Time evolution of two-dimensional quadratic Hamiltonians: A Lie algebraic approach

TL;DR

This paper introduces a Lie algebraic method to compute the evolution operator for two-dimensional quadratic Hamiltonians with time-dependent parameters, enabling systematic analysis of their quantum dynamics.

Contribution

It presents a general Lie algebraic framework applicable to a broad class of Hamiltonians with large dynamical algebras, demonstrated through electromagnetic field examples.

Findings

Derived explicit propagator for the 2D quadratic Hamiltonian.

Calculated Heisenberg picture operators for position and momentum.

Method applicable to various Hamiltonians with large dynamical algebras.

Abstract

We develop a Lie algebraic approach to systematically calculate the evolution operator of the generalized two-dimensional quadratic Hamiltonian with time-dependent coefficients. Although the development of the Lie algebraic approach presented here is mainly motivated by the two-dimensional quadratic Hamiltonian, it may be applied to investigate the evolution operators of any Hamiltonian having a dynamical algebra with a large number of elements. We illustrate the method by finding the propagator and the Heisenberg picture position and momentum operators for a two-dimensional charge subject to uniform and constant electro-magnetic fields.