Adiabatic Motion of a Quantum Particle in a Two-Dimensional Magnetic Field

TL;DR

This paper investigates the adiabatic behavior of a quantum charged particle in a 2D magnetic field, developing operators to separate fast and slow dynamics, and deriving an effective Hamiltonian that aligns with classical physics.

Contribution

It introduces a set of generalized operators to distinguish between fast and slow degrees of freedom in a quantum system under a magnetic field, enabling a second-order effective Hamiltonian derivation.

Findings

Operators for separating dynamics are constructed.

The Hamiltonian is expressed as a power series in magnetic length.

The effective guiding center Hamiltonian matches classical limits.

Abstract

The adiabatic motion of a charged, spinning, quantum particle in a two - dimensional magnetic field is studied. A suitable set of operators generalizing the cinematical momenta and the guiding center operators of a particle moving in a homogeneous magnetic field is constructed. This allows us to separate the two degrees of freedom of the system into a {\sl fast} and a {\sl slow} one, in the classical limit, the rapid rotation of the particle around the guiding center and the slow guiding center drift. In terms of these operators the Hamiltonian of the system rewrites as a power series in the magnetic length $\lb=\sqrt{\hbar c\over eB}$ and the fast and slow dynamics separates. The effective guiding center Hamiltonian is obtained to the second order in the adiabatic parameter $\lb$ and reproduces correctly the classical limit.