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Physical Review A

Semiclassical Moyal quantum mechanics for atomic systems

B. R. McQuarrie, T. A. Osborn, George C. Tabisz

Published Oct 1, 1998• 18 citations
Publisher

Abstract

The Moyal formalism utilizes the Wigner transform and associated Weyl calculus to define a phase-space representation of quantum mechanics. In this context, the Weyl symbol image of the Heisenberg evolution operator admits a generic semiclassical expansion that is based on classical transport and related $O({\ensuremath{\Elzxh}}^{2})$ quantum corrections. For two atom systems with a mutual pair interaction described by a spherically symmetric potential, the predictive power and convergence properties of this semiclassical expansion are investigated via numerical calculation. The rotational invariance and tensor structure present are used to simplify the semiclassical dynamics to the point where numerical computation in the six-dimensional phase space is feasible. For a variety of initial Gaussian wave functions and a selection of different observables, the $O({\ensuremath{\Elzxh}}^{0})$ and $O({\ensuremath{\Elzxh}}^{2})$ approximations for time dependent expectation values are determined. The interactions used are the Lennard-Jones potentials, which model helium, neon, and argon. The numerical results obtained provide a first demonstration of the practicality and usefulness of Moyal quantum mechanics in the analysis of realistic atomic systems.

DOI
10.1103/physreva.58.2944

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The Moyal formalism utilizes the Wigner transform and associated Weyl calculus to define a phase-space representation of quantum mechanics. In this context, the Weyl symbol image of the Heisenberg evolution operator admits a generic semiclassical expansion that is based on classical transport and related O(\ensuremath\Elzxh2)O({\ensuremath{\Elzxh}}^{2})O(\ensuremath\Elzxh2) quantum corrections. For two atom systems with a mutual pair interaction described by a spherically symmetric potential, the predictive power and convergence properties of this semiclassical expansion are investigated via numerical calculation. The rotational invariance and tensor structure present are used to simplify the semiclassical dynamics to the point where numerical computation in the six-dimensional phase space is feasible. For a variety of initial Gaussian wave functions and a selection of different observables, the O(\ensuremath\Elzxh0)O({\ensuremath{\Elzxh}}^{0})O(\ensuremath\Elzxh0) and O(\ensuremath\Elzxh2)O({\ensuremath{\Elzxh}}^{2})O(\ensuremath\Elzxh2) approximations for time dependent expectation values are determined. The interactions used are the Lennard-Jones potentials, which model helium, neon, and argon. The numerical results obtained provide a first demonstration of the practicality and usefulness of Moyal quantum mechanics in the analysis of realistic atomic systems.
The Moyal description of quantum mechanics, based on the Wigner-Weyl isomorphism between operators and symbols, provides a comprehensive phase space representation of dynamics. The Weyl symbol image of the Heisenberg picture evolution operator is regular in ħ and so presents a preferred foundation for semiclassical analysis. Its semiclassical expansion "coefficients," acting on symbols that represent observables, are simple, globally defined (phase space) differential operators constructed in terms of the classical flow. The first of two presented methods introduces a cluster-graph expansion for the symbol of an exponentiated operator, which extends Groenewold′s formula for the Weyl product of two symbols and has ħ as its natural small parameter. This Poisson bracket based cluster expansion determines the Jacobi equations for the semiclassical expansion of "quantum trajectories." Their Green function solutions construct the regular ħ ⇓ 0 asymptotic series for the Heisenberg-Weyl evolution map. The second method directly substitutes such a series into the Moyal equation of motion and determines the ħ coefficients recursively. In contrast to the WKB approximation for propagators, the Heisenberg-Weyl description of evolution involves no essential singularity in ħ, no Hamilton-Jacobi equation to solve for the action, and no multiple trajectories, caustics, or Maslov indices.
We formulate various semiclassical propagators for the Wigner phase space representation from a unified point of view. As is shown in several studies, the Moyal equation, which is an equation of motion for the Wigner distribution function, can be regarded as the Schrödinger equation of an extended Hamiltonian system where its "position" and "momentum" correspond to the middle point of two points of the original phase space and the difference between them, respectively. Then we show that various phase-space semiclassical propagators can be formulated just by applying existing semiclassical propagators to the extended system. As a result, a phase space version of the Van Vleck propagator, the initial-value Van Vleck propagator, the Herman-Kluk propagator, and the thawed Gaussian approximation are obtained. In addition, we numerically compare the initial-value phase-space Van Vleck propagator, the phase-space Herman-Kluk propagator, and the classical mechanical propagation as approximation methods for the time propagation of the Wigner distribution function in terms of both accuracy and convergence speed. As a result, we find that the convergence speed of the Van Vleck propagator is far slower than others as is the case of the Hilbert space, and the Herman-Kluk propagator keeps its accuracy for a long period compared with the classical mechanical propagation while the convergence speed of the latter is faster than the former.
In quantum mechanics, systems can be described in phase space in terms of the Wigner function and star-product 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 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 initial canonical coordinates and momenta. The reduction to a system of ordinary differential equations pertains rigorously at any fixed order in hbar\\hbarhbar. 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. The method of quantum characteristics admits the consistent incorporation of specific quantum effects, such as non-locality and coherence in propagation of particles, into semiclassical transport models.
In quantum mechanics, systems can be described in phase space in terms of the Wigner function and the star-product 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 ħ. 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 non-locality 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.
This chapter presents Wigner’s approach to quantum mechanics, based on the Wigner function in phase space. It explains Wigner–Weyl quantization, which makes it possible to associate functions on phase space to wave functions and operators, and it develops the technology to do quantum mechanics in this formalism. This includes the star product, Moyal evolution, and star-eigenvalue equations. It also develops semiclassical methods in this formulation, and it has a section on Berry’s semiclassical formula for the Wigner function in one-dimensional systems.
We present a phase space formulation of quantum mechanics in the Schrödinger representation and derive the associated Weyl pseudo-differential calculus. We prove that the resulting theory is unitarily equivalent to the standard "configuration space" formulation and show that it allows for a uniform treatment of both pure and mixed quantum states. In the second part of the paper we determine the unitary transformation (and its infinitesimal generator) that maps the phase space Schrödinger representation into another (called Moyal) representation, where the wave function is the cross-Wigner function familiar from deformation quantization. Some features of this representation are studied, namely the associated pseudo-differential calculus and the main spectral and dynamical results. Finally, the relation with deformation quantization is discussed.