The classical limit of quantum theory

Abstract

For a quantum observable $A_{\hslash }$ depending on a parameter $\hslash$ we define the notion ``$A_{\hslash }$ converges in the classical limit''. The limit is a function on phase space. Convergence is in norm in the sense that $A_{\hslash }\to 0$ is equivalent with $\Vert A_{\hslash }\Vert \to 0$. The $\hslash$-wise product of convergent observables converges to the product of the limiting phase space functions. ${\hslash }^{-1}$ times the commutator of suitable observables converges to the Poisson bracket of the limits. For a large class of convergent Hamiltonians the $\hslash$-wise action of the corresponding dynamics converges to the classical Hamiltonian dynamics. The connections with earlier approaches, based on the WKB method, or on Wigner distribution functions, or on the limits of coherent states are reviewed.

plain TeX, 33 pages, no figures

References in corpus (2)

• Ground states of the infinite q-deformed Heisenberg ferromagnet
• Dynamics by White-Noise Hamiltonians

Cited by in corpus (9)

• Optimal Cloning of Pure States
• Ground states of the infinite q-deformed Heisenberg ferromagnet
• Classical Limit of Quantum Dynamical Entropies
• Semi-Classical Dynamics in Quantum Spin Systems
• On the averaged quantum dynamics by white-noise Hamiltonians with and without dissipation
• The Narcowich-Wigner spectrum of a pure state
• Phase space trajectories in quantum mechanics
• A new application of non-canonical maps in quantum mechanics
• A Survey on the Classical Limit of Quantum Dynamical Entropies