The quantum theory of a harmonic oscillator with a time dependent frequencyarises in several important physical problems, especially in the study ofquantum field theory in an external background. While the mathematics of thissystem is straightforward, several conceptual issues arise in such a study. Wepresent a general formalism to address some of the conceptual issues like theemergence of classicality, definition of particle content, back reaction etc.In particular, we parametrize the wave function in terms of a complex number(which we call excitation parameter) and express all physically relevantquantities in terms it. Many of the notions -- like those of particle numberdensity, effective Lagrangian etc., which are usually defined using asymptoticin-out states -- are generalized as time-dependent concepts and we show thatthese generalized definitions lead to useful and reasonable results. Havingdeveloped the general formalism we apply it to several examples. Exact analyticexpressions are found for a particular toy model and approximate analyticsolutions are obtained in the extreme cases of adiabatic and highlynon-adiabatic evolution. We then work out the exact results numerically for avariety of models and compare them with the analytic results andapproximations. The formalism is useful in addressing the question of emergenceof classicality of the quantum state, its relation to particle production andto clarify several conceptual issues related to this. In Paper II(arXiv:0708.1237), which is a sequel to this, the formalism will be applied toanalyze the corresponding issues in the context of quantum field theory inbackground cosmological models and electric fields.
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