Generalized Ehrenfest Theorem for Nonlinear Schrodinger Equations

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It is shown that the Ehrenfest theorem can be generalized so that it is valid also for all space-localized solutions ψ of the nonlinear Schrödinger equations (in one or more space dimensions). Then it is shown that as a consequence, the motion of the localized ψ-field as a whole obeys the laws of classical mechanics and those of classical electrodynamics if the interaction of the ψ-field with an external electromagnetic field is defined by the rules of quantum mechanics applied to the nonlinear Schrödinger equation for ψ (in exactly the same manner as to the linear Schrödinger equation). This establishes the existence of a deep link between the nonlinear Schrödinger equations and classical mechanics and electrodynamics.

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