Classical limit
[1] The classical limit is used with physical theories that predict non-classical behavior.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.[2] More rigorously,[3] the mathematical operation involved in classical limits is a group contraction, approximating physical systems where the relevant action is much larger than the reduced Planck constant ħ, so the "deformation parameter" ħ/S can be effectively taken to be zero (cf.For example, if we consider something very large relative to an electron, like a baseball, the uncertainty principle predicts that it cannot really have zero kinetic energy, but the uncertainty in kinetic energy is so small that the baseball can effectively appear to be at rest, and hence it 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.The typical occupation numbers involved are huge: a macroscopic harmonic oscillator with ω = 2 Hz, m = 10 g, and maximum amplitude x0 = 10 cm, has S ≈ E/ω ≈ mωx20/2 ≈ 10−4 kg·m2/s = ħn, so that n ≃ 1030.It is less clear, however, how the classical limit applies to chaotic systems, a field known as quantum chaos.[5][6] In a crucial paper (1933), Dirac[7] explained how classical mechanics is an emergent phenomenon of quantum mechanics: destructive interference among paths with non-extremal macroscopic actions S » ħ obliterate amplitude contributions in the path integral he introduced, leaving the extremal action Sclass, thus the classical action path as the dominant contribution, an observation further elaborated by Feynman in his 1942 PhD dissertation.One simple way to compare classical to quantum mechanics is to consider the time-evolution of the expected position and expected momentum, which can then be compared to the time-evolution of the ordinary position and momentum in classical mechanics., the Ehrenfest theorem says[9] Although the first of these equations is consistent with the classical mechanics, the second is not: If the pairwere to satisfy Newton's second law, the right-hand side of the second equation would have read But in most cases, If for example, the potentialAn exception occurs in case when the classical equations of motion are linear, that is, whenFor general systems, the best we can hope for is that the expected position and momentum will approximately follow the classical trajectories.When the Planck constant is small, however, it is possible to have a state that is well localized in both position and momentum.The small uncertainty in momentum ensures that the particle remains well localized in position for a long time, so that expected position and momentum continue to closely track the classical trajectories for a long time.