Quantum mechanics of a general driven time-dependent oscillator

Chi‐Fai Lo

Carl J. Ballhausen

Zoltán Papp

H. Kanasugi, Hidehiko Okada

A systematic way of treating a general time-dependent harmonic oscillator in classical and quantum mechanics is given. By a general canonical transformation in classical mechanics, the time-dependent Hamiltonian can be transformed to a time-independent one (the Lewis-Riesenfeld invariant), explicitly separating a total time-derivative term. In quantum mechanics, one can obtain the phase of the wave function straightforwardly from this total-derivative term, together with the corresponding time-independent Hamiltonian. We solve analytically and algebraically the general time-dependent harmonic oscillator driven by a time-dependent inverse cubic force as an example.

Lin Zhang, Weiping Zhang

A variety of dynamics in this world can be approximately treated by a model: a driven and damped parametric oscillator. An intensive investigate of this time-dependent model from both classical and quantum mechanical points of view provides a dynamical insight to resolve the classical and quantum dynamics in order to understand the more complicated time-dependent evolutions not only happens in the macroscopic classical scale for the synchronized behaviors but also in the coherent microscopic scale for a quantum state control. By using a Lie transformation on a general time-dependent quadratic Hamiltonian, we exactly solve the dynamical behaviors of the driven and damped parametric oscillator to give a universal physical mechanism for the classical parametric control on the quantum state evolution in a dissipation system.