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

Abstract: We consider an isotropic two-dimensional harmonic oscillator with arbitrarily time-dependent mass M(t) and frequency Omega(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 = e^[iα_λ] ϕ_λ are solutions of the time-dependent Schroedinger 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 Omega(t) ≡ 0. By a mere redefinition of the parameters, these results can be applied also to the analogous models on the canonical non-commutative plane.