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Modern Physics Letters B

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

Hong-Yi Fan

Published Jun 10, 2001• 2 citations
Publisher

Abstract

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.

DOI
10.1142/s0217984901001914

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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.
We show that the newly found complete and orthonormal representation , which is constructed in terms of guiding centers and canonical momenta of an electron moving in a uniform magnetic field, also helps us to establish a systematic phase operator theory. The unitary phase operator, interpreted as a quantity canonically conjugated to the electron angular momentum, manifestly exhibits in the representation its phase behavior. As a byproduct, the phase distribution function, in the correct sense of probability in quantum mechanics, can also be defined by the representation.
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.
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 π⃗=mv⃗\vec{\pi}{=}m\vec{v}π=mv, v⃗\vec{v}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.