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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 [ 1 ] during the early development of quantum theory ; he used it to explain how quantized classical orbitals connect to quantum radiation. [ 2 ]
According to the correspondence principle, in certain limits the quantum equations of states must approach Hamilton's equations of motion.The latter state the following relation between the generalized coordinate q (e.g. position) and the generalized momentum p: {˙ = = {,}; ˙ = = {,}.
A heuristic postulate called the correspondence principle was introduced to quantum theory by Niels Bohr: in effect it states that some kind of continuity argument should apply to the classical limit of quantum systems as the value of the Planck constant normalized by the action of these systems becomes very small.
This means, in the case of Newton's second law, the right side would be in the form of , while in the Ehrenfest theorem it is in the form of . The difference between these two quantities is the square of the uncertainty in x {\displaystyle x} and is therefore nonzero.
In more advanced mathematics the monotone convergence theorem usually refers to a fundamental result in measure theory due to Lebesgue and Beppo Levi that says that for sequences of non-negative pointwise-increasing measurable functions (), taking the integral and the supremum can be interchanged with the result being finite if either one is ...
Moreover, Paul A. Samuelson's correspondence principle [3] [4] [5]: pp.122–123. states that stability of equilibrium has qualitative implications about the comparative static effects. In other words, knowing that the equilibrium is stable may help us predict whether each of the coefficients in the vector B − 1 C {\displaystyle B^{-1}C} is ...
We say that b corresponds to a if each point in b is contained in the image of the corresponding point in a. Upper hemicontinuity requires that, for any convergent sequence a in a domain, and for any convergent sequence b that corresponds to a, the image of the limit of a contains the limit of b.
The correspondence principle is a concept in quantum theory and relativity. Correspondence principle may also refer to: Correspondence principle (sociology), correspondence between social class and available education; In public finance, the principle that identifies the places where it is beneficial to provide public goods and services