The Poincare-Birkhoff theorem in Quantum Mechanics

TL;DR

This paper explores how the classical Poincaré-Birkhoff theorem manifests in quantum systems, showing that quantum states reflect classical resonant structures through quasiprobability functions.

Contribution

It demonstrates the quantum analogs of classical resonant tori and phase space structures, linking classical resonance phenomena to quantum state interactions.

Findings

Quantum states mirror classical resonant tori.

Quasiprobability densities reveal classical phase space structures.

Quantum manifestations occur at states differing by the resonance order.

Abstract

Quantum manifestations of the dynamics around resonant tori in perturbed Hamiltonian systems, dictated by the Poincar\'e--Birkhoff theorem, are shown to exist. They are embedded in the interactions involving states which differ in a number of quanta equal to the order of the classical resonance. Moreover, the associated classical phase space structures are mimicked in the quasiprobability density functions and their zeros.