Methods to solve problems where there is an explicit time dependence is an open subject nowadays.
The time dependent harmonic oscillator Hamiltonian reads It is well known that an invariant for this type of interaction has the form where
may be related to the time independent harmonic oscillator Hamiltonian via a unitary transformation of the form [2] as This allows an easy form to express the solution of the Schrödinger equation for the time dependent Hamiltonian.
The geometric meaning of this invariant can be realized within the quantum phase space.
[5] In 1966, Ralph Lewis rediscovered the invariant using Kruskal's asymptotic method.