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

Journal of the Physical Society of Japan

Wigner Representation of Quantum Operators and Its Applications to Electrons in a Magnetic Field

Ryogo Kubo

Published Nov 1, 1964• 144 citations
Publisher

Abstract

The Wigner representation F s ( p , q ) of a quantum operator F ( p , q ) is defined by F s ( p , q )\({=}h\Tr\) F ( p , q ) Δ ( p - p , q - q ) where Δ is a quantum analogue of a delta function in the phase space. This gives in particular the Wigner distribution function for a density operator. Basic theorems are summarized for the computation rules for quantummechanical operators in the Wigner representation. This is applied, in particular, to electrons in a magnetic field, for which a Wigner d. f. is introduced to describe the distribution of physical momenta \(\vec{\pi}{=}m\vec{v}\), \(\vec{v}\) being the velocity, and the position x . This description has the advantage to avoid the use of a vector potential and so to be gauge-independent. As examples of application, the diamagnetism and the Hall effect are briefly treated. Further applications of this treatment will be published later.

DOI
10.1143/jpsj.19.2127

Related papers

ANGULAR MOMENTUM-PHASE WIGNER OPERATOR FOR DESCRIBING ELECTRONS IN UNIFORM MAGNETIC FIELDS

Hong-Yi Fan

By introducing the angular momentum-phase Wigner operator both in <λ| representation and <l,r| representation, which are respectively the eigenstates of position and angular momentum of an electron in a uniform magnetic field, we establish the angular momentum-phase Wigner function whose marginal distributions are physically meaningful.

Modern Physics Letters B•Jun 10, 2001•2 citations

Gauge Invariant Generalization of the Wigner Function of Quantum System in a Nonuniform Magnetic Field

V. V. Dodonov, Vladimir I. Man’ko, D. L. Ossipov

As is well known, there are several formulations of quantum mechanics. One of them, closely connected with the Weyl representation of operators [ 1 ] and Wigner functions [ 2 ], use functions on the phase space to represent both observables and quantum states. For working in this representation it is necessary to replace operators by their Weyl symbols and density matrices by Wigner functions [ 3 , 4 ]. After fundamental works of Weyl, Wigner and Kirkwood [ 1 , 2 , 5 ] properties of the representation were studied and used to calculate the properties of various physical systems (see [ 6–23 ]), including systems with magnetic field [ 15–23 ].

CRC Press eBooks•Oct 31, 2024•0 citations

NEW REPRESENTATION FOR GAUGE-INVARIANT WIGNER OPERATOR OF AN ELECTRON IN A UNIFORM MAGNETIC FIELD

Hong-Yi Fan, Zhenshan S. Yang

By using the technique of integration within an ordered product (IWOP) of operators, we derive the normal-product form of the gauge-invariant Wigner operator of an electron in a uniform magnetic field and its expression in the newly constructed |λ> representation. The virtue of working in the |λ> representation lies in the fact that it is expressed in terms of the ladder operators Π±, K±(see Eq. (10)), thus the magnetic vector potential can naturally be included. On these bases we can easily obtain Wigner functions of some important electron states.

Modern Physics Letters A•Oct 30, 1998•10 citations

QUASI-PROBABILITY DISTRIBUTION FOR LAUGHLIN STATE VECTORS

Hong-Yi Fan, JINGXIAN LIN

Based on the gauge-invariant Wigner operator in <λ| representation (see Ref. 10), where the state |λ> can conveniently describe the motion of an electron in a uniform magnetic field, we provide an approach for identifying the corresponding state vector for Laughlin wave function and deriving the Wigner function (quasi-probability distribution) for the Laughlin state vector. The angular momentum-excited Laughlin state vectors are also obtained via <λ| representation.

Modern Physics Letters B•Jun 20, 2001•2 citations

Perturbation Theory in Terms of a Generalized Phase-Space Quantization Procedure

Omar Morandi, Luigi Barletti, Giovanni Frosali

A new approach to perturbation theory in the quantum phase-space formalism is proposed, in order to devise some approximated description of the quantum phase-space dynamics, which is not directly related to the usual semi-classical approximation. A general class of equivalent quasi-distribution functions based on the Wigner-Moyal formalism is considered and a first-order invariant formulation of the dynamics is obtained. The relationship between the various phase-space representation is expressed in term of a pseudo-differential operator defined by the Moyal product. In particular, our theory is applied to the sub-class of representations obtained by a first order perturbation of the Wigner representation. Finally the connection of our approach with some well established gauge-invariant formulation of the Wigner dynamics in the presence of an external magnetic field is investigated.

Bollettino Della Unione Matematica Italiana•Jan 1, 2011•57 citations

Phase-space formulation of quantum mechanics under phase-space transformations: Wigner functions of an oscillator system in crossed magnetic and electric fields

Lipo Wang

We give a rigorous proof for the following result that combines those discussed earlier by Kr\"uger and Poffyn [Physica 85A, 84 (1976)] and by Springborg [J. Phys. A 16, 535 (1983)]: The Wigner correspondence of a quantum operator can be obtained by simple substitutions under an arbitrary linear phase-space transformation. This property cannot be generalized to any other transformations or rules of phase-space correspondence. As an application of this result, the Wigner distribution functions of an anisotropic harmonic-oscillator system in crossed magnetic and electric fields are obtained.

Physical Review A•Feb 1, 1990•3 citations
The Wigner representation F s ( p , q ) of a quantum operator F ( p , q ) is defined by F s ( p , q )({=}h\Tr) F ( p , q ) Δ ( p - p , q - q ) where Δ is a quantum analogue of a delta function in the phase space. This gives in particular the Wigner distribution function for a density operator. Basic theorems are summarized for the computation rules for quantummechanical operators in the Wigner representation. This is applied, in particular, to electrons in a magnetic field, for which a Wigner d. f. is introduced to describe the distribution of physical momenta (\vec{\pi}{=}m\vec{v}), (\vec{v}) being the velocity, and the position x . This description has the advantage to avoid the use of a vector potential and so to be gauge-independent. As examples of application, the diamagnetism and the Hall effect are briefly treated. Further applications of this treatment will be published later.
As is well known, there are several formulations of quantum mechanics. One of them, closely connected with the Weyl representation of operators [ 1 ] and Wigner functions [ 2 ], use functions on the phase space to represent both observables and quantum states. For working in this representation it is necessary to replace operators by their Weyl symbols and density matrices by Wigner functions [ 3 , 4 ]. After fundamental works of Weyl, Wigner and Kirkwood [ 1 , 2 , 5 ] properties of the representation were studied and used to calculate the properties of various physical systems (see [ 6–23 ]), including systems with magnetic field [ 15–23 ].
By using the technique of integration within an ordered product (IWOP) of operators, we derive the normal-product form of the gauge-invariant Wigner operator of an electron in a uniform magnetic field and its expression in the newly constructed |λ> representation. The virtue of working in the |λ> representation lies in the fact that it is expressed in terms of the ladder operators Π±, K±(see Eq. (10)), thus the magnetic vector potential can naturally be included. On these bases we can easily obtain Wigner functions of some important electron states.
A new approach to perturbation theory in the quantum phase-space formalism is proposed, in order to devise some approximated description of the quantum phase-space dynamics, which is not directly related to the usual semi-classical approximation. A general class of equivalent quasi-distribution functions based on the Wigner-Moyal formalism is considered and a first-order invariant formulation of the dynamics is obtained. The relationship between the various phase-space representation is expressed in term of a pseudo-differential operator defined by the Moyal product. In particular, our theory is applied to the sub-class of representations obtained by a first order perturbation of the Wigner representation. Finally the connection of our approach with some well established gauge-invariant formulation of the Wigner dynamics in the presence of an external magnetic field is investigated.
We give a rigorous proof for the following result that combines those discussed earlier by Kr\"uger and Poffyn [Physica 85A, 84 (1976)] and by Springborg [J. Phys. A 16, 535 (1983)]: The Wigner correspondence of a quantum operator can be obtained by simple substitutions under an arbitrary linear phase-space transformation. This property cannot be generalized to any other transformations or rules of phase-space correspondence. As an application of this result, the Wigner distribution functions of an anisotropic harmonic-oscillator system in crossed magnetic and electric fields are obtained.