Calculation of the characteristic functions of anharmonic oscillators

U. Jentschura, Jean Zinn-Justin

Research output: Contribution to journalArticle

3 Citations (Scopus)

Abstract

The energy levels of quantum systems are determined by quantization conditions. For one-dimensional anharmonic oscillators, one can transform the Schrödinger 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 Bm(E,g)=n+12, 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 Am(E,g). The evaluation of Am(E,g) by the method of asymptotic matching is discussed for the case of the cubic oscillator of degree m=3.

Original languageEnglish
Pages (from-to)1332-1341
Number of pages10
JournalApplied Numerical Mathematics
Volume60
Issue number12
DOIs
Publication statusPublished - Dec 2010

Fingerprint

Logarithmic Derivative
Anharmonic Oscillator
Characteristic Function
Wave functions
Wave Function
Derivatives
Contour integral
Quantization
Evaluation
Instantons
Energy Levels
Quantum Systems
Electron energy levels
Pole
Poles
Transform
Higher Order
Energy
Form

Keywords

  • General quantum mechanics and problems of quantization
  • Semiclassical techniques including WKB and Maslov methods
  • Singular perturbations
  • Turning point theory
  • WKB methods

ASJC Scopus subject areas

  • Applied Mathematics
  • Computational Mathematics
  • Numerical Analysis

Cite this

Calculation of the characteristic functions of anharmonic oscillators. / Jentschura, U.; Zinn-Justin, Jean.

In: Applied Numerical Mathematics, Vol. 60, No. 12, 12.2010, p. 1332-1341.

Research output: Contribution to journalArticle

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