Calculation of the Characteristic Functions of Anharmonic Oscillators

Author

Ulrich D. Jentschura, Missouri University of Science and TechnologyFollow
Jean Zinn-Justin

Abstract

The energy levels of quantum systems are determined by quantization conditions. For one-dimensional anharmonic oscillators, one can transform the Schrö dinger equation into a Riccati form, i.e., in terms of the logarithmic derivative of the wave function. A perturbative expansion of the logarithmic derivative of the wave function can easily be obtained. The Bohr-Sommerfeld quantization condition can be expressed in terms of a contour integral around the poles of the logarithmic derivative. Its functional form is B_m(E,g)=n+1/2, where B is a characteristic function of the anharmonic oscillator of degree m, E is the resonance energy, and g is the coupling constant. A recursive scheme can be devised which facilitates the evaluation of higher-order Wentzel-Kramers-Brioullin (WKB) approximants. The WKB expansion of the logarithmic derivative of the wave function has a cut in the tunneling region. The contour integral about the tunneling region yields the instanton action plus corrections, summarized in a second characteristic function A_m(E,g). The evaluation of A_m(E,g) by the method of asymptotic matching is discussed for the case of the cubic oscillator of degree m=3.

Recommended Citation

U. D. Jentschura and J. Zinn-Justin, "Calculation of the Characteristic Functions of Anharmonic Oscillators," Applied Numerical Mathematics, vol. 60, no. 12, pp. 1332 - 1341, Elsevier, Dec 2010.

The definitive version is available at https://doi.org/10.1016/j.apnum.2010.03.015

Department(s)

Physics

Keywords and Phrases

Quantum Mechanics; Semiclassical Techniques; Singular Perturbations; Turning Point Theory; WKB Method; Communication Channels (Information Theory); Fading Channels; Mechanics; Oscillators (Electronic); Oscillators (Mechanical); Quantum Electronics; Quantum Interference Devices; Quantum Optics; Wave Functions; Function Evaluation

International Standard Serial Number (ISSN)

0168-9274

Document Type

Article - Journal

Document Version

Citation

File Type

text

Language(s)

English

Rights

© 2010 Elsevier, All rights reserved.

Publication Date

01 Dec 2010