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Many quantum mechanical [[Hamiltonian]]s are time dependent. How to solve problems where there is an explicit time dependence is an open subject nowadays. For these kind of problems it is of importance to look for constants of motion or [[Quantum invariants|invariants]]. For the (time dependent) [[harmonic oscillator]] it is possible to write several invariants, among them, the Ermakov–Lewis invariant which is developed below.

The [[time]] dependent [[harmonic oscillator]] Hamiltonian reads

: <math>\hat{H} =\frac{1}{2}\left[\hat{p}^2+\Omega^2(t)\hat{q}^2\right].</math>
It is well known that an [[Quantum invariant|invariant]] for this type of interaction
has the form

: <math>
\hat{I}=\frac{1}{2}\left[  \left( \frac{\hat{q}}{\rho}\right)
^{2}+(\rho\hat{p}-\dot{\rho}\hat{q})^{2}\right], </math>
where <math>\rho</math> obeys the Ermakov equation<ref>V.P. Ermakov, Univ. Izv. (Kiev) 20, 1 (1880)</ref>
:<math>  
\ddot{\rho}+\Omega^{2}\rho=\rho^{-3}.</math>
The above invariant is the so-called Ermakov-Lewis invariant.<ref>H.R. Lewis, Phys. Rev. Lett. 18, 510 (1967). http://link.aps.org/doi/10.1103/PhysRevLett.18.510</ref> It is easy to show that <math>\hat{I}</math> may be related to
the time independent harmonic oscillator Hamiltonian via a [[unitary transformation]] of the
form <ref>[[Héctor Manuel Moya Cessa|H. Moya-Cessa]] and M. Fernández Guasti,
Physics Letters A 311, 1 (2003). http://dx.doi.org/10.1016/S0375-9601(03)00461-4</ref>
:<math> \hat{T}=e^{i\frac{\ln\rho }{2}(\hat{q}\hat{p}+\hat{p}\hat{q}
)}e^{-i\frac{\dot{\rho}}{2\rho}\hat{q}^{2}}=
e^{i\frac{\ln\rho}{2}\frac{d\hat{q}^2}{dt}}
e^{-i\frac{\hat{q}^{2}}{2}\frac{d\ln\rho}{dt}},</math>
as
:<math>\frac{1}{2}\left[\hat{p}^2+\hat{q}^2\right]=\hat{T}\hat{I}\hat{T}^{\dagger}.</math>
This allows an easy form to express the solution of the [[Schrödinger equation]] for the time dependent [[Hamiltonian]].

The first [[exponential]] in the transformation is the so-called [[squeeze operator]].

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.

==References==
{{reflist}}

{{DEFAULTSORT:Ermakov-Lewis invariant}}
[[Category:Quantum mechanics]]