Lie algebraic approach to quadratic Hamiltonians and the bi-dimensional charged particle in time-dependent electromagnetic field

TL;DR

This paper employs a Lie algebraic method to analyze quadratic Hamiltonians, including a charged particle in time-dependent electromagnetic fields, enabling explicit calculation of evolution operators and propagators.

Contribution

The paper develops a Lie algebraic framework to systematically solve quadratic Hamiltonians and extends it to charged particles in electromagnetic fields.

Findings

Identified a closed Lie algebra for general quadratic Hamiltonians.

Derived explicit forms of evolution operators and propagators.

Applied the method to charged particles in time-dependent electromagnetic fields.

Abstract

We discuss the one-dimensional, general quadratic Hamiltonian and the bi-dimensional charged particle in time-dependent electromagnetic fields through the Lie algebraic approach. Such method consists in finding a set of generators that form a closed Lie algebra in terms of which it is possible to express the Hamiltonian and the therefore the evolution operator. The evolution operator is then the starting point to obtain the propagator as well as the explicit form of the Heisenberg picture position and momentum operators. First, the set of generators forming a closed Lie algebra is identified for the general quadratic Hamiltonian. This algebra is later extended to study the the Hamiltonian of a charged particle in electromagnetic fields, given the similarities between the terms of these two Hamiltonians.