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Journal of Physics A Mathematical and General

The quantum under-, critical- and over-damped driven harmonic oscillators

Kyu-Hwang Yeon, Suk-Sung Kim, Young-Mo Moon, Suc-Kyoung Hong, Chung‐In Um, Thomas F. George

Published Sep 10, 2001• 37 citations
Publisher

Abstract

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.

DOI
10.1088/0305-4470/34/37/321

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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.
For a damped harmonic oscillator forced by a time-dependent field, the exact wave function is obtained by three different methods: (i) path integral, (ii) second quantization, and (iii) dynamical invariant. The explicit form of the dynamical invariant involves a solution to a corresponding auxiliary equation. The coherent states, defined as eigenstates of a new destruction operator, form a nonorthogonal, overcomplete set and correspond to the minimum uncertainty states. These coherent states give the exact classical motion of the damped driven harmonic oscillator.
Using the invariant operator method and the unitary transformation method together, we obtained discrete and continuous solutions of the quantum damped driven harmonic oscillator. The wave function of the underdamped harmonic oscillator is expressed in terms of the Hermite polynomial while that of the overdamped harmonic oscillator is expressed in terms of the parabolic cylinder function. The eigenvalues of the underdamped harmonic oscillator are discrete while that of the critically damped and the overdamped harmonic oscillators are continuous. We derived the exact phases of the wave function for the underdamped, critically damped and overdamped driven harmonic oscillator. They are described in terms of the particular solutions of the classical equation of motion.
Using the equation of motion for a linearly damped harmonic oscillator as fundamental equation, a condition has been obtained that must be satisfied by its Lagrangian and a number of functional forms has been critically investigated. The Lagrangians that successfully describe a classical damped oscillator have been quantized using Feynman’s elegant path integral method. However, it is found that the expectation value of energy of the damped oscillator in all the cases becomes infinitely large for t→∞, ruling out the possibility of treating the damped quantum system as a one-body problem.
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.