Quantum dynamics in phase space: Moyal trajectories 2

Abstract

Continuing a previous paper [G. Braunss, J. Phys. A: Math. Theor. 43, 025302 (2010)10.1088/1751-8113/43/2/025302] where we had calculated ℏ2-approximations of quantum phase space viz. Moyal trajectories of examples with one and two degrees of freedom, we present in this paper the calculation of ℏ2-approximations for four examples: a two-dimensional Toda chain, the radially symmetric Schwarzschild field, and two examples with three degrees of freedom, the latter being the nonrelativistic spherically Coulomb potential and the relativistic cylinder symmetrical Coulomb potential with a magnetic field H. We show in particular that an ℏ2-approximation of the nonrelativistic Coulomb field has no singularity at the origin (r = 0) whereas the classical trajectories are singular at r = 0. In the third example, we show in particular that for an arbitrary function γ(H, z) the expression β ≡ pz + γ(H, z) is classically (ℏ = 0) a constant of motion, whereas for ℏ ≠ 0 this holds only if γ(H, z) is an arbitrary polynomial of second order in z. This statement is shown to extend correspondingly to a cylinder symmetrical Schwarzschild field with a magnetic field. We exhibit in detail a number of properties of the radially symmetric Schwarzschild field. We exhibit finally the problems of the nonintegrable Hénon-Heiles Hamiltonian and give a short review of the regular Hilbert space representation of Moyal operators.