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arXiv:hep-th/9808144 · Journal of the Korean Physical Society

Factorization and q-Deformed Algebra of Quantum Anharmonic Oscillator

Dongsu Bak A, S P Kim, S K Kim, Kwang‐Sup Soh, J. H. Yee

Published Aug 24, 1998• 2 citations
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Abstract

We have studied the underlying algebraic structure of the anharmonic oscillator by using the variational perturbation theory. To the first order of the variational perturbation, the Hamiltonian is found to be factorized into a supersymmetric form in terms of the annihilation and creation operators, which satisfy a q-deformed algebra. This algebraic structure is used to construct all the eigenstates of the Hamiltonian.

DOI
10.48550/arxiv.hep-th/9808144

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We have studied the underlying algebraic structure of the anharmonic oscillator by using the variational perturbation theory. To the first order of the variational perturbation, the Hamiltonian is found to be factorized into a supersymmetric form in terms of the annihilation and creation operators, which satisfy a q-deformed algebra. This algebraic structure is used to construct all the eigenstates of the Hamiltonian.
Deformation of the Heisenberg-Weyl algebra W s of creation-annihilation operators is studied and the problem of eigenvalues of the Hamiltonian for s deformed oscillators with interaction is solved within this algebra. At first, types of deformation are found for which solutions could be presented analytically and a simple q−deformation is considered by the means of the perturbation theory. Cases of reducing Hamiltonians which do not preserve the total particle number to that studied here are indicated. Investigations of new types of symmetries in different areas of mathematical physics by using the Inverse Scattering Method (ISM) and nonlinear differential equations, led to the appearance of notions ”Quantum Group”, ”Quantum Algebra” [1]. Nowadays, the deformed quantum statistics, parastatistics and corresponding algebras of creation-annihilation operators are widely studied. The common Heisenberg-Weyl algebra W s consists of generators ai, a + , ni, i =1 ,...,s ,
We discuss the superoscillators in thc supersymmetric quantum systems, and give the solution to the system. By making use of the annihilation and creation operators and their q -deformed operators, we also give the q -deformed quantum harmonic oscillator model in the supersymmetric quantum systems. The q -supercoherent states associated with the q -deformed supersymmetric quantum harmonic oscillators are constructed explicitly, and their properties are investigated. The uncertainty relations for q -supercoherent states are discussed as well.
In this paper, the q Euclidean space and the q deformed harmonic oscillators in this space are described by an algebraic system. In this system we can define the raising (creation) and the lowering (annihilation) operators. These operators form a quantum hyperplane and the corresponding covariant q partial derivatives. From the viewpoint of this quantum hyperplane, the operator q Schr\"odinger equation and the explanation of its solutions are discussed. In addition, some related q implicate-order structures are presented.
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.