Time evolution of quadratic quantum systems: Evolution operators, propagators, and invariants

We use the evolution operator method to describe time-dependent quadratic quantum systems in the framework of nonrelativistic quantum mechanics. For simplicity, we consider a free particle with a variable mass $M(t)$, a particle with a variable mass $M(t)$ in an alternating homogeneous field, and a harmonic oscillator with a variable mass $M(t)$ and frequency $\omega(t)$ subject to a variable force $F(t)$. To construct the evolution operators for these systems in an explicit disentangled form, we use a simple technique to find the general solution of a certain class of differential and finite-difference nonstationary Schrödinger-type equations of motion and also the operator identities of the Baker–Campbell–Hausdorff type. With known evolution operators, we can easily find the most general form of the propagators, invariants of any order, and wave functions and establish a unitary relation between systems. Results known in the literature follow from the obtained general results as particular cases.