The quantization of a classically ergodic system

Abstract

The motion of a charged particle with an anisotropic mass tensor is completely ergodic in two dimensions. Its periodic orbits can be mapped 1-to-1 into the binary sequences of even length. Their action integral is approximated very closely by a quadratic function of the binaries with only one empirical parameter and one scale factor. The trace of the semiclassical Green's function is a sum over all periodic orbits, and can be evaluated by a transformation which Kac used in the discussion of an Ising lattice with long range interaction. The poles of the trace as a function of the energy can be segregated by their discrete symmetry. The real parts agree very well with the quantum mechanical energy levels, while most of the imaginary parts are small, indicating sharp resonances. This is the first calculation of energy levels on the basis of classically ergodic trajectories. © 1982.