Factorization and q-Deformed Algebra of Quantum Anharmonic Oscillator

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. PACS number(s): 03.65.-w; 02.30.Mv; 11.80.Fv Typeset using REVTEX Electronic address: [email protected] Electronic address: [email protected] Electronic address: [email protected] Electronic address: [email protected] Electronic address: [email protected] 1 Quantum anharmonic oscillator has been frequently studied as a toy model for developing various approximation methods in quantum mechanics and quantum field theory [1–5]. Recently it has been used to develop various approaches to the variational perturbation theory [3,4], which enables one to compute the order by order correction terms to the well known variational approximation. More recently the model has been utilized to establish the Liouville-Neumann approach to the variational perturbation theory [5], where one constructs the annihilation and creation operators as perturbation series in the coupling constant whose zeroth order terms constitute those of the Gaussian approximation. However, the underlying algebraic structure of the anharmonic oscillator for its own sake has rarely been studied. In ref. [5], we have shown that to the first order of the variational perturbation the Hamiltonian is factorized as in the case of the simple harmonic oscillator, while the annihilation and creation operators satisfy the q-deformed algebra rather than the usual commutation relations. This is an interesting algebraic structure of the theory which may enable one to obtain more information on the theory. The connection between the q-deformed algebra and the quasi-exactly solvability has been found for certain type of potentials [6], and the possibility of a q-deformed quartic oscillator has also been suggested from the study of the energy spectra obtained by the standard perturbation method [7]. It is the purpose of this letter to study the algebraic structure of the anharmonic oscillator to the first-order variational perturbation, and to utilize this structure to obtain the general energy eigenstates of the system. It is the q-deformed algebraic structure of the theory that enables us to find the q-deformed Fock space [8]. We now consider the anharmonic oscillator described by the Hamiltonian, Ĥ = 1 2 p̂ + 1 2 ωx̂ + 1 4 λx̂, (1) where the mass is scaled to unity for simplicity. In the variational Gaussian approximation one searches for a simple harmonic oscillator whose energy eigenstates minimize the expectation value of the Hamiltonian (1). For this purpose, we introduce a set of operators, â and â, as linear functions of the dynamical variables x̂ and p̂: â = √ ΩG 2h̄ x̂+ i 1 √ 2ΩGh̄ p̂,