Exact evolution of the generalized damped harmonic oscillator

In this paper we use the algebraic and the invariant method to study the ti me-dependent damped harmonic oscillator from classical and quantum points o f view. The solution of the classical equation of motion and the wavefuncti on solving the time-dependent Schrodinger equation are found explicitly. We show that the original time-dependent quantum-mechanical problem is comple tely related to the well known time-independent harmonic oscillator. In add ition, we elucidate the intimate connection between the damped harmonic osc illator (DHO) and the generalized harmonic oscillator (GHO). More important ly, the evolution of the states of the DHO cannot be cyclic, in contradisti nction with the states of the GHO. Explicit expressions for both the dynami cal and the geometric angles and phases are deduced in the adiabatic limit. The coherent states describing the invariant-angle variables of the classi cal DHO are constructed; they allow us to recover the classical evolution a nd invariant from the quantum evolution.