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

Abstract

The knowledge of quantum phase flow induced under the 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's 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 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.

Semiclassical expansion of quantum characteristics for many-body potential scattering problem

“…The functional form of equations constructed with the use of the summation and the star-multiplication operations remains unchanged in all unitary equivalent coordinate systems. The star-product is not invariant under canonical transformations in general [19].…”

“…The formulation of quantum mechanics in phase space and the star-product are reviewed in [9][10][11][12][13][14][15]. The Stratonovich version [7] of the Weyl's quantization and dequantization is discussed in [14,[16][17][18][19][20]. Wigner functions have found numerous applications in quantum many-body physics, kinetic theory [21,22], collision theory, and quantum chemistry [23,24].…”

“…The notion of phase-space trajectories arises naturally in the deformation quantization through the Weyl's transform of Heisenberg operators of canonical coordinates and momenta [19,34,42,50]. These trajectories obey the Hamilton's equations in the quantum form and play the role of quantum characteristics in terms of which the time-dependent symbols of operators are expressed.…”

“…The lowest order quantum corrections to the Kepler periodic orbits are constructed. These corrections show the resonance behavior.In this work, we report the semiclassical expansion of quantum characteristics for many-body scattering problem, based on the method [19,34,42,50], and provide tools for calculation of average values of timedependent physical observables and scattering cross sections.We show that to any fixed order in it is sufficient to work with quantum characteristics, provided equations of motion and rules for calculation of the time dependent average values of physical observables are modified. The quantum evolution problem becomes thereby entirely identical with a statistical-mechanical problem.…”

“…In this work, we report the semiclassical expansion of quantum characteristics for many-body scattering problem, based on the method [19,34,42,50], and provide tools for calculation of average values of timedependent physical observables and scattering cross sections.…”