Optimized Perturbation Theory for Wave Functions of Quantum Systems

The notion of the optimized perturbation, which has been successfully applied to energy eigenvalues, is generalized to treat wave functions of quantum systems. The key ingredient is to construct an envelope of a set of perturbative wave functions. This leads to a condition similar to that obtained from the principle of minimal sensitivity. Applications of the method to quantum anharmonic oscillator and the double well potential show that uniformly valid wave functions with correct asymptotic behavior are obtained in the first-order optimized perturbation even for strong couplings. PACS numbers: 03.65.Ge,02.30.Mv,11.15.Bt,11.15.Tk Typeset using REVTEX 1 Naive perturbative expansions are usually divergent asymptotic series [1]. Thus various summation techniques such as the Borel and Padé methods have been developed [2]. In recent years, a new summation method “optimized perturbation theory (OPT)” has been studied. This combines straightforward perturbation with a variational principle and improves even the Borel non-summable series. Variants of OPT are called in various names, the (optimized) δ-expansion, the variational perturbation, the renormalized strong coupling expansion, and so on [3]. One of the key ingredients in the method is the principle of minimal sensitivity (PMS) [4], which says that physical quantities calculated in perturbation theory should not depend on any parameters absent in the original Hamiltonian. The method has been applied to improve the energy eigenvalues of the quantum anharmonic oscillator (AHO) and the double-well potential (DWP); AHO and DWP are the useful laboratories to test a new non-perturbative method. A rigorous proof has been also given for the convergence properties of the perturbation series for energy eigenvalues of AHO/DWP [5]. The method is also being extensively applied to perturbation series for energy and partition functions in quantum field theories [6]. To the best of our knowledge, the wave functions has never been explored in OPT even for simple quantum mechanical problems. Needless to say, wave functions have all information of the system in question, including the energy eigenvalues. The purpose of this Letter is to show a novel generalization of the idea of the optimized perturbation for studying the wave functions. Let’s take the following Schrödinger equation for one dimensional AHO/DWP; 1 2 (− d 2 dx ± x + λx4)ψn(x) = En(λ)ψn(x). (1) Naive Rayleigh-Schrödinger (RS) expansion for eigenvalues E(λ) = ∑ n λ Cn gives an asymptotic series with coefficients growing as Cn ∼ n! [7]. A scaled Hamiltonian in the simplest version of OPT (sometimes called the linear δ-expansion) reads Hδ = H0(Ω) + δ ·HI(Ω, λ) with H0(Ω) = 1 2 (− d 2 dx + Ωx), (2)