Hydrogen atom as a quantum-classical hybrid system

TL;DR

This paper investigates a hybrid quantum-classical model of the hydrogen atom, revealing limitations in proton dynamics and momentum conservation, and compares it to full quantum descriptions.

Contribution

It demonstrates the shortcomings of mean-field hybrid models in accurately capturing proton dynamics and highlights the lack of momentum conservation as a fundamental issue.

Findings

Electron dynamics closely matches full quantum results.

Proton behaves as a free particle in hybrid model.

Hybrid approach fails to conserve momentum.

Abstract

Hydrogen atom is studied as a quantum-classical hybrid system, where the proton is treated as a classical object while the electron is regarded as a quantum object. We use a well known mean-field approach to describe this hybrid hydrogen atom; the resulting dynamics for the electron and the proton is compared to their full quantum dynamics. The electron dynamics in the hybrid description is found to be only marginally different from its full quantum counterpart. The situation is very different for the proton: in the hybrid description, the proton behaves like a free particle; in the fully quantum description, the wave packet center of the proton orbits around the center of mass. Furthermore, we find that the failure to describe the proton dynamics properly can be regarded as a manifestation of the fact that there is no conservation of momentum in the mean-field hybrid approach. We expect that such a failure is a common feature for all existing approaches for quantum-classical hybrid systems of Born-Oppenheimer type.