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Modern Physics Letters B

QUASI-PROBABILITY DISTRIBUTION FOR LAUGHLIN STATE VECTORS

Hong-Yi Fan, JINGXIAN LIN

Published Jun 20, 2001• 2 citations
Publisher

Abstract

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.

DOI
10.1142/s021798490100194x

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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.
We use the complete and orthonormal ⟨λ| representation (Phys. Lett. A126 (1987) 150), constructed in terms of guiding centres [ K ± = M Ω/2 1/2 ( x 0 ∓i y 0 )] and kinetic momenta Π ± describing an electron in a uniform magnetic field, to identify that the Laughlin state vector is ∏ i < j N ( K + i - K + j ) m ||0⟩, where ||0⟩ is annihilated by (Π - , K - ). Following this, we further employ the ⟨λ| representation to derive the angular momentum of the Laughlin ground state and excited state in a direct and convenient way. This approach to studying some properties of the Laughlin state seems to be new.
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.
We review how to rely on the quantum entanglement idea of Einstein–Podolsky–Rosen and the developed Dirac's symbolic method to set up two kinds of entangled state representations for describing the motion and states of an electron in uniform magnetic field. The entangled states can be employed for conveniently expressing Landau wave function and Laughlin wave function with a fresh look. We analyze the entanglement involved in electron's coordinates (or momenta) eigenstates, and in the angular momentum-orbit radius entangled state. Various applications of these two representations, such as in developing angular momentum theory, squeezing mechanism, Wigner function and tomography theory for this system are presented. Thus the present review systematically summarizes a distinct approach for tackling this physical system.