Nonlinear dynamics from linear quantum evolutions

doi:10.1016/j.aop.2019.167957

DOE OSTI · 22852349

Nonlinear dynamics from linear quantum evolutions

Abstract

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 ϕ{sub t} 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 ϕ{sub t}. 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 ϕ{sub t}. Such a result, therefore, should be conceived as a generalization of the variational method already employed for stationary problems.

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71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS ANNIHILATION OPERATORS APPROXIMATIONS EIGENSTATES HILBERT SPACE LAGRANGIAN FUNCTION LIMITING VALUES NONLINEAR PROBLEMS QUANTUM MECHANICS QUANTUM STATES QUANTUM SYSTEMS VARIATIONAL METHODS VECTOR FIELDS

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2019-12-15. Nonlinear dynamics from linear quantum evolutions. https://doi.org/10.1016/j.aop.2019.167957

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