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Acta Sinica Quantum Optica

Studying Path Intergration Short-time Propgator of the Periodic Damp Harmonic Oscillator by Power Series Expansion Method

Xian-Ting Liang

Published Jan 1, 2009• 0 citations
OpenAlex

Abstract

In this paper we investigate the short-time propagator of the periodic damp harmonic oscillator by using Schwinger method and power series expansion technique which developed by Makri and her colleagues.It is found that the short-time propagators obtained by using the above two methods are nearly equal,but using power series expansion technique is much easier than Schwinger method.

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Physical Review Letters•Dec 11, 1995•12 citations
In this paper we investigate the short-time propagator of the periodic damp harmonic oscillator by using Schwinger method and power series expansion technique which developed by Makri and her colleagues.It is found that the short-time propagators obtained by using the above two methods are nearly equal,but using power series expansion technique is much easier than Schwinger method.
In this paper we obtain a propagator of path integral for a harmonic oscillator and a driven harmonic oscillator by using the power series expansion. It is shown that our result for the harmonic oscillator is more exact than the previous one obtained with other approximation methods. By using the same method, we obtain a propagator of path integral for the driven harmonic oscillator, which does not have any exact expansion. The more exact propagators may improve the path integral results for these systems.
The initial value problem method is formulated to calculate the propagator for time- dependent harmonic oscillators. The method is based on finding the initial position operator from Heisenberg equations. The investigated models in this paper are the damped harmonic oscillator, the harmonic oscillator with strongly pulsating mass, and the harmonic oscillator with mass growing with time. The comparison of the initial value problem method with Feynman path integral and Schwinger method is also described.
In this paper, the propagators for a particle moving in a time-dependent linear potential and a free particle with linear damping are calculated by the application of the integrals of the motion of a quantum system. The propagator for a charged harmonic oscillator is derived from the Feynman path integrals and the propagator for a damped harmonic oscillator is evaluated by the Schwinger method. The relation between the integrals of the motion, Feynman path integrals, and Schwinger method are also described.
In this paper, the propagator for a harmonic oscillator with mass obeying the function of m(t)=mtan2νt is derived by the Feynman path integral method. The wave function of this oscillator is calculated by expanding the obtained propagator. The propagator for a harmonic oscillator with strongly pulsating mass is evaluated by the Schwinger method. The propagator for a harmonic oscillator with mass rapidly growing with time is calculated by applying the integrals of the motion of quantum systems. The comparison between these methods are also discussed.
A simple framework for accurate solution of a general class of one-dimensional Fokker-Planck and/or Schr\"odinger equations is presented. The main idea is representing the propagator in the form P(x,t∣x0)=P0(x,t∣x0)exp[W(x,t∣x0)]P(x, t|{x}_{0})={P}_{0}(x, t|{x}_{0})\mathrm{exp}[W(x, t|{x}_{0})]P(x,t∣x0​)=P0​(x,t∣x0​)exp[W(x,t∣x0​)] and expanding the exponent WWW in a power series in a given function of ttt, where P0{P}_{0}P0​ is the exact solution of a reference harmonic-oscillator problem. The expansion coefficients are analytically evaluated from recursive relations. This approach is shown to be a dramatic improvement over the standard Taylor series expansion for the propagator in that just a few terms of the present expansion are sufficient to attain a very accurate description in the whole time domain.