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There are lots. +\geometry{letterpaper} % ... or a4paper or a5paper or ... +%\geometry{landscape} % Activate for for rotated page geometry +%\usepackage[parfill]{parskip} % Activate to begin paragraphs with an empty line rather than an indent +\usepackage{graphicx} +\usepackage{amssymb} +\usepackage{epstopdf} +\usepackage[comma,authoryear]{natbib} + +\DeclareGraphicsRule{.tif}{png}{.png}{`convert #1 `dirname #1`/`basename #1 .tif`.png} + +\title{The (?) Correspondence Principle} +\author{Robert Rynasiewicz\footnote{Department of Philosophy, +Johns Hopkins University; +E-mail: ryno@lorentz.phl.jhu.edu}} +\date{} % Activate to display a given date or no date + +\begin{document} +\maketitle + +\begin{abstract} +\noindent +One finds, even in texts by distinguished physicists, +diverse enunciations of the correspondence principle. +Typical is that quantum mechanics should agree with classical mechanics in some appropriate limit. Most commonly, the limit specified is that of high quantum numbers, +or of large masses and orbits of large dimensions. +But sometimes it is specified as mean behavior when large numbers quanta are involved, +or sometimes even as just the average of quantum mechanical variables. +Sometimes, the principle is even taken as a prescription for replacing the classical dynamical observables with an appropriate mathematical operator. +In 1918, however, Bohr proposed what he would later call the correspondence principle +as a way of deriving amplitudes and polarizations of emitted and absorbed spectral lines. +I will begin with Bohr's principle +and trace the evolution of correspondence considerations into the 1920s, +with a view as to whether in each case +it is supposed to play +the role of a theorem, an adequacy constraint, an inductive hypothesis or a heuristic. +Kramers, although a master of Bohr's uses of the principle, +appears to be the first to formulate it as a constraint in the limit. +\end{abstract} + +\noindent +{\bf Keywords:} +Correspondence principle; old quantum theory; atomic theory: dispersion; Bohr; Kramers; Pauli; Van Vleck. + +\section{Introduction} +Most physicists and physics students understand the correspondence principle +as the requirement that +the results of quantum physics go over to those of classical physics in some appropriate limit, +say of high quantum numbers, or of large numbers of quanta, +or as Planck's constant $h$ goes to zero. +This is what one typically finds in textbooks. +One can also find such formulations by participants of the quantum revolution. +For example in his 1933 lectures published under the title {\em Moderne Physik\/}, +Max Born cites Bohr's correspondence principle as the {\em Grundidee\/} +of the {\em provisorische Atommechanik\/} prior to the emergence of matrix mechanics. +According to Born, it demands that +``the new mechanics +goes over into classical mechanics +for the limit of large masses or large orbital dimensions.''\footnote{\citet{Born1933ModernePhysik}, +p.~67. +} +%{\em Moderne Physik\/} was translated into English in 1935 +%by John Dougall with the title {\em Atomic Physics\/}, +%because, as Born tells us in the Preface, +%the publisher already had in its series the title {\em Modern Physics\/}. +%Who was the publisher? +%Born gives Cambridge as the place where the Preface was written. +%Had he moved there or was he just visiting? +%N.B. I have not relied on the Dover reprint 8th edition +%since in the interim the text had undergone revisions +%at the hands of R.~J.~Blin-Stoyle and J.~M.~Radcliff. +%In particular, the date 1923 was inserted for Bohr's correspondence principle. +In 1925, at the very birth of the new quantum mechanics, +in ``The Fundamental Equations of Quantum Mechanics,'' +%\footnote{{\em Proceedings of the Royal Society of London\/}, +%Vol. 109, No. 752, (Dec. 1, 1925), pp. 642-653. +%Reprinted in B.~L.~van der Waerden, {\em Sources of Quantum Mechanics\/}. +%Mineola, NY: Dover, 1968. +%This collection was originally published by North Holland in 1967. +%} +Paul A.~M Dirac characterizes +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''\footnote{\citet{Dirac1925FundamentalEquations}, +p.~642. +} +According to these formulations, +the correspondence principle functions as a {\em constraint\/} on theorizing. + +This leaves us with a kind of puzzle. +Bohr's two fundamental postulates, +the principle of stationary states +and the Bohr frequency condition, +\begin{equation} +E^{\prime} - E^{\prime\prime} = h \nu , \label{eq: BohrFrequencyCondition} +\end{equation} +when applied to atomic systems, +yield only predictions about the frequencies of radiation emitted or absorbed +in transitions between stationary states. +They do not yield, on the one hand, any predictions about the state of polarization +of emitted radiation. +On the other, they do not make any +predictions about the emission and absorption probabilities. +This means they say nothing about which transitions are allowed or forbidden +and nothing about the intensity of the radiation emitted from an ensemble of systems. +The above formulation 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 low frequencies or large quantum numbers). +It cannot milk out of the two fundamental postulates any information +about transition probabilities or polarization. +Yet in the period prior to matrix mechanics +the correspondence principle was widely presumed to do just that. +As a ``rational'' generalization of classical electrodynamics into the quantum realm, +it served to give quantum physics predictive muscle, +something that can accrue to theory only by the annexation of substantive physical hypotheses. +The puzzle then +is that if the correspondence principle was introduced +to extend the quantum theory to make predictions not just about frequencies, +but about polarization and transition probabilities as well, +how did it come to pass that the correspondence principle +is now most commonly taken to constitute a constraint +on theory construction? + +In speaking of ``the'' correspondence principle, +we must be careful about what is to be understood. +In what follows I first develop the original strategy +associated with Bohr's correspondence principle +before attempting to identify a definitive formulation of the principle. +When looking for such a formulation, +we will then find that Bohr was not uniformly consistent in stating the principle. +Nonetheless, formulations can be found +that include the major predictive components of that doctrine +and thus cast the principle as making substantive assertions. +I will then turn to a brief survey of various ``extensions'' of the correspondence principle. +Apart from having interest in their own right, +they will lead us back to the reintroduction of the correspondence principle +within the Copenhagen School\footnote{By the `Copenhagen School' +I mean principally Bohr and Kramers. +} +explicitly as a constraint on quantum theorizing.\footnote{For further historical background, +see \citet{Darrigol1992c2q, Kragh2012Bohr}. +For a more pedagogical approach see \citet{Fedak+Prentis2002}. +} + + + \section{The Core Doctrine: Transition Probabilities and Polarization} +That the correspondence principle in the old quantum theory was intended +to extend Bohr's postulates in such a way as to provide predictions about +transition probabilities and polarizations +is certainly indicated by one of the terser formulations +that can be found in the literature. +According to Back and Land\'e, in their 1925 monograph +on the Zeeman effect and the multiplet structure of spectral lines: +%\footnote{Back, E. +%and Land\'e, A.: +%{\em Zeemaneffekt und Multiplettstruktur der Spektrallinien\/}, +%(Berlin: Julius Springer, 1925).} +\begin{quote} + The correspondence principle asserts that + the rate of occurrence (probability) of a quantum jump, + which is manifested by the intensity of the spectral emission line, + as well as the polarization of the emitted radiation, + is determined by + the intensity and polarization of the ``corresponding'' component oscillations + [{\em Einzelschwingungen\/}] + of the ersatz classical radiation + process.\footnote{\citet{Back+Lande1925Zeemaneffekt}, p.~17. + } +\end{quote} +For this to be helpful, though, +one needs to know +(a)~what are +``the `corresponding' component oscillations of the ersatz classical radiation process,'' +and +(b)~what is the determining relation. + +In order to flesh this out, +consider Bohr's address +``\"Uber die Serienspektra der Elemente'' +given to the {\em Deutsche Physikalische Gesellschaft\/} +in Berlin on April 27, 1920.\footnote{\citet{Bohr1920Ueber_die_Serienspektra}. +} +Although Bohr arguably first developed the correspondence principle +two years earlier in parts one and two of +{\em On the Quantum Theory of Line-Spectra\/}\footnote{\citet{Bohr1918QoL-I,Bohr1918QoL-II,Bohr1918-1922}. +} +(hereafter Q.o.L, following Bohr), +it is on this occasion that he first used the terminology +{\em Korrespondenzprinzip\/}. +He introduced it in the context of treating the hydrogen atom. + +Assume for simplicity the mass of the nucleus to be infinite + in comparison to that of the electron. +Then, as a Kepler problem due to the Coulomb attraction, +the latter revolves classically +with angular frequency $\omega$ around the former in an ellipse with semi-major axis $a$ with +\begin{equation} +\omega = \sqrt{\frac{2W^3}{\pi^2 e^4 m}} ,\; 2a = \frac{e^2}{W} , \label{eq: ellipse} +\end{equation} +where $e$ is the charge and $m$ the mass of the electron +and $W$ is the work required to remove the electron to infinity. +Bohr proceeds to use (what he calls) the Balmer formula +\begin{equation} + \nu = \frac{K}{{n^{\prime\prime}}^2} - \frac{K}{{n^{\prime}}^2} , \label{eq: Balmer} + \end{equation} +where $K$ is the Rydberg constant, +as a quantization condition. +Multiplying both sides by $h$ yields +an instance of the Bohr frequency condition~(\ref{eq: BohrFrequencyCondition}), +suggesting that the energy of the $n$th state is +$$ E_n = - \frac{Kh}{n^2}.$$ +Substituting this for $W$ in~(\ref{eq: ellipse}) yields +\begin{equation} +\omega_n = \frac{1}{n^3}\sqrt{\frac{2 h^3 K^3}{\pi^2 e^4 m}} ,\; 2a_n = \frac{n^2 e^2}{h K} + \label{eq: ellipse_n} +\end{equation} +for the orbital frequency and semi-major axis, respectively, of the $n$th stationary state. +Let $n^{\prime} > n^{\prime\prime}$ and $\tau = n^{\prime} - n^{\prime\prime}$. +Then, holding $\tau$ fixed, +the ratio $\omega_{n^{\prime\prime}} / \omega_{n^{\prime}}$ converges to unity +as $n^{\prime}$ goes to infinity. +The Balmer formula~(\ref{eq: Balmer}) can be rewritten +$$ \nu = \frac{K}{{n^{\prime\prime}}^2} - \frac{K}{{n^{\prime}}^2} += \tau K +\frac{n^{\prime} + n^{\prime\prime}}{(n^{\prime})^2 (n^{\prime\prime})^2} .$$ +When $n^{\prime}$ and $n^{\prime\prime}$ are large in comparison to their difference $\tau$, +we get from this and~(\ref{eq: ellipse_n}) the approximation +\begin{equation} +\nu \sim \tau \omega \sqrt{\frac{2 \pi^2 e^4 m}{K h^3}} , \label{eq: approx} +\end{equation} +where $\omega$ is the orbital frequency of one or the other of the two states.\footnote{Bohr's reasoning here +appears to be that if $n^{\prime}, n^{\prime\prime} >> \tau$, +then $n^{\prime}$ and $n^{\prime\prime}$ can be equated in the term +$$ +\frac{n^{\prime} + n^{\prime\prime}}{(n^{\prime})^2 (n^{\prime\prime})^2} +$$ +and +$$\frac{n + n}{n^2 n^2} = \frac{2}{n^3} . $$} + +Now $e$, $m$, $h$, and $K$ are all independently measurable physical constants. +It so happens that, to within experimental error, +$$ K = \frac{2 \pi^2 e^4 m}{h^3} . $$ +Thus, the root on the r.h.s. of~(\ref{eq: approx}), +as a matter of empirical fact, +is unity, +so that~(\ref{eq: approx}) reduces to +$$\nu \sim \tau \omega .$$ +Following Ehrenfest\footnote{\citet{Ehrenfest1923LePrincipe}. +} +and after him van Vleck,\footnote{\citet{VanVleck1924Extensions}. +} +we call this the {\em frequency theorem\/}. +Although here it follows from the {\em ad hoc\/} use of the Balmer formula +as a quantization condition +and the experimental identity $K = 2\pi e^4 m / h^3$, +it can be derived generally from the usual action-angle quantization conditions +for multiply-periodic systems, as Bohr did in Q.o.L.\footnote{Of course, the derivation +can be inverted to derive the equation for the Rydberg constant +from the requirement that $\nu\sim\tau\omega$ in the region of low frequencies, +as Bohr had done earlier for the special case $\tau = 1$ +in an address given before the Physical Society of Copenhagen in December of 1913. +See \citet{Bohr1922[1914]}, p.~13. +There Bohr says this is based on the {\em assumption\/} +``that it will be possible to calculate the emission of slow electromagnetic oscillations +on the basis of classical electrodynamics.'' +Moreover, ``this assumption is very strongly supported by the results of Lorentz's calculations \ldots .'' +In the Preface to the 1922 translation, +Bohr writes, ``The first germ of [the] correspondence principle may be found \ldots +in the deduction of the expression for the [Rydberg] constant \ldots.'' +See \citet{Bohr1922[1914]}, p.~v. +However, as \citet{Heilbron+Kuhn1969}, p.~251, suggest, +one finds the germ in the first paper of the 1913 trilogy, where Bohr explains, +``From the above considerations it will follow that, +taking the starting-point in the form of the law of the hydrogen spectrum +and assuming that the different lines correspond to a homogenous radiation +emitted during the passing between different stationary states, +we shall arrive at exactly the same expression for the [Rydberg] constant \ldots, +if we only assume (1)~that the radiation is sent out in quanta $h\nu$, +and (2)~that the frequency of the radiation emitted during the passing of the system +between successive stationary states will coincide with the frequency of revolution +of the electron in the region of slow vibrations.'' +See \citet{Bohr1913OnTheConstitution}, p.~14. +The point is that it {\em is\/} an {\em assumption\/}, although a plausible one as Bohr points out, +that classical electrodynamics gives the right results for slow vibrations.} +%Question: Does this mean that these quantization conditions together with the Coulomb %Hamiltonian +%(and the frequency condition) +%entail that $K = 2\pi e^4 m / h^3$? + +The physical significance is this. +For any periodic motion of frequency $\omega$ +(and thus for elliptical motion in particular) +the displacement $x$ in any given direction can be represented +as a function of time by a Fourier series of the form +\begin{equation} + x = \sum X_{\tau} \cos 2 \pi (\tau \omega t + c_{\tau}) , \label{eq: Fourier} +\end{equation} +where the sum is taken over all positive integer values of $\tau$. +Classically, a charged particle undergoing such motion +radiates at each of the overtones $\tau\omega$ of the fundamental frequency~$\omega$ +with an amplitude proportional to the square of $X_\tau$. +This is the {\em ersatz radiation process\/} referred to by Back and Land\'e above. +The {\em component oscillations\/} are the overtones $\tau\omega$. +The spectral line emitted quantum-mechanically +in the transition from the $n^{\prime}$th to the $n^{\prime\prime}$th state +has the same frequency as the $\tau$th overtone of the classical ersatz process +and hence that quantum transition +{\em corresponds to\/} +the $\tau$th component oscillation of the classical ersatz process. +Since I want to distinguish between the various components +of the strategy +associated with the correspondence principle, +let us call this correspondence the {\em correspondence mapping\/}. + +Before moving on, it should be stressed, as did Bohr, +that, although one arrives at this convergence in frequencies, +the {\em mechanisms\/} of radiation remain utterly distinct +even in the limit of slow oscillations. +Classically, a single atom radiates at all the overtone frequencies, +while quantum theoretically an atom will emit radiation +at a single frequency equal to the $\tau$th overtone +corresponding to the transition from state $n^{\prime}$ +to state $n^{\prime\prime}$. +An ensemble of atoms is required in the initial state $n^{\prime}$ +in order to replicate the spectrum of the classical radiation process. + + + +%QUESTION. The correspondence is established +%in the regime of $n^{\prime},n^{\prime\prime}>>\tau$ +%in virtue of the relation $\vu \sim \tau\omega$. +%But the relation fails outside this regime. +%So, how is the correspondence extrapolated to the general case? +%Note that Bohr wants to treat the (green) $H_{\beta}$ line +%as the ``octave'' (``in a certain sense'')of the (red) $H_{\alpha}$ line. +%The quantum numbers in question are $n^\prime = 4$, $n^{\prime\prime}=2$ +%in the first case and +%$n^{\prime}=3$, $n^{\prime\prime}= 2$ in the second. +%This suggests we first fix the Fourier expansion for the final state $n^{\prime\prime}= 2$, and +%then apply the numerology. +%However, in QoL, Bohr takes $n = 3$ to be the ground frequency. See p.~40. +%It appears that he IS taking the Fourier expansion for $n=2$, since then the ground frequency +%is given for $\tau=1$ which corresponds to the $3\rightarrow 2$ transition. +%Note that Sommerfeld (in the 4th ed. of Atombau?) first establishes {\em correspondence\/} +% and then shows that in the limit you get {\em coincidence\/}. + +%QUESTION. Here the correspondence is established +%for a particular model of the hydrogen atom. +%Yet Bohr immediately applies the selection rule derived from it to the harmonic oscillator. +%What legitimates this move? +%(Presumably, in QoL, the correspondence is established for generally periodic systems, +%so is this just a pedagogical simplification for the Vortrag?) +%Answer: this is a dumb question. The frequency theorem for the harmonic oscillator +%is trivial, viz., +%$\nu_{n^{\prime}\rightarrow n^{\prime\prime}} = +%(n^{\prime}\rightarrow n^{\prime\prime}) \omega . + +We have now given an answer to (a) above, viz., +the question, what are +``the `corresponding' component oscillations of the ersatz classical radiation process.'' +It remains to address~(b), viz., +how is the transition probability of a quantum jump and the polarization of the emitted radiation +determined by +the intensity and polarization of the radiation classically emitted +by the corresponding component of the classical radiation process. + +The frequency theorem leads immediately to an expectation concerning intensities. +As Bohr put it in his 1920 Berlin address: +\begin{quote} + 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 {\em intensities\/}; + an expectation that is equivalent to the proposition + that the relative {\em 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.\footnote{\citet{Bohr1920Ueber_die_Serienspektra}, p.~431. + Emphasis in original. + See also \citet{Bohr1922OnTheSeries}, p.~27. + } +\end{quote} +Assuming $n^{\prime}, n^{\prime\prime} >> \tau$, +the easily stated connection is that of identity +between probability amplitudes and amplitudes of oscillation. +In other words, for one dimension the connection is that +the probability of a transition from state $n^{\prime}$ to state $n^{\prime\prime}$ +is proportional to $X_{\tau}^2$. +For the three dimensional case, +we need to consider the Fourier expansions in the $y$ and $z$ directions +with amplitudes $Y_{\tau}$ and $Z_{\tau}$, respectively. +Then the probability of the transition is proportional to +$X_{\tau}^2 + Y_{\tau}^2 + Z_{\tau}^2$. +Call this the {\em slow-vibration hypothesis for transition-probabilities\/}. + +Given $Y_\tau$ and $Z_\tau$ in addition to $X_\tau$, +the state of polarization of the classical radiation component of frequency $\tau\omega$ +is fully determined. +Bohr made the additional assumption that, in the same regime, +the state of polarization of the radiation emitted +in the quantum transition from state $n^{\prime}$ to state $n^{\prime\prime}$ is the same +as for the corresponding classical radiation. +Call this {\em the slow-vibration hypothesis for polarization\/}. + +The slow-vibration hypotheses seem like eminently rational extensions of the frequency theorem. +There is, of course, no question of finding proofs of them, as one has for the frequency theorem. +For Bohr's theory, as mentioned earlier, simply makes no predictions about intensities and polarizations +of emitted light in the vast regime +in which $n^{\prime}$ and $n^{\prime\prime}$ are not both large +with respect to their difference~$\tau$. +So there is no place from which to start. +Rather, the entire point of the slow-vibration hypothesis is to provide a point of entry +to fill that lacuna. +Bohr's proposal is to start with the slow-vibration hypothesis +and then attempt to extrapolate downward +into the regime in which the frequency theorem fails, +i.e., to take the amplitudes of the Fourier components +as indicative in some way +of the probabilities of the corresponding quantum transitions +and the polarization of the light thereby emitted, +even if the the quantum numbers lie well below the slow-vibration range. +Call this the {\em downward extrapolation project\/}. +How one should extrapolate downward +is not straightforward. +As Sommerfeld put it in a way that might be construed as criticism: +\begin{quote} + This determination of the intensity and polarisation is not, + however, + fully unambiguous, + and this fact in itself characterises it as a process of approximation. + In calculating the [Fourier expansion] + are we to use as our basis the conditions of the initial orbit + or those of the final orbit, + or, perhaps, an intermediate orbit that is to be defined by taking the mean of both? + No answer is vouchsafed to this by the principle of correspondence. + It is easy to see in a general way, indeed, + that with the asymptotic condition [$n^{\prime}, n^{\prime\prime} >> \tau$], + the coefficients [$X_{\tau}$] that are obtained from the initial or final orbit, + or from an intermediate orbit must come out appreciably equal. + In the case of values of [$\tau$] that are comparable + with [$n^{\prime}$ or $n^{\prime\prime}$], + however, + the [$X_{\tau}$'s] in general become different for the initial and the final orbit + and hence a certain arbitrariness remains in applying + the principle of correspondence.\footnote{\citet{Sommerfeld1923AtomicStructure}, + pp.~581--582. + } +\end{quote} +Nonetheless, various averaging schemes were proposed and calculations performed, +most notably by H.~A.~Kramers in his 1919 doctoral dissertation +under Bohr\footnote{\citet{Kramers1919Intensities}. +} +and later by the American physicist +Frank Hoyt\footnote{\citet{Hoyt1923RelativeIntensity,Hoyt1924Relative}. +} +while visiting Copenhagen +as a National Research Fellow.\footnote{Van Vleck +also proposed a general scheme in \citet{VanVleck1924Extensions}, p.~334. +Hoyt was still working along these lines as late as October of 1925. +See \citet{Hoyt1925Application}. +} +Despite a measure of empirical success, +the problem of non-uniqueness remained a thorn. +As Max Born explained in his Winter 1925--1926 MIT lectures: +\begin{quote} + By this method Kramers has succeeded in representing satisfactorily + the results of observations in certain cases. + It is not satisfactory in principle that we should not find in quantum theory, + in the form here presented, a unique determination of the intensities. + This is one of the main reasons which led us to formulate our new quantum theory, + where the difficulty is overcome.\footnote{\citet{Born1926Problems}, + pp.~30--31. + } +\end{quote} + +There were, however, distinguished cases, +in which the non-uniqueness problem could be overcome +and the correspondence principle gain credibility by making correct predictions. +These cases are ones in which the Ritz combination principle fails: +certain transitions simply do not occur. +The correspondence principle identifies the reason for the ``forbidden'' transition +with the absence of the corresponding Fourier components. +Suppose, now, in the regime where $n^{\prime}, n^{\prime\prime} >> \tau$, +the amplitudes $X_\tau, Y_\tau, Z_\tau$ vanish for some particular value of $\tau$. +Then, according to the slow-vibration transition-probability hypothesis, +the intensity of the classical radiation emitted is identically zero, +and hence the probability of the transition from state $n^{\prime}$ to state $n^{\prime\prime}$ +is identically zero, and so the transition simply cannot occur. +This result can be unambiguously extrapolated downward if the amplitudes of the $\tau$-th component +are zero +not only for the initial state, but also for the final state and all intermediate states. +As Sommerfeld explained, +\begin{quote} + In this case we shall also have no scruples in inferring the value zero of the radiation. + The principle of correspondence then becomes specialised and condensed + into a {\em principle of selection\/}; + it forbids the occurrence of such spectral lines + the corresponding partial vibrations of which + do not occur in the series expansion.\footnote{\citet{Sommerfeld1923AtomicStructure}, + p.~82. + } +\end{quote} +The simplest example is that of a Planckian oscillator. +Since for each state no terms occur in the Fourier expansion other than the term for $\tau = 1$, +the selection rule $\Delta \tau = 1$ for emission and $\Delta \tau = -1$ for absorption +follows immediately, +and the frequency $\nu$ of the emitted radiation is just the mechanical frequency $\omega$. +In Q.o.L Bohr appealed to the absorption spectra of diatomic gases +as an empirical realization of this case. +The appearance in some such gases of a faint line at double the fundamental frequency, +he suggested, is explained +by assuming that the vibrations along the axis connecting the two atoms is not +strictly harmonic.\footnote{See \citet{Bohr1918QoL-I} or \citet{Bohr1918-1922}, p.~16. +} + +More sophisticated cases involved the spectra of hydrogen +subject to a constant, uniform external electromagnetic field. +These systems are not simply periodic but conditionally (or, multiply) periodic. +The Fourier expansions involve multiple fundamental frequencies, +in this case two, call them $\omega$ and $\sigma$:\footnote{Think of $\omega$ +as governing the original +elliptical motion and $\sigma$ the rate of precession of the perturbed orbit about the axis of symmetry of the field. +} +\begin{equation} +x = \sum X_{\tau,\kappa} \cos 2 \pi [(\tau\omega + \kappa\sigma)t + c_{\tau,\kappa}] , +\label{eq: multiperiodic} +\end{equation} +and similarly for $y$ and $z$, +where $\tau$ and $\kappa$ are summed over all integral values, positive and negative. +For the case of a magnetic field $H$ (Zeeman effect), +the effect is the superposition of a uniform rotation of frequency +$\sigma = e H / 4 \pi m c$ +on the original unperturbed orbit +with $\kappa$ restricted to $-1 \leq \kappa \leq 1$. +The selection rule for $\kappa$ is then $\Delta \kappa = -1,0,1$ +corresponding to the splitting of the unperturbed line into the classical triplet. +Moreover, the correct polarizations are predicted. +For $\kappa = 0$, the oscillations are rectilinear +parallel to the magnetic field with frequencies $\tau\omega$, +while for $\kappa = \pm 1$, the oscillations are in a plane perpendicular to the field +with frequencies $\tau\omega \pm \sigma$. + +For the case of a uniform electric field $F$ (Stark effect), +$\sigma = 3 e F / 8 \pi^2 m a \omega$ +and $\kappa$ is less restricted, +corresponding to the proliferation of new spectral lines. +Again, the correct polarizations are given. +When $\tau + \kappa$ is even, +Bohr argued, +the oscillations are linear parallel to the direction of the field, +and when odd, elliptical perpendicular to the field.\footnote{See +\citet{Bohr1920Ueber_die_Serienspektra} or \citet{Bohr1922OnTheSeries} for details. +} + +\section{The Scope of the Principle} +Thus, the correspondence principle, +or should we say, the strategy associated with it, +was not without its successes. +%\footnote{Bohr and Kramers claimed that +%the correspondence principle could also serve as a guide to atomic structure +%in the sense of building up shell structure +%in the construction of the periodic table. +%How this is supposed to work is less than clear. +%See +%\citet{Bohr1921AtomicStructure1,Bohr1921AtomicStructure2,Bohr1923Linienspektren,Kramers1923DasKorrespondenzprinzip}. +%``Atomic Structure,'' {\em Nature\/} vol.~107 no.~2682, March 24 (1921): 104--107; +%Bohr, ``Atomic Structure,'' {\em Nature\/} vol.~108 no.~2711, Oct.~13 (1921): 208--209; +%Kramers, +%``Das Korrespondenzprinzip und der Schalenbau des Atoms,'' {\em Naturwissenschaften\/} +%{\bf 27} (1923): 550--559. +%} +That strategy, we found, has a number of distinctly identifiable components: +\begin{itemize} + \item the frequency theorem + \item the correspondence mapping + \item the slow-vibration transition-probability hypothesis + \item the slow-vibration polarization hypothesis + \item the downward extrapolation project + \item the selection principle(s) +\end{itemize} +Which among these, +or rather, which subset of these, +constituted the correspondence principle? + +Keep in mind that I judiciously selected the Back and Land\'e quote +so as to motivate coverage of all these components of the doctrine. +But what right do we have to take Back and Land\'e to be any more authoritative +than Born or Dirac? +Can we not find a concise and definitive formulation in Bohr's writings? +Unfortunately, Bohr was not always clear nor necessarily consistent. +There are passages in which +the correspondence principle appears to be no more than a selection principle. +For example, +\begin{quote} + \ldots the possibility of the occurrence of a transition accompanied by radiation, + between two states \ldots is conditioned by the presence of certain harmonic components + in the expression [for] the electric moment of the atom. + \ldots We, therefore, call these the ``corresponding'' harmonic components in the motion, + and the substance of the above statement we designate + as the ``Correspondence Principle'' + for multiply periodic system.\footnote{\citet{Bohr1924Application}, p.~22. + BCW3 1976, p.~479. + } +\end{quote} +Such passages have led Alisa Bokulich +to insist that Bohr intended it to be no more than a selection principle.\footnote{See \citet{Bokulich2008Reexamining}, pp.~81--94 and \citet{Bokulich2010Bohr}. +} +But a selection principle is just a special case of zero probability, +and elsewhere Bohr connected the correspondence principle +explicitly with both probabilities and polarizations. +From the 7th Guthrie lecture, given 24~March 1922, +we find: +\begin{quote} + 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 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 polarisation 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.\footnote{\citet{Bohr1922-3Guthrie}, + p.~284. + BCW3 1976, p.~428. + } +\end{quote} +Moreover, Kramers' contribution +to the 1923 {\em Die Naturwissenschaften} Festschrift +for Bohr +claimed to speak on Bohr's behalf as follows: +\begin{quote} + The occurrence of a transition between two stationary states + accompanied by the emission of radiation + is uniquely correlated with one of the harmonic oscillations + into which the motion of the particle + (or more precisely: the electric moment of the atom) + can be decomposed. + This correlation requires + that + the probability for the occurrence of a transition depends on + the amplitude of the corresponding harmonic component, + and in fact so that + in the region of large quantum numbers + the intensity per unit time of the emitted radiation will be the same on average + as would be expected according to classical electrodynamics. + The polarization of the emitted radiation will display + a similar analogy with classical electrodynamics. + Thus, + if the corresponding harmonic oscillator in all states is a linear oscillation parallel + or a circular rotation perpendicular to a fixed line, + then the radiation will possess the same properties as that + which is emitted by an electron which executes an oscillation + of this type.\footnote{\citet{Kramers1923DasKorrespondenzprinzip}, + p.~552. + } +\end{quote} +Kramers added to this a remarkable footnote: +\begin{quote} + It seems + most appropriate to construe the correspondence principle thus formulated + as a postulate (or axiom), + in complete analogy with Bohr's two fundamental postulates. +\end{quote} +%This, however, does not jibe well with our earlier quote from the same article +%in which Kramers says the principle is a {\em heuristic\/}. +%Is it really possible to be both a fundamental postulate and a heuristic? +It is pretty clear that the correspondence principle +thus formulated was more than just a selection principle. +And construed as a postulate that yields substantive predictions beyond what is entailed +by Bohr's first two postulates, +it does not appear to have had the form of a constraint +on theory construction.\footnote{According to Bohr and Kramers, +the correspondence principle also did work in perturbation theory +and in the construction of the shell structure for the periodic chart. +For the former, see +\citet{Bohr1918QoL-II} and \citet{Bohr1920Ueber_die_Serienspektra}. +For the latter, see +\citet{Bohr1921AtomicStructure1}, \citet{Bohr1921AtomicStructure2}, and +\citet{Kramers1923DasKorrespondenzprinzip}. +} + +\section{Some ``Extensions'' of the Correspondence Principle} + +The correspondence principle thus understood +provided a guide to the determination of transition probabilities +for spontaneous emission and the polarization of the emitted radiation. +But what about transitions in the opposite direction +that do not involve radiation, +such as inelastic collisions in the bombardment of atoms with electrons, +as in the Franck-Hertz experiment?\footnote{Also, +what about electron capture under bombardment +with the the emission of X-rays? +For Kramers' attempt to calculate transition probabilities in this case, +see \citet{Kramers1923XRay}. +} +How could one estimate the transition probabilities? + +For such processes, Pauli introduced, +in his doctoral dissertation under Sommerfeld,\footnote{\citet{Pauli1922Ueber}. +} +what he called a ``mechanical'' correspondence principle, +which he attempted to craft in analogy with Bohr's electrodynamical principle. +He started with the observation that, +just as classical electrodynamics breaks down, +so should classical mechanics applied to inelastic collision processes. +For classically, the amount of energy $\Delta E$ +that an electron can give up to its target system varies continuously, +while quantum mechanically this is impossible, +since the target system can exist only in a discrete manifold of stationary states +and consequently $\Delta E$ can take on only discrete values. +In the limit of large quantum numbers, +however, +the energy states converge toward a continuum, +and thus one should expect the laws of classical mechanics to hold in the limit. +This can be taken to be the analogue of the frequency theorem, +but it is something assumed, not proven. +Nonetheless, it is like the frequency theorem in that +the agreement holds only for ensembles, +and not for individual transitions. +For the analogue of transition probabilities for high quantum numbers +Pauli proposed that one take the averages of the classical outcomes +when one fixes the initial energy and varies the other initial conditions +(direction of the initial velocity, +distance of the trajectory asymptote from the centre of gravity of the atom, +phase of the motion of the electron and of the atom bombarded). +These are to be regarded as uncontrollable. +For smaller quantum numbers, one ``extrapolates'' downward: +classical mechanics +``{\em must give an (at least approximate) measure for the relative frequency +of the various possible quantum energetic inelastic collisions +(with a given initial energy +for the colliding electron)\/}.''\footnote{\citet{Pauli1922Ueber}, p.~187. +Emphasis in original. +} +If a particular collision process is classically impossible, +then one has a {\em selection rule\/} forbidding that transition. + +Pauli realized that given the current state of the art this yields few if any predictions. +Nonetheless he held out hope that eventually +some of the calculational difficulties would be overcome. +At least one gets some definite consequences by considering that +the law of reversibility must continue to hold. +Thus, for every quantum mechanical process, the time-inverse process must also be possible. +This leads to so-called ``collisions of the second kind'' postulated by +Klein and Rosseland in Copenhagen +on the basis of considerations concerning thermal equilibrium +between atoms and free electrons.\footnote{\citet{Klein+Rosseland1921}. +} +For example, an electron should be able to experience an increase in velocity +when striking an atom in an excited state +which decays into a lower energy state. + +The existence of such inelastic collisions was confirmed\footnote{\citet{Cario+Franck1923}. +} +but P.~Tartakowsky subsequently took Pauli to task +for the manner of his formulation +of the ``mechanical'' correspondence principle.\footnote{\citet{Tartakowsky1924}. +} +According to Tartakowsky, +Pauli's point of departure was the requirement +that the results of quantum theory and classical mechanics +must coincide in the case of large quantum numbers. +That, however, was not the complaint. +It was rather that Pauli did not establish a {\em correspondence\/}, +in Bohr's sense of the word, +which, for Bohr, was specifically between +the components of the Fourier expansion of the motion in the stationary state +and the radiation emitted in quantum jumps. +In other words, Pauli had not established an analogue +of what I earlier called the {\em correspondence mapping\/}. +So Tartakowsky took it upon himself +to establish an analogous correspondence relation, in this case, +between the Fourier expansion for the motion and the energy +given up by one system to another +in the absence of emission. +Again, the self-described +point of departure was the requirement that +the results of quantum theory and classical mechanics +agree in the realm of high quantum numbers. +Tartakowsky then took up the problem, +not of electron scattering, +but of a radiation-free energy exchange between two quantum systems +$A$ and $B$ +that does not result in the production of additional kinetic energy, +in other words for which we have the energy exchange relation +$$E_A - {E^{\prime}}_A = - (E_B - {E^{\prime}}_B) ,$$ +where the primed quantities refer to final states. +He made the connection with the Fourier expansions by introducing +what he called a hidden coupling mechanism [{\em verborgener Koppelungsmechanismus\/}], +the details of which, however interesting, we do not have time to go into. +The important point here is that, +although Tartakowsky began with the constraint +that the results of quantum theory go over to those of classical mechanics +in the limit of large quantum numbers, +he did not take this to be the principle itself, +but instead only a preliminary to the formulation of a ``mechanical'' correspondence theorem. + +Contrast this with the tenor of the first part of Van Vleck's two-part 1924 paper, +which bears the subtitle +``Some Extensions of the Correspondence Principle.''\footnote{\citet{VanVleck1924Extensions}. +} +There Van Vleck made it clear that, in the case of emission, +whereas the asymptotic connection between quantum and classical physics for frequencies +is a mathematical consequence of the quantum conditions,\footnote{Including, of course, the Bohr frequency condition. +} +the existence of an analogous relation for intensities is a {\em hypothesis\/}. +This is what I referred to above as the transition-probability hypothesis for slow vibrations. +Van Vleck called it the correspondence principle for emission +and formulated it ``analytically'' +by explicitly calculating +the Einstein spontaneous emission probability ($A$-coefficient) in the classical limit +as a function of the amplitudes and frequencies of the Fourier expression +for the motion of the particle. +From this he derived a correspondence principle for absorption, +viz., that in the limit of high quantum numbers +the energy extracted from a non-zero radiation field +as determined by the Einstein $B$-coefficients for absorption and induced emission +agrees with the classical prediction for absorption. +Also demonstrated in the paper was a second extension of the correspondence principle, +this time +to a correspondence principle for orbital distortions: +in the limit of large quantum numbers +the orbital distortion resulting from a transition between nearby stationary states +agrees with that which would follow as a result of classical radiation +involving only the frequency component associated with the transition. + +It is evident from this that in extending ``the'' correspondence principle, +Van Vleck focused on the behavior of quantum systems in the limit of large quantum numbers. +He did consider briefly the downward extrapolation project, +but not under the rubric of a correspondence principle. +He did not discuss the topic of selection rules.\footnote{He did use the selection rule +for the harmonic oscillator as a consequence of ``the correspondence principle'' +but did not pause to consider which, +among those in his catalogue of correspondence principles, it comes from.} +And there was no overt construction of a correspondence mapping. +This is not to say these were beyond his ken. +For he spelled these out explicitly in his 1926 monograph-length report +for the National Research Council.\footnote{\citet{VanVleck1926Bulletin}. +There +he continued with the strategy of cataloguing correspondence principles. +The reader should not find it remarkable if some of these map +onto components of the doctrine associated with the correspondence principle +outlined above.} +Rather, they did not serve the development towards the principal result, +which was to show that Kramers' generalization of Ladenburg's quantum dispersion formula +goes over into the classical dispersion formula in the limit of large quantum numbers +``not just when the quantized system is a linear oscillator, +but also when it is the most general type +of non-degenerate multiply periodic orbit.''\footnote{\citet{VanVleck1924Extensions}, +p.~345. +} +In the 1926 report, +he called this {\em the correspondence principle for dispersion\/}. + +\section{The Correspondence Principle as Constraint} +That the quantum dispersion formula goes over to the classical formula in the limit +was claimed by Kramers in the letter to {\em Nature\/}, written in March of 1924, +in which he first introduced that formula. +\begin{quote} + It is, however, possible to establish a very simple expression \ldots, + which fulfills the condition, claimed by the correspondence principle, + that, in the region where successive stationary states of an atom + differ only comparatively little from each other, + the interaction between the atom and the field of radiation + tends to coincide with the interaction to be expected on the classical theory + of electrons.\footnote{\citet{Kramers1924Law}, p.~673. + } +\end{quote} +But note that here we do not have the framing of the proposition +that the quantum formula goes over to the classical one +in the region of high quantum numbers +as an {\em extension\/} of the correspondence principle. +Rather, +that it do so, and presumably any other law about the interaction of radiation and matter, +is taken to be a {\em requirement of the original correspondence principle\/}. +Here we have the implementation from within the Copenhagen School +of a new significance for ``the'' correspondence principle. +Whereas just a year earlier Kramers had characterized the correspondence principle +as something that could be viewed as a fundamental postulate +alongside the frequency condition and the postulate of stationary states, +it is now taken explicitly to function in a different capacity, viz., +as a constraint on further theory construction. +To my knowledge, this is the first appearance in the literature +of ``the'' correspondence principle as explicitly imposing a constraint. + +Its use as such caught on immediately, +especially in the literature on dispersion. +The exchange of letters in {\em Nature\/} between Gregory Breit and Kramers +that August was premised on the use of the correspondence principle +in this capacity.\footnote{\citet{Breit1924Quantum, Kramers1924Quantum}. +} +Born, in his ``\"Uber Quantenmechanik'' of 1924, +commented concerning his quantum formula for perturbation energy +that the first of its two most important properties is that +``it goes over for large $n_k$ (large in relation to $\tau_k$) +into the corresponding classical formula, +thereby satisfying the correspondence principle.''\footnote{\citet{Born1924QuantenMechanik}, +p.~389 and + \citet{vanderWaerden1968Sources}, p.~192. + } +And again, +``the [Kramers dispersion] formula in the limit of large quantum numbers +($n_k$ large compared to $\tau_k$) +goes over into the corresponding classical formula, +as required by the correspondence principle.''\footnote{\citet{Born1924QuantenMechanik}, +p.~390 and \citet{vanderWaerden1968Sources}, p.~193. +} +Not surprisingly, in their joint paper on dispersion of 1925 Kramers and Heisenberg wrote, +``In particular the principle requires that +in the region of high quantum numbers +the actual properties of the atom can be described asymptotically +with the help of the classical laws of electrodynamics.''\footnote{\citet{Kramers+Heisenberg1925}, +p.~684 and \citet{vanderWaerden1968Sources}, p.~226. +} +Even Bohr, in a paper dated November 1, 1924 +on a different topic +--- the polarization of fluorescent light --- +followed Kramer's lead. +Despite the absence in degenerate systems +of a direct connection between electron motion and polarization, +which is to be taken as analogous to the lack of a direct relation +between mechanical frequencies and spectral frequencies, +``the asymptotic agreement, +demanded by the correspondence principle, +of the consequences of the classical theory and the quantum theory +in the limit, +where neighboring stationary states deviate relatively little from one another, +is kept.''\footnote{\citet{Bohr1924ZurPolarisation}, p.~1117. +} + + +\section{Conclusion} +I do not mean to suggest that following Kramers' letter to {\em Nature\/}, +which appeared in May 1924, +all subsequent uses of the term ``correspondence principle'' +suddenly fell in line with this apparently new usage. +One has only to consider the quote from Back and Land\'e's 1925 monograph above. +Nor does there appear to be any hint of a sensed tension +between the use of the principle as constraint vs.~postulate. +Earlier in the letter in which Kramers first invoked the principle as constraint, +he cited it in its traditional capacity: +\begin{quote} + On Bohr's principle of correspondence, + the possibility for such transitions is considered as being directly connected + with the periodicity properties of the motion of the atom, + in such a way that + every possible transition between two stationary states is conjugated + with a certain harmonic oscillating component + in the motion.\footnote{\citet{Kramers1924Law}, p.~673 + and \citet{vanderWaerden1968Sources}, p.~178. + } +\end{quote} +And W.~Kuhn wrote in early 1925 concerning his sum rule: +\begin{quote} + In the region of high quantum numbers, + the above theorem on the $p$-summation + can be directly understood + as a requirement of the correspondence principle. + In the region of low quantum numbers, + in case the initial assumptions made about dispersion should prove correct, + our rule represents + a strengthening [Versch\"arfung]\footnote{\citet{Heisenberg1925Anwendung} + also proposed a {\em Versch\"arfung\/} + of the correspondence principle. + } + of the correspondence principle \ldots .\footnote{\citet{Kuhn1925Ueber}, + p.~410 and + \citet{vanderWaerden1968Sources}, p.~255. + } +\end{quote} +Of course, it takes only a minor adjustment in viewpoint in order to regard, e.g., +the slow-vibration transition-probability hypothesis +not as ``something we are led to expect'' as a matter of inductive generalization +from the frequency theorem, +but rather as a constraint to be imposed on any theoretical attempt +to calculate transition probabilities; +or in order to see the frequency theorem as proof +that the Bohr frequency condition satisfies the constraint on predicted frequencies. +Thus, the ``claim'' of the correspondence principle +when the theory is extended to other aspects of the interaction between radiation and matter. + +It is important, though, not to lose sight of the downward extrapolation project, +applied across the board, +and thus the introduction of empirical content beyond mere constraint, +as part of what ``the'' correspondence principle involved. +We have seen this in averaging schemes, selection principles, and polarization predictions. +There was also a tradition dating back to +\citet{Sommerfeld1921Atombau} +of exploiting a classical-quantum correlation in the guise of replacing +differential quotients with quotients of differences, +a procedure referred to by Born and Jordan as a correspondence transformation +[{\em korrespondenzm\"a{\ss}ige Umformung\/}].\footnote{\citet{Born+Jordan1925ZurQuantenmechanik}, p.~870 and +\citet{vanderWaerden1968Sources}, p.~290. +\citet{Duncan+Janssen2007} +have emphasized this as part of the lore +pertaining to the correspondence principle. +} +Born and Jordan also referred to Heisenberg's derivation of the law of multiplication +in the seminal {\em Umdeutung\/} paper\footnote{\citet{Heisenberg1925Umdeutung}. +} +as the application of +an ingenious correspondence consideration +[{\em geistreiche Korrespondenzbetrachtung\/}].\footnote{\citet{Born+Jordan1925ZurQuantenmechanik}, p.~859 and +\citet{vanderWaerden1968Sources}, p.~278. +} +Indeed, matrix mechanics can be claimed as the ultimate fulfillment of the downward project. +As Born, Heisenberg and Jordan put it in their famous {\em Dreim\"annerarbeit\/}: +\begin{quote} + 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 + [{\em Korrespondenzgedanken\/}].\footnote{\citet{Born+Heisenberg+Jordan1926}, + p.~558 and + \citet{vanderWaerden1968Sources}, p.~322. + } +\end{quote} +Without the need for any principle {\em outside\/} the theory, +i.e., any downward extrapolations, +what was left of ``the'' principle is the shell of the constraint.\\ +\ \\ +\noindent {\em Acknowledgements\/}: This paper was written as part +of a joint project in the history of quantum physics of the +{\em Max-Planck-Institut f\"ur Wissenschaftsgeschichte\/} +and the Fritz-Haber-Institut in Berlin. +I would like to thank Clayton Gearhart, John Heilbron, and Marius Stan +for useful comments on an earlier draft, +and Michel Janssen for much useful advice. +This research was supported in part by the +National Science Foundation under NSF Grant No.~SES-1027018. + +\renewcommand{\refname}{Bibliography} +\bibliographystyle{authordate1} +\bibliography{TheCorrespondencePrinciple} + +\vfill\eject + +\noindent Short Biography:\\ +\ \\ +\noindent {\bf Robert Rynasiewicz} is professor of philosophy +at the Johns Hopkins University, Baltimore, Maryland, USA. +He has held previous positions at Case Western Reserve University, +Cornell University, and the University of Pittsburgh. +Although he teaches advanced courses in logic and foundations of mathematics, +his main area of research is in the history and philosophy of physics +beginning with Newton. + +\end{document} \ No newline at end of file diff --git a/data/Wolfgang Pauli - Korrespondenz7.docx 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