Born-Oppenheimer approximation for an atom in constant magnetic fields

TL;DR

This paper develops a new reduction scheme for analyzing the quantum dynamics of an atom in constant magnetic fields, simplifying the Hamiltonian and confirming the atom's straight-line motion through asymptotic evolution analysis.

Contribution

It introduces a novel reduction method that yields a simplified Hamiltonian without vector potential terms, improving upon previous approaches.

Findings

Reduced Hamiltonian excludes vector potential terms.

Asymptotic expansion confirms straight motion of the atom.

Method provides a clearer understanding of atomic behavior in magnetic fields.

Abstract

We obtain a reduction scheme for the study of the quantum evolution of an atom in constant magnetic fields using the method developed by Martinez, Nenciu and Sordoni based on the construction of almost invariant subspace. In Martinez-Sordoni \cite{MaSo2} such a case is also studied but their reduced Hamiltonian includes the vector potential terms. In this paper, using the center of mass coordinates and constructing the almost invariant subspace different from theirs, we obtain the reduced Hamiltonian which does not include the vector potential terms. Using the reduced evolution we also obtain the asymptotic expantion of the evolution for a specific localized initial data, which verifies the straight motion of an atom in constatnt magnetic fields.