in
prima facie violations of this principle by particular classes of
physical systems. For example, Joseph Ford and Giorgio Mantica, in
their article, "Does Quantum Mechanics Obey the Correspondence
Principle? Is It Complete?" define the correspondence principle
as requiring that "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" (Ford and
Mantica 1992, p. 1087). Their concern is whether quantum mechanics can
account for the behavior of chaotic classical systems. Similarly,
Wojciech Zurek defines the correspondence principle as a demand for an
agreement between the "quantum and classical expectation
values" (Habib et al. 1998), arguing that without the phenomenon
of decoherence, the correspondence principle is violated.



While the contemporary physics literature's understanding of the
correspondence principle is largely in line with Max Born's
construal of this principle (given above), it was not what Bohr
himself meant by the correspondence principle. Although Bohr agreed
that quantum mechanics ought to be able to recover the empirically
confirmed predictions of classical mechanics, he explicitly rejected
this reading of the correspondence principle. Indeed Leon
Rosenfeld recounts Bohr's frustration at the continued
misunderstanding of his principle. When Rosenfeld off-handedly
suggested to Bohr that the correspondence principle was about the
asymptotic agreement of quantum and classical predictions, Bohr
emphatically protested and replied, "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 for
the theory" (Rosenfeld 1973, p. 690).


## 7. Generalized Correspondence Principle



Finally, it is important to distinguish between Bohr's
correspondence principle and what some philosophers of science have
referred to as the "generalized correspondence principle"
(alternately the "general correspondence principle").



In his classic essay "Correspondence, Invariance and Heuristics:
In Praise of Conservative Induction" Heinz Post defines this
principle as follows:



> 
> The most important heuristic restriction is the *General
> Correspondence Principle*. Roughly speaking, this is the
> requirement that any acceptable new theory \(L\) should account
> for the success of its predecessor \(S\) by
> 'degenerating' into that theory under those conditions
> under which \(S\) has been well confirmed by tests. (Post 1971,
> p. 228)
> 



The generalized correspondence principle here is seen both as a
constraint on the development of new theories and as an account of how
successor theories are related to their predecessors. Typically in
physics, Post's notion of 'degenerating' is cashed
out as the limit of some parameter whereby the equations of a
successor theory are recovered as the limit of the equations of
predecessor theory. Regarding the specific case of the
quantum-classical relation, Post argues that the general
correspondence principle fails:



> 
> Paradoxically, the only counterexample we have been able to find to
> the General Correspondence Principle is the paradigm example of the
> relation of quantum mechanics to classical mechanics. Contrary to the
> impression that may be given in some textbooks, it is not possible to
> reduce quantum to classical mechanics... This failure to
> establish a general correspondence between quantum mechanics and
> classical physics (in the limit where
> \(\Delta P\Delta Q \gg h\) should in our
> view be regarded as a shortcoming of quantum mechanics ... in its
> claim to the status of \(L\)-theory, rather than as a breakdown
> of the General Correspondence Principle. (Post 1971, pp.
> 233-234)[13]
> 



Although Post's generalized correspondence principle bears some
similarity to the physicist's conception of Bohr's
correspondence principle, it is quite different from Bohr's own
understanding. Indeed, according to Post, the generalized
correspondence principle does not even hold for the specific theory
pair that was Bohr's subject.



In contrast to Post, Radder (1991) has claimed that the generalized
correspondence principle does apply to the case of the
quantum-classical relation (p. 215), and that Bohr's
correspondence principle, which he interprets largely as a heuristic
tool, can be seen as an instance of the generalized correspondence
principle (p. 208).