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

Abstract

In quantum mechanics, systems can be described in phase space in terms of the Wigner function and the starproduct operation. Quantum characteristics, which appear in the Heisenberg picture as the Weyl's symbols of operators of canonical coordinates and momenta, can be used to solve the evolution equations for symbols of other operators acting in the Hilbert space. To any fixed order in the Planck's constant, many-body potential scattering problem simplifies to a statistical-mechanical problem of computing an ensemble of quantum characteristics and their derivatives with respect to the initial canonical coordinates and momenta. The reduction to a system of ordinary differential equations pertains rigorously at any fixed order in . We present semiclassical expansion of quantum characteristics for many-body scattering problem and provide tools for calculation of average values of time-dependent physical observables and cross sections. The method of quantum characteristics admits the consistent incorporation of specific quantum effects, such as nonlocality and coherence in propagation of particles, into the semiclassical transport models. We formulate the principle of stationary action for quantum Hamilton's equations and give quantum-mechanical extensions of the Liouville theorem on conservation of the phase-space volume and the Poincaré theorem on conservation of 2p-forms. The lowest order quantum corrections to the Kepler periodic orbits are constructed. These corrections show the resonance behavior.

Wigner Distribution Functions and the Representation of Canonical Transformations in Time-Dependent Quantum Mechanics

“…A subsequent Wigner transformation shows explicitly that for the timeindependent problems considered by this method, essentially the classical results are reproduced. This agrees with the fact that at least for quadratic Hamiltonians, the Wigner function evolves as W (x , p , t) = W (x M (x , p , −t), p M (x , p , −t), 0), where x M and p M are the Moyal time evolution of position and momentum which, again for quadratic Hamiltonians, coincide with the classical evolution [2,3,4,5].…”

Comment on “Dynamics of nuclear fluid. VIII. Time-dependent Hartree-Fock approximation from a classical point of view”

“…The first-order solutions are helpful to determine the phase-space Green function in the framework of the phase-space path integral method. Quantum trajectories exist in the de Broglie -Bohm theory [1,2,3] and appear in the framework of the deformation quantization [4,5,6,7,8,9] as the Weyl's symbols of the Heisenberg operators of canonical coordinates and momenta [10,11,12,13,14]. In the de Broglie -Bohm theory, the particle trajectories play an important role in the interpretation of measurements, whereas within the deformation quantization framework quantum trajectories have properties assigned to characteristic lines of first-order partial differential equations (PDE).…”