In physics, a correspondence principle is any one of several premises or assertions about the relationship between classical and quantum mechanics.
The physicist Niels Bohr coined the term in 1920 during the early development of quantum theory; he used it to explain how quantized classical orbitals connect to quantum radiation.
Modern sources often use the term for the idea that the behavior of systems described by quantum theory reproduces classical physics in the limit of large quantum numbers: for large orbits and for large energies, quantum calculations must agree with classical calculations. A "generalized" correspondence principle refers to the requirement for a broad set of connections between any old and new theory.
History
Max Planck was the first to introduce the idea of quanta of energy, while studying black-body radiation in 1900. In 1906, he was also the first to write that quantum theory should replicate classical mechanics at some limit, particularly if the Planck constant h were taken to be infinitesimal. With this idea, he showed that Planck's law for thermal radiation leads to the Rayleigh–Jeans law, the classical prediction (valid for large wavelength). In 1913, he provided the first postulates of what is now known as old quantum theory.
Arnold Sommerfeld refined Bohr's theory leading to the Bohr-Sommerfeld quantization condition. Sommerfeld referred to the correspondence principle as Bohr's magic wand (), in 1921.
Bohr's correspondence principle
The seeds of Bohr's correspondence principle appeared from two sources. First Sommerfeld and Max Born developed a "quantization procedure" based on the action angle variables of classical Hamiltonian mechanics. This gave a mathematical foundation for stationary states of the Bohr-Sommerfeld model of the atom. The second seed was Albert Einstein's quantum derivation of Planck's law in 1916. Einstein developed the statistical mechanics for Bohr-model atoms interacting with electromagnetic radiation, leading to absorption and two kinds of emission, spontaneous and stimulated emission. But for Bohr the important result was the use of classical analogies and the Bohr atomic model to fix inconsistencies in Planck's derivation of the blackbody radiation formula.
Bohr used the word "correspondence" in italics in lectures and writing before calling it a correspondence principle. He viewed this as a correspondence between quantum motion and radiation, not between classical and quantum theories. He writes in 1920 that there exists "a far-reaching correspondence between the various types of possible transitions between the stationary states on the one hand and the various harmonic components of the motion on the other hand." was in 1923, in a summary paper entitled (in the English translation) "On the application of quantum theory to atomic structure". In his chapter II, "The process of radiation", he defines his correspondence principle as a condition connecting harmonic components of the electron moment to the possible occurrence of a radiative transition. In modern terms, this condition is a selection rule, saying that a given quantum jump is possible if and only if a particular type of motion exists in the corresponding classical model.
Sommerfeld had been skeptical of the correspondence principle as it did not seem to be a consequence of a fundamental theory; Kramers' work convinced him that the principle had heuristic utility nevertheless. Other physicists picked up the concept, including work by John Van Vleck and by Kramers and Heisenberg on dispersion theory. The principle became a cornerstone of the semi-classical Bohr-Sommerfeld atomic theory;
Bohr's 1922 Nobel prize was partly awarded for his work with the correspondence principle. Since quantum mechanics operates in a discrete space and classical mechanics in a continuous one, any correspondence will be necessarily fuzzy and elusive. or in a limit where the Planck constant in the quantum formula is reduced to zero, <math>\hbar \rightarrow 0</math>. For example, Albert Einstein in his 1905 work on relativity noted that classical mechanics relied on Galilean relativity while electromagnetism did not, and yet both work well. He produced a new theory that combined them in a way that reduced to these separate theories in approximations.
Ironically the singular failure of this "generalized correspondence principle" concept of scientific theories is the replacement of classical mechanics with quantum mechanics.