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

TL;DR

This paper develops invariants for a 2D time-dependent harmonic oscillator in a magnetic field, providing explicit solutions and basis states, with applications to Landau problems and non-commutative planes.

Contribution

It introduces a method to find invariant observables and explicit solutions for a general time-dependent 2D harmonic oscillator in a magnetic field, including Landau and non-commutative models.

Findings

Derived two commuting invariants for the system.

Constructed explicit orthonormal basis of solutions.

Applicable to Landau and non-commutative plane models.

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 $\phi_\lambda$ 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 Schr\"odinger 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 $\Omega(t) \equiv 0$. By a mere redefinition of the parameters, these results can be applied also to the analogous models on the canonical non-commutative plane.