Phase space formulations of the quantum electronic motion in a magnetic field

P. Filipowicz, Jan Mostowski

Katherine Newton, J.D.F. Franklin

A. O. Barut, Anthony J. Bracken

For a free Dirac electron, the Heisenberg equations define an internal dynamical system in the rest frame, isomorphic to a finite three-dimensional oscillator with a compact SO(5) phase space, such that the spin of the electron is the orbital angular momentum of the internal dynamics (Barut and Bracken 1980, 1981a). In the present work, the change in this internal dynamics due to an external magnetic field is studied. In order that the internal motion can be distinguished from the centre of mass motion, the solutions of the corresponding Hamilton and Heisenberg equations for the relativistic classical motion and the relativistic quantum mechanical spinless motion are also presented. The solutions for the electron exhibit the effect of the spin terms both in the internal motion and external motion, and we are able to identify the properties of the Zitterbewegung in the external field.

Anatoli Polkovnikov

M. Brack, Rajat K. Bhaduri

This chapter explains why Hamilton's principal function plays such an important role in the quantum propagator. It motivates the reader by touching on the fundamental aspects, as well as by describing some of the exciting experiments in the field. The chapter describes the classical motion of the electron in the magnetic field, displaying its ergodic character on what is called a 'Poincaré surface of section' of the phase-space. It discusses three recent experiments involving the transport properties of electrons in high-mobility semiconductor microstructures that show the imprint of chaotic orbits in conductance measurements. The connection of the old quantum theory to quantum mechanics is limited to the role of the periodic orbits in the quantization condition. In post-Bohr quantum mechanics, formulated by the matrix mechanics of Heisenberg and the wave equation of Schrödinger, the connection to classical periodic orbits is not at all apparent.

T. S. Monteiro

The first study of the quantum dynamics for a highly excited ('Rydberg') H 2 molecule in a magnetic field is presented and compared with the well studied atomic case. Wavefunctions from an accurate quantal calculation have been used to obtain the corresponding Wigner functions by means of a transformation to a basis where the latter are analytical. Unlike the atom, the dynamics of the excited electron in the molecular case experiences two, interfering, energy regimes. This study consider the case near electronic scaled energies -0.5 and -2.3, which would correspond to predominantly regular motion in the atomic case. In the molecular case there is evidence for chaotic behaviour in this energy range. Quantum states for the molecule may divide into those where the underlying dynamics is a perturbed version of the hydrogen atom case, and others where different configuration-space and phase-space structures are apparent.