The classical limit is the ability of a physical theory to approximate or "recover" classical mechanics when considered over special values of its parameters. The classical limit is used with physical theories that predict non-classical behavior. A postulate called the correspondence principle was introduced to quantum theory by Niels Bohr; it states that, in effect, some kind of continuity argument should apply to the classical limit of quantum systems as the value of Planck's constant tends to zero.
In quantum mechanics, due to the Heisenberg'suncertainty principle, an electron can never be at rest; it must always have a non-zero kinetic energy, a result not found in classical mechanics. For example, if we consider something very large relative to an electron, like a baseball, the uncertainty principle predicts that it cannot have zero kinetic energy, but the uncertainty in kinetic energy is so small that the baseball can appear to be at rest, and hence appears to obey classical mechanics. In general, if large energies and large objects (relative to the size and energy levels of an electron) are considered in quantum mechanics, the result will appear to obey classical mechanics.
In general and special relativity, if we consider flat space, small masses, and small speeds (in comparison to the speed of light), we find that objects once again appear to obey classical mechanics.
The conditions under which quantum and classical physics agree are referred to as the correspondence limit, or the classicallimit.
The correspondence principle limits the choices to those that reproduce classical mechanics in the correspondence limit.
For this reason, Bohm has argued that classical physics does not emerge from quantum physics in the same way that classical mechanics emerges as an approximation of special relativity at small velocities; rather, classical physics exists independently of quantum theory and cannot be derived from it.
But, this limit is just the characteristic function of a standard normal distribution, N(0,1), and the central limit theorem follows from the Lévy continuity theorem, which confirms that the convergence of characteristic functions implies convergence in distribution.
Thus the central limit theorem can be interpreted as a statement about the properties of density functions under convolution: the convolution of a number of density functions tends to the normal density as the number of density functions increases without bound, under the conditions stated above.
There are some theorems which treat the case of sums of non-independent variables, for instance the m-dependent central limit theorem, the martingale central limit theorem and the central limit theorem for mixing processes.
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