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Physical Review A

Mechanism for extension of the wave function in quantum dynamics

L. E. Reichl

Published May 1, 1989• 13 citations
Publisher

Abstract

In a simple nonlinear quantum-dynamical system, described by a two-resonance Hamiltonian, we show that the extension of the wave function in Hilbert space, as parameters of the system are varied, is caused by the overlapping of higher-order nonlinear quantum resonances and is therefore associated with the local destruction of quantum numbers which appears to result from this overlap.

DOI
10.1103/physreva.39.4817

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In a simple nonlinear quantum-dynamical system, described by a two-resonance Hamiltonian, we show that the extension of the wave function in Hilbert space, as parameters of the system are varied, is caused by the overlapping of higher-order nonlinear quantum resonances and is therefore associated with the local destruction of quantum numbers which appears to result from this overlap.
We study the quantum dynamics of diatomic molecule driven by a circularly polarized resonant electric field. We look for a quantum effect due to classical chaos appearing due to the overlapping of nonlinear reso-nances associated to the vibrational and rotational motion. We solve the Schrödinger equation associated with the wave function expanded in term of proper stationary states, |n> |lm> (vibrational angular momentum states). Looking for quantum-classic correspondence, we consider the Liouville dynamics in the two dimensional phase space defined by the coherent-like state of vibrational states. We consider the rela-tionship between the overlapping of the classical resonances and the mixing of the quantum states, and it is found some similarities when the quantum dynamics is pictured in terms of number and phase operators.
We investigate the quantum signatures of classical nonlinear resonances by making the analytic connection between the quantum wave functions and the classical periodic orbits for the uncoupled systems. It is found that the highly efficient extension of the localized coherent states within the classical caustics is an intriguing phenomenon in mesoscopic systems with nonlinear resonances. With the theoretical analysis, we experimentally demonstrate that the laser resonator with an intracavity saturable absorber can be employed to visualize the wave patterns analogous to the quantum wave functions associated with Fermi resonance.
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In this paper we analyze two different functional formulations of classical mechanics. In the first one the Jacobi fields are represented by bosonic variables and belong to the vector (or its dual) representation of the symplectic group. In the second formulation the Jacobi fields are given as condensates of Grassmannian variables belonging to the spinor representation of the metaplectic group. For both formulations we shall show that, differently from what happens in the case presented in paper I, it is possible to endow the associated Hilbert space with a positive definite scalar product and to describe the dynamics via a Hermitian Hamiltonian. The drawback of this formulation is that higher forms do not appear automatically and that the description of chaotic systems may need a further extension of the Hilbert space.