Hamiltonian point of view of quantum perturbation theory

TL;DR

This paper establishes a direct connection between quantum perturbation theory and classical Hamiltonian systems, enabling new computational methods for quantum geometric phases through classical tools.

Contribution

It demonstrates that Van Vleck-Primas quantum perturbation theory can be exactly reformulated as a classical perturbation problem for finite-dimensional systems, without relying on conceptual similarities.

Findings

Quantum perturbation theory can be recast as a classical Hamiltonian problem.

New methods for calculating quantum geometric phases are introduced.

The approach bridges quantum and classical perturbation theories explicitly.

Abstract

We explore the relation of Van Vleck-Primas perturbation theory of quantum mechanics with the Lie-series-based perturbation theory of Hamiltonian systems in classical mechanics. In contrast to previous works on the relation of quantum and classical perturbation theories, our approach is not based on the conceptual similarities between the two methods. Instead, we show that for quantum systems with a finite-dimensional Hilbert space, the Van Vleck-Primas procedure can be recast exactly into a classical perturbation problem. As a non-obvious consequence, this approach gives a new way of calculating the geometric phase of quantum systems using tools from the theory of classical canonical transformations