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American Journal of Physics

Note on the time-dependent damped and forced harmonic oscillator

P. G. L. Leach

Published Dec 1, 1978• 16 citations
Publisher

Abstract

A Hamiltonian for the time-dependent damped and forced harmonic oscillator is derived. A simple time-dependent linear canonical transformation transforms the Hamiltonian to one whose solution is readily obtained. The wave function for the corresponding quantum mechanical problem is given.
DOI
10.1119/1.11388

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The time-dependent Schrdinger equation for the harmonic oscillator with time-dependent frequency ω=ω(t) and mass m=m(0)ω(0)ω(t)exp[2c∫t0ω(t)dt] can be solved by means of simple transformation and separation of variables.The wave functions are obtained for c21,c2=1 and c21,respectively.When |x|→∞,the wave function for c21 tends to zero but it is not so for c2≥1.Finally,some special cases are given for the time-dependent harmonic oscillator and one of them is similar to a damped harmonic oscillator.