Classical, semiclassical, and quantum mechanics of a globally chaotic system: Integrability in the adiabatic approximation

Abstract

We examine the classical, semiclassical, and quantum mechanics of the Hamiltonian H= 1/2 (p2x+p2y+x2y2). The dynamics of this system are globally chaotic. However, the classical and quantum mechanical problems can be solved analytically by assuming an adiabatic separation of the x and y motion. We construct the canonical transformation to adiabatic action–angle variables and investigate the connection between this integrable approximation and the exact dynamics. In addition, we present a simple semiclassical formula that predicts energy levels in excellent agreement with the exact energy spectrum. The quantum adiabatic potential curves of this system have a very unusual structure—infinitely many curves cross at one point.