Phase-space propagation and stability analysis of the 1-dimensional Schr\"odinger equation for finding bound and resonance states of rotationally excited H$_2$

TL;DR

This paper employs a phase-space approach to analyze the Schrödinger equation for H₂, revealing insights into bound and resonance states through geometric and topological properties of phase-space trajectories.

Contribution

It introduces a phase-space representation for the 1D Schrödinger equation to identify and analyze bound and resonance states of rotationally excited H₂, linking dynamics to quantum states.

Findings

Phase-space trajectories distinguish bound and resonance states.

Energy eigenvalues are obtained from phase-space trajectory properties.

Global energy-momentum diagram provides comprehensive state visualization.

Abstract

A mathematical phase-space representation of the 1-dimensional Schr\"odinger equation is employed to obtain bound and resonance states of the rotationally excited H$_2$ molecule. The structure of the phase-space tangent field is analyzed and related to the behavior of the wave function in classically allowed and forbidden regions. In this phase-space representation, bound states behave like unstable orbits meanwhile resonance states behave similarly to asymptotically stable cycles. The lattice of quantum states of the energy-momentum diagram for H$_2$ is calculated allowing to have a global view of the energy as function of the quantum numbers. The arc length and winding number of the phase-space trajectories, as functions of the energy, are used to obtain the energy eigenvalues of bound and resonance states of H$_2$