Bohr's Correspondence Principle
As many people emphasized, what is called the "Correspondence Principle" played an important role in the development of quantum mechanics. Bohr, in his 1925 paper, describes this as follows:
Nevertheless, the visualization of the stationary states by mechanical pictures has brought to light a far-reaching analogy between the quantum theory and the mechanical theory. This analogy was traced by investigating the conditions in the initial stages of the binding process described, where the motions corresponding to successive stationary states differ comparatively little from each other. Here it was possible to demonstrate an asymptotic agreement between spectrum and motion. This agreement establishes a quantitative relation by which the constant appearing in Balmer's formula for the hydrogen spectrum is expressed in terms of Planck's constant and the values of the charge and mass of the electron. The essential validity of this relation was clearly illustrated by the subsequent test of the predictions of the theory regarding the dependence of the spectum on the nuclear charge. ...
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 correspondng 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 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's description here is rather abstract (and you may have recognized Bohr's peculiar "style"), but what he meant was illustrated, more specifically and intelligibly, by Dr. Tomonaga, on pp. 137-153 of his book. "Asymptotic agreement" means roughly this: if the quantum number n (we have not defined this yet, but imagine, e.g., the number of energy quanta filling a large box, as was illustrated in Energy Quantum) involved in the problem in question is large enough (so that results obtained by classical mechanics are expected to hold, at least approximately), then classical results and quantum results will converge, as far as numerical calculation is concerned.
If I may add a word here, there is a probability for each stationary state to change into another stationary state (although these states are usually stable), according to quantum mechanics; and if some energy is supplied or extracted by some cause, this probability changes (which can be calculated by means of Schroedinger equation, which Bohr did not know yet). Now, the question is "why probability here?"; and this question may illuminate the character of the correspondence principle. Dr. Tomonaga's exposition (pp. 142-5) should be quite helpful. In short, the strength of light emitted from an atom is related with this probability. On the calssical theory, we can compute this strength, but according to quantum theory, the notion of the strength is meaningless, as long as we focus our attention on a single event of emission; an atom either emit light (corresponding to a single spectral line, monochromatic light ray) or not, and that's all. On this view, the strength makes sense only when we count many transitions and count the events with the same spectral line; thus, it makes sense only when the frequency of the same spectral line in many events involving the same atom, or in many events involving a number of the same kind of atom. Then, this frequency can be related with the notion of probability in question. What we measure in our experiment is the strength of light, interpreted as an expression of this frequency among the multitude of similar events. Since the emission of light from an atom is connected with a transition, and the classical notion of its strength can be related with its frequency, we have to introduce the notion of probability (interpreted as an average frequency in a unit time) of a state transition, on quantum theory. In this way, we can bridge the gap between the classical theory and the quantum theory, at least tentatively. This is the job of the correspondence principle.
Bohr's correspondence principle is quite interesting, also from a philosophical point of view: How is a new theory formed in relation to older theories? Is a heuristic in such a process merely a matter of psychology, or are there any logical or rational elements in it? What is abandoned, and what is innovated in such a process of theory change? Bohr's use of the correspondence principle nicely illustrates these questions and answers to them, in the process of the formation of quantum mechanics. Indeed, Heisenberg seems to have proceeded along this line and have reached his Matrix Mechanics (this is mentioned in Bohr's second paper).
Bohr, N. (1934), Atomic Theory and the Description of Nature, Ox Bow Press (reprint)
Tomonaga, S. (1969) 量子力学 I、みすず書房。
See also a useful site on Microphysics, at Kyushu University: http://www2.kutl.kyushu-u.ac.jp/seminar/MicroWorld/MicroWorld.html
