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» This article discusses quantum theory. For other uses, see Correspondence principle (disambiguation).

In physics, the correspondence principle is a quantitative tool in the old quantum theory, explicitly formulated by Niels Bohr in 1923. It says that the behavior of quantum mechanical systems reproduce classical physics in the limit of large quantum numbers.

Quantum Mechanics

The rules of quantum mechanics are highly successful in describing microscopic objects, atoms and elementary particles. But macroscopic systems like springs and capacitors are accurately described by classical theories like classical mechanics and classical electrodynamics. The laws of physics should be independent of the size of the physical objects being described, so there must be some limit in which quantum mechanics reduces to classical mechanics. Bohr's principle demands that classical physics and quantum physics give the same answer when the systems become large.
   The conditions under which quantum and classical physics agree are referred to as the correspondence limit, or the classical limit. Bohr provided a rough prescription for the correspondence limit: it occurs when the quantum numbers describing the system are large. A more elaborated analysis of quantum-classical correspondence (QCC) in wavepacket spreading leads to the distinction between robust "restricted QCC" and fragile "detailed QCC". See and references therein. "Restricted QCC" refers to the first two moments of the probability distribution and is true even when the wave packets diffract, while "detailed QCC" requires smooth potentials which vary over scales much larger than the wavelength, which is what Bohr considered.
   The post-1925 new quantum theory came in two different formulations. In matrix mechanics, the correspondence principle was built in and was used to construct the theory. In the Schrödinger approach classical behavior isn't clear because the waves spread out as they move. Once the Schrödinger equation was given a probabilistic interpretation, Ehrenfest showed that Newton's laws hold on average--- The quantum statistical expectation value of the position and momentum obey Newton's laws.
   The correspondence principle is one of the tools available to physicists for selecting quantum theories corresponding to reality. The principles of quantum mechanics are broad - they say that the states of a physical system form a complex vector space, but they don't say which operators correspond to physical quantities or measurements. The correspondence principle limits the choices to those that reproduce classical mechanics in the correspondence limit.
   Because quantum mechanics only reproduces classical mechanics in a statistical interpretation, and because the statistical interpretation only gives the probabilities of different classical outcomes, Bohr has argued that classical physics doesn't emerge from quantum physics in the same way that classical mechanics emerges as an approximation of special relativity at small velocities. He argued that classical physics exists independently of quantum theory and can't be derived from it. His position is that it's inappropriate to understand the experiences of observers using purely quantum mechanical notions such as wavefunctions because the different states of experience of an observer are defined classically, and don't have a quantum mechanical analog.
   The relative state interpretation of quantum mechanics is an attempt to understand the experience of observers using only quantum mechanical notions. Neils Bohr was an early opponent of such interpretations.

Other Scientific Theories

The term "correspondence principle" is used in a more general sense to mean the reduction of a new scientific theory to an earlier scientific theory in appropriate circumstances. This requires that the new theory explain all the phenomena under circumstances for which the preceding theory was known to be valid, the "correspondence limit".
   For example, Einstein's special relativity satisfies the correspondence principle, because it reduces to classical mechanics in the limit of velocities small compared to the speed of light (example below). General relativity reduces to Newtonian gravity in the limit of weak gravitational fields. Laplace's theory of celestial mechanics reduces to Kepler's when interplanetary interactions are ignored, and Kepler's reproduces Ptolmey's equant in a coordinate system where the Earth is stationary. Statistical mechanics reproduces thermodynamics when the number of particles is large. In biology, chromosome inheritance theory reproduces Mendel's laws of inheritance, in the domain that the inherited factors are protein coding genes.
   In order for there to be a correspondence, the earlier theory has to have a domain of validity--- it must work under some conditions. Not all theories have a domain of validity. For example, there's no limit where Newton's mechanics reduces to Aristotle's mechanics because Aristotle's mechanics, although academically viable for many centuries, doesn't have any domain of validity.

Examples

Bohr Model

If an electron in an atom is moving on an orbit with period T, the electromagnetic radiation will classically repeat itself every orbital period. If the coupling to the electromagnetic field is weak, so that the orbit doesn't decay very much in one cycle, the radiation will be emitted in a pattern which repeats every period, so that the fourier transform will have frequencies which are only multiples of 1/T. This is the classical radiation law: the frequencies emitted are integer multiples of 1/T.
   In quantum mechanics, this emission must be of quanta of light. The frequency of the quanta emitted should be integer multiples of 1/T so that classical mechanics is an approximate description at large quantum numbers. This means that the energy level corresponding to a classical orbit of period 1/T must have nearby energy levels which differ in energy by h/T, and they should be equally spaced near that level:
»

Delta E_n= m_0 v^2
which is the Newtonian expression for kinetic energy.

Correspondence Principle in Sociology

The correspondence principle is a sociological theory that serves to observe the relationship between social standing and the educational system. Particularly, the aim of the correspondence principle is to examine the relationship between a person’s social standing and the type of education that's received at school. In its most basic form the principle states that the social relations of the school can be directly related to those in the work-place, meaning that educational institutions prepare students for their future work roles . Apart from the curriculum that's offered by the school, the correspondence principle argues that the structure of the school and also the personal experience given to each student is more important to their future socialization. Furthermore, the principle also emphasizes that there's a strong relationship between the child’s education and the interaction they've with their parents at home. There is a significant intergenerational replication of consciousness and socialized inequality via the linkages among the authority relations experienced by fathers at work, transferred to childrearing styles, and replicated in school interaction with teachers .
   Typically, the correspondence principle is considered a branch of the conflict theory which was made famous by Karl Marx. Marx’s conflict theory states that there's a division in society, one side being represented by the capitalists and the other by the workers. A very small percentage of the population can be defined as the “capitalists,” the group which owns the money and means of production. The “workers” are the remaining part of society who is willing to sell their labor to the capitalists. The correspondence principle advances a neo-Marxist argument about the specific nature of the institutional linkages in the family, school, and work "chain" comprising the socio- economic life cycle . Many sociologists who support this principle argue that education is just a means of maintaining social class boundaries. Many argue that schools in capitalistic societies are geared toward giving children different types of education based solely on their social standing rather than by their inherent skills. Under this principle schools are believed to give lower class children a different type of education compared to their upper class counterparts. Typically, it's said that lower class children are put on an educational track that will prepare them for working class or “blue collar” jobs. It is thought that the education of lower class children is different because it prepares them to enter the work force directly after high school. Schooling teaches working class children to sit quietly at their desk, obey the teacher’s authority, and also acquaints them with becoming familiar with repetitive tasks. Similarly, the education of upper class children is thought to be geared toward upper class or “white collar” professions. With upper class children, instead of focusing on preparing them to enter the workforce, there's added emphasis is on preparing them to move on to four year colleges and universities after high school. Here they're trained to be professionals and capitalists by teaching them how to think critically and instilling in them a sense of responsibility and authority.
   

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