Canonical Transformations for Time-Dependent Harmonic Oscillators

Abstract

A canonical transformation changes variables such as coordinates and momenta to new variables preserving either the Poisson bracket or the commutation relations depending on whether the problem is classical or quantal respectively. Classically canonical transformations are well established as a powerful tool for solving differential equations. Quantum canonical transformations have been defined and used relatively recently because of the non-commutativeness of the quantum variables. Three elementary canonical transformations and their composite transformations have quantum implementations. Quantum canonical transformations have been mostly used in time-independent Schrödinger equations and a harmonic oscillator with time-dependent angular frequency is probably the only time-dependent problem solved by these transformations. In this work, we apply quantum canonical transformations to a harmonic oscillator in which both angular frequency and equilibrium position are time-dependent.

Quantum features of a charged particle in ionized plasma controlled by a time-dependent magnetic field

“…We, in this work, may need to deal the problem of a time-dependent Hamiltonian system (TDHS) which is not easy to handle. There are several mathematical techniques available for rigorous quantum treatment of TDHSs, such as invariant operator method [5,6], reduction method [14], propagator method [15], and canonical transformation method [16]. Among them, we will use invariant operator method as mentioned in the introductory part.…”