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Journal of Physics A Mathematical and General

Lie algebraic approach to the coupled-mode oscillator

M. A. Al-Gwaiz, M. Sebawe Abdalla, Sharief Deshmukh

Published Feb 21, 1994• 13 citations
Publisher

Abstract

The coupling of electromagnetic fields for the construction of light amplifiers is described by a certain Hamiltonian which can be expressed as the sum of three differential operators. Following the method of Steinberg (1977) and Dattoli et. al. (1990), the solution of Schrodinger's equation for the Hamiltonian of the system is constructed from the solutions of the corresponding equations for the components of the Hamiltonian.

DOI
10.1088/0305-4470/27/4/020

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The coupling of electromagnetic fields for the construction of light amplifiers is described by a certain Hamiltonian which can be expressed as the sum of three differential operators. Following the method of Steinberg (1977) and Dattoli et. al. (1990), the solution of Schrodinger's equation for the Hamiltonian of the system is constructed from the solutions of the corresponding equations for the components of the Hamiltonian.
We study the known coherent states of a quantum harmonic oscillator from the standpoint of the original developed noncommutative integration method for linear partial differential equations. The application of the method is based on the symmetry properties of the Schrödinger equation and on the orbit geometry of the coadjoint representation of Lie groups. We have shown that analogs of coherent states constructed by the noncommutative integration can be expressed in terms of the solution of a system of differential equations on the Lie group of the oscillatory Lie algebra. The solutions constructed are directly related to irreducible representation of the Lie algebra on the Hilbert space functions on the Lagrangian submanifold to the orbit of the coadjoint representation.
We study the known coherent states of a quantum harmonic oscillator from the standpoint of the originally developed noncommutative integration method for linear partial differential equations. The application of the method is based on the symmetry properties of the Schrödinger equation and on the orbit geometry of the coadjoint representation of Lie groups. We have shown that analogs of coherent states constructed by the noncommutative integration can be expressed in terms of the solution to a system of differential equations on the Lie group of the oscillatory Lie algebra. The solutions constructed are directly related to irreducible representation of the Lie algebra on the Hilbert space functions on the Lagrangian submanifold to the orbit of the coadjoint representation.