Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1307.8068 · Annales Henri Poincaré

Semiclassics for Particles with Spin via a Wigner–Weyl-Type Calculus

Omri Gat, Max Lein, Stefan Teufel

Published Jan 7, 2014• 5 citations
arXivHTMLPDF
DOI
10.1007/s00023-013-0294-0

Related papers

Mixed Weyl symbol calculus and spectral line shape theory

T. A. Osborn, M. F. Kondrat’eva, George C. Tabisz +1 more

A new and computationally viable full quantum version of line shape theory is obtained in terms of a mixed Weyl symbol calculus. The basic ingredient in the collision-broadened line shape theory is the time-dependent dipole autocorrelation function of the radiator-perturber system. The observed spectral intensity is the Fourier transform of this correlation function. A modified form of the Wigner-Weyl isomorphism between quantum operators and phase space functions (Weyl symbols) is introduced in order to describe the quantum structure of this system. This modification uses a partial Wigner transform in which the radiator-perturber relative motion degrees of freedom are transformed into a phase space dependence, while operators associated with the internal molecular degrees of freedom are kept in their original Hilbert space form. The result of this partial Wigner transform is called a mixed Weyl symbol. The star product, Moyal bracket and asymptotic expansions native to the mixed Weyl symbol calculus are determined. The correlation function is represented as the phase space integral of the product of two mixed symbols: one corresponding to the initial configuration of the system, the other being its time evolving dynamical value. There are, in this approach, two semiclassical expansions - one associated with the perturber scattering process, the other with the mixed symbol star product. These approximations are used in combination to obtain representations of the autocorrelation that are sufficiently simple to allow numerical calculation. The leading O( 0 ) approximation recovers the standard classical path approximation for line shapes. The higher-order O( 1 ) corrections arise from the noncommutative nature of the star product.

arXiv:physics/9811047•Journal of Physics A Mathematical and General•Jan 1, 1999•10 citations

Semiclassical quantization of spin

Paul Houle

•Jan 1, 1999•1 citation

SEMICLASSICAL QUANTUM MECHANICS: A PATH-INTEGRAL APPROACH

Barry R. Holstein

WORLD SCIENTIFIC eBooks•Jun 1, 2001•0 citations

Semiclassical Moyal quantum mechanics for atomic systems

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

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.

Physical Review A•Oct 1, 1998•18 citations

From the Weyl quantization of a particle on the circle to number–phase Wigner functions

Maciej Przanowski, Przemysław Brzykcy, Jaromir Tosiek

arXiv:1311.7335•Annals of Physics•Oct 26, 2014•11 citations

The Weyl‐Wigner‐Moyal Formalism on a Discrete Phase Space. I. A Wigner Function for a Nonrelativistic Particle with Spin

Maciej Przanowski, Jaromir Tosiek, Francisco J. Turrubiates

Abstract The Weyl‐Wigner‐Moyal formalism for quantum particle with discrete internal degrees of freedom is developed. A one to one correspondence between operators in the Hilbert space and functions on the phase space is found. The expressions for the Stratonovich‐Weyl quantiser, star product and Wigner functions of such systems for arbitrary values of spin are obtained in detail. As examples the Landau levels and the corresponding Wigner functions for a spin nonrelativistic particle as well as the magnetic resonance for a spin nonrelativistic uncharged particle are analysed.

arXiv:1812.07325•Fortschritte der Physik•Dec 1, 2019•11 citations
A new and computationally viable full quantum version of line shape theory is obtained in terms of a mixed Weyl symbol calculus. The basic ingredient in the collision-broadened line shape theory is the time-dependent dipole autocorrelation function of the radiator-perturber system. The observed spectral intensity is the Fourier transform of this correlation function. A modified form of the Wigner-Weyl isomorphism between quantum operators and phase space functions (Weyl symbols) is introduced in order to describe the quantum structure of this system. This modification uses a partial Wigner transform in which the radiator-perturber relative motion degrees of freedom are transformed into a phase space dependence, while operators associated with the internal molecular degrees of freedom are kept in their original Hilbert space form. The result of this partial Wigner transform is called a mixed Weyl symbol. The star product, Moyal bracket and asymptotic expansions native to the mixed Weyl symbol calculus are determined. The correlation function is represented as the phase space integral of the product of two mixed symbols: one corresponding to the initial configuration of the system, the other being its time evolving dynamical value. There are, in this approach, two semiclassical expansions - one associated with the perturber scattering process, the other with the mixed symbol star product. These approximations are used in combination to obtain representations of the autocorrelation that are sufficiently simple to allow numerical calculation. The leading O( 0 ) approximation recovers the standard classical path approximation for line shapes. The higher-order O( 1 ) corrections arise from the noncommutative nature of the star product.
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.
Abstract The Weyl‐Wigner‐Moyal formalism for quantum particle with discrete internal degrees of freedom is developed. A one to one correspondence between operators in the Hilbert space and functions on the phase space is found. The expressions for the Stratonovich‐Weyl quantiser, star product and Wigner functions of such systems for arbitrary values of spin are obtained in detail. As examples the Landau levels and the corresponding Wigner functions for a spin nonrelativistic particle as well as the magnetic resonance for a spin nonrelativistic uncharged particle are analysed.