Quantum dynamics in phase space: Moyal trajectories 2

Journal Article · Tue Jan 15 00:00:00 EST 2013 · Journal of Mathematical Physics

DOI:https://doi.org/10.1063/1.4773229· OSTI ID:22162705

Braunss, G. ^[1]

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1. Mathematisches Institut der Justus-Liebig-Universitaet, Arndtstrasse 2, D 35392 Giessen (Germany)

Continuing a previous paper [G. Braunss, J. Phys. A: Math. Theor. 43, 025302 (2010)] where we had calculated Planck-Constant-Over-Two-Pi {sup 2}-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 Planck-Constant-Over-Two-Pi {sup 2}-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 Planck-Constant-Over-Two-Pi {sup 2}-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 {gamma}(H, z) the expression {beta}{identical_to}p{sub z}+{gamma}(H, z) is classically ( Planck-Constant-Over-Two-Pi = 0) a constant of motion, whereas for Planck-Constant-Over-Two-Pi {ne} 0 this holds only if {gamma}(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 Henon-Heiles Hamiltonian and give a short review of the regular Hilbert space representation of Moyal operators.

• OSTI ID: 22162705
• Journal Information: Journal of Mathematical Physics, Vol. 54, Issue 1; Other Information: (c) 2013 American Institute of Physics; Country of input: International Atomic Energy Agency (IAEA); ISSN 0022-2488
• Country of Publication: United States
• Language: English

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
APPROXIMATIONS
COULOMB FIELD
DEGREES OF FREEDOM
HAMILTONIANS
HILBERT SPACE
MAGNETIC FIELDS
PHASE SPACE
POLYNOMIALS
POTENTIALS
QUANTUM MECHANICS
RELATIVISTIC RANGE
REVIEWS
SCHWARZSCHILD METRIC
SINGULARITY
SYMMETRY
TRAJECTORIES
TWO-DIMENSIONAL CALCULATIONS