Nonlinear dynamics from linear quantum evolutions
Annals of Physics(2019)
摘要
Linear dynamics restricted to invariant submanifolds generally gives rise to nonlinear dynamics. Submanifolds in the quantum framework may emerge for several reasons: one could be interested in specific properties possessed by a given family of states, either as a consequence of experimental constraints or inside an approximation scheme. In this work we investigate such issues in connection with a one parameter group ϕt of transformations on a Hilbert space, H, defining the unitary evolutions of a chosen quantum system. Two procedures will be presented: the first one consists in the restriction of the vector field associated with the Schrödinger equation to a submanifold invariant under the flow ϕt. The second one makes use of the Lagrangian formalism and can be extended also to non-invariant submanifolds, even if in such a case the resulting dynamics is only an approximation of the flow ϕt. Such a result, therefore, should be conceived as a generalization of the variational method already employed for stationary problems.
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关键词
Unitary maps,Quantum states,Lagrangian mechanics,Generalized coherent states,Classical-like maps,Unfolding,Reduction,Variational principles
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