Exact Solutions of Non-Autonomous Quantum Systems with Quadratic Hamiltonian**The project supported in part by National Natural Science Foundation of China and the Doctoral Education Fund of the State Education Commission of China, and the China Nuclear Industry Research Fund

Abstract

Based on algebraic dynamics, we present an algorithm to obtain exact solutions of the Schrödinger equation of non-autonomous quantum systems with Hamiltonian expressed in quadratic function of creation and annihilation operators of bosons. The Hamiltonian is treated as a linear function of generators of a symplectic group. Similar to the canonical transformation of classical dynamics, we employ a set of gauge transformations to gradually transform the Hamiltonian to a linear function of Cartan operators. The exact solutions are obtained by inverse gauge transformations. When the system is autonomous, this algorithm can obtain the normal mode of the Hamiltonian, as well as the eigenstates and eigenvalues.