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Journal of Physics A Mathematical and General

The quantum damped driven harmonic oscillator

Chung‐In Um, Kyu Hwang Yeon, W. H. Kahng

Published Feb 21, 1987• 48 citations
Publisher

Abstract

Using the Caldirola-Kanai Hamiltonian with an external driving force for the damped driven harmonic oscillator as the quantum dissipative system, the authors have exactly evaluated the propagator, wavefunction, uncertainty relation and the transition amplitudes by the Feynman path integral method.
DOI
10.1088/0305-4470/20/3/024

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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.
The cases of under-, critical- and over-damping are treated for the quantum driven harmonic oscillator. Following a survey of the classical version, the quantum invariant operator for each case is constructed and their eigenvalues are evaluated. Using these eigenvalues and the Schrödinger equations, the wavefunctions are obtained for each case. From formal path integral theory, their propagators are evaluated and are checked with those obtained by the closed property formed by the complete set of wavefunctions.