ABSTRACT
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.
