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Physics & Astronomy International Journal

Propagators for a particle in a time-dependent linear potential and a free particle

Surarit Pepore, R Biswas

Published Oct 19, 2021• 1 citation
Publisher

Abstract

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.

DOI
10.15406/paij.2021.05.00238

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The application of the integrals of the motion of a quantum system in deriving Green function or propagator is established. The Greenfunction is shown to be the eigenfunction of the integrals of the motion which described initial points of the system trajectory in the phasespace. The explicit expressions for the Green functions of the damped harmonic oscillator, the harmonic oscillator with strongly pulsatingmass, and the harmonic oscillator with mass growing with time are obtained in co-ordinate representations. The connection between theintegrals of the motion method and other method such as Feynman path integral and Schwinger method are also discussed.
The application for the integrals of the motion of a quantum system in deriving Green function or propagator is presented. The Green function is shown to be the eigenfunction of the integrals of the motion which described initial points of the system trajectory in the phase space. The exact expressions for the Green functions of the dual damped oscillators and the coupled harmonic oscillators are evaluated in co-ordinate representations. The relation between the integrals of the motion method and other methods such as Feynman path integral and Schwinger method are also presented.
Using the Caldirola-Kanai Hamiltonian for the quantum dissipative system, the author expresses the propagator as the modified Feynman path integral in the configuration space, which can then be evaluated for the damped harmonic oscillator by Montroll's method. The propagator of the damped harmonic oscillator can also be calculated beyond and at caustics with the help of Horvathy-Feynman formula. The new results are confirmed by investigating the classical paths joining two fixed end-point positions, Finally the author obtains the time-dependent wavefunctions from the propagator of the dynamical system.