Class of invariants for the 2D time-dependent Landau problem and harmonic oscillator in a magnetic field

We consider an isotropic two dimensional harmonic oscillator with arbitrarily time-dependent mass $M(t)$ and frequency $Ω(t)$ in an arbitrarily time-dependent magnetic field $B(t)$. We determine two commuting invariant observables (in the sense of Lewis and Riesenfeld) $L,I$ in terms of some solution of an auxiliary ordinary differential equation and an orthonormal basis of the Hilbert space consisting of joint eigenvectors $ϕ_λ$ of $L,I$. We then determine time-dependent phases $α_λ(t)$ such that the $ψ_λ(t)=e^{iα_λ}ϕ_λ$ are solutions of the time-dependent Schrödinger equation and make up an orthonormal basis of the Hilbert space. These results apply, in particular to a two dimensional Landau problem with time-dependent $M,B$, which is obtained from the above just by setting $Ω(t) \equiv 0$. By a mere redefinition of the parameters, these results can be applied also to the analogous models on the canonical non-commutative plane.

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