On Time-Dependent Quadratic Quantum Hamiltonians

Abstract

We apply the techniques of canonical transforms to equations of the type \[\left[ {A( t )\mathbb{P}^2 + B( t )\{ \mathbb{P}\mathbb{Q} + \mathbb{Q}\mathbb{P} \} + C( t )\mathbb{Q}^2 + D( t )\mathbb{Q} + E( t )\mathbb{P} + F( t )\mathbb{1}} \right]\psi ( q,t ) = - i\partial _t \psi ( q,t ),\]where $\mathbb{Q}$ and $\mathbb{P}$ are the quantum position and momentum operators. The time-dependent parameters of the ${\text{W}} \wedge {\text{SL}}( {{\text{2,}}R} )$ evolution operator are found through linear differential equations. In terms of these we give explicitly the Green’s function, all separating coordinates and similarity solutions of the equation. We analyze the behavior of Gaussian and coherent-state initial conditions in closed form and present a new interpretation of all the Lewis-Riesenfeld constants of motion.