Time-dependent coupled harmonic oscillators

In this paper we use a canonical and a unitary transformation along with the Lewis and Riesenfeld invariant method to obtain the quantum states of a general system of two time-dependent statically coupled harmonic oscillators. The wave function obtained is written in terms of a c-number quantity, which is solution of the Milne-Pinney equation. We consider three different systems of time-dependent coupled oscillators, and for each one we solve the respective Milne-Pinney equation and discuss how the uncertainty product evolves with time.