Classical limit

The regime in which a quantum theory reproduces classical physics, typically when Planck’s constant is negligible relative to relevant scales or quantum numbers are large.

Ehrenfest theorem: A relation connecting the time evolution of quantum expectation values to classical equations of motion. It shows when average quantum position and momentum approximately obey Newton’s laws.
Planck constant: The fundamental constant setting the scale of quantum action and the size of quantum effects. The classical limit is often expressed as action becoming large compared with this constant.
Quantum-classical transition: The physical and conceptual transition between quantum behavior and the effective classical world. It is the broader problem that includes, but is not identical to, taking a formal classical limit.
Geometric optics: The approximation describing light as rays when its wavelength is small relative to relevant structures. It is the classical ray limit of wave optics, derived through the same short-wavelength logic.
WKB approximation: A semiclassical method that approximates wavefunctions using slowly varying amplitudes and rapidly varying phases. Its phase follows classical action when the action greatly exceeds Planck’s constant.
Commutator: An operator measuring the failure of two operators to commute under composition. Position and momentum noncommutation weakens in appropriately scaled classical limits.
Quantum measurement problem: The problem of explaining how definite measurement outcomes arise from quantum state evolution. A classical limit of equations alone does not settle why individual outcomes occur.
Rydberg atom: An atom with an electron excited to a state of very high principal quantum number. Its large quantum numbers produce dynamics with pronounced classical correspondences.
Stationary phase approximation: An asymptotic method in which oscillatory integrals are dominated by points where the phase is stationary. In path integrals, it selects classical paths as Planck’s constant becomes small.
Poisson bracket: A mathematical operation on functions in classical phase space that governs Hamiltonian time evolution. It is the classical counterpart approached by suitably scaled quantum commutators.

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