Solving the von Neumann equation with time-dependent Hamiltonian. Part I: Method

TL;DR

This paper introduces a Lie-algebraic method for explicitly constructing unitary operators in quantum systems with time-dependent Hamiltonians, clarifying the roles of Magnus and Wei-Norman expansions and offering simpler rotation representations.

Contribution

It develops a Lie-algebraic framework for solving the von Neumann equation with time-dependent Hamiltonians, connecting known expansions to rotation group representations.

Findings

Magnus and Wei-Norman expansions correspond to different rotation group representations

Euler angle representation simplifies the construction of unitary operators

Framework applies to SU(2) and SU(1,1) Lie algebras

Abstract

The unitary operators U(t), describing the quantum time evolution of systems with a time-dependent Hamiltonian, can be constructed in an explicit manner using the method of time-dependent invariants. We clarify the role of Lie-algebraic techniques in this context and elaborate the theory for SU(2) and SU(1,1). We show that the constructions known as Magnus expansion and Wei-Norman expansion correspond with different representations of the rotation group. A simpler construction is obtained when representing rotations in terms of Euler angles. The many applications are postponed to Part II of the paper.