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Acta Physica Sinica

AN EXACT WAVEFUNCTION OF DAMPED HARMONIC OSCILLATOR

Ling Rui-liang, Feng Jin-fu, 江苏省常熟高等师范专科学校物理系，常熟 215500

Published Jan 1, 1998• 2 citations
Publisher

Abstract

A damped harmonic oscillator was studied using canonical tramsformation and starting from the method of path integrals.An exact wavefunction for the damped harmonic oscillator has been obtained from a Gaussiantype propagator.The fluctuations of displacement and momentum at zero-point are also given.
DOI
10.7498/aps.47.1952

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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.
Assuming a damping force proportional to the velocity, it is shown that the usual quantum condition between the physical variables, namely the displacement and the momentum, has to be modified by the damping. However, it is possible to change to new variables which are canonical. Then we can apply the usual quantum mechanics and use the usual physical interpretation through the statistical average values. The case of a one-dimensional damped harmonic oscillator under the action of an arbitrary external force has been worked out as an example. No difficulty arises and the results are reasonable.