# correspondence principle

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## correspondence principle

**correspondence principle,**physical principle, enunciated by Niels Bohr in 1923, according to which the predictions of the quantum theory must correspond to the predictions of the classical theories of physics when the quantum theory is used to describe the behavior of systems that can be successfully described by classical theories. Technically this principle means that the results of a quantum theory analysis of a problem that involves the use of very large quantum numbers must agree with the results of a classical physics analysis. Such correspondence is known as the classical limit of the quantum theory. Ordinarily the quantum theory is used to describe the behavior of bodies that are so small that they cannot be seen under an optical microscope, while the theories of classical physics are used to analyze the behavior of large-scale bodies. The correspondence principle provided an important theoretical basis for the development of a detailed correlation between the newer quantum theory and the classical physics that preceded it.

## Correspondence principle

A fundamental hypothesis according to which classical mechanics can be understood as a limiting case of quantum mechanics; or conversely, many characteristic features in quantum mechanics can be approximated on the basis of classical mechanics, provided classical mechanics is properly reinterpreted. This idea was first proposed by N. Bohr in the early 1920s as a set of rules for understanding the spectra of simple atoms and molecules.

The classical motions in simple dynamical systems can be understood as composed of independent partial motions, each with its own degree of freedom. Each degree of freedom accumulates its own classical action-integral. The frequency of the classical motion for any particular degree of freedom is given by the partial derivative of the energy function with respect to the corresponding action. Bohr noticed that this classical result yields the correct quantum-theoretical result for the light frequency in a transition from one energy level to another, provided the derivative is replaced by the difference in the energies. Moreover, precise information about the possibility of such transitions and their intensities is obtained by analyzing the related classical motion. This information becomes better as the quantum numbers involved be come larger. The apparent inconsistencies in Bohr's quantum theory are thereby overcome by a set of rules that came to be called the correspondence principle. *See* Action

After 1925, the success of the new quantum mechanics, particularly wave mechanics, reduced the correspondence principle to a somewhat vague article of faith among physicists. However, the appeal to classical mechanics is still convenient for some rather crude estimates such as the total number of levels below a given energy. Such estimates help in finding the approximate shape of large atoms and large nuclei in the Thomas-Fermi model.

The correspondence principle, however, has assumed a more profound significance. Experimental techniques in atomic, molecular, mesoscopic, and nuclear physics have improved dramatically. High-precision data for many thousands of energy levels are available where the traditional methods of quantum mechanics are not very useful or informative. However, the basic idea behind the correspondence principle must still be valid: Quantum mechanics must be understandable in terms of classical mechanics for the highly excited states, even in difficult cases like the three-body problem, where the overall behavior seems unpredictable and chaotic. The wider application of Bohr's correspondence principle allows many basic but difficult problems to be seen in a new light. *See* Atomic structure and spectra, Chaos, Mesoscopic physics, Molecular structure and spectra, Quantum mechanics

*The Great Soviet Encyclopedia*(1979). It might be outdated or ideologically biased.

## Correspondence Principle

a postulate of quantum mechanics requiring that in the limiting case of large quantum numbers the physical consequences of quantum mechanics coincide with the results of classical theory. The correspondence principle reflects the circumstance that quantum effects are significant only in the consideration of very small objects, where the values of action are comparable to Planck’s constant *h*. If, however, the quantum numbers characterizing the state of a physical system, such as the orbital angular momentum quantum number *l*, are large, then the quantity *h* may be ignored and the system obeys classical laws to a high degree of accuracy. From the formal point of view, the correspondence principle means that as ħ → 0, the quantum-mechanical description of physical objects becomes equivalent to the classical description.

The correspondence principle is often understood in the following more general sense: any new theory that claims to present a deeper description of physical reality and to have a broader range of applicability than the old theory must include the old theory as a limiting case. Thus, relativistic mechanics (*see*RELATIVITY, THEORY OF) is equivalent to classical mechanics in the limit of low speeds *v*—that is, when *v* ≪ *c*, where *c* is the speed of light in a vacuum. Formally, the classical limit is approached when *c* → ∞.

When the fundamental axioms of a theory have already been formulated, the correspondence principle is of primarily illustrative value: the principle draws attention to the continuity of theoretical constructs. In a number of cases, the correspondence principle is helpful in developing approximate methods of problem solving. For example, if in a given physical problem *h* can be regarded as small, this situation is equivalent to the quasiclassical approximation to quantum mechanics. The nonrelativistic Schrödinger wave equation in this case results in the classical Hamilton-Jacobi equation as *h*— → 0. When, however, the principles of a new theoretical discipline that is taking shape are still largely unclear, the correspondence principle has independent heuristic value.

The correspondence principle was advanced in 1923 by N. Bohr in the old quantum theory, which preceded quantum mechanics. Bohr applied the principle to the problem of atomic emission and absorption spectra. Subsequently, after a consistent quantum mechanics had been created, the characteristics of atomic spectra were explained on a deeper foundation, with the essential features of the mathematical apparatus being determined by the correspondence principle.

The importance of the correspondence principle, however, goes far beyond the framework of quantum mechanics. The principle is widely used in quantum electrodynamics and elementary particle theory and undoubtedly will be an integral part of any new theoretical scheme.

### REFERENCES

Bohr, N.*Tri stat’i o spektrakh i slroenii atomov*. Moscow-Petrograd, 1923. (Translated from German.)

Blokhintsev, D. I.

*Osnovy kvantovoi mekhaniki*, 3rd ed. Moscow, 1961.

Schiff, L.

*Kvanwvaia mekhanika*. Moscow, 1957. (Translated from English.)

O. I. ZAVIALOV