Class of invariants for the two-dimensional 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 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 phi(lambda) of L, I. We then determine time-dependent phases alpha(lambda) (t) such that the psi(lambda) (t) = e(i alpha lambda)phi(lambda) are solutions of the time-dependent Schrodinger 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) equivalent to 0. By a mere redefinition of the parameters, these results can be applied also to the analogous models on the canonical non-commutative plane. (C) 2011 American Institute of Physics. [doi: 10.1063/1.3653486]