Many quantum mechanical Hamiltonians are time dependent. How to solve problems where there is an explicit time dependence is an open subject nowadays. For problems of this kind it is of importance to look for constants of motion or invariants. For the (time dependent) harmonic oscillator it is possible to write several invariants, among them, the Ermakov–Lewis invariant which is developed below.
It is well known that an invariant for this type of interaction has the form
where obeys the Ermakov equation
The above invariant is the so-called Ermakov–Lewis invariant. It is easy to show that may be related to the time independent harmonic oscillator Hamiltonian via a unitary transformation of the form 
This approach may allow to simplify problems such as the Quadrupole ion trap, where an ion is trapped in a harmonic potential with time dependent frequency. The transformation presented here is then useful to take into account such effects.
- V.P. Ermakov, Univ. Izv. (Kiev) 20, 1 (1880)
- H.R. Lewis, Phys. Rev. Lett. 18, 510 (1967). http://link.aps.org/doi/10.1103/PhysRevLett.18.510
- Moya-Cessa, H.; Guasti, M. Fernández. "Coherent states for the time dependent harmonic oscillator: the step function". Physics Letters A. 311: 1–5. doi:10.1016/S0375-9601(03)00461-4.