Weyl's symbols of Heisenberg operators of canonical coordinates and momenta as quantum characteristics

Journal Article · Tue May 15 04:00:00 EDT 2007 · Journal of Mathematical Physics

DOI:https://doi.org/10.1063/1.2735816· OSTI ID:20929701

Krivoruchenko, M I; Faessler, Amand ^[1]

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1. Institute for Theoretical and Experimental Physics, B. Cheremushkinskaya 25, 117259 Moscow (Russian Federation) and Institut fuer Theoretische Physik, Tuebingen Universitaet, Auf der Morgenstelle 14, D-72076 Tuebingen (Germany)

The knowledge of quantum phase flow induced under Weyl's association rule by the evolution of Heisenberg operators of canonical coordinates and momenta allows to find the evolution of symbols of generic Heisenberg operators. The quantum phase flow curves obey the quantum Hamilton equations and play the role of characteristics. At any fixed level of accuracy of semiclassical expansion, quantum characteristics can be constructed by solving a coupled system of first-order ordinary differential equations for quantum trajectories and generalized Jacobi fields. Classical and quantum constraint systems are discussed. The phase-space analytic geometry based on the star-product operation can hardly be visualized. The statement 'quantum trajectory belongs to a constraint submanifold' can be changed, e.g., to the opposite by a unitary transformation. Some of the relations among quantum objects in phase space are, however, left invariant by unitary transformations and support partly geometric relations of belonging and intersection. Quantum phase flow satisfies the star-composition law and preserves Hamiltonian and constraint star functions.

• OSTI ID: 20929701
• Journal Information: Journal of Mathematical Physics, Journal Name: Journal of Mathematical Physics Journal Issue: 5 Vol. 48; ISSN JMAPAQ; ISSN 0022-2488
• Country of Publication: United States
• Language: English

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
CANONICAL DIMENSION
DIFFERENTIAL EQUATIONS
GEOMETRY
HAMILTONIANS
HILBERT SPACE
MATHEMATICAL EVOLUTION
PHASE SPACE
QUANTUM MECHANICS
SEMICLASSICAL APPROXIMATION
TRANSFORMATIONS