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Annals of Physics

The exact solution for the generalized time-dependent harmonic oscillator and its adiabatic limit

Xiao-Chun Gao, Jing-Bo Xu, Tiezheng Qian

Published Oct 1, 1990• 2 citations
Publisher
DOI
10.1016/0003-4916(90)90035-m

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Journal of Physics A Mathematical and General•Aug 25, 2000•25 citations
The Lewis' invariant and exact solution for the driven generalized time-dependent harmonic oscillator is found by making use of the Lewis-Riesenfeld quantum theory. Then, the adiabatic asymptotic limit of the exact solution is discussed and the Berry's phase for this system is obtained. We then proceed to use the exact solution to construct the coherent state and calculate the corresponding exact classical phase angle. This phase angle can give the Hannay's angle in the adjabatic limit. The relation between the exact Lewis' phase and the corresponding classical phase angle is discussed.
In this paper we use the algebraic and the invariant method to study the time-dependent damped harmonic oscillator from classical and quantum points of view. The solution of the classical equation of motion and the wavefunction solving the time-dependent Schrödinger equation are found explicitly. We show that the original time-dependent quantum-mechanical problem is completely related to the well known time-independent harmonic oscillator. In addition, we elucidate the intimate connection between the damped harmonic oscillator (DHO) and the generalized harmonic oscillator (GHO). More importantly, the evolution of the states of the DHO cannot be cyclic, in contradistinction with the states of the GHO. Explicit expressions for both the dynamical and the geometric angles and phases are deduced in the adiabatic limit. The coherent states describing the invariant-angle variables of the classical DHO are constructed; they allow us to recover the classical evolution and invariant from the quantum evolution.