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Physical Review A

Algebraic view to the quantum anharmonic oscillator

G. Dattoli, A. Torre

Published Mar 1, 1988• 15 citations
Publisher

Abstract

We use the harmonic-oscillator realization of the SU(1,1) group to study the time evolution of a quantum state undergoing an anharmonic interaction. The result is then applied to the propagation of an initially Gaussian beam through an optical fiber with a quartic aberration.
DOI
10.1103/physreva.37.1571

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We present a new Lie algebraic method to study the eigenvalues and eigenfunctions of quantum anharmonic oscillators. We consider the Hamiltonians for the simple harmonic and anharmonic oscillator as the two generators of a Lie algebra, whose other generators may be found exactly or up to any order of the parameter involved. Speciflcally, the closed commutator algebra for the quartic anharmonic oscillator is established in a perturbation sense. An element of this Lie group, turning out to be the four-photon operator, transforms the quartic anharmonic oscillator Hamiltonian to the harmonic one in a perturbation sense; thus, facilitating the calculation of the eigenvalues and eigenfunctions of the former.
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Operator algebra techniques are employed to derive the quantum evolution operator for the harmonic oscillator. The derivation begins with the construction of the annihilation and creation operators and the determination of the wave function for the coherent state as well as its time-dependent evolution, and ends with the transformation of the propagator in a mixed position-coherent-state representation to the desired one in configuration space. Throughout the entire procedure, besides elementary operator manipulations, it is only necessary to solve linear differential equations and to calculate Gaussian integrals.