If quantum mechanics were to be applicable to macroscopic objects, there must be some limit in which quantum mechanics reduces to classical mechanics. But macroscopic systems, like springs and capacitors, are accurately described by classical theories like classical mechanics and classical electrodynamics. The rules of quantum mechanics are highly successful in describing microscopic objects, atoms and elementary particles. Ĭlassical quantities appear in quantum mechanics in the form of expected values of observables, and as such the Ehrenfest theorem (which predicts the time evolution of the expected values) lends support to the correspondence principle. This concept is somewhat different from the requirement of a formal limit under which the new theory reduces to the older, thanks to the existence of a deformation parameter. The term codifies the idea that a new theory should reproduce under some conditions the results of older well-established theories in those domains where the old theories work. The principle was formulated by Niels Bohr in 1920, though he had previously made use of it as early as 1913 in developing his model of the atom. In other words, it says that for large orbits and for large energies, quantum calculations must agree with classical calculations. In physics, the correspondence principle states that the behavior of systems described by the theory of quantum mechanics (or by the old quantum theory) reproduces classical physics in the limit of large quantum numbers. The considered approach naturally leads to the notion of momentum operator, fundamental commutation relations, density matrix and Liouville equation construction etc.For other uses, see Correspondence principle (disambiguation). The solution of the operator equation may be considered as the construction of Hamiltonian of the system and a move to the Schrodinger representation. Matrix equation of Heisenberg leads to an operator equation. Root model leads to a consistent condition that connects eigenvectors and eigenvalues of a mechanical system that is described by a matrix equation of Heisenberg. Constructing of a root multi-parametric statistical model leads to obtaining such frequencies and base functions in Fourier decomposition so that classical laws of motion are fulfilled in average. From all possible multi-parametric statistical models the root model is most remarkable. Statistical conformities in quantum mechanics are fundamental and are not due to the incomplete information about the system. It is demonstrated that quantum mechanics may be considered as a rational statistical generalization of classical mechanics. It is shown that new relativistic laws of dynamics make it necessary to change kinematical relations of classical mechanics such as the law of velocity composition, coordinate transformation etc. The difference of relativistic mechanics from classical mechanics is due to the new definition of momentum that is proportional to energy and velocity. A comparative description of relativistic and classical mechanics is given using three main principles: the definition of momentum, the main law of dynamics (Newton's second law) and the law of conservation of energy. Based on the Bohr's correspondence principle it is shown that relativistic mechanics and quantum mechanics may be considered as generalizations of classical mechanics.
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