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

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 fast and a 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 = √ h̄c 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. short title: Adiabatic motion in a 2-dimensional magnetic field PACS: 03.65.-w, 02.90.+p