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Chinese Physics Letters

New Applications of the ⟨λ| Representation in Studying the Angular Momentum of the Laughlin State Vector

Fan Hong-Yi, Lin Jing-Xian

Published Mar 21, 2001• 2 citations
Publisher

Abstract

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.

DOI
10.1088/0256-307x/18/4/301

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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.
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.
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.