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Correspondence Principle THE (?) CORRESPONDENCE PRINCIPLE ROBERT RYNASIEWICZ 1. 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 Moderne Physik, Max Born cites Bohr's cor- respondence principle as the Grundidee of the provisorische Atommechanik prior to the emergence of matrix mechanics. According to Born, it demands that \die neue Mechanik f¨urden Grenzfall großer Massen oder großer Bahndimensionen in die klassische Mechanik ¨ubergeht." [20, p. 67] In 1925, at the very birth of the new quantum mechanics, in \The Fundamental Equations of Quantum Mechanics," P. A. M Dirac characterizes the corre- spondence 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."[25, p. 642] According to these formulations, the correspondence principle functions as a 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, (1) E0 − E00 = hν; 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 slow 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 1 2 ROBERT RYNASIEWICZ 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? This also means that, in speaking of \the" correspondence principle, we must be careful about what is to be understood. In what follows I first develop the original doctrine, or strategy associated with Bohr's correspondence principle before attempting to identify a definitive formulation of the principle. In then looking for such a formulation, we'll 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'll then turn to a brief survey of various \extensions" of the correspondence principle. Apart from interest in their own right, they will lead us back to the reintroduction of the correspondence principle within the Copenhagen school explicitly as a constraint on quantum theorizing. 2. 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: The correspondence principle asserts that the rate of occurrence (probabil- ity) 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 deter- mined by the intensity and polarization of the \corresponding" component oscillations [Einzelschwingungen] of the ersatz classical radiation process. ([1], p. 17) For this to be helpful, though, one needs to know (a) what are \the `corresponding' compo- nent 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 Deutsche Physikalische Gesellschaft in Berlin on April 27, 1920 [5]. Although Bohr is supposed to have first developed the correspondence principle two years earlier in parts one and two of On the Quantum Theory of Line-Spectra [2, 3, 4] (hereafter Q.o.L, following Bohr), it is on this occasion that he first used the terminology Korresponden- zprinzip. He introduces 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 ! around the former in an ellipse with semi-major axis a with r 2W 3 e2 (2) ! = ; 2a = ; π2e4m W THE (?) CORRESPONDENCE PRINCIPLE 3 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 K K (3) ν = − ; n002 n02 where K is the Rydberg constant, as a quantization condition. Multiplying both sides by h yields an instance of the Bohr frequency condition (1), suggesting that the energy of the nth state is Kh E = − : n n2 Substituting this for W in (2) yields r 1 2h3K3 n2e2 (4) ! = ; 2a = n n3 π2e4m n hK for the orbital frequency and semi-major axis, respectively, of the nth stationary state. 0 00 0 00 Now assume n > n and τ = n − n . Then, holding τ fixed, the ratio !n00 =!n0 converges to unity as n0 goes to infinity. The Balmer formula (3) can be rewritten K K n0 + n00 ν = − = τK : n002 n02 (n0)2(n00)2 When n0 and n00 are large in comparison to their difference τ, we get from this and (4) the approximation r 2π2e4m (5) ν ∼ τ! ; Kh3 where ! is the orbital frequency of one or the other of the two states.1 Now e, m, h, and K are all independently measurable physical constants. It so happens that, to within experimental error, 2π2e4m K = : h3 Thus, the radical on the r.h.s. of (5), as a matter of empirical fact, is unity, so that (5) reduces to ν ∼ τ!: Following Ehrenfest [27] and after him van Vleck [47], call this the frequency theorem. Although here it follows from the ad hoc use of the Balmer formula as a quantization condition and the experimental identity K = 2πe4m=h3, it can be derived generally from 1Bohr's reasoning here appears to be that if n0; n00 >> τ, then n0 and n00 can be equated in the term n0 + n00 (n0)2(n00)2 and n + n 2 = : n2n2 n3 4 ROBERT RYNASIEWICZ the usual action-angle quantization conditions for multiply-periodic systems, as Bohr did in Q.o.L. The physical significance is this. For any periodic motion of frequency ! (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 X (6) x = Xτ cos 2π(τ!t + cτ ); where the sum is taken over all positive integer values of τ. Classically, a charged par- ticle undergoing such motion radiates at each of the overtones τ! of the fundamental frequency ! with an amplitude proportional to the square of Xτ . This is the ersatz radi- ation process referred to by Back and Land´eabove. The component oscillations are the overtones τ!. The spectral line emitted quantum-mechanically in the transition from the n0th to the n00th state has the same frequency as the τth overtone of the classical ersatz process and hence that quantum transition corresponds to the τth component oscillation of the classical ersatz process. Since I want to distinguish between the various components of the doctrine associated with the correspondence principle, let us call this correspondence the correspondence mapping. Before moving on, it should be stressed, as did Bohr, that, although one arrives at this convergence in frequencies, the 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 τth overtone corresponding to the transition from state n0 to state n00. An ensemble of atoms is required in the initial state n0 in order to replicate the spectrum of the classical radiation process. We have now given an answer to (a) above, viz., the question, what are \the `corre- sponding' 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 polariza- tion of the emitted radiation determined by the intensity and polarization of the radiation classically emitted by the corresponding component of the classical radiation process.
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