Calculation of the Characteristic Functions of Anharmonic Oscillators
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 Bm(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 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.
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
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
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Article - Journal
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