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Science in China Series A-Mathematics Physics Astronomy & Technological Science

EXTENDED FEYNMAN FORMULA FOR HARMONIC OSCILLATOR AND ITS APPLICATION

Liang Jiu-Qing, Guo Ben--Hong, G. Morandi

Published Nov 20, 1991• 0 citations
Publisher

Abstract

We report on a new derivation of the extended Feynman propagator for the harmonic oscil-lator at and beyond caustics using the explicit eigenfunction expansion for the propagator.It is shown that the Maslov corrections beyond caustics are directly determined by the parityof the eigenfunctions. The phase correction and related effect of the propagator of a chargedparticle in constant magnetic field and in the presence of Chern-Simon backgraound gaugefield are studied.

DOI
10.1360/ya1991-34-11-1346

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We report on a new derivation of the extended Feynman propagator for the harmonic oscil-lator at and beyond caustics using the explicit eigenfunction expansion for the propagator.It is shown that the Maslov corrections beyond caustics are directly determined by the parityof the eigenfunctions. The phase correction and related effect of the propagator of a chargedparticle in constant magnetic field and in the presence of Chern-Simon backgraound gaugefield are studied.
Abstract The Maslov correction to the wave function is the jump of (−π2) \left( { - \frac{\pi } {2}} \right) (−2π​) in the phase when the system passes through a caustic. This can be explained by studying the second variation and the geometry of paths, as conveniently seen in Feynman’s path integral framework. The results can be extended to any system using the semiclassical approximation. The 1-dimensional harmonic oscillator is used to illustrate the different derivations reviewed here.
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.