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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.
More specifically, it quantifies the "amount of information" (in units such as shannons , nats or hartleys) obtained about one random variable by observing the other random variable. The concept of mutual information is intimately linked to that of entropy of a random variable, a fundamental notion in information theory that quantifies the ...
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.
In contrast, a variable is a discrete variable if and only if there exists a one-to-one correspondence between this variable and a subset of , the set of natural numbers. [8] In other words, a discrete variable over a particular interval of real values is one for which, for any value in the range that the variable is permitted to take on, there ...