949 A systematic way of treating a general time-dependent harmonic oscillator in classical and quantum mechanics is given. By a general canonical transformation in classical mechanics, the time-dependent Hamiltonian can be transformed to a time-independent one (the Lewis-Riesenfeld invariant), explicitly separating a total time-derivative term. In quantum mechanics, one can obtain the phase of the wave function straightforwardly from this total-derivative term, together with the corresponding

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... me-independent Hamiltonian. We solve analytically and algebraically the general time-dependent harmonic oscillator driven by a time-dependent inverse cubic force as an example. § 1. Introduction The harmonic oscillator is a simple and useful physical system widely applicable in both classical and quantum mechanics. Almost all textbooks of classical mechanics treat harmonic oscillators even with damping effects taken into account (see, e.g., Ref. 1)). On the other hand, in quantum mechanics, cases more general than time-independent harmonic oscillators are not usually treated at the textbook level. In some advanced texts, however, for example, Ref. 2), this topic is treated. There still remains much debate concerning the applicability of the phenomenological one-body quantum damped harmonic oscillator (Refs. 3)~5) for related literature), but in this paper we do not touch upon this topic. We also do not discuss the case of harmonic oscillators interacting with a fluctuating background, although it is very interesting and important (see, e.g., Refs. 6) and 7))_ The problem of a quantum harmonic oscillator with time-dependent frequency (see Ref. 8) for earlier references) was solved by Husimi 9 > in 1953 using a Gaussian ansatz for the wave function. In 1969, Lewis and Riesenfeld 10 >,u> also solved this problem with an ansatz for a particular invariant (sometimes called the Lewis-or Lewis-Riesenfeld invariant). This invariant can be shown to be essentially an integral of motion in terms of suitable dynamical variables. Such invariants are also investigated using the Lie theory of the symmetry group (for the equation of motion) of harmonic oscillators 12 H 4 > or using the N oether Theorem. 15 > Solvability of these harmonic oscillators is closely related to the Lie algebrac-su(1, !)-structure of the system (see, for instance, Ref. 16)). Generalizations of these invariants are proposed in, for example, Ref. 17) (see Ref. 18) for the references of the invariants). Authors of these papers found that these invariants correspond mathematically to the Ermakov-invariant for a system of coupled differential equations. We shall make it clear (in § 2) that, within the context of physics, the Ermakov-invariant and the Downloaded from https://academic.oup.com