## Introduction

Analogical thinking usually works with a touch of blindness: formal relations of a given theory are tentatively applied to new objects, and if the operation is empirically successful, the concepts originally underlying these relations are assumed to extend to these objects. The eventual need for a reinterpretation of the extended theory in terms of new concepts appears only at a later stage. In the previous chapter, we saw a good example of this typical process in Planck's formal adaptation of Boltzmann's discretization of mechanical states. Planck's procedure preserved the continuity of energy exchanges between resonators and radiation, as Boltzmann's original procedure presupposed the continuity of the dynamics of gas molecules; the necessity of quantum discontinuity appeared only a few years later.

With the correspondence principle Niels Bohr has given us a most remarkable counterexample: a principle of analogy which never concealed the contrast between the old and the new theory. In this instance, the old theory was "ordinary" electrodynamics, while the new one was an atomic theory that from the start flatly contradicted some basic principles of electrodynamics. The analogy was explicitly formal and was certainly never intended to include the old theory in the new one.

However, Bohr's description of atomic phenomena retained classical concepts like electromagnetic field, electrons' position, momentum, and energy. This could give the impression that his quantum theory was self-contradictory, drawing its success from clever empirical considerations amidst a cloud of illusory depths.

In this chapter I will document the opposite thesis. Bohr was never a narrow empiricist (and never became a positivist either). His quantum theory, far from being contradictory, provided at any stage an analysis of its relation to classical theory that conciliated the persisting recourse to classical concepts with quantum discontinuity. Most important, Bohr realized that certain fundamental concepts could still be used in the quantum theory because they could be defined through an application of classical theory, *within its accepted range of validity* . For instance, the frequency of the emitted radiation could be defined through a legitimate application of wave optics to spectrometers, and the energy of stationary states could be defined through an application of ordinary mechanics (according to the adiabatic principle) to slow deformations of atomic systems.

Quantum-theoretical relations like "*D* E = *hv* " were allowed to relate classically defined *concepts* ; but they could not be explained in terms of a mere extension of classical *laws* , which would have brought contradiction. For instance, the mechanism of quantum transition had to be left undetermined, at least until proper quantum concepts could be built. This is why Bohr insisted on the incompleteness of his theory. Any further recourse to classical concepts or laws in the atomic realm had to be of a "formal" nature. So was the recourse to electronic orbits in stationary states, since these orbits did not interact with radiation according to ordinary electrodynamics. In other words (not Bohr's), there was no valid optical theory providing a means of observation of electrons at the atomic scale.

There was, moreover, no warranty that the formal use of classical concepts or laws would remain a lasting feature of quantum theory. For instance, the application of classical mechanics to electronic orbits could be only approximate and provisional, since it disregarded radiative corrections to the Coulomb forces. This is why Bohr tried to isolate the assumptions of his theory that could be formulated without appeal to formal classical laws. After a period of hesitation he reached this aim in 1917. The resulting "postulates" were expressed in terms of purely quantum-theoretical concepts (like stationary state) or in terms of well-defined classical concepts, that is to say, concepts defined through an application of a classical theory within its range of validity. For this reason Bohr believed his theory to be solidly anchored; and he proved to be right, since Heisenberg's matrix mechanics of 1925 was based on exactly the same postulates as Bohr's 1918 essay "On the quantum theory of line spectra": the postulate of stationary states and the relation *D* E = *hv* .

Conversely, Bohr often insisted on the provisional character of additional assumptions. In light of new empirical or formal developments he was always ready to reconsider his and others' preconceptions about the motion in stationary states. He successively considered strictly periodic motions obeying ordinary mechanics (1913-1916), motions of multiperiodic systems also subjected to ordinary mechanics (1917), multiperiodic motions of not necessarily multiperiodic systems still subjected to ordinary mechanics (1918-1922), multiperiodic motions eluding ordinary mechanics (1922-1925); and, finally, in spring 1925 he completely gave up the notion of definite electronic orbits.

Bohr's theory was deliberately incomplete and systematically open to revision. Around the stable pillars of the quantum postulates it needed metatheoretical "principles" that could direct constructive developments. The main principle was the correspondence principle, a procedure for deriving quantum analogues of relations between motion and radiation based on classical electrodynamics. In 1917 the initial successes of this adaptation convinced Bohr of the possibility of a "rational generalization" of classical electrodynamics based on the quantum postulates. However, the precise expression and the scope of the correspondence principle depended on the assumptions made about the electronic motion. Whenever this motion was a priori determined, the "correspondence" aided in *deducing* properties of emitted radiation. In the opposite case, characteristics of the electronic motion could be *induced* from the observed atomic spectra. This ambiguity made the correspondence principle a very flexible tool that was able to draw the most from the permanent inflow of empirical data.

In the gradual process of freeing atomic motion from classical preconceptions, the deductive side of the correspondence principle shrank, until nothing seemed to be left of it, at least in the eyes of Bohr's most open critic, Wolfgang Pauli. The heuristic power of this principle, however, was not yet exhausted. Even before the final collapse of the motion of electronic orbits, Bohr's closest disciples had started a symbolic translation of classical mechanical relations into purely quantum-theoretical ones, that is to say, a translation in terms of the basic quantities entering Bohr's postulates: energies of stationary states, quantum numbers, and transition probabilities. Heisenberg's matrix mechanics was, in fact, the ultimate result of this extended process, a symbolic system naturally and automatically integrating the formal analogy expressed in the correspondence principle.

Hence, tar from being a naive or irrational extension of classical concepts, the correspondence principle allowed for the development of formal structures that could fill the conceptual void created by the breakdown of classical laws. The later interpretation of these structures within the framework of "complementarity" (which I will not describe here) fulfilled Bohr's early hope of a "rational generalization" of classical electrodynamics.